Georg Glaeser

Geometry and
its Applications
in Arts, Nature
and Technology
Geometry and its Applications in Arts,
Nature and Technology
Georg Glaeser

Geometry and its

Applications in Arts,
Nature and Technology
Second Edition
Georg Glaeser
Department of Geometry
University of Applied Arts Vienna
Vienna, Austria

Originally published by Springer Wien New York, 2012

ISBN 978-3-030-61397-6 ISBN 978-3-030-61398-3 (eBook)

https://doi.org/10.1007/978-3-030-61398-3

Mathematics Subject Classification (2020): 51, 53, 68, 78, 85, 92, 97

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Preface
A very personal foreword
Ever since I was young, I have been fascinated by geometric forms – be they
abstract, forms in arts, or forms that occur in nature or technology.
Despite having always been a “half biologist”, I decided to study mathemat-
ics and geometry, because I realized that they are mighty tools to describe
nature.

At the age of 17 (that is, in 1972), I had – amid numerous books about
animals – three books by Walter Wunderlich in my bookshelf: Descriptive
Geometry I and II and Planar Kinematics ([27], [28]). I was fascinated by
perspective drawings of bridges and accurate constructions of snail shells,
as well as by the geometry of motion. Back in those days, there were no
computers on the market, and everything had to be drawn by hand, which
required not only patience but also extensive knowledge.
vi Preface

In the course of the many years that followed, I have realized that this has
been of great advantage to me: Since learning to program a computer is not
hard for mathematicians – you only need to apply the structured thinking
that determines your whole approach to the tackled problems – comput-
ers have become a new and very efficient tool to support the solution of the
tasks – no more, no less. The underlying brainwork, paired with spatial imag-
ination, still remains the most important issue. Mainly during the 1990s, I
developed graphics tools for the fast visualization of geometric problems by
means of the computer, working in part with my colleagues Hellmuth Stachel
and Hans-Peter Schröcker ([3], [4], [5]).
A predecessor of this book was published in German in 2005, with two more
editions in 2007 and 2014 ([6]). In between those publications there was one
English edition by Springer Verlag Wien-New York in 2012 ([7]). Each edition
had more applications and new topics, thus eventually increasing the number
of pages from 400 to 700. Some of the topics are described in more detail in
[8] and the two “compendia” about conics and quadrics ([16], [22]) which I
wrote together with Hellmuth Stachel and Boris Odehnal.
Self-sufficient geometry?
Geometry is usually considered to be one branch of mathematics. Indeed, the
mental processes required for geometry are closely related to mathematical
logic. Furthermore, since geometry is a very comprehensive field, it is often
thought of as an independent science. In any case, geometry and mathematics
complement each other quite nicely. Nevertheless, no special mathematical
knowledge is required in order to read and understand this book. Whatever
mathematical equipment we need will arise by itself.
Geometry can be conducted in several ways, but the kind of geometry that we
use in this book seeks to utilize the reader’s three-dimensional imagination as
often as possible. Such imagination is present to varying degrees in different
people, but research has shown that a preoccupation with geometry can
improve one’s spatial imagination considerably. You will sometimes find notes
in small print about analytical geometry or calculus, but these instances will
only serve as reinforcement of ideas that should already have been established
by your three-dimensional imagination.
A sixth sense
Since geometry not only utilizes but also improves this “sixth sense” of spatial
imagination, younger readers often find it easier to progress in this discipline
compared to other branches of mathematics. The latter require very solid
foundational knowledge, as errors that may be overlooked by inexperienced
eyes may lurk in every line of calculation. The kind of geometry with which we
are concerned, however, always yields results which can, to some extent, be
verified by our imagination. It is for this reason that geometric (“synthetic”)
proofs are usually preferred to strictly mathematical (“analytic”) ones. It
is by no means uncommon for these proofs to contain such deep levels of
Preface vii

understanding that new or additional results inevitably come to the attentive

mind.

Descriptive geometry and more

My concept is based on the ideas of my academic teacher Walter Wunder-
lich and my predecessor Erich Frisch, who both have, in traditional ways,
constructed a simple yet thorough educational framework concerning “de-
scriptive geometry”. Yet, time never stands still, and due to the revolution
of geometry brought about by the triumph of computing, their concepts
needed to be considerably modified and expanded upon. In this book, I wish
to maintain a strong connection to the related fields of computer graphics
and computational geometry, which both owe their roots to classical geom-
etry. Furthermore, the geometry that is described in this book shall not be
limited to mere depiction, especially since the methodology for the creation
of images has so profoundly changed in the digital age.

Acknowledgments – the network

I would like to especially thank my co-workers Boris Odehnal for his cor-
rections of the manuscript and for fruitful geometric discussions, and Franz
Gruber – who has passed away too early – who contributed to the book by
providing a lot of wonderful interactive animations that allow a deeper in-
sight in geometric processes. My former co-workers Christian Perrelli, Hans-
Peter Schröcker, Gerhard Karlhuber, and Thomas Backmeister have also
contributed images to this book.
Further help came from Wilhelm Fuhs, Herbert Löffler, and Michael Schrott.
Several illustrations were created by Harald Andreas Korvas, Zorica Nicolic,
Otmar Öhlinger, Markus Roskar, Paulo Tosold, and Stefan Wirnsperger.
The final proofreading of this second English edition was done by Eugenie
Maria Theuer, Tamara Radak, and Irene Karrer.
I was assisted in this endeavor by several colleagues at the Technical Univer-
sity of Vienna (Fritz Manhart and Walter Jank). Walter Hofmann made a
valuable contribution to the topic of sundials. Andreas Rüdinger, who rep-
resented the German publisher on this project, has been very constructive
throughout this book’s development.
I am privileged to teach at the University of Applied Arts in Vienna, where
I can count renowned artists like Zaha Hadid, Hans Hollein, Oliver Kar-
tak, Bernhard Leitner, Greg Lynn, Paolo Piva, Wolf D. Prix, Hani Rashid,
Cristina Díaz Moreno, and Efrén García Grinda among my (partly former)
colleagues. Several examples in this book are inspired by their creative work.
I am also grateful to have enjoyed further assistance by my colleagues Klaus
Bollinger, Ernst Maczek-Mateovits, Roland Burgard, and Marcus Bruck-
mann.
Furthermore, I would like to thank the students at the University of Applied
Arts Vienna, who, in a generous interplay of give and take, were always keen
viii Preface

to contribute ideas, several of which eventually came to fruition. Some of

their work is also presented in this book.
Finally, my then 12-year-old daughter Sophie has contributed several draw-
ings (e.g., Fig. 3.64) and several models (Fig. A.4). Due to my unceasing
enthusiasm for photography this book has become a sort-of “picture book
of geometry”. Good photos rarely emerge “by themselves”, or require the un-
wieldy power of expensive photographic equipment. What is needed is a lot of
patience, a good eye, and – as I will show in the appendix – an understanding
of geometry.
Vienna, June 2020

This picture series of the Moon was taken a month before the completion of this edition, in the
midst of the “Corona crisis”. Since almost all airplanes were grounded and there was virtually no
pollution during the crisis, the sky was clearer than it had ever been in the years before. Due to
the different day times – or better night times – at which the pictures were taken, the colors of
the Moon and the sky vary. The last image is a composition of a waxing crescent and a waning
gibbon.
Table of Contents

1 An idealized world made of simple elements 5

1.1 Points, lines, and circles in the drawing plane . . . . . . . . . . . . . . 6
1.2 Special points inside the triangle . . . . . . . . . . . . . . . . . . . . 13
1.3 Elementary building blocks in space . . . . . . . . . . . . . . . . . . . 25
1.4 Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5 Polarity, duality and inversion . . . . . . . . . . . . . . . . . . . . . . 34
1.6 Projective and non-Euclidean geometry . . . . . . . . . . . . . . . . . 45
2 Projections and shadows: Reduction of the dimension 53
2.1 The principle of central projection . . . . . . . . . . . . . . . . . . . . 54
2.2 Parallel projection and normal projection by means of restriction . . . . 58
2.3 Assigned normal projections . . . . . . . . . . . . . . . . . . . . . . . 66
2.4 Principal lines and the Law of Right Angles . . . . . . . . . . . . . . . 75
2.5 The difference in technical drawing . . . . . . . . . . . . . . . . . . . 83
3 Polyhedra: multi-faced and multi-sided 87
3.1 Congruence transformations . . . . . . . . . . . . . . . . . . . . . . . 88
3.2 Convex polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.3 Platonic solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.4 Other special classes of polyhedra . . . . . . . . . . . . . . . . . . . . 109
3.5 Planar sections of prisms and pyramids . . . . . . . . . . . . . . . . . 118
4 Curved but simple 125
4.1 Planar and space curves . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.2 The sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.3 Cylinder surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.4 The ellipse as a planar intersection of a cylinder of revolution . . . . . 164
5 More about conic sections and developable surfaces 175
5.1 Cone surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.2 Conic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.3 General developables . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.4 Maps and “sphere developments” . . . . . . . . . . . . . . . . . . . . 217
5.5 Reflection in a circle, a sphere, and a cylinder of revolution . . . . . . 226
6 Prototypes 233
6.1 Surfaces of degree two . . . . . . . . . . . . . . . . . . . . . . . . . . 234
6.2 Three types of surface points . . . . . . . . . . . . . . . . . . . . . . 259
6.3 Surfaces of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . 268
6.4 The torus as a prototype for all other surfaces of revolution . . . . . . 278
6.5 Pipe and channel surfaces . . . . . . . . . . . . . . . . . . . . . . . . 287
7 Further remarkable classes of surfaces 293
7.1 Ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
7.2 Helical surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
7.3 Different types of spiral surfaces . . . . . . . . . . . . . . . . . . . . . 315
7.4 Translation surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
7.5 Minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
8 The endless variety of curved surfaces 337
8.1 Mathematical surfaces and free-form surfaces . . . . . . . . . . . . . . 338
8.2 Interpolating curves and surfaces . . . . . . . . . . . . . . . . . . . . 347
8.3 Bézier- and B-spline-curves . . . . . . . . . . . . . . . . . . . . . . . 351
8.4 Bézier- and B-spline surfaces . . . . . . . . . . . . . . . . . . . . . . 354
8.5 Surface design done differently . . . . . . . . . . . . . . . . . . . . . 358
x Table of Contents

9 Photographic image
and individual perception 363
9.1 The human eye and the pinhole camera . . . . . . . . . . . . . . . . . 364
9.2 Different techniques of perspective . . . . . . . . . . . . . . . . . . . 367
9.3 Other perspective images . . . . . . . . . . . . . . . . . . . . . . . . 381
9.4 Geometry at the water surface . . . . . . . . . . . . . . . . . . . . . . 401
10 Kinematics: Geometry in motion 415
10.1 The pole around which everything revolves . . . . . . . . . . . . . . . 416
10.2 Different mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 423
10.3 Ellipse motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
10.4 Trochoid motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
11 Spatial motions 459
11.1 Motions on the sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 460
11.2 General spatial motion . . . . . . . . . . . . . . . . . . . . . . . . . . 468
11.3 What is the position of the Sun? . . . . . . . . . . . . . . . . . . . . 473
11.4 Minute-precise sundials for identifying the mean time . . . . . . . . . . 493
11.5 Kinematics in the animal kingdom . . . . . . . . . . . . . . . . . . . . 503
12 A variety of filling patterns 511
12.1 Ornaments and symmetry groups . . . . . . . . . . . . . . . . . . . . 512
12.2 Periodical tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
12.3 Non-periodical tilings . . . . . . . . . . . . . . . . . . . . . . . . . . 522
12.4 Non-Euclidean tilings . . . . . . . . . . . . . . . . . . . . . . . . . . 530
13 The nature of geometry and the geometry of nature 533
13.1 Basic geometric forms in nature . . . . . . . . . . . . . . . . . . . . . 534
13.2 Evolution and geometry . . . . . . . . . . . . . . . . . . . . . . . . . 541
13.3 Planetary paths and fish swarms . . . . . . . . . . . . . . . . . . . . . 546
13.4 Scaling behavior in nature . . . . . . . . . . . . . . . . . . . . . . . 552
13.5 Musical harmony through the eyes of geometry . . . . . . . . . . . . . 556
14 More applications 559
14.1 Multiple reflections, illusions, and impossibles . . . . . . . . . . . . . . 560
14.2 More about inversions in space . . . . . . . . . . . . . . . . . . . . . 574
14.3 Voronoi structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
14.4 Hexagons on a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 589
14.5 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
A Geometrical freehand drawing 607
A.1 Normal view vs. oblique view . . . . . . . . . . . . . . . . . . . . . . 608
A.2 Don’t be afraid of curved surfaces . . . . . . . . . . . . . . . . . . . . 615
A.3 Shadows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
A.4 Perspective sketching . . . . . . . . . . . . . . . . . . . . . . . . . . 624
B Geometry and photography 637
B.1 Focal lengths and viewing angles . . . . . . . . . . . . . . . . . . . . 638
B.2 3D images in photography? . . . . . . . . . . . . . . . . . . . . . . . 642
B.3 Stereoscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
B.4 When should we use which focal length? . . . . . . . . . . . . . . . . 658
B.5 Primary and secondary projection . . . . . . . . . . . . . . . . . . . . 665
B.6 From below or from above? . . . . . . . . . . . . . . . . . . . . . . . 670
B.7 Collinear distortion of a photo . . . . . . . . . . . . . . . . . . . . . . 675
B.8 The problem with the rolling shutter . . . . . . . . . . . . . . . . . . 678
Bibliography 685
Introduction
Where geometry matters
Naturally, there exist many cross connections between geometry and tech-
nical disciplines such as mechanical engineering, architecture, and civil en-
gineering. The fields of fine arts and design are likewise influenced by or
dependent on geometry, to say nothing of physics, astronomy, geography,
chemistry, biology, and molecular biology. All of these fields utilize geometry
to derive new insights or to visualize existing ones. Even harmonic relation-
ships in music can be depicted in purely geometrical terms.

• When should geometric questions be pursued at a lower dimension? The

rotation about a straight line can be imagined much more effectively if the
line appears as a point. If the underlying procedure is properly understood,
many problems can be simplified.

• When, on the other hand, should we look at a problem from a “higher

level”? The planar conic sections – when spatially considered – are barely
different. Beings inhabiting higher dimensions would, thus, consider many
of our differentiations to be superfluous.

Conic sections Creation of a geoglyph

• How could the ancient Peruvians have created their enormous geoglyphs
without having had to invent the equivalent of a hot-air balloon? A com-
bination of central projection and central dilatation made it possible.

Light refraction on the eye How many photos are required?

• How does vision actually work, and how is it reproduced by a camera?

What conditions need to be met so that the image in a photograph seems
2 Introduction

“natural” and not distorted or flattened? Five hundred years ago, Leonardo
da Vinci and Leone Battista Alberti , who worked with exact spatial pro-
jections to planes, already asked these questions.

• When can a distorted photo be “rectified”? How many photos are required
in order to reconstruct the true shape of an object? Such questions are not
only interesting to architects, but also to the designers of 3D scanners.

• What does a map where the shortest distances between pairs of points
on the globe appear as straight lines or as circular arcs look like? From
what maps can the course angle be directly deduced? Such problems arise
from the fact that spheres cannot be unfolded into planes without certain
distortions. All of cartography depends on such geometrical insights.

Maps and shortest distances Good triangulation

• Modern architects today are fond of designing curved buildings. Can ge-
ometry be used to minimize the potentially horrendous construction costs
resulting from this artistic choice?

Constant velocity joint Motion of the Earth about the Sun

• How can the rotation about an axis be transmitted to another axis, and
how can this transition be accomplished in a smooth manner? This ques-
tion arises frequently in mechanical engineering.

• Why does it not always take the equal amount of time for the Sun to
reach its point of culmination? This question from the field of astronomy
is partially related to physics, but it involves geometry to an even greater
degree.

• “Through each point, there exists exactly one parallel to a predefined

line.” What happens if one removes this unprovable geometrical axiom?
It results in the outlandish field of “non-Euclidean geometry”, the insights
of which are even relevant to modern physics!
Introduction 3

Hyperbolic geometry Snails and horns

• The helical structure of DNA has become the symbol of modern biotech-
nology. What can be deduced about the growth of snails from the general
shape of their shells?

Extreme perspectives The trick with tilt-shift lenses

• Using simple geometrical tricks, how can you create pictures which would
otherwise require an expensive tilt-shift lens? Why is the correct selec-
tion of focal distance not only a matter of “visibility”? When are ultra-
perspectives not only allowed but necessary? Why does sharpness depend
on a camera’s aperture?

These and many other questions will be broached and answered in this book.
Contemporary students have the advantage of computer programs which
allow them to visualize graphics that they would otherwise have to painstak-
ingly draw by hand. Therefore, the balance of geometrical education should
always tilt in favor of geometrical comprehension and the analysis of its many
processes. Such a change in the curriculum allows teachers to address topics
that, while relevant to the future professional experience of students, could
not have been addressed otherwise due to time restrictions.
Level of detail
I have made a great effort to describe laws and rules as simple as possible,
before delving into additional details, exceptions, and special cases.
For example, I first state the following (p. 62): “If two lines are parallel in
two different parallel projections, then they are, in general, also parallel in
three-dimensional space.” Exceptions are addressed subsequently – in this
important case, in considerable detail.
Further examples: If a curve is “not bent”, it means that it has a mathematical
curvature of zero. Or: “Each planar section of a sphere is a circle.” Of course,
if the plane merely touches the sphere, then the circle shrinks to a so-called
“null circle”, which can also be considered a point. But what if the plane
4 Introduction

does not intersect the sphere? Mathematicians can prove that this yields an
“imaginary” circle. But how can such an abstract idea be expressed without
overstraining the layman? If such exceptions are left unsaid, the interested
reader will still understand the idea, and the fundamental assertion remains
concise and to the point!
The website www.uni-ak.ac.at/geometrie
This book is accompanied by a website. There you will find updates, weblinks
to many specialized topics, and further examples of geometry in action. Fi-
nally, a plethora of interactive programs is provided that let you visualize
many complex sequences of motion and physical simulations.
The advantage of such additional material is obvious: First of all, the website
is constantly updated, and the material is continually enriched and expanded
upon without having to change the book to which it refers. I hope that you
will find a lot of useful material on the website and provide me with feedback
from which I can further improve this service!

The geometrical net . . .

1 An idealized world
made of simple elements

In this chapter, we will learn about the most important building blocks of
the geometric world: points, straight lines, and circles in the two-dimensional
drawing plane, as well as planes, circles, spheres, cylinders, and cones of
revolution in three-dimensional space.
Points in geometry are “infinitely small” and, therefore, have no expansion. In
the same sense, straight lines are always “infinitely thin” – as well as “infinitely
long”. Analogies can be made for planes, cylinders and so on: Mathematically
idealized versions simplify how we deal with these objects in our imagination.
In geometry, we are not afraid to use the term infinity. Likewise, the abstract
concept of dimension is very frequently invoked: A set of planes in 3-space
forms a three-dimensional plane space. The theoretical foundation for this
statement is given by projective geometry, with which we shall deal in the
last section.
First, we will define certain terms in the drawing plane and further deepen
our understanding by considering some remarkable examples of the geometry
of triangles. This should provide a good basis for a smooth transition to
spatial geometry. During this process, we shall gain numerous insights by
generalizing our results from planar geometry.
© Springer Nature Switzerland AG 2020 5
G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_1
6 An idealized world made of simple elements

1.1 Points, lines, and circles in the drawing plane

Zero-dimensional spheres
Points in geometry (and anywhere in mathematics) are “infinitely small” and,
therefore, have no expansion. As such, they can be seen as zero-dimensional
objects. When we label points, we will adhere to European convention and
utilize Latin capital letters such as A, B, P , Q, etc.
Rays without beginning or end
By stringing together an infinite number of points – in a given direction – we
get an infinitely thin and infinitely long one-dimensional straight line. It can
be said that a straight line is “generated” by translating a point. According to
European convention, straight lines are labeled using Latin lower-case letters
such as a, b, g, h, etc.
Straight lines are not bounded segments: Each line segment P Q lies on an
infinitely long carrier straight line g = P Q.
In order to get accustomed to geometrical language, we will now review some
terms that we have already used so far.
Circles
Circles are so-called lines of constant distance from a point. All points on
the circle k have the same constant distance from the midpoint M , namely
the radius r. In this book, we will usually notate circles using the letter
k. When we speak of more than one circle, they are notated as k1 , k2 , . . . or
kA , kB , kM , . . . (given the midpoints A, B or M ). Circles possess the attribute
of being uniformly curved and, like all curves, they are also infinitely thin.
The most natural curve

Fig. 1.1 Water waves

Circles are very popular elements of construction because it is comparatively

easy to work accurately with them. Straight lines are rarely found in nature.
1.1 Points, lines, and circles in the drawing plane 7

Circles in the plane or spheres in space are frequent by comparison. One

only needs to throw stone pebbles into a lake to be amazed by the concentric
rings that are then formed by the wave peaks. All types of waves, including
light waves and sonic waves, expand in a circular manner in the plane and
spherically in three-dimensional space.
It would be impossible to draw long straight lines on a desert island. A forked
twig or crotch, however, will let us draw circles rather easily. The alleged last
words of Archimedes (about 287–212 B.C.) come to mind: “Do not disturb
my circles!” (He was referring to his constructions in the sand, directing his
statement at the Roman soldier who, after the conquest of Syracuse, stabbed
him to death despite an order to spare his life.)

Fig. 1.2 It is difficult to find straight lines – especially triangles – in nature. This elongated puddle
of water froze overnight, forming remarkable geometric shapes.

Triangles
A triangle is a union of three points A, B, C that are not on one line. It
can equally be described by the three line segments that are formed by the
three points: AB = c, BC = a, and CA = b. All other parameters, such as the
angles α = ∢BAC, β = ∢ABC, and γ = ∢ACB, can thus be calculated or
constructed. This book prefers construction to calculation because, instead
of yielding mere numbers, this method produces a “residual” that may be
useful for further contemplation.
Congruent and similar triangles
Let us construct a triangle ABC where the lengths of its sides are given: We
can orient the side AB in any way we like (such as horizontally) and draw
two circles kA and kB with the radii AC and BC. If the given lengths of
the triangle’s sides are mathematically possible (the sum of the length of two
sides must exceed the third), we will get two solutions for the third vertex
(C1 and C2 , Fig. 1.3). These two solutions do not differ by size (hence, they
are called congruent) but by “orientation”: They are “reversely congruent”.
By drawing the parallel to AB through C, we will also notice that:
The sum of angles in a triangle equals 180○ .
8 An idealized world made of simple elements

Fig. 1.3 Congruent solutions Fig. 1.4 Similar triangles, theorem of intersecting lines

⊕ Remark : Fig. 1.5 shows both sides of a peacock’s feather (see also Fig. 6.65). Their
distinct pattern is not caused by color pigmentation but is actually the effect of a
complex “twisting” of single hairs. The two sides are not identical, but their out-
lines are certainly similar (the figures in the image are reversely congruent). When
considered spatially, it nevertheless remains the same object, which, interestingly
enough, is directly congruent. ⊕

Fig. 1.5 Peacock’s feather (front and back side) – reversely congruent?

The term similarity generalizes the notion of congruence. Two triangles are
similar if they possess identical angles. The side lengths may change, but a
relationship can immediately be established using a factor of similarity. Here,
the theorems of intersecting lines are applicable. These simple statements are
surprisingly difficult to prove: For this, we need so-called axioms, which are
unprovable but commonsense statements that must be accepted as true.
Let us consider a triangle ABC, whose side lengths a, b, c are multiplied by
a constant factor k (the factor of similarity), resulting in a similar triangle
A1 B1 C1 (Fig. 1.4): a1 = k a, b1 = k b, c1 = k c. Triangles with a constant
aspect ratio are considered similar.
In addition, the following applies:

AB ∶ A1 B1 = AC ∶ A1 C1 = BC ∶ B1 C1
1.1 Points, lines, and circles in the drawing plane 9

If we translate one triangle into the other in such a way that the legs of an
angle of the first triangle overlap the respective angle of the second triangle,
then the remaining two sides are parallel. Thus, we have the

Theorems of intersecting lines: If an angle is intersected by two parallel lines,

then the ratio of the distances AB and AB1 on the first leg and the respective
distances AC and AC1 on the second leg are equal. Furthermore, we have the
same ratio on the parallel sections BC and B1 C1 .

One of the oldest theorems of geometry

Contemporary geometry, with its strict requirements for proof, dates back
to the ancient Greeks, whose science dominated the southern Mediterranean
(from Sicily through Asia Minor right down to northern Egypt). Alexandria
(in the Nile delta) was seen as their scientific capital. Even before Egypt
became hellenized, it was visited by Thales of Milet (625–546 B.C., Asia
Minor), who gained much geometric knowledge from there.
The realization that a logical proof should be required for every geometric
theorem was a great breakthrough for geometry and mathematics. Among
other things, Thales proved the famous theorem that bears his name but was
known long before him:

Thales’s theorem: All angles in a semicircle are right angles. Put in a different
way: From each point on the circle, the diameter appears at a right angle.

Proof : Let us consider an arbitrary rectangle ABCD, as in Fig. 1.6. By definition,

rectangles only have right angles at each vertex (AB ⊥ BC). The diagonal AC has
the length d. It is obvious that this rectangle possesses a “circumcircle” with radius
d/2, the midpoint of which lies at the intersection of the diagonals. Thus, if seen
from the point B (as well as D), AC appears at a right angle. If we now look at all
rectangles with an equally long diagonal d, we get the same result every time. Of
all these rectangles, we can always overlay both the diagonal and the circumcircle.
B and D thus pass through the circumcircle. ⊙

This theorem forms the basis for many further proofs.

Thales’s theorem is merely a special case of the following situation:

Theorem of the angle of circumference: From each point of a circle, a fixed

chord of the circle is always seen at the same angle or from its supplementary
angle.

Proof : Let ABM be an isosceles triangle with ∢AM B = ϕ. Then ∢BAM =

∢ABM = ψ = 1/2(180○ − ϕ). Let us now draw a circle around M through A
and B and pick an arbitrary point C on it. We now have two further isosce-
les triangles AM C and BM C. If δ and ε describe the same angles of these tri-
10 An idealized world made of simple elements

Fig. 1.6 Thales’s theorem Fig. 1.7 Theorem of the angle of circumference

angles, then γ = ∢ACB = ε − δ. The sum of all angles in a triangle ABC is

[ψ − δ] + [ε + ψ] + [ε − δ] = 180○ . Thus, we get 2(ε − δ) = ϕ or γ = ϕ/2 (= con-
stant). If C resides on the other side of AB, then the following is true for the angle
sum of ABC: [δ − ψ] + [δ + ε] + [ε − ψ] = 180○ and thus γ = δ + ε = 180○ − ϕ/2. ⊙

As a by-product of this proof, we can derive the

Modification of the theorem of the angle of circumference: From the circum-

center of a triangle, each triangle side is visible under a central angle which is
twice the inscribed angle in the opposite vertex.

⊕ Remark : Proving Thales’s theorem was not necessary: it is merely a special case
of the theorem of the angle of circumference. However, it is always useful to practice
different techniques of proof. In [9] you can find a proof for this theorem that is
“kinematic” in nature – much like the proof of Thales’s theorem: If one parameter is
changed, the point traces a curve, which in this case is a circle. We will particularly
benefit from such considerations in the chapter about kinematics. ⊕

The circle steps out of line

Circles play a fundamental role in the classical geometry of Euclid . They
intersect a straight line twice at the most. If the straight line merely touches
the circle, the points are actually infinitely close to each other. Likewise, two
distinct circles never share more than two points. This is as self-evident as it
is unusual – at a later point, we will find out why this is the case. At any rate:
Most classical constructions can be subdivided in such a way that they can
be reassembled merely by intersecting straight lines and circles with other
straight lines and circles. Such problems are then either totally linear (if no
circles are required) or quadratic or even more than that. The construction
of a triangle whose side lengths are given is quadratic because two (reversely
congruent) solutions are obtained by intersecting circles.
1.1 Points, lines, and circles in the drawing plane 11

Directly congruent figures, twisted into each other

Fig. 1.8 Directly congruent figures. Right side: With two-sided and two-colored candles, it is not
immediately evident that both parts were cast using the same mold.

We will need the following elementary theorem later:

Two directly congruent figures (same orientation) in the plane can always be
transformed into each other by an unambiguously defined translation and/or
rotation.

Proof : Let us pick two directly congruent triangles ABC and A∗ B ∗ C ∗ (Fig. 1.8 on
the left) on top of the two congruent figures and try to overlap them via rotation. In
order to determine the center of rotation, we will need to employ the perpendicular
bisector . This is the straight line that runs through the midpoint of P Q, at a right
angle to P Q. It is the locus of all points which are equidistant to P and Q. If ABC
and A∗ B ∗ C ∗ describe two congruent triangles, then the center Z of rotation has to
be equidistant to A and A∗ , i. e., on the perpendicular bisector of AA∗ . The same
applies to B and B ∗ , C and C ∗ .

Fig. 1.9 Reflection in a point or in a straight line, rotation by 180○

The question is whether all perpendicular bisectors actually run through the center
of rotation Z. Let Z be the intersection between the first two bisectors. Therefore,
we can imagine Z as being connected to the straight line AB. The auxiliary triangle
12 An idealized world made of simple elements

ABZ can be rotated into the triangle A∗ B ∗ Z about the point Z. However, C is also
fixed to AB, that is, the distance CZ does not change during the rotation. Thus,
Z must also lie on the perpendicular bisector of C and C ∗ . This remains true for
all respective point pairs on the congruent figures. ⊙
If the figure f is rotated by 180○ about the center Z, it becomes a directly
congruent figure f ∗ and can equally be seen as the reflection of f . Point re-
flections in the plane are, therefore, directly oriented. Reflections at a straight
line, on the other hand, are reversely oriented (Fig. 1.9).
Similar figures, twisted into each other

Fig. 1.10 The leaves of this plant are arranged in such a way that there are always
two opposite leaves at each stem node. The next pair of leaves is then rotated by
90○ ; at the same time the size of the leaves is scaled. The scaling factor differs from
node to node.

Fig. 1.10 shows an example which at first glance seems to be a spatial exercise:
How are the leaves of the plant distributed along the stem? In this simple
case, the leaves lie in parallel planes perpendicular to the stem. Two leaves
of equal size are always in the same parallel plane, opposite to each other
(rotation of about 180○ ). The next pair of leaves is of a different size and is
rotated by about 90○ . The change in size is worth looking at in more detail
(see Chapter 13).
1.2 Special points inside the triangle 13

1.2 Special points inside the triangle

Every “real” triangle (A, B, and C do not lie on a straight line) possesses
an orthocenter, a centroid, and a circumcenter. This may be familiar to most
people, but it is not at all trivial to prove. The circumcenter U is the easiest
to see.
A circle through three points – the circumcenter

A circle can be defined by three points. To put it differently: Every triangle

possesses a circumcircle.

Proof : Let us first construct the center of the circumcircle as the intersection of
two perpendicular bisectors: The locus of all points that are equidistant to A and
B is a straight line perpendicular to AB at the midpoint MAB of the line segment
AB. Furthermore, the locus of all points that are equidistant to B and C is a
straight line that is perpendicular to BC at the midpoint MBC of the line segment
BC. Both perpendicular bisectors intersect at U . Now we have U A = U B, and
also U B = U C, from which we can conclude U A = U C. This means that the third
perpendicular bisector must also pass through U . Consequently, U is the center of
the circle through A, B, and C. ⊙
The centroid: a point of physical significance
Centroids (or barycenters) play a significant role in physics, as they simplify
many considerations. The existence of an unambiguous centroid S in a trian-
gle is not trivial to prove. A vector-based proof for its existence can be found
in [9]. Its construction, on the other hand, is very simple: One merely needs
to connect the vertices of the triangle to the midpoints of the opposite sides
and intersect these so-called medians.
⊕ Remark : The term centroid has its roots in physics. There, a distinction is made
between vertex centroids, edge centroids, and volume centroids. For arbitrary figures
or polygons, these centroids are not identical [9]. For triangles, edge centroids and
volume centroids coincide, and for polygons with central symmetry, all centroids
coincide. This is why it is so useful to work with triangles for technical applica-
tions. In three-dimensional space, these extraordinary properties are inherited by
tetrahedra or polyhedra with central symmetry. ⊕
The orthocenter: a point of geometric significance
The altitude of a triangle is the straight line that is orthogonal to a triangle
side and passes through the opposite vertex. The following observation is
important for many later considerations and – once again – not at all trivial:

All three altitudes of a triangle pass through a single point, the orthocenter.
14 An idealized world made of simple elements

Fig. 1.11 Circumcenter, centroid, orthocenter, Euler line

Proof : Let us consider Fig. 1.11 on the left: The midpoints of the triangle sides form
the medial triangle that is similar to the original triangle, because all of its sides are
parallel to the respective sides of the original triangle. The perpendicular bisectors
of the original triangle intersect at a single point, as we have already proven, but
they can also be interpreted as the altitudes of the newly created medial triangle,
because they are (according to the definition of an “altitude”) perpendicular to their
triangle side and pass through the opposite vertex. ⊙
This proof is not only elegant but gives us a further advantage: it immedi-
ately implies the existence of the Euler line. (This is precisely what we mean
when we say that synthetic proofs are more useful than the purely analytical
ones.)
All three points are guaranteed to lie on a straight line
Fig. 1.11 on the left shows that the side lengths of the original triangle ABC
relate to the resulting pink medial triangle MAB MBC MCA at a ratio of 2 ∶ 1.
What is more, as the respective vertices lie on rays through a fixed center –
the centroid S – they can be transferred into each other by means of central
similarity. Given this central similarity, the orthocenter H of the big triangle
(the anti-complementary triangle) is transformed into the orthocenter of the
medial triangle, which, in turn, is also the circumcenter U of the original
triangle. The straight line through the center S that connects H and U is
the Euler line. Furthermore, we have HS ∶ SU = 2 ∶ 1.

The incenter – or: a circle on three tangents

The question about a circle that touches all three triangle sides leads us
to the incircle. Its center is determined by the intersection of at least two
angle bisectors. Each circle that touches two straight lines has to be centered
on such a bisector. If, for instance, we have intersected the angle bisectors
through A and B, then we have found the center of a circle that touches both
AC and AB, but also BC and BA, and thus, all three sides. Its midpoint
must, therefore, also belong to a third angle bisector – this means that all
bisectors pass through a single point.
Thus, three straight lines (tangents) can define a circle unless they pass
through a single point. However, this is not unambiguous: we have ignored
1.2 Special points inside the triangle 15

Fig. 1.12 Incircle center, three excenters

that there always exists a pair of angle bisectors that forms a right angle.
How many points do these three pairs define? Once again, it is possible to
use just two pairs for construction (the third pair provides no new solution).
All in all, there exist four solutions to this construction, namely, the incircle
and the three excircles. The question of finding a circle tangent to the side
lines of a triangle is, thus, ambiguous. At least there exists a “finite number”
of solutions! By the way: in general, neither the incenter nor the centers of
the excircles (the excenters) lie on the Euler line.

Two ways of defining a circle are still missing . . .

Fig. 1.13 Secant and tangent of a circle . . .

The following theorem is useful for some circle constructions:

16 An idealized world made of simple elements

For all secants of a circle through a fixed point P , the product of the distance
of P to the intersection points remains constant. This is called the power of
the point with regard to the circle.

Proof : We use the notations of

Fig. 1.14. According to Pythagoras,
the following is true: m2 +(r2 −x2 ) =
d2 , or m2 − x2 = d2 − r2 = constant.
However, m2 −x2 = (m+x)⋅(m−x) is
exactly the product of the distances
to the intersections (the power). If
P lies outside the circle, there ex-
ist touching secants whose intersec-
tion points move together. In such
cases, the product of the distances
Fig. 1.14 The power of a point
becomes the square of the tangent
distance. ⊙

▸▸▸ Two nontrivial circle constructions

Find a circle k that passes through two points P1 and P2 and furthermore
touches a straight line t, or alternatively, a circle k that passes through a
point P and touches two given straight lines t1 and t2 .
Solution:
Let us start with the first task (Fig. 1.15): We intersect the line P1 P2 of
the given
√ points with the given tangent t (Fig. 1.15) and determine the
power ZP1 ⋅ ZP2 of the intersection Z, which can easily be constructed
using Thales’s theorem. We have now found the contact point T on t. The
desired center of the circle lies on t’s normal at the contact point T and
the perpendicular bisector of P1 and P2 . We should not overlook the second
solution: The tangent section can be determined on both sides.

Fig. 1.15 2 points + 1 tangent Fig. 1.16 1 point + 2 tangents

1.2 Special points inside the triangle 17

Now for the second problem (Fig. 1.16): The locus of the centers of all circles
that touch both tangents t1 and t2 is the pair of angle bisectors (Fig. 1.16).
The solution circles lie in the same sector as P that is formed by t1 and t2 .
Consequently, the centers lie on one particular “half-line”. Let us draw an
arbitrary circle k touching t1 and t2 , with the center M located on this half-
line. Relative to Z, it is similar to the touching circles. The point P is thus
related, due to similarity, to two possible points P1 or P2 at the intersection
of k with the ray ZP . The rays M P1 and M P2 are transformed into the
parallel rays M2 P and M1 P . M1 and M2 are the desired centers of the two
solutions k1 and k2 . ◂◂◂

If you have gained an appetite for more oddities related to triangles and
circles, you are invited to ponder over three additional but no less intriguing
theorems by Feuerbach, Kiepert, and Morley. You may also jump forward to
the next section, where we will leave the drawing plane for a short excursion
into Euclidean space in order to explain how these topics apply in three
dimensions.
A circle through nine (!) points in the triangle
If we look at the triangle ABC, we may notice that the centroid S subdivides
the medians at a ratio of 2 ∶ 1. According to our knowledge about the points
on the Euler line, the ratio SU ∶ SH = 1 ∶ 2 holds for the orthocenter H,
the centroid S, and the circumcenter U of a triangle ABC. If we reflect the
triangle at S and shrink it at a ratio of 1 ∶ 2, we get a directly congruent
triangle A2 B2 C2 , the vertices of which are the midpoints of the triangle sides
(Fig. 1.17).
If we also transform the circumcircle, we get a circle k that passes through
the midpoints of the triangle edges. The circumcenter U corresponds to the
midpoint U2 = F of the smaller circle. With F S = s, we have SU = 2s, F U =
3s, and HU = 6s. Thus, F is the midpoint of the line segment HU . Given
these circumstances, Feuerbach had the following brilliant idea: shrinking the
circumcircle from the orthocenter H at ratio 1 ∶ 2 (without reflection) results
in the same circle k, because U travels towards F .
After this type of shrinking, the original triangle ABC corresponds to a
triangle A3 B3 C3 of half the size. The circle k contains these points as well.
The triangles A3 B3 C3 and A2 B2 C2 are congruent due to the same scaling
factors and have mutually parallel sides.
The six points are, thus, not arbitrarily distributed on the circle, but they
are mutually opposite to each other. Now we still want to show that the foot
of each altitude A1 , B1 , and C1 of the triangle ABC lies precisely on k: By
definition AA1 ⊥ BC and thus A1 A3 ⊥ A1 A2 , which means that k is Thales’s
circle over A3 A2 and thus A1 also lies on k. The analogue is true for B1 and
C1 . Thus, we have found no fewer than nine points on the circle! For the
purpose of illustration: Combinatorics tells us that if we pick any 9 points, 3
18 An idealized world made of simple elements

Fig. 1.17 Nine-point circle through the foot of each altitude, the side bisectors, and the vertices
of the medial triangle

from this set can be combined into circles in (9 ⋅ 8 ⋅ 7)/(3 ⋅ 2 ⋅ 1) = 84 different

ways.
The beer mat method
From the proof above, the following remarkable addition can be derived (as
an “encore”, so to speak): If we scale the nine-point circle k from C by the
factor 2, the bisectors A2 and B2 pass over to A and B, as well as C3 into
H. Therefore, there exists a circle that is congruent to the circumcircle and
contains two vertices of the triangle as well as the orthocenter. This statement
is true for all triangle points. So, the next time you find yourself waiting for
your meal at a restaurant, with three circular beer mats within your reach,
mark three points ABC at the edges of a single beer mat (which then becomes
the circumcircle) and position the other two mats so that they contain two
of these points each. The remaining intersection of the two beer mats is the
orthocenter.
A point of high importance
Let us now imagine, as in Fig. 1.18, that we have drilled three tiny holes into
a disk through which we have pulled three strings of equal length that are
tied together at the end G. (The triangle that is formed by the holes must
not have an angle greater than 120○ .) To the other ends of the strings, we
attach three equal weights. The weights will immediately stretch the strings
to form angles of 120○ : the three equal forces are then in equilibrium.
In sum, the system tries to place the weights at the lowest possible location.
With a given string length d and the distances x, y, and z of the point G
1.2 Special points inside the triangle 19

from the triangle points A, B, and C. The weights hang at depths d − x,

d − y, and d − z. The sum of these depths is, therefore, 3d − (x + y + z). If the
weights have to hang as low as possible, then the expression x + y + z must
be a minimum. It is also possible to define G as the point where the sum of
distances to the triangle vertices is minimal.

Fig. 1.18 Fermat point

The point G is ascribed to both Fermat and Steiner . It can be constructed

by intersecting the two circles or by intersecting at least two circles of cir-
cumference or, much more simply, by extending every triangle side outwards
into equilateral triangles and connecting the new points X, Y , and Z with
the opposite triangle points A, B, and C: The circumcircle of the triangle
ABZ is the circle of circumference of the segment AB, from whose points
the segment on the side of Z is visible at 60○ , and the segment on the side
G at the supplementary angle of 180○ − 60○ = 120○ .
In this special case, it should be obvious that the three lines actually intersect
at a point: If a point G is found as the intersection of two lines, then two
triangle sides can be seen under 120○ from G – and the third triangle side
must automatically be 120○ , since the sum of these angles must add up to
360○ .
Let us now discuss a generalization of this method.
How can an arbitrary triangle be converted into an equilateral
triangle?
Two triangles ABC and XY Z are perspective to a point if the connecting
straight lines AX, BY , and CZ pass through one point. A theorem that can
be traced back to Kiepert now states the following (Fig. 1.19, without proof):

Kiepert’s theorem: If isosceles triangles are constructed above the sides of an

arbitrary triangle ABC and they each share a common base angle ϕ, then the
new vertices XY Z form a triangle that is perspective to the original triangle.
For ϕ = 30○ , XY Z is an isosceles triangle.
20 An idealized world made of simple elements

Fig. 1.19 Kiepert triangle and Escher lizard

▸▸▸ An inventive tessellation

Fig. 1.20 Tessellation using the Escher lizard

A plane can be “tessellated” using equilateral triangles. If one shapes the

originating tiles in a particularly ingenious fashion – as M.C. Escher certainly
did – and uses them to fill the figure in Fig. 1.19, one gets a remarkably non-
trivial tessellation, as in Fig. 1.20. More information on tessellation will be
provided in Chapter 12. ◂◂◂

Are there other methods to convert an arbitrary triangle into an

equilateral triangle?
Triangle geometry is an extremely “mature” discipline of science. To make
a sensational discovery in this field in our day and age would come close
to a miracle. In the year 1904, the American mathematician Frank Morley
discovered the following: a theorem whose aesthetics recalls the elegant proofs
in Euclid’s Elements – despite the fact that the theorem is not fundamentally
geometrical, as we shall see soon.

Morley’s theorem: If the interior angles of an arbitrary triangle are trisected,

the intersections of the trisecting straight lines that are adjacent to the triangle
sides form an equilateral triangle.

Proof : The proof that follows originates, in essence, from D.J. Newman (1996).
We shall put the cart before the horse, so to speak, and begin with the result –
an equilateral triangle XY Z with side length 1. We then measure off, as in 1.21,
1.2 Special points inside the triangle 21
4π
the angles u, v, and w twice each. We require that u + v + w = . The resulting
3
auxiliary triangles should not overlap. If we can show that s = s and t = t∗ , then we
∗

are almost done, because by this “relabeling”, we obtain equality for the remaining
angles. First, we show that s + t = s∗ + t∗ .

Fig. 1.21 Morley’s theorem

2π
This is a result of s = π − u − w, t = π − u − v ⇒ s + t =
3
π 2π
and s∗ + t∗ = π − (2π − − (v + w)) = − u.
3 3
Moreover, by applying the law of sines in the triangles XZB and XY C, we obtain
1 d 1 e sin s e
= , = ⇒ = ,
sin s sin u sin t sin u sin t d
and by applying the law of sines in BCX, we get

d e sin s∗ e
= ⇒ = .
sin t∗ sin s∗ sin t∗ d
We may now conclude (because the sine function is monotonic) that

sin s∗ sin s
s + t = s ∗ + t∗ ∧ = ⇒ s = s ∗ ∧ t = t∗ .
sin t∗ sin t

⊙
22 An idealized world made of simple elements

The dilemma of angle trisection

As we have already mentioned, Morley’s theorem is not fundamentally ge-
ometrical. This means that it cannot be verified or “replicated” by compass
and ruler alone. Why do such approaches fail?
The task requires the division of an arbitrary angle into three equal parts.
From a mathematical standpoint, this involves the solution of an algebraic
equation of three. Despite our best efforts, our compass and ruler are not
enough to provide a solution.
And yet, the problem seems so simple! How would you trisect an angle?
With a set square and some quick mental arithmetic? Archimedes provided
a solution that requires some constructional agility but allows us to find a
mathematically correct solution:

Fig. 1.22 Trisection according to Archimedes

Let α be an angle (Fig. 1.22) through an isosceles triangle AM B. Let us

draw an auxiliary circle k (radius r = AM ) around M . Now we can mark
two points P and Q on our ruler at a distance r and move the ruler in such
a way that P moves along the elongated triangle leg AM and Q moves on
the circle k. If the elongation of P Q passes through B, then the section M Q
encompasses, along with the triangle leg AM , an angle of ϕ = α/3.
Proof : The proof of Archimedes is as ingenious as it is simple: We recognize two
isosceles triangles M BQ and P QM (the equilateral legs have length r). From ϕ =
∢P M Q it follows that ∢M P Q = ϕ. The angle ∢M QB is twice as large (2ϕ) because
the third angle in the triangle P M Q measures 180○ − ϕ − ϕ. This also means that
∢M BQ = 2ϕ, and furthermore that ∢P M B = 180○ − ϕ − 2ϕ. The supplementary
angle α is, therefore, 3ϕ. ⊙

However, can this be considered a simple compass-and-ruler-based solution?

Well, it is not as flawless as it may appear: Our meticulous fitting of the ruler
is not a precise construction but rather a matter of “trial and error”. Let us
try to be more precise and draw all positions of the ruler that pass through
B. We now get a so-called pencil of rays. Each ray possesses an intersection
Q with the circle k. From there, we can determine a point P at distance r. If
1.2 Special points inside the triangle 23

P lies on the elongation of AM , we were successful. What, then, is the locus

of all points P ? If the locus were a circle or an arbitrary conic section, then
everything would be simple enough – the intersection of a conic section and
a straight line can be determined with the help of ruler and compass. Alas:
the curve is of a higher degree – namely, it is a so-called conchoid, here a
limaçon (Fig. 1.23, left).

Fig. 1.23 A limaçon and an astroid

Let us give it a second try: Let us choose P on the elongated leg AM . We then
determine a point P on k at distance r (this yields two symmetrical solutions)
and we get Q. In the general case, the elongation P Q does not intersect k
at B, but we can try this many times. This leads to P Q enveloping a curve.
The right solution is the tangent from B to this. Yet, maybe this curve is
“simple”? No such luck! This time, it is a so-called astroid (Fig. 1.23, right),
which is also of a higher degree.
Many a courageous mathematician has “bloodied his nose” on angle trisection
– despite the fact that it has long been proven that there can be no solution
by ruler and compass.

Points at infinity
Points can be infinitely far away. In such cases, mathematicians speak of
points at infinity. (Should you find it difficult to imagine this concept, try
imagining a distance of something like a light year, which, for all modest
purposes, amounts to roughly the same thing.) Points at infinity are often
labeled by the subscript u: Thus, Pu is, if not otherwise stated, a point at
infinity. This may sound simpler than it is: We shall soon discover the mental
stumbling blocks and geometrical advantages that it implies.
Every straight line has a point at infinity. But why only one and not two?
Is it not true that straight lines proceed towards infinity in two directions?
Are these two, by any chance, the same point? Can two points on a straight
24 An idealized world made of simple elements

line be identical if one of the points is located one light year away in one
direction and the other is another light year away in the opposite direction?
Let us take some time and remember the following simple statement:

• Two different straight lines in the plane always possess an intersection.

Fig. 1.24 Parallel lines intersect, by definition, exactly once – but only “at infinity”!

If we do not allow for points at infinity, we must immediately make an excep-

tion: Two straight lines in the plane always possess an intersection, provided
that they are not parallel. If, on the other hand, we decide to be pragmatic
and allow for points at infinity, we can state the following: Two parallel
straight lines share one point at infinity (and not two, otherwise we would
once again encounter an exception). We shall soon discover that there is a
very reasonable explanation for this if we look at the problem “from a higher
perspective”. For now, let us remember:

• A point at infinity is determined by the direction of the straight line.

Parallel straight lines determine the same point at infinity.

This whole issue appears much simpler when considered “from a higher per-
spective”. Yet, in order to do this, we need to know more about planes, which
are discussed in the next section.
1.3 Elementary building blocks in space 25

1.3 Elementary building blocks in space

Let us now venture from the drawing plane into three-dimensional space.
This is not a dangerous step, because our bodies naturally occupy three-
dimensional space. Our spatial imagination will permit us to comprehend
the necessary trains of thought.
In space, points remain “infinitely small”, having no extent whatsoever. But
this is more of a philosophical statement. The stars of our universe could
hardly be called infinitely small; on a photo from a great distance, however,
they appear as points. Speaking of which: what appear to be stars in Fig. 1.25
are the relics of fireworks in the sky on New Year’s Eve . . .

Fig. 1.25 “Points” and “straight lines” in space

Points may, once again, be located at infinity, and the familiar u in the sub-
script is used for their notation. We will soon explain the mental difficulties,
as well as the advantages, of this type of thinking.
In the drawing plane, an infinite number of points results – if arranged in a
certain way – in an infinitely thin and infinitely long one-dimensional straight
line. We can also say that a straight line is “generated” by the translation of
a point.
Straight lines should be thought of as unbounded: Every line segment P Q
possesses an infinitely long straight carrier line g = P Q.
In space, however, we are still missing a third basic element:
Surfaces without curvature
In everyday speech, “plane” describes something horizontal: The High Plane
of Tibet, the Pannonian Plane, etc. Sloped, uncurved surfaces are commonly
referred to as “sloping sites,” but not so in geometry: A plane is formed
when we array an infinite number of parallel straight lines along another
straight line. It can be said that a plane is “generated” by the translation of
a straight line along another straight line. The generating straight line and
the directional straight line of the translation both “generate the plane.”
26 An idealized world made of simple elements

Fig. 1.26 A “plane” . . .

Fig. 1.27 . . . and a “sloped plane” in colloquial speech

In Europe, planes are denoted by lowercase Greek letters, such as α, β, ε,

ϕ, etc. This sometimes leads to problems with the notation of angles, which
are usually also named in this manner. For example: image planes are often
called π, while having nothing to do with the circle circumference to diameter
ratio π = 3.141592653589793 . . . It is, nevertheless, important to stick to this
convention, since the alternative is even more confusing (many Greek capital
letters, for instance, are identical to Latin capital letters, leading to planes
being possibly mistaken for points).
Let a be the generating straight line and b the straight line pointing in the
direction of the translation. The straight lines a and b intersect at S = a ∩ b
(read “a intersected with b”). The generated plane shall be called ε = ab (read
“a joined with b”). Every straight line g that intersects a and b lies entirely
in ε, since it intersects every parallel of a in ε. One can therefore write g ⊂ ε
(read “g in ε”).
Thus, an infinite number of straight lines lies in a plane. Each of these straight
lines also implies a point at infinity. What, then, is the locus of all of these
points? If we intersect all straight lines in the plane with another straight line
u in the same plane that is “almost infinitely far away”, then all intersections
lie on this straight line. Therefore, it makes sense to conclude:
• All points at infinity of a plane lie on a straight line, the line at infinity
of the plane.
We now have the following theorem without exception:
1.3 Elementary building blocks in space 27

• Two different planes always have a straight line of intersection.

Every plane stance, thus, defines a line at infinity, and since an infinite num-
ber of such orientations are possible in space, the following question arises:
what is the locus of all lines at infinity? Now we know how to proceed in an
analogous way: we intersect all planes of space with an “almost infinitely dis-
tant plane” ω and get an infinite number of almost infinitely distant straight
lines that, by definition, lie in ω, which thus becomes the plane at infinity.
Mysterious? Not at all! The next chapter deals with projections, and in it,
we shall again consider planes at infinity, but from a higher position. What
may seem confusing right now will by then seem completely normal.
The following theorems should already be perfectly comprehensible to us:

• A straight line g always shares an intersection point S with a plane ε. (If

the straight line is located entirely in the plane, then every point on the
straight line is an “intersection point.”)

• Three different planes ε, ϕ, ψ always intersect in one point S = ε ∩ ϕ ∩ ψ.

(If the point is a point at infinity, then the intersection lines of two planes
are mutually parallel.)

• If two parallel planes ε ∥ ϕ are intersected by a third arbitrary plane ψ,

then both intersection lines of ψ are parallel.
As a “proof,” the following could obviously be stated:
The chain of evidence is as follows: The three planes ε, ϕ, ψ share a point
at infinity. All intersecting lines of any two planes must pass through
this point and must, therefore, be parallel. By the way: the third, not-so-
evident intersection line ε ∩ ϕ is the line at infinity of the planes.
28 An idealized world made of simple elements

1.4 Euclidean space

We live in three-dimensional space and are, in general, more or less used
to three-dimensional thinking. We call this space the “space of perception.”
Distances and angles are of great importance to us. A sheet of paper can be
seen as the two-dimensional subspace of the space of perception. We are par-
ticularly familiar with geometry in two dimensions, as it is easy to – literally
and figuratively – keep an “overview” of its limited boundaries. However, if
the sheet of paper takes on astronomical proportions, it becomes more com-
plicated. After all, we are located on the globe, on a surface on which straight
lines do not even exist.
Perfectly straight lines do not exist in nature either. From physics we know
that not even light travels in perfectly straight directions: It is diverted by
the mass of the stars and star systems in whose vicinity it passes. The term
gravitational lensing is used to describe this phenomenon.
However, let us stay “on Earth”, just like Euclid , one of the greatest thinkers
in history. More than 2, 300 years ago, he operated a school of mathematics
in Alexandria and condensed, in his 13-chapter Elements (which are treated
as separate “books”), the entire mathematical and geometric knowledge of
his time. The Elements survived the turmoils of history (including two fires
at the Library of Alexandria) and were later translated – from Arabic into
Latin – and published in Europe, where they have henceforth formed the
basis for geometry right up until the 19th century.
Basic terms and axioms
First of all, Euclid defined two basic terms, namely, “points” and “straight
lines,” which can be put into geometrical relations by means of words like
“lie on,” “between,” “congruent,” etc. He then defined the so-called axioms
of geometry: sensible, immediately obvious, and non-contradictory rules that
can be assumed but cannot be proven, such as: “For two points, there exists
exactly one straight line on which they both lie,” or “Of three points on a
straight line, there is always one that lies between two others.”
Now he defines new entities (circles, etc.) about which assertions can be made,
and all of these assertions can be proven entirely by logical combinations of
the aforementioned axioms. Thus, gradually, one brick is laid upon the other.
It is essential that all definitions and proofs are based on secure knowledge
that can be traced back to the axioms.
This method was groundbreaking for its day – moreover, it made it possible
to verify the existing empirical rules and procedures of the Egyptians and the
Babylonians. Some of these rules could actually be proven and were, thus,
promoted to “theorems,” while others were, nevertheless, determined to be
useful approximations.
1.4 Euclidean space 29

A pillar of little support

One of Euclid ’s axioms says that it is possible to draw a unique parallel to a
straight line through a point outside of it. Mathematicians have been trying
for many centuries to prove this axiom by means of others, thus promoting
it to the status of a “theorem.” By the mid-19th century, it was proven that,
while geometry is possible without this axiom, several well-known theorems
would have to be discarded.
In the last section, we will learn more about non-Euclidean geometry. This
particular axiom of parallels is problematic when considering the number of
points at infinity of a straight line: If we do not want to accept that there
exists only one point at infinity for each direction of a straight line, then
much of space of perception geometry must inevitably collapse.
The sum of distances

Fig. 1.28 Distance circle Fig. 1.29 String construction of an ellipse

If we look for all points P in the plane that are at given distances d1 and
d2 from points F1 and F2 , then we must intersect the circles k1 (F1 ; d1 ) and
k2 (F2 ; d2 ). As we know, this yields (in general) two solutions, or no (real)
solution. If the circles merely touch, we speak of a double solution.
Thus, we can construct an ellipse as the locus of all those points P whose
sum of distances from two fixed points F1 and F2 is constant (= 2a).
The principal vertices A and B of the curve are found when the distance
circles touch each other. The auxiliary vertices C and D can be found when
both radii equal a. F1 and F2 are called focal points of the ellipse. For the
hyperbola, the difference of the distances from two fixed points is constant.
You can find out more about conic sections in Chapter 5.
The Apollonian circles
The search for all points P whose distances from two fixed points F1 and
F2 form a constant ratio lead to the Apollonian circles. If this ratio is 1 ∶ 1,
the circle “degenerates” into the perpendicular bisector. A straight line could,
thus, be interpreted as a circle with an “infinitely large radius” and “a center
at infinity.”
The perpendicular bisector is the set of points that is equidistant to two given
points. While this may be obvious, Apollonius’s more complex assertion needs
to be proven. For this purpose, a “full-blooded mathematician” would draw a
30 An idealized world made of simple elements

Fig. 1.30 Apollonian circles Fig. 1.31 Pair of angle bisectors

coordinate system that is defined by the following fixed points: Let F1 (0, 0)
be the point of origin and F2 (d, 0) a position on the abscissa. The condition
for a point of the plane P (x, y) is then as follows:
√ √
P F1 ∶ P F2 = x2 + y 2 ∶ (x − d)2 + y 2 = c = constant.

By squaring the equation and proceeding with a slight rearrangement of the

terms, one gets the equation of a circle with the midpoint on the x-axis.
This solution may be elegant but it would have been entirely unknown to
Apollonius of Perga (born in Asia Minor 262 B.C., died in Alexandria 190
B.C.). Let us not forget that Apollonius not only proved this theorem but also
authored eight tomes of fundamental work on the “geometry of positions,”
which deal with conic sections (Conica) in particular.
He first considered the following theorem:

The angle bisectors of a triangle divide the opposite side in the ratio of the
adjacent sides.

Proof : The theorem is immediately obvious, given an appropriate sketch (Fig. 1.31).
The two angle bisectors C form a right angle because they subdivide the angle γ
and the supplementary angle 180○ − γ, and therefore γ/2 + (180○ − γ)/2 = 90○ . Let
us now draw the parallel p through a triangle vertex B towards the inner bisector
that is simultaneously orthogonal to the outer bisector. The straight line p inter-
sects the extension of AC in a point B ∗ , and the triangle BCB ∗ is isosceles. On
the side AC, CA = b, and CB ∗ = a. By means of the inner angle bisector and p, this
affine ratio is transferred onto AB according to the theorem of intersecting lines:
AT1 ∶ T1 B = b ∶ a. The point B ∗ thus arrives at B while the point C at the division
point T1 on the inner angle bisector A remains fixed. According to the theorem of
intersecting lines, the following is, indeed, true: AT1 ∶ T1 B = b ∶ a. ⊙
Harmonic points
The exterior angle bisector divides the line segment by analogous reasoning
but with an inverted sign:

AT2 ∶ BT2 = b ∶ a or AT2 ∶ T2 B = b ∶ (−a).

1.4 Euclidean space 31

The pair (A, B) is said to separate the pair (T1 , T2 ) harmonically. Harmonic
pairs of points play a fundamental role in projective geometry, which is the
subject of a later chapter.
However, let us return to Apollonius. The theorem that involved the angle
bisectors might also have been proven by means of the sine law [9], of which
he was probably aware – but this was not necessary, as he proceeded in his
preferred way of “elementary geometry.”

Fig. 1.32 Proof of Apollonius

We are getting really close to the circle of Apollonius (Fig. 1.32): We have the
triangle ABP and both angle bisectors that cut out the division points T1
and T2 from AB, the position of which is exclusively dependent on the ratio
a ∶ b. The right angle that is formed by both angle bisectors almost “cries
out” for Thales’s theorem: C lies on the Thales circle above T1 T2 . Now we
shall change the triangle in such a way that a ∶ b remains constant. Thus, T1
and T2 also remain constant, and the Thales circle, likewise, remains fixed. C
moves on the Thales circle, which we may now call the circle of Apollonius.
All that remains is the product
Next to the ellipse (constant sum of distances to two fixed points) and the
hyperbola (absolute value of the difference of the distances is constant), Apol-
lonius also managed to fit the circles (constant quotient) into this scheme.
He presumably also tried to attain useful results from the product of the dis-
tances, but they simply do not yield such “beautiful” results (they are fourth-
order curves, which would have caused problems for the ancient Greeks, as
there were no computers that help us to make sense of them nowadays).
If the product of the distances to two fixed points F1 and F2 should be con-
stant for the points P , we are dealing with a so-called Cassini oval , which we
will encounter again when discussing the intersections of a torus (Fig. 6.80).
From the plane into three-dimensional space
All prior reflections about distances are equally applicable in space if we
rotate the locus curves about the axis F1 F2 . Circles, the centers of which
lie on the axis, turn into spheres. At their limiting cases (infinite radii), the
planes appear orthogonal to the axis by rotation.
32 An idealized world made of simple elements

Fig. 1.33 Cassini’s curves, defined as the locus Fig. 1.34 . . . and a stunning similarity to the
of all points with a constant product of the dis- cross-section of a pine tree trunk. Consider also
tances from two fixed points . . . Fig. 1.8, right.

During the rotation, the intersection points of two such circles move in planes
perpendicular to the axis. Ellipses trace ellipsoids, and arbitrary curves gen-
erate arbitrary surfaces of revolution. We will discuss these surfaces in great
detail.
Analogies on the sphere
The equivalent solutions on the sphere are equally interesting. For instance,
let us ask for the points on the Earth’s surface that are equidistant from two
fixed points: we then have to intersect two circles on the sphere. An analogous
ellipse construction leads to a spherical conic section. For all points P on such
a curve, the sum of minimal (spherical) distances from F1 over P to F2 is
equal.
Other distance lines and distance surfaces
Apart from circles and spheres, there exist other important distance lines
and distance surfaces.
What does the curve of points P in the plane look like if they have a fixed
orthogonal distance d from a straight line f in this plane? This question
yields a parallel pair of straight lines. By means of rotation of this object
about the axis f , we get a cylinder of revolution with radius d, whereas the
translation yields a distance plane to the given plane ϕ through f .

Fig. 1.35 Parallel pair and cylinder Fig. 1.36 Angle bisector and cone

Given two fixed non-parallel straight lines f1 and f2 in the plane, we are
looking for all points P that have the same orthogonal distance to both
1.4 Euclidean space 33

lines. A pair w1 , w2 of angle bisectors can be found. The two angle bisectors
w1 and w2 are always perpendicular: Let α = ∢f1 f2 be the mutual angle of
both given straight lines. Since these are not oriented, one could also have
measured β = 180○ − α (the supplementary angle). For the angle ω = ∢w1 w2 ,
the following is true: ω = α/2 + β/2 = α/2 + (180○ − α)/2 = 90○ .
In this case, a rotation about an axis does not result in an analogous theorem
in spatial geometry. Nevertheless: If we rotate about one of the bisectors, such
as w1 , the straight lines f1 and f2 will lie on the same cone of revolution. The
circle P that touches both straight lines becomes an inscribed sphere that
touches the cone, and the rays through the contact points form an additional
cone of revolution that intersects the first cone perpendicularly along a circle.
The translation of the planar situation in the direction of the normal to the
carrier plane is much more interesting. It yields a pair of angle bisector planes,
on which all points are located that are equidistant to two given planes.
We can conclude that it is often possible to derive spatial analogies from
two-dimensional situations.
34 An idealized world made of simple elements

1.5 Polarity, duality and inversion

Infinity can mean many things

Until now, we have used the term dimension many times. A point, for
instance, is zero-dimensional – it has no extention. A straight line is one-
dimensional – with an extention in precisely one direction. A plane is two-
dimensional. It can be seen as both a collection of points and a collection of
straight lines. Let us attempt a comparison: are there more points or straight
lines in a plane?
The question may strike you as somewhat insane. First of all: why should we
even want to find out? Secondly: it should be obvious that a plane contains
an infinite number of points, as well as an infinite number of straight lines.
Whatever could we hope to differentiate with a term like infinity?
The answer to this question leads to unexpected connections. Pondering such
supposed absurdities is, indeed, worthwhile.
A close relationship
We define the following geometrical operations in the plane that we shall call
polarity with regard to the circle:

Fig. 1.37 Polarity in the plane

Let us choose an arbitrary but fixed circle k with the midpoint M (Fig. 1.37).
If a point P lies outside this circle, then we have two tangents at k that touch
k at two points, Q and R. The connecting straight line p = QR is assigned
to the point P : P ↦ p. It is perpendicular to the line M P : p ⊥ M P . The
inverse mapping is also unambiguous: If an arbitrary straight line p intersects
the circle at the points Q and R, then we can define tangents through these
points that intersect at a point P : p ↦ P . Once again, M P ⊥ p applies. The
assignment p ↔ P is, thus, invertible, and we speak of pole P and polar p.
Even points at infinity are captured: Their polars are the diameters of k.
1.5 Polarity, duality and inversion 35

The case of P lying on the circle k poses no additional difficulties. In such

cases, p is the tangent of k in P .
If P lies inside k, we define its polar as follows: We draw an auxiliary straight
line p∗ ⊥ M P through P , whose pole P ∗ is the foot of the normal of the
straight line p from M that is to be determined. In the reverse case, if p
passes through k, we determine the foot of the normal P ∗ of M on p, whose
polar p∗ intersects M P ∗ in the pole P . Even the line at infinity of the plane
can be determined in this way: It is equal to the center M of the circle.
Therefore, the polarity of the circle consistently guarantees the unambiguous
relationship between points and straight lines in the plane, from which it
follows that the plane contains precisely as many points as straight lines.
It is, therefore, correct to speak of the plane as a two-dimensional point space
as well as a two-dimensional line space. These two spaces are mutually dual
spaces.
This circumstance can be further illuminated mathematically: The definition
of a point in the plane requires two numbers (the x-coordinate and the y-
coordinate). The same is true for the definition of a straight line (e. g., the
y-value on the y-axis and the slope angle).
Let us, once again, attempt to derive a spatial analogy (Fig. 1.38). If we
rotate our plane about the axis M P , then the circle k generates the sphere
κ, the tangent on the circle becomes a cone of revolution, the contact points
Q and R become contact circles of the cone and κ, and their carrier plane is
the polar plane π of the point P . The assignment π ↦ P works in the same
way. The polarity with respect to the sphere κ, once again, guarantees the
consistent assignment between points and planes of space (as for the plane at
infinity, it is the midpoint of the sphere). We can, thus, conclude that space
contains as many points as it contains planes.

Fig. 1.38 Polarity in space

Thus, Euclidean space can be interpreted as both a three-dimensional point

space as well as a three-dimensional plane space. Both spaces are mutually
dual spaces.
⊕ Remark : Mathematically speaking, a point in space is defined by three numbers
(e. g., through its Cartesian coordinates). The same is true for a plane (which is,
36 An idealized world made of simple elements

e. g., determined by the normal distance from the origin and the inclination angles
with respect to the xy-plane and the xz-plane). ⊕

Back to the two-dimensional case: If r is the radius of the circle, then we

have a simple relation between the pole P and its polar p ∋ P ∗ according to
Euclid’s theorem (Fig. 1.37):

M P ⋅ M p = r2 .

With this, we can prove the following theorem:

If a point A in the plane moves on a straight line b (Fig. 1.39), then its polar a
traverses a pencil of straight lines about the pole B of b. The assignment
pole↔polar is, thus, invertible, and pencils correspond to ranges of points, and
vice versa.

Fig. 1.39 The proof Fig. 1.40 The anti-inverse point

Proof : Let B be a pole and b ∋ B ∗ its polar (Fig. 1.39); then it follows that
M B = r2 /M b. Further, let A be a point on b. For its polar a ∋ A∗ , M A = r2 /M a.
a intersects the ray M B at the distance x. The triangles M BA∗ and M AB ∗ are
similar; it is also true that

M A ∶ M B ∗ = x ∶ M a ⇒ x = r2 /M B ∗ = M B.

⊙
⊕ Remark : The construction of polars of a point with respect to a circle so far
involves tangents of circles. We have to differentiate between points inside and
outside the circle. There exists a very simple substitute construction that always
works: As in Fig. 1.40, simply construct a right-angle hook, that yields a point Pa
that is “anti-inverse” to P . According to the altitude theorem, M P ⋅ M Pa = r2 .
Therefore, the point only needs to be reflected at the center of the circle in order
to lie on the polar. ⊕
It appears that while making the leap from the plane into space, we have
somehow lost track of straight lines; so, let us catch up on that. A straight
1.5 Polarity, duality and inversion 37

line g in space can be interpreted as the connection of two points A and

B (Fig. 1.38 on the right). The polar planes α and β correspond to these
points. Let us now consider the intersection line g ∗ = α ∩ β. If we are able to
show that a polar plane γ through g ∗ (γ ⊃ g ∗ ) can, indeed, be assigned to
every point C on g, then it makes sense to define g and g ∗ as polars. In that
case, straight lines correspond to straight lines, and therefore, line space is
self-dual!
Proof : Let us interpret the planar situation spatially. The circle can then be in-
terpreted as a sphere contour. The line g = AB lies in the image plane, the corre-
sponding polar planes α and β are then “projecting” – this means that they appear
as straight lines. Their intersection line g ∗ is perpendicular to the image plane and
appears as a point at the distance r2 /M g. This distance is equal for all points
of g. Thus, the linearity of planar polarity is transferred into three-dimensional
space! ⊙

t T∗
c∗ t T∗ c∗

P
P

c
c
k k

Fig. 1.41 Polarization of a curve c at a circle k: The tangents t of c are transformed into the points
T ∗ of the resulting curve c∗ . To the right: Conics (ellipses, parabolas, hyperbolas) are transformed
into conics.

With polarization as our tool, we can transform not only points but entire
curves (Fig. 1.41). When we polarize a curve c (as a set of points) at a circle k,
the result is a curve c∗ that is enveloped by a set of tangents. We can also
polarize the tangents of c and get the points of c∗ .
The whole transformation is reversible. That is to say, if we polarize c∗ at k,
the result is c. Remarkably, the simplest curve (of algebraic degree two),
namely a conic, is transformed into another conic.
Fig. 1.42 illustrates the polarization of a space curve at a sphere. The set of
all tangents t of the curve c is transformed into the set of all tangents t∗ of a
corresponding space curve c∗ . The process does, however, not transform the
curve pointwise. In Chapter 7 we will discuss the depicted special case (c is
a helix) .
38 An idealized world made of simple elements

t∗
P∗
c∗
c∗

c P c
t
Σ

Fig. 1.42 Polarization of a space curve c at a sphere Σ: The tangents t of c are transformed into
the tangents t∗ of the resulting curve c∗ . Note that the contact points P and P ∗ only correspond
indirectly to each other.

The principle of duality

We now notice the following: By polarization with respect to a circle in
the plane, the terms point and straight line are interchanged, as are the
operations connect and intersect. Thus, if a theorem contains only such terms
and operations, it can easily be converted into a correct dual theorem.
Let us consider a simple example:
• The connection of two points is a straight line ↔ The intersection of two
straight lines is a point.
The analogue is also possible in space: By polarization with respect to a
sphere in space, the terms point and plane are interchanged, as are the op-
erations connect and intersect. In this case, the term straight line remains
unchanged. Thus, if a theorem of spatial geometry contains only such terms
and operations, it can easily be converted into a correct dual theorem.
Here are some simple examples that illustrate this:
• The connection of two points is a straight line. ↔ The intersection of two
planes is a straight line.
• Three intersecting planes define a point. ↔ Three “connected” points de-
fine a plane.
• If two straight lines lie in a plane, then they intersect at a point. ↔ If two
straight lines intersect at a point, then they lie in a plane.
The following theorem may not be as self-explanatory:
• Space curves can be generated by a moving point as well as by a moving
plane.
1.5 Polarity, duality and inversion 39

We will often use this theorem: It holds the key to the so-called developable
surfaces, which play an increasingly important role in modern architecture.

An infinite amount and infinitely more

Polarity has shown that space contains as many planes as it does points.
From mathematics, we know that in a Cartesian coordinate system, every
point can be described by three coordinates. This is why we speak of three-
dimensional space (which should, more accurately, be referred to as a point
space), and for the same reason a three-dimensional plane space exists as a
dual counterpart of point space. But is the line space also three-dimensional?

Let us consider all straight lines determined by a fixed point P and an arbi-
trary point in a horizontal plane β (P ∈/ β) (Fig. 1.43). There exist as many
lines as there are points in this plane, because any point in β can be joined
with P .
If we now move the vertex of the pencil P on an arbitrary straight line g ∥ β,
we get the same number of connecting lines for each of the infinite positions
of P . This number alone – of all straight lines that intersect g – is as large
as the number of points in space! It would appear that there are many more
straight lines in space than there are points.

g∥β

Fig. 1.43 Variation of the vertex of the pencil. Through each point P , there exist as many straight
lines as there are points in the plane.

Now, the vertex of the pencil P should be able to vary all over the plane γ ∥ β.
There, this point should have the two coordinates (u, v). The points in the
base plane β also possess two-dimensional coordinates (x, y). Each choice of
the four parameters (u, v, x, y) yields a new straight line that is not identical
to any other. These parameters could be interpreted as coordinates in a four-
dimensional coordinate system (however it may look: we are not capable of
imagining it visually). We therefore say that line space is four-dimensional.
40 An idealized world made of simple elements

⊕ Remark : That may be easy to state, but it is quite something to consider that our
space of perception should have many more straight lines than points and planes!
Naturally, such an idea has consequences, and thus, we must not be surprised if,
from a geometrical point of view, many things that involve straight lines require
more effort.
For instance, we will see that it is not at all easy to view an arbitrarily positioned
straight line in such a way that it appears as a single point (in other words, that it
is “projecting”). Moreover, if we want to imagine rotations about arbitrary axes, we
have to be “geometrically trained,” and pretty well-trained at that. How simple it
is, by comparison, to imagine a rotation about a vertical axis!
It is all the more important to visualize complicated situations through specific
points of view, which is the task of descriptive geometry. We call this the “reduction
of dimension,” and the training of this skill is one of the highest goals of geometry
education. ⊕
⊕ Remark : In algebraic geometry, dimensions are counted by “degrees of freedom”:
Three points in space define a plane. Each point in the plane has two degrees of
freedom. Thus, the dimension of the plane space (considered as a space of points)
can be calculated by 3 ⋅ 3 − 3 ⋅ 2 = 3. By analogy, two spatial points define a straight
line where each point on the straight line possesses 1 degree of freedom. Thus, the
dimension of the line space is 2 ⋅ 3 − 2 ⋅ 1 = 4. ⊕

Inversion – not every transformation is linear

The polarity with respect to a circle (M ; r) in the plane defines a linear
relation between points and straight lines on a plane. The pole P corresponds
to the polar line p ∋ P ∗ where M P ⋅ M P ∗ = r2 . We can also assign the point
P ∗ to the point P . This relation is called inversion or reflection in a circle.
While this correlation is invertible, it is, intriguingly enough, no longer linear:
When P moves in a straight line, P ∗ moves on a circle.
The proof for this is mathematically easy to derive and can be found in [9].
Without going too much into detail (we will talk about it later): The inverse
of an algebraic curve of degree n is (generally) a curve of degree 2n. The
straight line is of degree one, the circle degree two. A conic section (usually)
corresponds to a curve of degree four, etc. The inversion is, thus, called a
quadratic transformation.
An allegedly inexplicable “anomaly”
One exception is particularly notable during inversion: a circle does not cor-
respond, as we would expect, to a quartic curve, but once again to a circle –
and if the circle contains the circle midpoint M , it corresponds to a straight
line (the proof for this will be given on p. 225)! Such things are irritating at
first: are we not dealing with a quadratic transformation after all?
In order to explain this occurrence, we would have to make an adventurous
“excursion” into a non-real world – but this would exceed the scope of this
book. Rest assured, the supposed anomaly can be brought under mathemat-
ical control.
1.5 Polarity, duality and inversion 41

Fig. 1.44 Inversion in a circle, Peaucellier -Lipkin inverting linkage

A mechanism for the realization

The fact that circles correspond to circles is not at all self-evident, but it can
be used to construct a mechanism that automatically executes the inversion
(Fig. 1.44).
Here, it is justified to ask the question, “Why do I need this?” Let us ex-
plain: By using this inversion, a mechanism can be built that converts a
pure rotation into an exact straight translation. This problem is not to be
underestimated in the field of mechanics: a rotation is something very prac-
tical – every engine rotates a driveshaft in one form or another. Using the
mechanism, a point is to be moved in a straight line without having to be
directly guided on something like a straight track. A certain H. Hart solved
this problem through inversion . . .
Spherical reflection
The inversion can be performed equally well in three-dimensional space. The
inversion circle is replaced by an inversion sphere (M ; r). The points P and
P ∗ are, once again, linked via the relation M P ⋅ M P ∗ = r2 . P ∗ lies in the
polar plane of P . The transformation P ↦ P ∗ is quadratic: Thus, planes are
mapped to spheres (usually). The proof can be made by rotating the planar
situation straight line ↦ circle about the normal of the straight line through
the point M .
Surfaces of degree n (see p. 169) are transformed into surfaces of degree
2n. Planes are also considered as spheres with “infinitely large radius” and
a “center at infinity.” Spheres and planes are transformed into spheres or
planes, i.e., “inversions preserve spheres”; see also Fig. 6.106).
42 An idealized world made of simple elements

Fig. 1.45 Reflection in a sphere Fig. 1.46 Inversion of a rotational surface in a sphere

By analogy to circle reflection, we may, at this point, speak of reflection in

a sphere. By the way, the physical reflection in a spherical mirror is, as a
concept, very different from inversion: It transfers space onto the surface of a
sphere. This means that three-dimensional space must be mapped onto a two-
dimensional surface, and this loss of dimension implies a simultaneous loss of
information: the situation in space cannot be reconstructed from the image
on the sphere. On the other hand, the inversion in the sphere is an invertible
relation between points in space that does not cause a loss of information.
Purism
The ancient Greeks considered a problem to be “exactly solvable” if it could
be constructed by compass and ruler. This is a justifiable demand for peo-
ple without computers. For many centuries, however, it has been possible to
exactly solve not only quadratic problems but also those of degree three and
four – by a detour into complex algebra. All other methods are approxima-
tions at best, including the computer-assigned ones, even if they claim to be
precise to 15 decimal places.
It is possible to be stricter still. Which problems are only solvable by com-
pass? It can be shown that every problem that is solvable by compass and
ruler is also solvable by compass alone. Of course, the separate steps become
more elaborate. The following example is said to have been devised according
to a suggestion from Napoleon Bonaparte and is famous for this reason.
The allegedly simple problem reads as follows: How can we find the center
K of a given circle k without the assistance of a ruler?
By compass and ruler, it can be solved in only a few seconds: One simply
has to choose two arbitrary points A and B on the circle k and use them
as centers to draw two circles a and b of equal size. The line joining the
intersections of a and b has to pass through the center. By repeating the
1.5 Polarity, duality and inversion 43

Fig. 1.47 One of the most useful tools until the end of the 20th century

Fig. 1.48 Napoleon’s assignment, including instructions

construction with a third arbitrary point C instead of B, the final center can
be found. It is, of course, the circumcenter of the triangle ABC.
Without a ruler, it is not that much more difficult. Six circles are needed in
order to find the midpoint – this is quite easy if you follow the instructions
(Fig. 1.48). However, the proof for this is pretty elaborate; it requires many
properties that we have come to know from inversion.

▸▸▸ A proof that isn’t: appearances can be deceiving . . .

In this chapter, we have proven several theorems in an elementary way, rely-
ing only on our imagination and our common sense. Consider the following
“proof” that “every angle is a right angle.” Even experienced geometers often
struggle to find the error in the proof.

Assertion: Every angle in the plane is a right angle.

Proof : Let ABCD be a quadrangle with the following properties (Fig. 1.49, upper
left):
(1) ∢ABC = 90○ , (2) ∢BCD = α =/ 90○ , (3) AB = CD.
44 An idealized world made of simple elements

Fig. 1.49 Our quadrangle, the point M , congruent triangles, M outside.s

The perpendicular bisectors m1 and m2 on BC and AD are not parallel due to (2)
and (3) and thus intersect at a point M (Fig. 1.49 lower left). Let us now compare
the triangles ABM and DM C (Fig. 1.49 upper right). According to precondition
(3) overlineAB = CD and thus BM = CM (M lies m1 ) as well as DM = AM (M
lies on m2 ). Both triangles are, thus, congruent and have, thus, the same angles:
α1 = ∢M CD = ∢ABM = α1′ . Since M BC is isosceles, it follows that α2 = α2′ and
finally α1 + α2 = α = α1′ + α2′ = 90○ .
Thus, the proof appears to be finished if M lies inside the quadrangle ABCD. This
point may still lie outside of the quadrangle (Fig. 1.49 lower right). In this case,
we get congruent triangles ABM and DCM as before and thus the equal angles
∢DCM = ∢ABM as well as ∢BCM = ∢CBM (the triangle M BC is isosceles).
This time, if we calculate the difference of the angles, we also get ∢BCD = α =
∢ABC = 90○ . ⊙
As we can see, appearances can be deceiving – even in two dimensions! Can
you find the error in the proof? ◂◂◂
1.6 Projective and non-Euclidean geometry 45

1.6 Projective and non-Euclidean geometry

In this section, the geometrical circumstances will be presented from a dif-
ferent point of view. This way, projective geometry becomes the basis for
many new insights – and the explanation for otherwise perplexing oddities
of geometry.
Why, for example, is a circle defined by three points and a parabola by
four, but it takes five points to define an ellipse and a hyperbola? Why do
two circles possess a maximum of two intersections, but two arbitrary conic
sections up to four?
The solution lies in two-fold arrangements: Firstly, the line at infinity should
not be considered as unreachable. Secondly, non-real elements of the drawing
plane, and complex conjugate points and straight lines in particular, should
also be accepted.
Working with elements at infinity can actually be made plausible through a
“dimensional jump” from the drawing plane into space. The “complex exten-
sion” of the plane requires a jump into four-dimensional space which cannot
be visually imagined.
Interestingly enough, geometries that deal with higher dimensional spaces
actually exist and achieve important results for “regular” geometry.
Let us discard lengths and angles
Projective geometry is a very important field of geometry – it limits regular
geometry insofar as it completely discards all congruence relations. Terms
such as the length of a line segment or the size of an angle become irrelevant.
Hence, the concept of parallelism has to be changed. Pythagoras’s theorem
and the Law of Right Angles (see page 75) from descriptive geometry cannot
be stated in projective geometry.
The incidence relation becomes all the more important: Do three points lie on
a straight line? Do three planes intersect in a straight line? The conclusions
can be directly carried over to Euclidean geometry, even if it allows for points
at infinity, which we have defined as the intersections of two parallel straight
lines.
Projective geometry makes use of terms such as point, straight line, and
plane – as well as incidence relations such as “lies on” or “intersects”. This
immediately recalls the duality principle that we have already discussed.

For each theorem of projective geometry, the corresponding dual theorem is

automatically correct and does not need to be proved.

The proof of the dual statement can be omitted because the original state-
ment merely needs to be dualized, i. e., the terms points, lines, planes have
46 An idealized world made of simple elements

to be changed to planes, lines and points respectively; ∪ and ∩ have to be

interchanged, while incidence relations are retained.
The dimensional jump
Let us consider the Euclidean plane πe , this time “embedded” in space: It
could, for example, be the locus of all points with height zero. Now let us
project all elements of this plane from a center point Z that does not lie in
πe , such as a point on the z-axis other than the origin.

Fig. 1.50 We escape into space

Each point gives rise to a line (ray) that is the connection of a particular
point with Z. Thus, lines become planes if projected from Z. The new objects
are elements of a star with apex Z and planar geometry thus becomes the
geometry in a star in space.
In order to understand the situation planimetrically, let us intersect the pen-
cil with an arbitrary plane π that does not pass through Z. Now, any point
in πe corresponds to a point in π and any straight line in πe corresponds to
a straight line in π.
Ranges of points and pencils of straight lines
If a point Pe in πe moves on a straight line ge , then the corresponding pencil
of rays moves in a projection plane γ – thus, the corresponding point P in
π moves on a straight line g = π ∩ γ. This functions unambiguously in both
directions. The (dual) analogy may be drawn for pencils of straight lines in
πe and π.
The mapping π ↦ πe is a perspectivity with axis a = πe ∩ π and center Z.
Projectivity = likewise linearity
In a last step, let us, once again, rid ourselves of the idea that we have escaped
to space. Let us “merge” both planes by arbitrarily positioning them next to
each other. The alternating linear relationship (collineation) between both
fields is preserved – but in general, we now have neither a center nor an axis
of collineation. If we are working in the Euclidean plane, we are performing
1.6 Projective and non-Euclidean geometry 47

Euclidean geometry, and if we work in the projective plane, we are performing

projective geometry.

A conic section always remains a conic section

Fig. 1.51 Intersections of a cone of revolution

In the pencil through Z, a circle ke in the Euclidean plane πe corresponds to

a circular cone that is (generally) not right, i. e., it is not a cone of revolution.
This is what we call a quadratic cone. Common sense might suggest that the
cone should be elliptic, but if the section plane π is “steep enough,” it may
yield a hyperbola as intersection curve or, as a special case, a parabola. We
would then have to speak of a hyperbolic or a parabolic cone – though it
is the same cone. Let us, therefore, simply state that the cone is quadratic,
which relates to its algebraic property that it always possesses two intersec-
tion points with an arbitrary straight line (if mere touching counts as two
intersections and non-real solutions are allowed).

Fig. 1.52 Affinely distorted cones

In Euclidean space there exist no elliptic, hyperbolic, or parabolic cones – only

quadratic cones. It is, therefore, not necessary to distinguish between the various
types of conic sections in projective geometry.
48 An idealized world made of simple elements

The line at infinity is suddenly “completely normal”

Is it really impossible to tell these types of conic sections apart? What is the
difference, planimetrically speaking, between an ellipse, a hyperbola, and a
parabola? The number of points at infinity is obviously important: A hyper-
bola has exactly two, a parabola exactly one, and an ellipse none (in real
space).

Fig. 1.53 Arbitrary quadratic cones

The Euclidean plane has an infinite amount of “points at infinity” – any point
at infinity belongs to all straight lines parallel to a certain direction. In the
first chapter, we have decided that each straight line only has one point at
infinity even though we could have also assumed two. This will now prove to
have been very reasonable.
If we now draw a straight line of each direction from Z, then they form a
plane that is parallel to πe . It intersects π in a “completely regular” line u
(if π is not parallel to πe ), namely, the image of the line at infinity of πe .
The distance to an arbitrary point at infinity Ue of the Euclidean plane is,
according to the construction, parallel to the pencil straight line ZUe , which
intersects the projective plane π in a point U on the image of the line at
infinity u (Fig. 1.55).

Fig. 1.55 This is how a line at infinity can be

Fig. 1.54 Three types of conic sections derived.

Fig. 1.54 now shows symbolically how we can distinguish the three types of
conic sections: by means of their relative position to a regular straight line u.
1.6 Projective and non-Euclidean geometry 49

Non-Euclidean geometry
In the previous sections, we have discussed the geometry of Euclid , which
is essentially equivalent to the kind of geometry that we are familiar with
and use every day. It is based on a system of axioms, a collection of self-
explanatory assumptions that are not further proven.
Theoretically, it is now possible to create new geometries that are based on
other axioms. However, this entails the enormous restriction that the entire
system of axioms must be completely non-contradictory. It should, therefore,
be impossible to derive two theorems from the system that prove mutually
opposing things.
Moreover, a system of axioms should, if possible, be independent and com-
plete, even though this aspect is not essential.
In fact, such systems of axioms have already been found. The corresponding
geometries can be seen – like the Euclidean – as some kind of playing around,
with terms and axioms as rules of the game. As fascinating as this game
may be: if the geometry in question stands in strong contradiction to our
usual understanding, and if it does not produce any results that are worth
mentioning, then it is doomed to be insignificant.
Euclid ’s system of axioms had a real “bone of contention,” namely the

Parallel postulate: Through each point P in a plane there exists exactly one
line that is parallel to a given line not through P .

For many centuries, nobody doubted this self-evident assertion. It has, how-
ever, been impossible to prove it by means of the other axioms, thus neces-
sitating its status as an axiom of its own.
Gauss, Lobatschewskij , and Bolyai realized almost at the same time (but
independently of each other) that it could be replaced by contrary axioms
without causing the entire system of axioms to become contradictory. The
thus emerging geometries were called non-Euclidean geometries.
It is basically possible to differentiate between two types of non-Euclidean
geometries: hyperbolic geometry, where there exists an infinite number of
parallels through each point, and elliptic geometry, which was discovered
somewhat later by Riemann and contains no parallels at all.

What’s so fascinating about this?

What is fascinating about these geometries is that they play around with the
concept of parallels, and thus, with the concept of infinity. “Locally speak-
ing,” “our” Euclidean geometries are barely different from their non-Euclidean
cousins. Thus, any creature that inhabits its own small space and is not con-
cerned by far-away objects would not be aware of the inherent difficulties. No
wonder that modern physics, which concocts theories about the formation
and the expansion of the universe, is very interested in these geometries.
50 An idealized world made of simple elements

We will further see that things appear “entirely normal” if we look at the
issue “from a higher perspective”: If we project the objects of the elliptic
plane from a point off the plane, we get Euclidean geometry in a star that
should be quite familiar to us by now. If we now intersect the rays of the star
with a sphere centered at the apex of the star, we are, once again, dealing
with geometry in a sphere.

Fig. 1.56 Left: sphere model of the elliptic plane (measurement of angles). Right: On a sphere, you
can construct in a similar way as on a plane. This image specifically illustrates the so-called “string
construction,” which yields a spherical conic section. It shows that, as in a plane, the tangents of
the curve bisect the angle between the lines to the foci. The points S1 and S2 can, therefore, also
be identified as focal points: the pencil of rays through one point is reflected by a pencil of rays
through the other point.

If the elements of the hyperbolic plane are projected onto a surrounding

sphere, then they can be thought of in much more traditional Euclidean
terms. A hyperbolic creature can never leave the boundary of the sphere.
A straight line of the hyperbolic plane corresponds to a projected circle on
the sphere. It possesses two intersections with the world boundary, and thus,
two “points at infinity.” Through an arbitrary point on the sphere, there now
exist two different projecting circles through these points at infinity. Every
point on the straight line that lies outside the sphere and plays no role in
hyperbolic geometry. Therefore, it can be stated that all straight lines that
occupy a whole sector are mutually parallel in the hyperbolic sense.
If we now dare a dimensional jump and take advantage of the conditions in
three-dimensional non-Euclidean space, we have almost arrived at the ideas
of Stephen Hawking, who speaks of unimaginable cosmic phenomena that, if
seen from a higher dimension, are not at all unusual.

Geometry on the sphere

Let us perform non-Euclidean geometry on the surface of a sphere. We define
“line segments” as the shortest connections between points (great circles). The
equator and all circles of longitude are, thus, “straight lines”; arbitrary circles
of latitude are not. By means of such straight lines, it is now possible to
define objects such as triangles.
1.6 Projective and non-Euclidean geometry 51

Fig. 1.57 On a sphere, you can construct in a similar way as on a plane. This image specifically
illustrates the so-called “string construction,” which yields a spherical conic section. It shows that,
as in a plane, the tangents of the curve bisect the angle between the lines to the foci. The points
S1 and S2 can, therefore, also be identified as focal points: the pencil of rays through one point is
reflected by a pencil of rays through the other point.

It is thus possible to describe geometry on the sphere just as well as in

the plane. It must be remembered that the sum of angles of such a spherical
triangle always exceeds 180○ . All Euclidean axioms are still applicable, except
for the parallel postulate: On a sphere, two different straight lines (great
circles) always share two common points and can, thus, never be “parallel.”

Fig. 1.58 The lengths a, b, and e at planar and spherical conics

Fig. 1.57 and Fig. 1.58 shall illustrate how constructions of the Euclidean
plane can be transformed onto the sphere. Angles and lengths have to be
interpreted differently. The lengths a, b, and e in the Euclidean plane corre-
spond to the arc lengths of great circles on the sphere – and, thus, to angles.
52 An idealized world made of simple elements

The curvature of space

Fig. 1.59 In the vicinity of large masses, space appears to be curved – and thus, non-Euclidean.

Going one dimension higher, we have reached the ambient space. Just as the
“inhabitant of a sphere” cannot immediately detect that “their” supposedly
straight lines are actually curved (see also application p. 264), we do not
really know the (eventual) trajectory of light rays (Fig. 1.59)! We know from
modern physics that light is deflected by large mass (and even “swallowed”
by black holes). In the vicinity of large masses, space appears to be curved
– and thus, non-Euclidean. Since Einstein’s general theory of relativity, we
have to look at things differently.

Fig. 1.60 Curved spaces

⊕ Remark : The question about the large-scale Euclidean nature of space is not yet
conclusively answered, but as soon as astrophysicists surmount this challenge, they
will also have answered a seemingly unrelated issue: Will our universe, which was
created in a Big Bang, collapse in the far future in a Big Crunch?
Interestingly enough, the answer to these questions appears to lie in the determi-
nation of the mass of Neutrinos. Neutrinos are inconceivably small particles that
travel through space at nearly the speed of light and effortlessly penetrate mass
on their way. Thus, the question about the largest seems intimately linked to the
question about the smallest . . . ⊕
2 Projections and shadows:
Reduction of the dimension

This chapter deals with projections from space onto a plane. Such a projec-
tion is the basic idea of perspective, which was first explored in the Renais-
sance. While this kind of projection makes things more vivid, it does not
make them easier. Classical Descriptive Geometry, which emerged at the end
of the 18th century, uses special kinds of projections.
In particular, these are the normal projections in the “principal directions.”
This deliberate dimensional jump can often simplify the solution of problems.
Shadows do not only make images more interesting, they also offer addi-
tional information: While a spatial situation can never be unambiguously
reconstructed from one image, shadows make this task much easier.
One of the goals of Descriptive Geometry is to describe spatial objects by
means of two “associated” normal projections. In these, it is then possible to
manipulate the images with well-developed techniques in order to measure
lengths and angles, and if necessary, to reconstruct the object in space.

© Springer Nature Switzerland AG 2020 53

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_2
54 Projections and shadows: reduction of the dimension

2.1 The principle of central projection

The points in space are pushed into a plane
Let us imagine an arbitrary fixed projection plane or image plane π in space
and a fixed center of projection Z that does not lie in π (Z ∈/ π). A straight
line p = ZP through an arbitrary point P in space shall be called a projection
ray. P must, therefore, not coincide with the center of projection (P =/ Z).
The intersection of p with the image plane results in the projection P c (the
image point) of P .
It does not matter if the point is in front of the image plane, behind it, or
if it is even behind the center of projection. All points that lie on the same
projection ray have the same image point. How else could the entire space
be squeezed into a single plane?

Fig. 2.1 Central projection of points with a viewing cone or a viewing pyramid

Photography is (essentially) a central projection. The eye (the optical center

of the lens) lies between the object and the image plane (the light-sensitive
surface).
Due to the limited viewing angle of the lens, only points within the “viewing
cone” are actually pictured. Furthermore, due to the rectangular form of the
negative (or the light-sensitive chip), this cone is further reduced to a frustum
(“viewing pyramid”).
Points at infinity are absolutely normal
If we are lucky, the projection ray p will intersect the image plane in a “proper
point” P c = p ∩ π. This rule is equally valid for points at infinity. Points at
infinity Pu are given by the direction of a straight line: the projection ray p
is, then, simply parallel to this direction and intersects the image plane at
a point Puc , which is no longer at infinity (Fig. 2.2 left). Points at infinity in
the image plane are exceptions as they lie in π and thus do not have to be
projected. Their image points are, therefore, also points at infinity (Fig. 2.2,
right).
2.1 The principle of central projection 55

Fig. 2.2 After central projection, points at infinity become proper points (see left). An exception:
the points at infinity in the image plane are identical to their projections, and thus remain fixed
(see right).

Points can “disappear”

In unlucky cases, the projection ray p may lie parallel to the image plane.
The image point thus resides at infinity even though the point itself does not
need to be a point at infinity: it only needs to lie in the particular plane πv
that contains the center Z and is parallel to π (πv ∥ π, πv ∋ Z). Such a point
is called a vanishing point and the plane πv is called the vanishing plane
(Fig. 2.2, right).
We thus see: all points in space (including the points at infinity) that do not
lie in the vanishing plane have image points that are not points at infinity.

What happens to straight lines?

Let us now map a straight line in space, which is, of course, done by mapping
all of its points. You will now correctly assume that it is only necessary to
map two points A and B of a straight line g in order to get the connecting
straight line of the image points as an image of the entire straight line:
g c = Ac B c . This is easy to prove: the projection rays through all points of g
form the plane Zg, which, indeed, intersects the image plane π in a straight
line: g c = (Zg) ∩ π.

A linear mapping
Central projection is, thus, “line preserving.” This should not come as a sur-
prise, as we know (from photography) that straight edges of buildings or
floor ledges also appear as straight lines. Every day, we see hundreds of pho-
tographs in books and in magazines, and therefore, this observation appears
“natural” to us. In a separate chapter concerning perspective images, we will
discuss what exactly is so “natural” about it.
In any case: central projection is “simple” because it is linear, or line pre-
serving. We have already dealt with a linear mapping, namely, the polarity
with respect to a circle or a sphere; however, while in those cases, points
were transformed to straight lines or planes, central projection maps points
to points and straight lines to straight lines.
56 Projections and shadows: reduction of the dimension

The mapping only works in one direction

There still remains a big difference to polarity: the latter works in both
directions, whereas central projection is not invertible. In other words: a
spatial object cannot be “reconstructed” from one photograph (Fig. 2.3, left).
Additional data is needed; for instance, it is essential to know whether or not
the spatial object in question is a cuboid. If that is the case, then the spatial
object can, indeed, be reconstructed true to scale (Fig. 2.3, right).

Fig. 2.3 Two pairs of objects with identical projections

“3D scanning” has become a highly relevant problem recently, and one that
has not yet been solved to complete satisfaction. Laser beams are often used
to detect objects – a process that not only determines the geometric projec-
tion but also the distances from the projection center. However, this method
is problematic for very large objects (such as reconstructions from the air) or
microscopically small objects. In order to guarantee an accurate reconstruc-
tion of complicated objects, one usually needs a larger number of high-quality
photographs.
From my perspective . . .
Central projection (usually called perspective) will be a major concern in
Chapter 9. Perspective images are of great importance in painting and archi-
tecture, and most people find “interesting perspectives” appealing. This may
be due to the fact that perspective images convey a very subjective and per-
sonal impression. After all, it is often said that “from my perspective, things
seem such and such.”
A closer examination of objects at infinity
Central projection can also serve as a linking tool between Euclidean and
projective geometry, where different rules need to be established.
The mere fact that a central projection allows us to pull points and lines at
infinity into the image plane and thus within our reach is a fascinating fact,
isn’t it? The usefulness of assigning only one point at infinity to a straight
line once again becomes apparent (in central projection, it becomes a regular
point on the straight line). The term “line at infinity” in a plane now becomes
comprehensible: if one projects all points at infinity that lie in a plane ε that
is not parallel to the image plane, then all projection rays lie in a plane ε
through the center Z that is parallel to ε. It intersects the image plane in a
“proper” straight line.
2.1 The principle of central projection 57

Central projection of planes

The term plane at infinity is not as easily understood using central projection
– although the plane at infinity can be mapped point by point, all these image
points put together fill up the entire image plane. The planes that contain
the center Z are exceptions: they contain an infinite number of projection
rays that only have a single image point, and thus, the planes through Z
appear as straight lines.

Fig. 2.4 Parallel lighting (roof of a tent, seen from below)

⊕ Remark :
Central lighting
Tacitly, we have always considered an “eye point” or a center of sight instead of a
general projection center. It should be noted that central projections often occur in
relation to shadows. The light source need not be infinitely far away (even though
this is often assumed for the Sun and the Moon). Besides, many “finitely far” light
sources can only be simplified as points in a broad sense.

Fig. 2.5 Two-sheeted hyperboloid of revolution in theory and in practice

Not even a conic spotlight provides light rays that exclusively originate in one point.
This would need a so-called two-sheeted hyperboloid of revolution (Fig. 2.5), which
we will discuss later. Still, it is true that every finite point-shaped light source
produces a central projection from space to the shadow-receiving surface (which is
often a shadow-receiving plane). ⊕
58 Projections and shadows: reduction of the dimension

2.2 Parallel projection and normal projection by

means of restriction
Roughly speaking, central projection is comparable to the photography of
space. The farther the center of the lens (which, in geometric terms, means
the projection center, the eye point or the light source), the more likely it
is that the viewing rays (projection rays or light rays) are parallel. Fig. 2.8
shows a photo from the air, where the relative distance is already very large.
Fig. 2.7 shows a normal projection, i. e., the center of the lens is infinitely
far away, whereas Fig. 2.6 shows a general (oblique) parallel projection.

Fig. 2.6 Oblique parallel projection . . . Fig. 2.7 . . . and normal projection

If the center of the projection (eye point or position of the light source) is
a point at infinity, the rules of central projection still apply. However, the
viewing rays or light rays are parallel; this implies an important property:

Parallel projection preserves the parallelity of lines: If two straight lines in space
are parallel, then their parallel projections are also parallel (unless they appear
as points).

Proof : Let a ∥ b be two parallel straight lines in space. All projection rays through
the points of a and b form two parallel planes α and β. These two planes are to
be intersected with the projection plane π in order to find the images of a and b.
The three planes α, β and π share a point at infinity U , which remains fixed during
projection as it already lies in π. The image lines ap = α ∩ π and bp = β ∩ π pass
through U and are, therefore, parallel. ⊙
However, the converse of the theorem does not hold:
If two straight lines have parallel images, we may not conclude that the cor-
responding lines in space are parallel.
From the preservation of parallelity, we can immediately infer the preserva-
tion of affine ratios under parallel projection:

Parallel projection preserves affine ratios, i. e., it preserves the ratio of lengths
of segments on the same (or a parallel) straight line.
2.2 Parallel projection and normal projection by means of restriction 59

Proof : Let g be a straight line with three points P , Q, and R on it. The trace point
G = g ∩ π of the straight line remains fixed during (parallel) projection (⇒ g p ∋ G).
Both legs g and g p lie in the plane and are now intersected by parallels (projection
rays) through P , Q, and R. From the intercept theorem, we can conclude P Q ∶
QR = P p Qp ∶ Qp Rp . ⊙
Preservation of parallelity and preservation of affine ratios are invaluable
when drawing an “oblique projection” or when reconstructing a spatial situ-
ation from an oblique projection. We consider a Cartesian coordinate system
(and its origin U ) to be attached to an object (such as a cuboid). The three
coordinate axes x, y, and z are mutually perpendicular lines in space and
concurrent in U . We fix the unit points on the axes.
▸▸▸ A simple depiction by means of a horizontal projection
In practice, one type of parallel projection is widely used: the horizontal
projection (also called horizontal axonometric projection). In this projection,
the image of the x-axis is chosen arbitrarily (for example, at an angle α ≠ 0○
to the horizontal, Fig. 2.9). The image of the y-axis is perpendicular to the
chosen image of the x-axis, and the image of the z-axis “looks upwards.”
x- and y-values remain undistorted while the z-values may be arbitrarily
shortened (although it is also possible to leave them undistorted).

Fig. 2.8 Normal projection . . . Fig. 2.9 . . . and horizontal projection

Fig. 2.9 demonstrates how pictures of some descriptive quality can be “con-
jured up” using only a small number of lines. This procedure is especially
useful for objects that have a complicated floor plan but whose construc-
tion barely changes with height (such as prisms and cylinders of revolution).
Obviously, this applies in particular to city maps:
All streets and floor plans can thus be depicted and remain undistorted
(except for the scale); it is, therefore, possible to measure lengths and angles
in horizontal planes. The oblique view presents information in the same image
about the relative height and appearance of the buildings. ◂◂◂
60 Projections and shadows: reduction of the dimension

Fig. 2.10 Oblique or . . . Fig. 2.11 . . . normal projection: informative

Fig. 2.12 Perspective views with horizontal axes come close to normal projections with a hori-
zontal axis. On the left: An architectural model by Lukas Galehr. On the right: Since it can be
assumed that the coral fishes are similar to each other, multiple views are combined in a single
image.

Fig. 2.11 and Fig. 2.12 show (good approximations to) normal projections
of a scene. These images allow conclusions that are much more difficult to
deduce based on central projections. It can, for instance, immediately be said
whether or not the three objects in Fig. 2.11 are positioned uniformly or if
the layer of snow on all three chairs is equally thick.
▸▸▸ Reconstruction of a spatial situation based on shadows
How high above the ground is the cat in Fig. 2.13?
Solution:
The picture shows two near-parallel projections (long telescopic lens) of a cat
“galloping” on a slightly inclined hill. Let us pretend to be Sherlock Holmes
for a moment: The picture on the right almost seems like a composite pho-
tograph: Only on close inspection do we recognize a possible shadow on the
right side. The left image (taken only moments later) confirms our assump-
tion.
The sunlight, therefore, falls at a very flat angle. This means that the scene
must have taken place only shortly after sunrise or shortly before sunset. The
dew is still glittering on the grass – so it must have been morning. So then,
is it possible to say how high the “flying” cat in the right image might be?
2.2 Parallel projection and normal projection by means of restriction 61

Fig. 2.13 How high above the ground is the cat? The shadow provides the answer!

Let us be a little broad in our assumptions: The hill is obviously an inclined

plane. The light comes quite precisely from the side. One anticipates the
shadow P s of the left back paw P . We can now draw a triangle P P ′ P s ,
whose side P ′ P s is parallel to the hill direction (P P ′ is perpendicular). The
distance P P ′ indicates the distance of the jump: If the cat’s pelvis height is
assumed to be 30 cm (the animal appears to be relatively young), then the
height of the jump would be almost half of that – perhaps about 12 cm.

Fig. 2.14 A scarab beetle is taking off from a car roof. Shadows and reflections provide useful
information about the spatial situation.

Fig. 2.14 shows another situation where you can train your skills. Here, we
have reflections in the base plane (a car roof) in addition to the cast shadow
and therefore three projections in one image.
⊕ Remark : In our considerations, we have used the preservation of parallelity under
normal projection and also the preservation of affine ratios under parallel shadows.
On an actual photograph taken with a physical lens, we would need to proceed
62 Projections and shadows: reduction of the dimension

with much greater precision. We will later return to this problem and address it in
greater detail. ⊕ ◂◂◂

The following theorem can sometimes be very useful:

Two straight lines in space are parallel if they appear parallel in two parallel
projections. They are guaranteed to be parallel if they appear parallel in three
“independent” parallel projections.

Proof : If the images of two straight lines a and b are parallel under a parallel
projection, then they lie on two parallel planes α1 ∥ β1 that appear as straight lines.
If the straight lines are parallel in a second parallel projection, then (as a rule) they
also lie on two further parallel planes α2 ∥ β2 , and then, a = α1 ∩ α2 ∥ b = β1 ∩ β2 .
However, it is possible that a and b are skew, and by complete chance, both projec-
tion directions are perpendicular to the common normal n of a and b. In this case,
only a parallel projection that is not perpendicular to n is helpful. The three pro-
jection rays through a point in space must, therefore, not lie on a single plane. ⊙
▸▸▸ Radar corner reflectors and cat’s eyes
A construction as in Fig. 2.15 is often found on smaller ships. It is to be
shown that this reflector, composed of three mutually perpendicular squares,
reflects all incoming rays (and radar rays in particular) “back to the sender.”
The retroreflectors on bicycles (“cat’s eyes”), which reflect the headlights of
cars, work in the same way.

Fig. 2.15 Dangling radar corner Fig. 2.16 “Cat’s eye”: different magnifications of a retrore-
reflector on a ship flector on wheel spokes

Solution:
The two-dimensional analogue is known much better: A billiard ball is re-
flected by the “double cushion” and, at least theoretically, returns parallel to
the original direction. The proof for this can be seen on the left of Fig. 2.17:
If one takes into account the law of reflection (“angle of incidence = angle of
reflection”), then the ray returns, with the respective denomination of angles,
at an angle of 90○ + (ε − 90○ ) = ε.
And now the jump into space: Let us look at the situation such that two of the
three reflecting planes appear as straight lines. Such a principal view produces
2.2 Parallel projection and normal projection by means of restriction 63

Fig. 2.17 Reflection of a billiard ball

precisely the aforementioned billiard problem. The reflection in the projected

planes corresponds to the planar double reflection, and the reflection on the
third – frontal – plane is not apparent: The reflection normal appears as a
point, and both the incoming ray and the reflected ray lie on the same plane
that appears as a straight line (Fig. 2.18 center, left, and right)!

Fig. 2.18 Reflection in the corner of a cube (center: plan view and front view)

Relative to the direction of incidence, we can, thus, see the ray as parallel
in all principal views. However, if two straight lines appear parallel in three
independent parallel projections, then they are also parallel in space! ◂◂◂

Fig. 2.17 on the right resolves the following problem: How must a billiard ball
be hit if it should return to the starting point P ? The path of the ball (the
angle of incidence) is first reflected by the first cushion. Whichever direction
it is, it must pass through the mirrored starting point P ∗ . After having been
reflected once, the ray must pass through the finishing point Q = P . It must
also pass through the point Q∗ , which was mirrored by the second cushion.
This finally and unambiguously defines the once-reflected ray.
▸▸▸ Mirror optics instead of refractive lenses
Almost all animals use the refractive properties of lenses or crystals in order
to focus light on their retina. Crayfish as well as some shrimp and lobster
species employ a different trick: They have special compound eyes where the
inside of each facet is a mirror cubic prism. How does this work?
64 Projections and shadows: reduction of the dimension

countless thin direction of

quadratic prisms ray

normal top view retina

projection

Fig. 2.19 Left: In a rectangular prism that is specular inside, one part of the incoming light
rays is reflected exactly once “from the cushion” in such a manner that the ray exits in a parallel
course when viewed from the top. Right: When many such thin prisms are arranged on a spherical
surface, the parallel light rays are partially focused on a concentric sphere.

Solution:
The underlying geometry was discovered by Klaus Vogt in 19801 : when a
light ray enters such a specular cubic prism, there is, depending on the angle
and direction of incidence, a lower or higher possibility that the ray will get
to the other side “into the open air” after being mirrored exactly twice at
two consecutive faces of the prism. (Some light rays are reflected more often,
others exit the prism without any reflection.)
When we look in the direction of the prism axis (top view), the light ray
appears to bounce “off the cushion” like a billiard ball (Fig. 2.19 on the left).
Its inclination towards the prism axis is thereby retained. If the prism is very
thin, the double reflection will look to a good approximation like a simple
reflection on a virtual plane ε (projecting in axis direction) that is orthogonal
to the top-view direction of the light ray. The incident and reflected light rays
form the same angle with ε.
We will now squeeze thousands of such thin prisms onto the surface of a
sphere (Fig. 2.19 on the right) and illuminate this system in the given di-
rection s⃗. Let us now look at the prisms whose axes are inclined to s⃗ at the
same angle α (they cluster around a small circle on the sphere). All light rays
that undergo the special double reflection are then reflected as though they
were reflected by hundreds of virtual planes ε that touch a cone of revolution
with a half angle of aperture α. Accordingly, the reflected rays form a cone
of revolution. The distance between that cone’s tip and the sphere center is
– as one can figure out easily – approximately half the sphere radius.
In every light-ray direction s⃗, there are many light rays focused on a point
that lies on a concentric sphere. Loosely speaking, we could say that light
rays from points that are “farther away” in space are focused with great light

1 K. Vogt: Die Spiegeloptik des Flusskrebsauges, J. comp. Physiol. 135: 1–19

2.2 Parallel projection and normal projection by means of restriction 65

intensity onto a point of a concentric sphere (the retina). This is the same
principle that we know from geometrical optics. The sole difference to a lens
eye is that no lenses are needed here!

Fig. 2.20 Reflector eyes can be found in crayfish and several shrimp and lobster species. They can
be recognized by their rectangular pseudo-pupils (black spots that change their position depending
on the position of the spectator.)

As can be seen in Fig. 2.20, the reflector eyes of crayfish and shrimps are not
spheres but ellipsoids, to which the above considerations can also be applied.
Eyes based on mirror optics can be recognized by their “pseudo-pupils”, which
are almost rectangular. Inside these rectangles, the light rays coming from the
direction of the spectator land on the retina in the manner described above.
Their quadrangular shape is due to the quadrangular form of the individual
facets. The non-square shape results from the fact that the retina is not a
perfect sphere. The facets appear black because no light is reflected back to
the spectator. That is why the position of the pseudo-pupils changes with
the position of the spectator. ◂◂◂
66 Projections and shadows: reduction of the dimension

2.3 Assigned normal projections

Fig. 2.21 Two projections convey much more information than just one.

In order to capture an object unambiguously, we need, in general, at least

two projections. Fig. 2.22 shows (almost) synchronous photographs (central
projections) of a cat in mid-flight. Under ideal conditions (and given com-
pletely synchronous photos), it would be possible to reconstruct the positions
of all visible body parts from several of such pictures.

Fig. 2.22 Two different (almost) simultaneous projections

Normal projections are much better suited for this than photographs. If, in
addition, the projections are not picked arbitrarily but perpendicular to each
other, it further simplifies the reconstruction.
2.3 Assigned normal projections 67

In most cases, we can easily attach a Cartesian coordinate system to an object

(Fig. 2.21, Fig. 2.23). The normal projections to the coordinate planes thus
become the principal view s. The projection towards the z-axis becomes the
plan view (or top view ), the one towards the y-axis becomes the front view,
and the one towards the y-axis becomes the right side view (or profile view ).
If a point in space is defined by the coordinates P (x, y, z), then its principal
views have the coordinates P ′ (x, y), P ′′ (y, z), and P ′′′ (x, z). Any two out
of the three views share one coordinate, so that they can be assigned to each
other. If the plan view is drawn underneath the front view, then there exists
an ordering line from P ′ to P ′′ (Fig. 2.24). The coordinates of each point
can be read from plan view and front view alone. The double y-coordinate
defines the ordering line.

Fig. 2.23 On the left: the three principal views (normal projections on pairwise perpendicular
planes). On the right: A famous study by Leonardo da Vinci, almost 300 years before Gaspard
Monge (Fig. 2.24). The two views pictured side-by-side are assigned normal projections. The
undistorted front view of the head above is an additional assigned auxiliary view.

Fig. 2.24 The creation of plan view, front view, and right side view (based on a principle by
Gaspard Monge)
68 Projections and shadows: reduction of the dimension

Likewise, there exists an ordering line between front view and right side view,
and between plan and side view, as in Fig. 2.30, left and middle. (For a better
understanding, a general normal projection is additionally depicted on the
right side.) Both assigned normal projections of points in space always lie on
ordering lines. The position of the coordinate origin – and thus, the position
of the axis projections – is not essential for many applications. The relative
position to the planes of reference parallel to the coordinate planes are much
more important.
▸▸▸ Which object is it?
What does a spatial object look like when its contour in the front view is a
square, an isosceles triangle in a profile view, and a circle in the plan view?
This may be quite a challenge for a beginner. Even when provided with the
principal views of the wedge-shaped object in Fig. 2.25 – where, next to the
contours, an additional edge and half an ellipse are visible –, it is not easy to
deduce its actual appearance. The solution is a cylinder of revolution which
has been cut symmetrically twice. This procedure yields, as we will soon see
(on p. 164), ellipses as edges. ◂◂◂

Fig. 2.25 Principal views and a Fig. 2.26 Construction of the unfolding with the help of
general normal projection the principal views (parallel rotation of AS)

▸▸▸ Simple construction of the unfolding (Fig. 2.26)

Given a quadratic pyramid, its unfolding (development) is to be constructed.
Solution:
Let us position the object in the most simple way. The base edges in the
plan view appear undistorted. If we rotate an edge about the object axis into
a plane parallel to the front plane (such a rotation can be seen in the plan
view), then the edge will appear undistorted in the front view. ◂◂◂

▸▸▸ Assignment problems?

One often deals with two or more special views of an object that are put
into arbitrary positions. The object is much more easily imagined, however,
if these views are brought into assigned position. By the way: This is an
exquisite way to train one’s spatial ability.
2.3 Assigned normal projections 69

Fig. 2.27 A sepia in two (almost) assigned projections.

Fig. 2.27 shows two photos of a sepia (squid) that were taken in two almost
mutually perpendicular directions. Though they are not normal projections,
the pictures can be assigned in a certain way, and together they help to
comprehend the complicated shape of the animal. Note the bio-luminescence
(a flashlight was shone onto the animal to trigger this reaction). ◂◂◂

Fig. 2.28 Left, center: two special views. Right: assigned position.

Fig. 2.28 illustrates this principle by means of two images of a snail shell,
each in mutually perpendicular directions. Despite the fact that these are
photographs – and not strict normal projections –, both pictures can easily
be assigned (right). This is immediately helpful in the detection of assigned
points.
Fig. 2.29 shows computer generated images (clams have “mathematical shapes”,
see Chapter 7). Here it is perfectly possible to assign different views.
The American norm speaks of the plan view, front view, and left side view
and right side view. In contrast to Monge’s model, the bottom views and back
views are often invoked – a custom which can occasionally cause confusion.
Computer rendering (with its capacity to provide arbitrary spatial views
70 Projections and shadows: reduction of the dimension
general normal
projection front view left side
view

top view
Fig. 2.29 A clam in three assigned projections and a general normal projection.

with little to no additional effort) has, however, eroded these norms to some
extent.

Fig. 2.30 Plan view (center) and the assigned normal projection (left) of a scene (right)

▸▸▸ Shadows on the coordinate-parallel screen planes

In the plan view of Fig. 2.30, the parallel shadow on the horizontal ground
plane is to be determined with the help of an assigned side view. The direction
of the light ray is given in the plan view and right side view. If this direction
is offset through the assigned points P ′ and P ′′′ , then the image Ps′′′ of the
intersection point Ps is already found in the right side. Its plan view Ps′ lies
on the ordering line.
⊕ Remark : Without discussing the example in further detail (we have not yet dis-
cussed the shadow of spheres on planes or the shadow of straight lines of spheres),
the bounding lines can be found immediately by looking at the light rays tangential
to the sphere in the principal views. ⊕ ◂◂◂
2.3 Assigned normal projections 71

▸▸▸ What is broken or dislocated?

In Fig. 2.31, the image at the center shows a clear right side view (multiview
projection) of the left lower arm: ulna and elbow overlap. Fig. 2.31 shows

Fig. 2.31 Three X-ray photographs – right: front view; center: multiview projection. In the left
image, the position is changed by not quite 90○ .

three X-ray photographs of a left lower arm and hand. The images show
a wrist fracture, or distal radius fracture. However, one of the carpal bones
(the lunate bone) also seems to be “whirring around” (left-most image), which
suggests a joint dislocation (luxation).
In this image, the position of the aforementioned carpal bone seems fairly
“normal” (because there is no luxation). The fracture in the radial bone is
clearly visible in the multiview projection. Based on this image, we can also
give a good estimate of the angle by which the bone must be realigned for
treatment. ◂◂◂

Side views are often utilized in Descriptive Geometry. The witty freehand
sketch in Fig. 2.32 (found on a paper fragment) should demonstrate the
principle: If one wishes to create a third view to two assigned normal views
(such as the plan and front view), which should be assigned to one of the
two given views (the front view, in this case), one simply picks a projecting
base plane ε in the other (dropped) view perpendicular to the direction of
the ordering line (green). In the new view, it also appears as a point and as
perpendicular to the ordering line. On the ordering lines in the new view, it
is possible to transcribe the distances of ε from the dropped view (blue). Side
views will seldom be used to produce coherent images – rather, special views
will need to be generated where (for instance) the planes appear as straight
lines ([?]).
72 Projections and shadows: reduction of the dimension

Fig. 2.32 Additional side view Fig. 2.33 Principal projections as shadows

▸▸▸ Shadows provide additional information

Geometrically speaking, shadows do not differ from projections. If shadows
can be detected in an image, then they provide the same kind of additional
information as additional views. Fig. 2.33 shows a mathematically defined
heart.2 Due to the shadows on the coordinate planes (they are equivalent
to the principal views), the overall shape of the object can be deduced with
some accuracy.

Fig. 2.34 Shadows provide additional information. ◂◂◂

The so-called determination of visibility is a frequent problem. Fig. 2.35 il-

lustrates that the position of apparent collision points can sometimes not be
judged from a photo. Yet, in some cases, the solution is provided by mere
coincidence – in this case, through the shadow (Fig. 2.34) or with the help of
a photo taken an instance earlier oder later (Fig. 2.36). Let us ponder what
is needed for such a determination in more general terms.
▸▸▸ Determination of visibility for skew straight lines
Two straight lines have a point of intersection if they lie in a plane. If they
do not lie in a plane, they are “skew.”

2 The formulas were found at www.mathematische-basteleien.de/herz.htm.

2.3 Assigned normal projections 73

Fig. 2.35 Visibility is often difficult to determine. In the scene in the left image, the orca throws
the “stuntman” into the air. Don’t worry: no collision with the column actually took place. In the
right image, it can be gathered from some details that the parachute does not become entangled.

Fig. 2.36 A seagull doing a pedicure?

If we imagine straight lines as materialized rods, then one straight line is

hidden by the other skew straight line at the apparent intersection point in
every view, or it is itself hidden by the latter. In order to make the decision,
we need two projections – ideally, assigned normal views.
Fig. 2.37 shows a plan view and a front view of two straight lines a and b.
The intersection in the front view does not correspond to the intersection in
the plan view, otherwise the points would have to lie on an ordering line.
The lines a and b are, thus, skew. In the plan view, a point 1 on a and
another point 2 on b seem to coincide. If we draw the ordering line through
the apparent point of intersection 1′ = 2′ and look for the assigned points
1′′ ∈ a′′ and 2′′ ∈ b′′ , we can decide which point “lies higher” – and is, thus,
visible in the plan view. By analogy, the projections 3′ ∈ a′ and 4′ ∈ b′ can be
determined for the overlapping points 3′′ ∈ a′′ = 4′′ ∈ b′′ . The point that lies
further ahead is visible in the front view. We thus remember:

Visibility decisions in a normal projection are done in an assigned normal pro-

jection.

⊕ Remark : A draftsman suggests the overlapping of two straight lines by a slight

interruption of the overlapped line – thus creating an “aura” around each straight
74 Projections and shadows: reduction of the dimension

Fig. 2.37 Determination of visi- Fig. 2.38 . . . and practice: The leg positions are confusing
bility in theory . . . enough, but in what way do the ropes pass?

line. Such subtle tricks contribute a lot towards helping our spatial imagination. On
the other hand, errors in such determinations of visibility have a negative effect on
spatial imagination, and can even “destroy” the mental image of a three-dimensional
object. ⊕ ◂◂◂

The following important theorem occurs in many deliberations:

A straight line that is perpendicular to a plane is also perpendicular to each

straight line in that plane (whether intersecting or not).

Proof : Let us look at a right angle with the legs a and b (vertex S = a ∩ b). If we
leave a fixed and rotate b about a, then b generates the normal plane ν ⊥ a. Each
arbitrary straight line in ν is parallel to one of the positions of b, and thus also
forms a right angle with a. ⊙

Fig. 2.39 Practical application of the constant right angle: the axis of rotation of the excavator
scoop drawn in red is perpendicular to the desired ground plane. The excavator can now flatten
the incline with the scoop by a rotation of the jib.

Fig. 2.39 illustrates the theorem: an excavator scoop, which rotates about a
normal to the skew ground plane, can thus be used to flatten ground soil.
Another example can be found in application p. 75.
2.4 Principal lines and the Law of Right Angles 75

2.4 Principal lines and the Law of Right Angles

If a straight line is parallel to the image plane and thus perpendicular to the
direction of the normal projection, it is called the principal line. Segments
on the principal line are not distorted in a normal projection. For principal
lines, the following theorem holds:

Law of Right Angles: A right angle appears as a right angle in a normal projec-
tion if, and only if, at least one of the legs is a principal line and no leg appears
as a point.

Proof : Let a and b be the legs once more, S = a ∩ b the apex of the right angle, and
π the image plane. During rotation about a, b generates the normal plane ν ⊥ a. If
a ∥ π is now parallel, then the plane ν appears as a straight line perpendicular to a
in the image.
If a ∥/ π, then ν does not appear as a straight line. The normals n ⊥ a then form
different angles with π. It follows that the special normal b ∥ π is perpendicular to
every normal in the image, including a.
This law can also be proven elegantly by using vector calculus: We use an adapted
coordinate system with a z-axis that is perpendicular to the image plane. Two
vectors a⃗ and ⃗b shall be orthogonal in space (⃗ a ⋅ ⃗b = ax bx + ay by + az bz = 0.) The
vectors are orthogonal in a normal projection only if az bz = 0, that is, if az = 0 or
bz = 0, which means that either a ⃗ or ⃗b is parallel to the xy-plane. ⊙
▸▸▸ Non-trivial right angles in a cuboid

Fig. 2.40 Application of the theorem of right angles: A cuboid is displayed in different ways in
a normal projection so that the drawn right angle can be seen“undistorted”, i.e. as a right angle.
The direction of the projection has to be perpendicular to the edge 12, which can be done in an
infinite number of ways.

A good exercise for understanding the theorem is Fig. 2.40: In a cuboid one
can find quite a few right angles. One non-trivial example is the triangle 123.
Since 13 is perpendicular to the side plane that has 12 as diagonal, it is also
perpendicular to 12. Now consider a normal projection of the cuboid. Due to
76 Projections and shadows: reduction of the dimension

our theorem we can see the right angle in 1 if and only if either 12 or 13 is
perpendicular to the projection rays. ◂◂◂

▸▸▸ An abundance of right angles . . .

Fig. 2.41 shows two fundamentally different pictures of tower cranes (wide-
angle and telescopic lens photo). The supporting structures of such cranes
form a substantially right angle. Why do all the right angles of the cranes on
the right also appear as such in the image?

Fig. 2.41 An eternal construction site: the Sagrada Familia in Barcelona (Antoni Gaudí) has been
surrounded by cranes for years (this is part of the concept). The crane towers and their jibs form
a right angle, which never appears as such in wide-angle photographs (left). In a telescopic-lens
photograph (approximately a normal projection) with a horizontal optical axis, on the other hand,
the crane structures will form a right angle, because all the towers are the principal lines.

Solution:
The photograph on the left is a classic example of central projection, where
the Law of Right Angles only applies in special cases: Both legs of the triangle
must be parallel to the image plane (that is, they must be principal lines).
The photograph on the right is a nearly normal projection. When the optical
axis is horizontal (and the image plane = sensor plane is thus perpendicular),
then all the towers of the cranes in the photographs are principal lines, and
the Law of Right Angles applies: regardless of their orientation, the jibs
always appear horizontally in a photograph. ◂◂◂

A direct conclusion of the Law of Right Angles is the following fundamental

rule of drawing that we shall discuss in the appendix about freehand drawing:
2.4 Principal lines and the Law of Right Angles 77

During normal projection, a circle appears as an ellipse whose principal axis is

perpendicular to the image of the circle’s axis. The diameter of the circle equals
the length of the principal axis.

Fig. 2.42 In a normal projection, the image of a circle c is an ellipse cn . The minor axis of cn is
identical with the image an of the axis of the circle a. The semimajor axis equals the radius r of
the circle.

Proof : We assume beforehand that the image cn of a circle is an ellipse (we will
show this later). Among all diameters of c, only the one on the principal line appears
undistorted (Fig. 2.42). Due to all other diameters appearing shortened, the main
apices of the image ellipse lie on the principal line. The minor axis of the image
ellipse is, therefore, the normal projection an of the circle axis. ⊙
▸▸▸ If and only if

Fig. 2.43 At the indicated positions, there is a right angle in space. In the photo to the right, the
helicopter blades appear orthogonally whenever one of the blades is parallel to the sensor plane.

In mathematics, iff is a logical connective between statements, where either

both statements are true or both are false. Applied to the theorem of right
angles, this can also mean: If we know that we have a right angle in space
and we have verified that the angle in a normal projection is also 90○ , then
we know that at least one of the legs is a principal line.
78 Projections and shadows: reduction of the dimension

Fig. 2.43 shows two photos of different scenes, in which we know that there
is a right angle at the indicated positions. In the photo this angle appears to
be under 90○ . (In the photo to the right this only applies to some extent, but
let’s say it was an accurate right angle.) The photograph to the left is clearly
a central perspective, where the subsequent circular arcs and both legs of the
right angle are in planes parallel to the image plane. Obviously, the theorem
in such a special case holds also for central projections.
The end points of the rotor blades create a circle. Its axis is the rotation
axis of the system. In this case, the photograph closely resembles a normal
projection which is due to the fact that a strong telescopic lens was used.
The image of the axis of the rotating system is thus perpendicular to the
rotor blade that appears undistorted.
◂◂◂

▸▸▸ Luggage X-ray scans – are they normal projections?

During luggage controls, smaller luggage bags are screened with X-rays by
moving the bag on a conveyor belt and scanning them as orthogonal cross-
sections (Fig. 2.44). What kind of projection do we get in such a scan?

Fig. 2.44 A photographer’s hand luggage moves through the X-ray tube. The image is not a
photograph (= central projection) but a scan which corresponds to an almost orthogonal parallel
projection. The Law of Right Angles thus applies to some degree. On the right: a photograph of
the same suitcase for further illustration.

Solution:
After analysing the left images in Fig. 2.44, we can formulate the following hy-
pothesis: The cross-sections are scanned by moving the X-ray beam through
the tube almost orthogonally to the conveyor direction (and thus always in
a parallel course). The result is saved onto a “pixel bar” that is orthogonal
to the X-ray beam’s orientation, which generates a normal projection of the
cross-section. The overall result as the bag moves on the conveyor belt is a
parallel projection, but this parallel projection comes very close to a normal
projection.
2.4 Principal lines and the Law of Right Angles 79

To consolidate this hypothesis, we can test it on the basis of various criteria.

The preservation of parallel lines and of affine ratios obviously apply – as ver-
ified, amongst others, by the external hard-drive that is circled in Fig. 2.44.
When a normal projection is given, the Law of Right Angles must also apply.
This is clearly shown by the rim of the case (dashed coordinate axes), and
also in a non-trivial way by the long photographic objectives attached to the
cameras: the different annuli of the objectives are positioned orthogonally to
the optical axis. The elliptical images of circles should be exactly perpendic-
ular to the image of the axis – which is, indeed, fairly precisely the case here.
◂◂◂

Fig. 2.45 The Law of Right Angles is responsible for the phenomenon that the crescent appears
“tilted” at sunset.

▸▸▸ The crescent does not seem to be facing the Sun (Fig. 2.45)
At sunset, the Sun disappears below the horizon. If we have waxing moon,
it is still above the horizon. One would expect the crescent to be horizontal,
as it “faces the Sun.” Yet, it is tilted. Why is that?
80 Projections and shadows: reduction of the dimension

Solution:
When we focus on the Moon and zoom in, we have a near-orthogonal pro-
jection, which means that the Law of Right Angles applies. The contour of
the waxing moon is composed of the contour c of the Moon (a half circle)
and the visible half of the “terminator” t, i. e., the circle that separates the
illuminated and the dark side of the Moon (see also Application 4.17).
In order to get the minor axis of the image ellipse of the terminator, the axis s
of the circular terminator t has to be projected on the image plane π. The
line s can also be interpreted as the light ray through the center M of the
Moon. Fig. 2.45 illustrates how that can be done: Let Z be the spectator and
S∞ be the Sun (almost) at infinity. The projection plane π is perpendicular
to the principal projection ray M Z. The axis s = M S∞ of the terminator is
intersected with π, which leads to an auxiliary point H with H = H n . The
projected ray sn = M n H n = M n H obviously has a non-vanishing inclination
angle ϕ. The higher the Moon stands, the larger is this alleged “error”. You can
find out more about the topic at http://www.grad.hr/hdgg/kog_stranica/
kog13gif/kog13_glaeser.pdf. ◂◂◂

▸▸▸ When exactly do spring and the other seasons begin (Fig. 2.46)?
The beginnings of spring and fall are characterized by the fact that day and
night have the same duration (equinox ). Summer and winter begin when the
nights are longest or shortest respectively. How can we find their exact points
of time?

Fig. 2.46 The Law of Right Angles helps us to find the beginnings of the seasons. Left: general
normal projection, right: plan view. (The shape of the elliptic orbit is exaggerated to show the
construction more clearly.)

Solution:
Due to Kepler’s law, the orbit of the Earth (ecliptic) is an ellipse with the
Sun in one focus. The axis direction of the Earth stays constant during the
year. If we consider a plan view onto the plane of the ecliptic, the sun rays
that reach the Earth are always principal lines.
2.4 Principal lines and the Law of Right Angles 81

The equinox happens when the terminator (see previous application) of the
Earth passes through the poles. In this moment, the Earth’s axis and the sun
rays are perpendicular.
If we consider a top view – that is, a normal projection – of the elliptical
trajectory, all the sun rays that reach the Earth are principal lines, because
they lie on the projection plane. Since a normal projection is given, the Law
of Right Angles applies: The desired right angle between the sun rays and
the oblique earth axis can be seen – provided that it is located in space – in
the top view, because one leg of the angle is a principal line! The two points
at which day and night have the same duration (beginnings of spring and
fall) thus lie on a normal with respect to the direction of the top view of the
Earth’s axis.
Perpendicular to this direction (that is, in the direction of the earth axis
in top view), we can find – in a manner that is by no means trivial — the
beginnings of summer and winter. We must show that the angle σ between
the sun rays and the axis direction is extreme in this position. To prove this,
we once again need the Law of Right Angles (Fig. 2.47):

Fig. 2.47 The extreme angle between the sun rays and the earth axis

Let E be the position of the Earth where the plan view of the axis direction
runs through the Sun S. A is a fixed point on the axis (EA is constant). A′ is
its top view, and C is the foot of the perpendicular from A to the straight line
ES. The value σ is determined by the right triangle EAC, and C is its foot
on s. The triangle appears in the given position because A′ = C projecting.
In any other neighboring position E ∗ or E ∗∗ before or after the special event,
the foot C can be constructed by looking for the planar foot from A′ to the
corresponding light ray s∗ or s∗∗ , because every light direction is a principal
line. This means that both E ∗ C ∗ and E ∗∗ C ∗∗ before and after the special
position are shorter than EC, and σ is thus an extreme value in E. We will
elaborate on this topic in Chapter 11. ◂◂◂

▸▸▸ Topographic projection

In many areas of application, such as the building of roads (Fig. 2.48, Fig. 5.54
on the right), a single normal projection – the plan view – is sufficient if
one also works with “height markings.” This procedure is called topographic
projection.
82 Projections and shadows: reduction of the dimension

perspective image

An agricultural road rising

beyond ground level
point of view

Fig. 2.48 Topographic projection, from which a central view (perspective intersection method, see
p. 369) can easily be constructed (at the top). The planned road is adjusted to a fault scarp, with
the cuttings and slopes ideally balancing each other out (redeployment of material). The curved
road that steadily rises by 10% has helical margins. The slopes are thus helical developables (see
p. 310).

This makes it possible to solve the most common problems (including the
involved complex surfaces) in disciplines such as street planning or carto-
graphy. ◂◂◂
2.5 The difference in technical drawing 83

2.5 The difference in technical drawing

Classical Descriptive Geometry and technical drawing share many of their
methods despite the fact that technical drawing utilizes many “drawing short-
cuts” in addition to different “rules.” This has enormous advantages dur-
ing drawing, but requires certain foreknowledge from the reader, such as an
awareness about the nature of a section (which we will discuss shortly). To
imagine “genuinely” sectioned objects is to commit a fundamental mistake!
Technical drawing engineers tend to be very pragmatic. Rules are “sugges-
tions” that should, if possible, be considered. Owing to the impossibility of
adhering to all rules simultaneously, an engineer is, therefore, mindful of
comments such as “if possible,” “if there is enough space,” etc.
▸▸▸ Special additional views
In technical drawing, one also speaks of plan views, front views, left and right
views. Usually, as many views are drawn as are necessary.
⊕ Remark : “Devices” (a term occasionally used for such objects) can be viewed in
any number of special and general views. Such images aid the spatial imagination
of the beholder, but still, it is rare that these devices can actually be built from the
images alone (Fig. 2.49, right).

SCHNITT A B

view X

Fig. 2.50 Section through a pipe elbow.

Fig. 2.49 For the device pictured here, the front Whatever lies behind the section plane is
view and an additional assigned view are enough. drawn in.

Fig. 2.49 on the left shows a device pictured in a typical technical drawing: the
front view and an additional assigned view are enough to determine the shape and
describe all necessary lengths and angles undistortedly. In technical drawing, thin
dot-dashed lines are used for different purposes. They generally represent axes of
rotation, but they may also be used for axes of symmetry or – in the form of
coordinate axes – to mark the center of a circle. ⊕ ◂◂◂
84 Projections and shadows: reduction of the dimension

▸▸▸ Virtual sections

Technical drawing often confronts us with the problem of displaying objects
that, even if given many projections, are still very difficult to picture and
to dimension. Object sections were invented for this purpose – an imaginary
decomposition along a section plane that is usually perpendicular to a body
axis or contains the axis. Whatever is contained in the section plane is usually
crosshatched (excepting longitudinal sections of braces, ribs, screws, nuts,
flat washers, etc.). Whatever lies in front of the section plane is omitted –
whatever lies behind is drawn.
Fig. 2.50 on the left shows a section through an elbow. In this case, a simple
section (AB) is sufficient to depict the location of drilling holes and screws.
The arrows refer to the viewing direction. ◂◂◂

▸▸▸ Multiple virtual sections

The drawings in Fig. 2.51 serve to illustrate the necessity for more than one
section even when dealing with supposedly simple objects. The following rule
holds: With multiple sections, every section is done without reference to the
other sections.
B D

A C Section AB Section CD
Fig. 2.51 Multiple sections of a single object: The left-most drawing is a half-section. The other
drawings show a left-side view and two additional sections through the object.

On the left, we see a front view of the object (a nozzle). Its symmetrical
structure allows us to achieve two goals at the same time: one half of it – right
up until the semi-colon axis of symmetry – is depicted as a view (whether
it is the top or the bottom half depends on the norm involved). Since the
other half is symmetric, it can be pictured as a so-called half-section in order
to illustrate the – not all too complicated – interior of the nozzle. Here, the
important rule must be noted that the section axis must not be drawn as a
continuous line: the section is only virtual!
If all parts of the nozzle were cylindrical, we would already be done. This is
due to a common assumption in technical drawing that all objects are auto-
matically cylindrical unless otherwise noted. A view from the left (usually
drawn from the right side of the front view) shows, however, that at least the
head of the nozzle is square. Whatever lies behind it may still be cylindrical
or square. In the example, the part between A and B is cylindrical, whereas
the thickening to the right of it is square. This can be shown properly in a
2.5 The difference in technical drawing 85

section from A to B (“full section”). Finally, a full section from C to D should

indicate that the rest of the nozzle is, once again, cylindrical. Given enough
space, overlapping sections are drawn “in sequence.” ◂◂◂

▸▸▸ Dimensioning
Devices should primarily be dimensioned in places where it is expected (where
the measurements are not shortened) and secondarily in a way that can
be measured on the physical workpiece. For this reason, the diameters of
circles (rather than their radii) are usually dimensioned – if necessary with
a “measurement across space” (Fig. 2.52 on the left). The diameter sign ∅ is
added if the circle appears as an edge.
25 2

45´
o/ 20

o/ 38
o/ 14

o/ 20

o/ 26
39.5
6
10
10 28
15 30
215

Fig. 2.52 Dimensioning a workpiece

Fig. 2.53 The aforementioned workpiece, drawn with real sections on the left (wrong!) and trans-
parently in spatial view on the right

In order to dimension invisible drilling holes (and similar features), one works
with full sections or half-sections. It is often enough to “rip” the object apart
locally – in other words, to perform a partial section. If the object is too long
to be displayed in full length, it can be halted by a wavy line or an eight-loop.
External dimensions are always important. Closed loops of dimensions are
to be avoided. If possible, intersecting dimension lines are to be avoided.
As simple as the workpiece in Fig. 2.52 may be, it is relatively difficult to
create a comprehensible sketch displaying all details without ambiguity. The
computer allows us to use transparency in the drawing. If a quarter of the
object is taken out, it might raise the clarity of the drawing, but it can also
be misunderstood! ◂◂◂
86 Projections and shadows: reduction of the dimension

▸▸▸ Sections in construction drawings

Construction drawings may differ considerably from technical drawings. Even
the “norm” is different (1 cm instead of 1 mm). Furthermore, lines are used
instead of dimensioning arrows. As with technical drawings, virtual sections
are essential.

Fig. 2.54 Building with one horizontal and two vertical sections. Despite the computer simulation,
the model remains very important for comprehension. The character of the outer layer is partic-
ularly apparent – it holds two different pencils of straight lines! The mirrored floor contributes
additional effects to the presentation, but does not exactly improve one’s capacity to imagine the
object.

Fig. 2.54 shows a considerably complicated building (architect: Cornelia

Faißt) with three assigned sections. Without going into too much detail:
nobody can pretend to be able to imagine the object based on the three sec-
tions. Surprisingly, some architects have developed the remarkable capability
of gaining deep insights into virtually incomprehensible scenes by a series of
sections. Still, it is important for every architect to build physical models.
⊕ Remark : In anticipation of Chapter 6, let it be said that the outer wall of the
building is formed by so-called HP shells (hyperbolic paraboloids), which hold two
different pencils of straight lines (Fig. 6.20, 7.8). The planar sections in Fig. 2.54 of
such surfaces are – as the name implies – parabolas and hyperbolas. ⊕ ◂◂◂
3 Polyhedra:
multi-faced and multi-sided

Planes are non-curved spatial surfaces. Surfaces which are composed entirely
of planar polygons are called polyhedra. Polyhedra have been (and still re-
main) key to many geometric insights. In recent times, they have also become
essential in dealing with computational representations of curved surfaces.
These are “triangulated” or subdivided into triangles. Whenever computers
deal with arbitrary surfaces (no matter how complicated they are), an ap-
proximating polygonal surface is considered – the closer it comes to the true
shape of the surface, the finer the triangulation is.
As is often the case in geometry, special cases play an important role. For
example, there are a few polyhedra that are bounded by congruent polygons
– most of them are the so-called Platonic solids. If a polyhedron contains two
types of polygons, we get a further class of polyhedra, called the Archimedean
solids. Both types of solids appear frequently in nature, and in the microcosm
in particular. What was investigated by the ancient Greeks more than two
thousand years ago now appears under our microscopes.

© Springer Nature Switzerland AG 2020 87

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_3
88 Polyhedra: multi-faced and multi-sided

3.1 Congruence transformations

Before dealing with polyhedra, we should discuss the important topic of
congruence transformations. They are defined as a spatial transformations
where angles and lengths (of the moving object) are preserved (Fig. 3.1).

Fig. 3.1 A congruence transformation, except for the wings

Among the congruence transformations, there are translations and rotations.

Both are directly congruent, i. e., they preserve orientations.

Fig. 3.2 Rotations of the magic cube according to Rubik

Fig. 3.2 shows repeated rotations about intersecting axes, not of the entire
“magic cube” but of specific layers. In such cases, the theorem of “replacing
multiple rotations by a single one” obviously does not apply, not to mention
that it is quite a challenge to reverse all of these rotational steps. A computer
simulation has an unfair advantage: the program simply records all arbitrary
rotations taken by the user and, when the magic cube is to be “solved,” simply
runs the steps in reverse (with negative rotation angles).
3.1 Congruence transformations 89

Fig. 3.3 Three (non-uniform) rotations about intersecting and parallel axes

Fig. 3.3 shows three independent rotations about intersecting and parallel
axes. They are non-uniform (they are harmonic oscillations), which does not
matter from a geometric point of view. By the way: the outermost right axis
almost appears as a point.

If one combines two rotations about intersecting axes, one gets a rotation about
a third axis that runs through the intersection of the two given axes.

By the same token, it is possible to split an arbitrary spatial rotation into

several rotations about certain axes. This result, discovered by Euler , is ex-
tremely important in analytic geometry and kinematics.

Fig. 3.4 Rotation about an general axis

Fig. 3.4 shows a rotation about an general straight line. More specifically, it
is the line about which – given this specific type of bicycle – the front wheel
can be rotated (let us ignore the fact that this axis slightly varies in the real
world during the motion). The position of the Sun in this scene is such that
the shadow of the axis of rotation almost appears as a point. If one views
the rotation in the direction of the axis of rotation (in the special case, from
the position of the Sun), then the motion appears as an “entirely ordinary”
two-dimensional rotation.
90 Polyhedra: multi-faced and multi-sided

Reflections in a plane, on the other hand, are indirectly congruent, i. e., they
alter orientations. Fig. 3.5 shows an everyday reflection in an upright mirror.

Fig. 3.5 Reflections in a vertical plane. The peacock’s feather on the right touches the non-re-
flecting glass surface at the front (pushed in fibers).

Such a reflection does not alter left and right, as it is commonly believed, but
front and back, which is especially evident in Fig. 3.5 on the right. From a
geometric point of view, this statement can be further generalized: A mirror
inverts the sign of the distance of a point from the mirror plane.

Fig. 3.6 Reflections in a horizontal plane

Unlike reflections in a plane, reflections in a point are indirectly congruent!

In three-dimensional space, orientation can only be defined by orienting a
tripod (e. g., three concurrent edges of a cube).
At first, Fig. 3.7 gives the impression that it is a reflection. Upon closer
inspection, one notices the presence of two separate but essentially directly
3.1 Congruence transformations 91

congruent animals. They can be made to overlap using a simple rotation of

180○ . The axis of rotation is roughly equivalent to the midline of both parallel
body axes.

Fig. 3.7 Left: reflection in a plane? Right: reflection in several planes, but not quite easy to
visualize (the jellyfish are located behind the reflecting planes.)

Fig. 3.8 Such figures result from . . . Fig. 3.9 . . . reflections in a trihedron.

The photo in Fig. 3.8 is remarkable: It appears to be a convex polyhedron.

Fig. 3.9, however, shows the situation from another point of view: Just a tiny
part of the alleged polyhedron is inserted into a spherical “mirror triangle” (in
the photo this part is not yet in its final position). Multiple reflections at the
sides of the triangular mirror reproduce the part-polyhedron in such a way
that it appears as one large closed polyhedron when observed in direction of
the rotation axis of the triangular mirror.
Fig. 3.10 shows the transformation of a skeleton from a crouching position
into a stretched position. In practice, a “composite” of many rotations and
translations is required. According to an idea by Hannes Kaufmann, all tran-
sitional positions can be efficiently approximated by interpreting them as
linear combinations of the starting position and the final position.
Computer graphics is particularly interesting for transformations that can be
used to animate living creatures in a very realistic way. Professional programs
employ such techniques particularly well, and in order to understand them,
92 Polyhedra: multi-faced and multi-sided

Fig. 3.10 This transformation is strictly mathematical

Fig. 3.11 More than congruent transformations

Fig. 3.12 Realistic computer simulation of the above motion

considerable knowledge of kinematics is required. Chapters 10 and 11 of this

book are dedicated to this topic. Fig. 3.11 and Fig. 3.12 shall illustrate this.
3.2 Convex polyhedra 93

3.2 Convex polyhedra

Loosely speaking, a closed polyhedron is convex if it has no dents. If we connect

two arbitrary vertices by a line (chord), then this segment must lie either entirely
inside the polyhedron or in one of its faces.

Such solids play an important part in our world. They have geometric prop-
erties that make them particularly “simple.”
Prisms and pyramids
A prism is formed by parallel translation of a polygon (which does not nec-
essarily have to be planar). We can also say: one “connects” a generating
polygon with a point at infinity. The second definition leads to a pyramid as
a connection of a generating polygon with an arbitrary point. However, this
point must not lie in the carrier plane of the generating polygon (if it is a
planar polygon) and must not be a point at infinity.

Fig. 3.13 Still huge, even from far away

Seen this way, prisms are simply special cases of pyramids: the apex (tip) of
the pyramid is, then, a point at infinity.
By analogy, a planar polygon is convex if every possible chord lies within
the boundary of the polygon. The following theorem should be immediately
apparent:

Connecting a planar convex polygon with a fixed point (or a point at infinity)
that does not lie in the carrier plane results in a convex pyramid (or a convex
prism).

The ancient Egyptians have left us the most beautiful examples for regu-
lar pyramids. The construction of these colossal objects must have been as
difficult as their appearance is obvious. Given a quadratic base with a side
94 Polyhedra: multi-faced and multi-sided

Fig. 3.14 Convex or non-convex polygons? With some imagination, one can find two “almost
convex” positions. The corresponding “almost stretched-out” legs would need to be completely
stretched.

Fig. 3.15 From a multistage Mastaba to perfection

length of over two hundred meters, even the leveling of the foundation (ac-
complished to centimeter accuracy) is a true masterpiece. In principle, step
pyramids were built first (Fig. 3.15 on the left) and then their surface was
flattened. (The peak of the Great Pyramid of Giza is still very well pre-
served.) 4.500 years (!) ago, immediately after the construction was finished,
the appearance must have been even more impressive than it is today.
A series of simple properties
As indicated, convex polyhedra possess certain properties that can be very
useful for certain geometric considerations. For instance, the following theo-
rem is true:

The shadow of a convex polyhedron on an arbitrary plane (as well as its con-
tours) are convex polygons.
3.2 Convex polyhedra 95

The proof uses the “chord-definition” of convexity and it is straightforward:

if we imagine such an object as being filled by tense “fibers,” then these also
cast shadows that cannot lie outside the shadow contours of the object.

Fig. 3.16 Shadows of convex polyhedra

▸▸▸ Shadows of prisms and pyramids

Fig. 3.16 shows the shadows of three simple and convex objects. Shadows of
prisms are often bounded by shadows of the generators of the prisms – the
shadows of edges of the base and top surfaces form the rest. If the prism
stands on the plane on which the shadow falls, then the edges in question
are easily found: one only needs to project the light source in the direction of
the edges to the shadow-receiving plane and, once there, draw the “tangents”
towards the base polygon. Architecture mostly deals with straight prisms
and parallel lighting (from the Sun). In this case, the prism edges in question
can be found by moving the plan view in direction of the light rays towards
the base polygon.
For pyramids, we can proceed in a similar way: In this case, all edges run
through a point – this is also true for their shadows. Therefore, we get the
shadow of the pyramid apex and, from this point, place the “touching straight
lines” towards the base polygon. They delineate the shadow polygon. ◂◂◂

A further practical property of convex bodies that also follows from their
definition is the following:

During projection, every face of a convex body is either completely visible,

completely hidden, or it appears as an edge.

By analogy – if one considers lighting instead of viewing – the following is

true:
96 Polyhedra: multi-faced and multi-sided

Convex objects do not throw shadows on themselves. Their faces are either
completely lit, completely shaded, or the light rays through their edges touch
along the whole face.

⊕ Remark : Computer geometry takes advantage of this fact. A normal vector n ⃗ of

the face of a convex body can always be oriented in such a way that it points towards
the outside. If n
⃗ forms an acute angle with the projection ray (or light ray) oriented
towards the projection center (or light center) p⃗, then the entire face is visible (or
illuminated). It is merely necessary to determine the sign of the dot product n ⃗ ⋅ p⃗.
This also holds true with central projections (or central illuminations): In this case,
the projection rays (or light rays) usually differ for the points of the face, but the
decision whether or not the angle is acute is not affected by this. ⊕
We can derive the following rule from the aforementioned statements:

Rule of Visibility : In the projection (image) of a convex body, only fully visible
or fully hidden edges go through a vertex that lies completely within the body
contours. The contour polygon is fully visible.

Combination of several convex bodies

In practice, we rarely depict one single convex body. The question is: Which
of the previously stated practical rules can be applied to scenes that consist
of many such building blocks?
Let us discuss the procedure using a certain example. Fig. 3.19 shows a
non-convex body – upon closer inspection, however, one notices that it is
“harmlessly non-convex”: It can easily be split into three convex building
blocks: two regular eight-sided prisms and one cuboid extension.
While visibility determination for such a body is “trivial” for our imagina-
tion, the drawing of the shadows is much less so and requires a disciplined
approach.

Fig. 3.17 Systematic procedure: consider the bodies one by one . . .

In a first step, all three partial convex bodies are considered one by one
according to the aforementioned rules. In particular, the shadows on the base
plane can be determined separately. The entire shadow on the base plane
emerges from the “combination” of the separate convex shadow polygons.
The thus emerging shadow polygon can easily be non-convex.
⊕ Remark : A hint for freehand drawers (for us, freehand drawing is an important
goal to which a whole appendix is dedicated): The shadow on a surface “no longer
3.2 Convex polyhedra 97

knows to which partial body it belongs.” All shadows on a surface adapt the tex-
ture of the surface on which they lie. One should, furthermore, not try to highlight
the separate convex shadow polygons. Shadows tend to have a “surface-like appear-
ance.” Not even the boundaries of the entire shadow polygon should be particularly
stressed: shadow boundaries are usually not sharply delimited – not even in photo-
graphs. This is due to the fact that there are no genuine point light-sources in
practice. Even the sun on the firmament appears “disc-like” and is to be considered
as a light cone with an aperture of half a degree. ⊕

Fig. 3.18 . . . and bring them into mutual rela- Fig. 3.19 Normal view of three convex poly-
tion. hedra, including the shadow

By subdividing a scene into convex parts, we can determine all surfaces that
are entirely shaded (“dark”). The following important rule holds true for
them:

Faces that are entirely dark cannot receive additional cast shadows.

This simplifies our work, which now concerns the determination of the shad-
ows of the subdivided bodies on each other. We now have to consider:

Two convex bodies never mutually cast shadows on each other. If there exist
shadows in between them at all, then only one body casts shadows on the other.

Fig. 3.20 The detail shows the inflection points of the shadow boundary.
98 Polyhedra: multi-faced and multi-sided

⊕ Remark : This approach is a huge time-saver and can also prevent many possible
errors. One can now derive a kind of priority list that, in most cases, is easily
sortable: disregarding certain convoluted exceptions, there is always one “frontmost”
building block on which no shadows are cast by other bodies. This is followed by the
building blocks on which only the first body casts shadows, and so on. Even in the
rare cases where the sorting of such a list is, at first, not possible, the challenge can
be surmounted by dividing single (often very long) convex blocks into two separate
ones. ⊕
▸▸▸ Acceleration of computer programs

If one considers all advantages that can be gained by subdividing a scene

into convex partial bodies, then one can get fairly realistic images of complex
scenes such as Fig. 3.21. This can prove to be important: for example, if one
has to develop a “chess program,” which should primarily play in an intelligent
manner, but in addition give the player a realistic image of the situation on
the chess board, it would be a pity to waste much processing power on the
highly realistic “rendering” of that image. A “real-time” visualization (at least
thirty frames per second) is desirable and, in this case, easily achievable.

Fig. 3.21 Scenes composed of . . . Fig. 3.22 . . . convex polyhedra

Fig. 3.23 Interpretation as full and hollow bodies

3.2 Convex polyhedra 99

In [3], I have described many computer algorithms that make such fast ren-
dering of realistic images possible. Despite the age of the algorithms – es-
pecially by computer graphics standards – they still allow computer graph-
ics fast as lightning with correctly calculated shadows. Even the scene in
Fig. 3.22 (depicting the Commodore Amiga, a groundbreaking minicomputer
in its day, but completely antiquated in our time) can be animated in real-
time. The algorithms are a typical example of the fruitful application of
theoretical geometric knowledge in other disciplines. ◂◂◂

Fig. 3.23 on the left shows complex, “rounded” polyhedra. The shadow bound-
aries of this body (on which all shadow polygons are based) are still com-
paratively simple. In Fig. 3.23 on the right side, the biggest polygons were
removed from the polyhedron. This, of course, means that the polyhedron is
no longer convex. At the same time, one may still derive some advantages
from the fact that the “hull” of the polyhedron is convex: thus, either all faces
potentially throw a shadow on the base plane, or none of them do. Further-
more, all front faces of the polyhedron are either completely illuminated or
completely dark. Equipped with this knowledge, the rendering speed for the
scene can be significantly increased.
Multiple light sources

Fig. 3.24 Penumbra shadows and umbra shadows of a convex object

Dealing with artificial light but also with regular sunlight and reflecting sur-
faces leads to penumbra shadows and umbra shadows: areas in penumbra
shadows are illuminated by at least one light source each, or none at all.
Areas in umbra shadows are not illuminated by any light source.
The result may look complicated, but geometrically speaking, it can easily be
accomplished by treating each light source “separately.” For convex bodies,
one gets two or more convex shadow polygons. The umbra shadow lies where
all polygons overlap. The other areas are more or less illuminated, depending
on how many light sources are visible from their positions.
100 Polyhedra: multi-faced and multi-sided

What is true for convex polyhedra holds true for their refinements to smooth
surfaces. In Fig. 3.25, for instance, one can see an elongated half-ellipsoid
(Chapter 6) as part of a chandelier. The shadow outlines with respect to
each light source are ellipses, which are convex curves. The penumbras are
in general not convex. The area that is in shadow with respect to all light
sources is convex.

Fig. 3.25 Penumbra shadows and umbra shadows of a convex object (half ellispoid)

▸▸▸ Shadow profiles

One can analyze shadow casting of buildings during the day by calculating
the shadows of the scene at regular intervals (e.g., each hour).

Fig. 3.26 Shadow profiles of a skyscraper (summer and winter half-year)

Fig. 3.26 shows how it can then be determined for how many hours single
points remain shaded on any given day. Such questions are highly relevant
during constructional changes in dense urban areas.
◂◂◂
3.2 Convex polyhedra 101

Fig. 3.27 By means of modern computer hardware, shadow profiles can be calculated within
seconds. This is enormously useful for architects, who are, by law, obliged to make sure that their
buildings do not cast too much shadow on their surroundings. Thus, it is possible to determine
the shadow duration for each visible point in the scene (program development by Günter Wallner,
www.uni-ak.ac.at/geom/schattenprofile/).

When do we speak of a polyhedron?

Fig. 3.28 Two of the three objects do not represent a polyhedron: For real polyhedra, all polygons
must be closed, and two adjacent polygons in the plane are “merged”.

Fig. 3.28 shows some objects built with Geomag1 that are not polyhedra in
the strict sense: Concerning the model on the left, the rotors of the helicopter
are not polygons. Concerning the object in the middle, the base and top
surfaces (regular hexagons) are divided into equilateral triangles that lie in
one and the same plane. By “extending” the midpoints of the hexagons from
the plane, we can construct a regular six-sided pyramid, whose side surfaces
are merely isosceles and not equilateral.

Euler characteristic
For closed polyhedra without “holes” – and therefore especially for convex
polyhedra –, Leonhard Euler discovered a simple relation between the num-
ber e of vertices, the number k of edges, and the number f of faces:

e+f = k +2.

1 Geomag is the brand name of a toy construction system consisting of nickel-plated steel balls
and connecting rods with a magnet on each end.
102 Polyhedra: multi-faced and multi-sided

For example, for the cube with e = 8 vertices, k = 12 edges, and f = 6 faces,
we have 8 + 6 = 12 + 2.
Proof : An elegant and brief proof can be provided by applying the principle of
mathematical induction, which is often useful in mathematics. The trick lies in
proving the theorem for polyhedra from which we remove a face. We call this type
of polyhedron “open”. The number of vertices and edges of a closed and open poly-
hedron are the same, whereas the opened one has one face less. Thus, it is to be
shown that for open polyhedra we have e + f = k + 1.
Let us first start the induction: The theorem is true for the simplest case, a single
polygon (f = 1) with e vertices and as many edges.
Now we show that the theorem is true for f + 1 faces if it is true for f faces: Let
us assume that the Euler characteristic is correct for all open polyhedra of up to
f faces. If this allows us to prove that the theorem is also true for open polyhedra
with f + 1 faces, then this conclusion can also be applied to all f .

Fig. 3.29 Figure accompanying the proof

Let us consider an open polyhedron with f faces. Let us subdivide this object into
two open polyhedra along an arbitrary, closed but non-planar polygon composed of
m edges, exactly two of which should belong to the “top polygon” (Fig. 3.29). In sum,
both parts still provide f faces, but there are m − 2 vertices and m − 1 edges more.
For both partial polyhedra, according to the induction assumption, e1 + f1 = k1 + 1
and e2 + f2 = k2 + 1 hold true. This means (e1 + e2 ) + (f1 + f2 ) = (k1 + k2 ) + 2. Due to
e1 +e2 = e+m−1, f1 +f2 = f and k1 +k2 = k+m−2, we get (e+m−1)+f = (k+m−2)+2
and thus, indeed, e + f = k + 1. ⊙

Fig. 3.30 Stellations of kites (middle picture: rhombic triacontahedron): 60 faces, 120 edges and
62 vertices

Fig. 3.30 shall serve as an example for Euler’s formula (in this case it is
easiest to count the faces (12 ⋅ 5) and the edges (12 ⋅ 5 + 30 ⋅ 2).
3.3 Platonic solids 103

3.3 Platonic solids

About 2500 years ago, the ancient Greeks started to classify spatial geometric
bodies systematically. It seemed obvious to ask for all polyhedra with con-
gruent faces and equal angles along the edges. Several of these were already
known, and it was soon realized that only one was missing: the dodecahedron,
which has twelve regular pentagons for its faces.
As simple as it gets
The cube – a regular hexahedron – is the best-known example of a Platonic
solid even though it does not have the minimum number of vertices, edges,
and faces: It has eight vertices, twelve edges, and six congruent square faces.
What makes the cube so simple is the fact that all of its edges are mutually
perpendicular. It is, therefore, a special cuboid, and cuboids, from which
many day-to-day objects are derived, are in any case easy to imagine.

Fig. 3.31 The shadow of a cube is a hexagon as a rule.

⊕ Remark : Fig. 3.31 shows a cube that is illuminated by two light sources. In
general, a hexagon belongs to each light source in the shadow-receiving base plane.
Both hexagons share four points as long as the cube is placed on the plane. The
umbra shadow appears where they both overlap, and the penumbra shadow where
only one of the hexagons affects the surface. In Fig. 3.31 on the right, one light
source produces “grazing light”. In such exceptional cases, the shadow polygon is a
rectangle if the light rays are parallel to a face of the cube. ⊕
⊕ Remark : Pyrite (Fig. 3.32) is one of the most frequent sulfidic minerals in nature.
It crystallizes in a cubic system and often occurs in the form of well-developed,
cube-shaped crystals that exhibit characteristic streaks. Sometimes it also appears
as a dodecahedron. ⊕
If a sketch of a cube is to be drawn, most people first draw a square and then
attach four parallel line segments of equal length (but shorter than the edges
of the square). This yields the vertices at the back, behind the square. Such a
sketch is quite informative, even though cubes in nature cannot be visualized
104 Polyhedra: multi-faced and multi-sided

Fig. 3.32 Pyrite (“Fool’s gold”) with cuboid crystals

like this. It conveys what we know about the cube. A technical drawer would
depict two squares above or next to one another and say: This is the front
view, the other one is a top or a side view. This is even less ambiguous, though
it may not be immediately apparent to a novice. Everyone understands it.
Let us unfold the model of a cube into a plane. The result is often called
the development of the cube. We now have the need to align the six squares
in such a way that they occupy the least possible space (if, for instance,
we wish to build many such cube models). Furthermore, it is our intention
to minimize the number of edges that have to be glued together (i.e., the
number of double edges in the unfolded model). If, as in Fig. 3.33, we only
wish to develop the rest of the cube after a planar section, we have to be
careful not to end up having several parts in our hands.

Fig. 3.33 Development of a cube including the planar section – badly chosen, since it “falls apart”.

Three objects from equilateral triangles

There can only be one Platonic solid whose faces are equilateral quadrangles
(or squares), because only three squares can meet in every vertex. If there
were four, then 4 × 90○ = 360○ would complete the full angle, and the vertex
pyramid would not be spatial anymore.
This is the reason why we cannot assume equilateral hexagons as the faces
of a Platonic solid: At least three faces meet at each vertex, and 3 × 120○
(the angle between two adjacent edges of a hexagon) already adds up to
3.3 Platonic solids 105

Fig. 3.34 Scenes of a development (tetrahedron, octahedron, cube)

360○ . Regular polygons with more than six edges are, thus, excluded from
the beginning.
Thus, we can only derive new Platonic solids from equilateral triangles and
pentagons. Equilateral triangles (with angles enclosed by edges of 60○ ) offer
three such possibilities: A spatial corner can be formed by three, four, and
even five triangles. In all three cases, the sum of all angles in a corner is less
than 360○ .
If a vertex pyramid is formed by three triangles, then the polyhedron is
already complete: This functions with four equilateral triangles in the form
of a regular tetrahedron. Fig. 3.34 on the left shows how it can be constructed
from its development.

Fig. 3.35 Scenes of a development (dodecahedron, icosahedron)

If a vertex pyramid is formed by four triangles, then we get the regular

octahedron. Fig. 3.34 in the middle shows a possible development and the
way it is folded. The octahedron can also be interpreted as a double square
pyramid with the same base. Both pyramids have half of the diagonal of the
base square for their height.
If a vertex pyramid is formed by five triangles, then we get the regular icosa-
hedron (from icosi =twenty). Fig. 3.35 on the right side shows a possible
development and its reconstitution into a considerably complicated struc-
ture.
106 Polyhedra: multi-faced and multi-sided

Fig. 3.36 Platonic solids constructed by using Geomag. On the right: a regular icosahedron. The
spheres and rods of Geomag are uniform in size, but not so in the image: The tetrahedron is
slightly enlarged in comparison to the dodecahedron.

This concludes the possibilities of creating regular polyhedra from equilateral

triangles. If our systematic approach is used, only one body remains where
three regular pentagons (angle between edges of 108○ ) form a corner (3 ×
108○ < 360○ ).
The last Platonic solid whose beauty defies description

Fig. 3.37 Development of a dodecahedron

Half a dodecahedron (from a pentagon = five vertices and dodecahedron =

twelve faces) remains if one attaches a movable, congruent pentagon to each
side of a regular base pentagon, and rotates the attached objects about the
edges until a polyhedron emerges (Fig. 3.37 on the left).
Two thus emerging, mirror-symmetrical dodecahedra fit together when one
half is rotated by 36○ about the perpendicular axis (Fig. 3.37 on the right).
Fig. 3.35 on the left shows a practical suggestion for the construction of a
dodecahedron.
The condition that all faces of a polyhedron have to be congruent is obvi-
ously very limiting. The frequent wish by architects that an arbitrary surface
be approximated by small, congruent triangles (a cost-saving solution, see
Fig. 6.53), already fails in principle: One always gets the edges of a small
number of polyhedra, among which are the regular tetrahedra, octahedra,
and icosahedra.
3.3 Platonic solids 107

Fig. 3.38 The “finished” dodecahedron in an oblique view and in the principal views

Fig. 3.39 stellated dodecahedron as a non-convex regular polyhedron (see also Fig. 3.58)

In practice, there exists a marginally useful cop-out: We abandon the neces-

sity of convexity of the body and/or the faces themselves. It is then possible
to obtain four additional types of Platonic solids: the star polyhedra. Fig. 3.39
shows these unusual structures that were discovered relatively late (by Kepler
in 1615 and Poinsot in 1809).
The “small stellated dodecahedron” – first described by Kepler – is con-
structed by twelve five-cornered stars (pentagrams) intersecting each other
(Fig. 3.39 on the right). For better understanding, one of the non-convex
pentagons is colored blue. In the case of the great stellated dodecahedron
(Fig. 3.39 left), 20 three-sided pyramids stick on an icosahedron in such a
way that we again obtain non-convex pentagons. Twenty regular triangles in-
tersect one another on the great icosahedron (Fig. 3.39, middle left); twelve
regular pentagons form the great dodecahedron (Fig. 3.39, middle right).
Crafting a pentagonal dodecahedron is easy on a computer. Fig. 3.40 shows
the “making of” a stable model. In practice – for example, when using the
Geomag magnetic building blocks – it helps to “strengthen” the pentagons
by inserting “panels.” Only five panels have been inserted on the left, and the
construction is still not stable. A sixth panel (image center left) is required
for the dodecahedron to be absolutely stable.
108 Polyhedra: multi-faced and multi-sided

Fig. 3.40 How many panels do we need to build a stable model of a pentagonal dodecahedron?

▸▸▸ Dualization of Platonic solids

Due to their regularity, it is possible to inscribe an insphere and circumscribe
a circumsphere for every Platonic solid.
If such a polyhedron is polarized at such a sphere, then the vertices corre-
spond to the faces of the polar polyhedron (and vice versa). The new poly-
hedron is once again regular. If the sphere is the insphere, then the faces
correspond to their points of contact with the sphere and, thus, to their mid-
points. The polyhedron that emerges from these surface midpoints is once
again platonic!

Fig. 3.41 Polarization of the Platonic solid: The tetrahedron is self-dual, the cube is dual to the
octahedron, and the dodecahedron is dual to the icosahedron.

The tetrahedron (4 vertices, 4 faces) is “dual to itself”. Cubes (8 vertices,

6 faces) and octahedra (6 vertices, 8 faces) form a dual pair, just as the
icosahedron (12 vertices, 20 faces) and dodecahedron (20 vertices, 12 faces)
do. ◂◂◂

Fig. 3.42 The five Platonic solids, and almost a sixth one

Fig. 3.42, once again, shows the Platonic solids next to each other. The
surface of the sixth solid (third from the left) is ideally composed of ten equi-
lateral triangles. (It can be divided into two five-sided pyramids.) However,
the angles of the faces (edge angles) are not all equal.
3.4 Other special classes of polyhedra 109

3.4 Other special classes of polyhedra

Archimedean solids

Fig. 3.43 Different Archimedean solids

A more generous condition that allows at least two different types of congru-
ent surfaces presents many more possibilities (13 to 15 depending on more
or less sophisticated counting). Fig. 3.43 shows several such solids that were
already studied by Archimedes.
The soccer ball is by far the most familiar Archimedean solid (or rather, a
spherical variant of it). It can be “developed” from a dodecahedron and thus
possesses 12 regular pentagons and 20 regular hexagons.

Fig. 3.44 Two Archimedean solids derived from a dodecahedron

▸▸▸ Tiling of space

If the vertices of an octahedron are cut (Fig. 3.45 on the left), a new Archimedean
solid that can be used to fill a space without leaving any gaps emerges.

Fig. 3.45 On the left: truncated octahedron (intersection with a cube). Center: exact tiling using
such solids. On the right: tiling using irregular dodecahedra.
110 Polyhedra: multi-faced and multi-sided

Such tilings of space are not at all easy to accomplish when regular solids are
used. It is impossible with the sole use of regular dodecahedra (see Fig. 3.45
on the right). The proof requires a calculation.
Only quite recently (1993), it was discovered by Denis Weaire and Robert
Phelan that irregular dodecahedra (with 12 faces) and tetradecahedra (with
14 faces) of equal volume – very similar solids possessing two hexagonal faces
– can be arranged in groups of eight in order to fill space without leaving
any gaps.2

Fig. 3.46 Tiling of space with irregular 12-sided and 14-sided solids according to Weaire-Phelan.
On the left and in the middle, there are two views of the basic building block, composed of eight
equal-volume polyhedra.

Fig. 3.47 More or less dense sphere packing in theory and practice

The easiest way to tile three-space is, of course, by using cubes. Spheres are
not well-suited for this purpose. The gaps caused by spheres comprise almost
half of the volume of the arrangement in Fig. 3.47: The ratio of the volume
of a sphere to the volume of the circumscribed cube equals (4π/3) ∶ 8 ≈ 0, 52.
The translation of the positions results in a slightly better utilization of space,
but even in the optimal arrangement, it only covers about 3/4.
In this case, nature reveals itself – as it so often does – to be rather pragmatic:
A little bit of squeezing and shoving, and space is, once again, optimally
utilized (see Fig. 3.47 on the right). ◂◂◂

2 More information at http://en.wikipedia.org/wiki/Weaire-Phelan_structure or – in greater

detail and with more emphasis on mathematics – http://torus.math.uiuc.edu/jms/Papers/
foams/forma.pdf.
3.4 Other special classes of polyhedra 111

▸▸▸ Different designs of a soccer ball

Fig. 3.48 Different soccer ball designs

Fig. 3.49 First row: design for the Champions league final in Munich, 2012. Second row: a design
based on the cube.

Fig. 3.50 Other variations of a soccer ball

112 Polyhedra: multi-faced and multi-sided

The classic form of the soccer ball is derived from the Archimedian solid with
12 pentagons and 20 hexagons. Every once in a while a new soccer ball is
designed (Fig. 3.48). Take a look at the, hopefully, self-explanatory series of
images in Fig. 3.49. Fig. 3.50 shows some more variations. ◂◂◂

Rhombic polyhedra

Fig. 3.51 Left block: Five Platonic solids lead to two new interesting solids: the rhombic do-
decahedron (12 faces) and the rhombic triacontahedron (30 faces). Right: Due to the fact that
the rhombic dodecahedron can be generated by means of a cube, it follows that it is suitable for
“packing space”

.Let us go back again to the Platonic solids, the faces of which are regular n-
gons. Through each edge e, we have a plane ε perpendicular to the connection
of the edge with the center C of the solid (Fig. 3.51). All planes ε that contain
the sides of a face form a regular n-sided pyramid. Two neighboring pyramids
touch along a normal plane ε so that the two adjacent equilateral triangular
pyramid-faces form a planar rhombus.

Fig. 3.52 Packing space with rhombic dodecahedra: a series of stroboscopic images during rotation

We would now expect five new solids that consist of as many congruent
rhombuses as the adjacent Platonic solid has edges. However, it turns out
that the cube and the octahedron (with 12 edges each) lead to the same result
3.4 Other special classes of polyhedra 113

– the rhombic dodecahedron (Fig. 3.51 upper middle and left). The same is
true for the regular dodecahedron and the icosahedron (with 30 edges each).
They generate the rhombic triacontahedron (triaconta = 30, Fig. 3.51 lower
images). Finally, the regular tetrahedron with its six edges leads to a trivial
result: the cube. In this case, the rhombic faces are squares.

Fig. 3.53 The projection of Platonic solids onto the circumsphere leads to three different remark-
able rhombic tilings on the sphere.

Since a rhombic dodecahedron can be created from a cube and cubes can
be stapled to fill space, we have found another non-trivial example of tiling
space (application p. 109, Fig. 3.51 right).
The upper images in Fig. 3.53 illustrate that the apices of the above-mentioned
regular cones through the edges of the Platonic solids lie above the centers
of faces. If we now project all points to the circumsphere, we get three dif-
ferent rhombic tilings of the sphere with 6, 12, and 30 congruent spherical
rhombuses and 4 + 4 = 8, 8 + 6 = 14, and 20 + 12 = 32 grid points. We will learn
more about the regular distribution of points on the sphere later on.

▸▸▸ More polyhedra consisting of congruent rhombuses?

Fig. 3.54 on the left shows a polyhedron with 30 rhombuses (a somewhat
“distorted version” of our rhombic triacontahedron) in the principal views.
In the plan view, the sides of the rhombuses have equal length due to the
fact that they have equal length in space plus their inclination is constant.
Therefore, the height differences of all sides are constant (Pythagoras), and
all vertices of the triacontahedron lie in equally distributed horizontal carrier
planes (to be seen in the front view).
In the plan view, the six rhombuses that appear as edges form a regular
hexagon around the projection A′ of the apex A. Non-trivially, the six vertices
1′ , 2′ , 3′ , . . . of three connected rhombuses are also equally distributed on a
circle c′ (drawn in red) through A′ . We could also take a circle, distribute
six points A, 1, 2, 3, . . . equally, and define that these points have constant
height differences. Then, we rotate the points six times about the vertical
axis through A by 60○ and get a polyhedron consisting of 30 rhombuses.
114 Polyhedra: multi-faced and multi-sided

Fig. 3.54 Generalization of polyhedra consisting of rhombuses. The polyhedron on the left is a
rhombic triacontahedron – its height can be varied by choosing the constant height difference of
the horizontal carrier planes of the vertices.

This method can be generalized to get more vertices (Fig. 3.54 middle). If we
only distribute four points on the red circle, we get a rhombic dodecahedron.
Since the height differences can still be chosen, the polyhedron’s height can
vary.
Fig. 3.54 on the right shows a smoothened ornamental realization of the
polyhedron shown in the middle. Later on, when discussing helices, we will
once again come back to this example (application p. 324). ◂◂◂

Catalan solids
In 1865, Eugéne Catalan described solids which are the duals of the Archi-
medean solids. The faces of a Catalan solid are congruent non-regular poly-
gons (Fig. 3.55). The vertices and faces correspond to faces and vertices of
the dual type.3 Once again, we encounter pure rhombic solids: the already
well-known rhombic dodecahedron and the rhombic icosahedron (20 faces).

3 http://en.wikipedia.org/wiki/Catalan_solid provides a good survey and all the names of

the solids.
3.4 Other special classes of polyhedra 115

Fig. 3.55 Polarization of Archimedean solids at a sphere around the center. Six (blue-shaded)
Archimedean solids result in (yellow-shaded) solids with congruent triangles, two (gray-shaded)
result in solids with congruent rhombuses, and the four orange-shaded ones result in polyhedra
with congruent pentagons or quadrilaterals.

▸▸▸ Folding

Fig. 3.56 One of the stelae of the University of Applied Arts Vienna (Oliver Kartak)

Stelae (standing slabs) are sometimes used as landmarks. For the University
of Applied Arts Vienna (Fig. 3.56) the designer created staelae that resemble
origami. This Japanese art of paper folding is a process by which a flat square
116 Polyhedra: multi-faced and multi-sided

or rectangular sheet of paper is transformed into a specific shape or sculpture

through folding and sculpting techniques.

Fig. 3.57 A crease pattern in accordance with the theorem of Kawasaki

Fig. 3.58 Similar origamis: On the left we see birds made by my daughter Sophie, on the right
two great stellated dodecahedrons put together.

A criterion for an origami construction to be flat is given by the theorem

of Kawasaki. It states that a given crease pattern can be folded into a flat
origami if the angles α1 , α2 , . . . in the crease points fulfill certain simple con-
ditions (we will not go into detail here, Fig. 3.56 shows a practical example).
The folding process is illustrated in Fig. 3.57, and the designer of the metal
3.4 Other special classes of polyhedra 117

stele in Fig. 3.56 did also use this theorem in order to fold the stele from a
flat piece of metal.
The practice of folding origami is very advanced, sometimes rather mag-
nificent. On the Internet one can find a lot of different instructions for the
creation of remarkable sculptures out of paper. Fig. 3.58 shows two examples.
The picture to the right shows two great stellated dodecahedrons of different
size (see Fig. 3.39). Since they are similar, it is possible to stack the smaller
one on top of the other. ◂◂◂

▸▸▸ Foldable flower vases and pots (Fig. 3.59)

Fig. 3.59 Smoothly adjustable flower vases made of Tetra Pak carton

Packaging material can be waterproof (Tetra Pak carton packages). Steffen

Kehrle has used such material to design “adjustable” vases, which can be
opened and closed quite smoothly – almost like a flower – by means of a
simple elastic band. ◂◂◂
118 Polyhedra: multi-faced and multi-sided

3.5 Planar sections of prisms and pyramids

General pyramids are constructed by connecting a polygon that is not nec-
essarily planar with a fixed point (the apex of the pyramid). If the apex is a
point at infinity, then we speak of a prism.

Fig. 3.60 From the spatial situation to a construction in the plane: parallel perspectivity

If we intersect a prism or a pyramid with two arbitrary planes ε and ε which

intersect all edges but do not contain the apex, then both intersected figures
are related by central perspectivity (in the case of a pyramid) or parallel
perspectivity (in the case of a prism).
When (spatial) parallel perspectivity is interpreted in a parallel projection
(e.g., on drawing paper), it represents perspective affinity. Accordingly, spatial
central perspectivity appears in the image as perspective collineation.
With parallel perspectivity (Fig. 3.60) and central perspectivity (Fig. 3.62),
corresponding points lie on parallel perspectivity rays or perspectivity rays
through the apex. Corresponding sides of the section polygons intersect each
other on the straight intersection line a = ε ∩ ε: The respective face ϕ of the
prism or pyramid intersects ε and ε along the straight carrier lines ε ∩ ϕ and
ε ∩ ϕ of the respective edges, and these have to pass through the intersection
point ε ∩ ε ∩ ϕ = a ∩ ϕ. The straight line a is called the axis of perspectivity.
3.5 Planar sections of prisms and pyramids 119

Fig. 3.61 Perspective affinity of two point fields

If one planar section polygon of a prism or a pyramid is known, then each

further section can be determined very efficiently by just using the axis a
of perspectivity and only one point of the new intersection. It is then pos-
sible to restrict oneself to a “two-dimensional” construction: Let P1 Q1 be
an edge of the first section figure and P2 the already known point on the
edge through P1 . Q2 can then immediately be determined since P1 Q1 and
P2 Q2 intersect on a. This construction works for each projection and is truly
two-dimensional.
It is even possible to free oneself from spatial interpretation and to define
the following:

If two planar connected fields of points are related in such a way that cor-
responding points lie on rays that are either parallel or pass through a fixed
center, and if additionally the corresponding connecting straight lines of two
points intersect on a fixed axis a, then the relation is a planar perspective affinity
or a planar perspective collineation. The connecting rays of the corresponding
points are called the rays of affinity or rays of collineation. The axis is called
the axis of affinity or axis of collineation, and the center is called the center of
collineation.

⊕ Remark : The connecting straight lines of respective points pass through a center,
which may also be a point at infinity. The points are “mutually related via perspec-
tive.” This concept was already mentioned in connection with Kiepert’s Theorem
(p. 19). Therefore, one speaks of perspective affinity and collineation. ⊕
The preservation of affine ratios and parallelity further applies to perspective
affinity: This follows from the two aforementioned properties (assigned points
120 Polyhedra: multi-faced and multi-sided

Fig. 3.62 From the spatial situation to a task in the plane: central perspectivity

Fig. 3.63 Spatial perspectivity can be applied to all views. The parallel projection on the left and
the central projection on the right show how parallel lighting is imaged. The left image shows a
planar perspective affinity, the right image a perspective collineation.

lie on parallels, assigned straight lines intersect on the axis of affinity) and
from the intercept theorem.
In this way, the following entities are perspectively affine / collinear in space
during each parallel projection / central projection:

• two planar sections of a prism / pyramid;

• each planar polygon and its parallel / central projection in a plane;

• each planar polygon and its parallel shadow / central shadow on a plane.

• each planar polygon and its parallel shadow / central shadow on a plane.

In anticipation of the next chapter, prisms / pyramids can also be “refined” by

choosing a base curve instead of a basal polygon. In such cases, the following
entities are perspectively affine / collinear:
3.5 Planar sections of prisms and pyramids 121

Fig. 3.64 Perspective affinity (space) ↦ perspective collineation (plane)

• Two planar sections of a cylinder / cone.

Fig. 3.65 The affine relation between a circle and an ellipse (de la Hire’s construction, sometimes
called the Proklus construction, uses this fact and is indirectly illustrated in Fig. 11.12 on the
right.)

⊕ Remark : Mathematically speaking, an affine relation between two fields of points

P (x, y) and P ∗ (x∗ , y ∗ ) is expressed by a linear relation between the respective
coordinates:

x∗ = A x + B y, y∗ = C x + D y (A, B, C, D ∈ R).

In matrix notation [9], one writes p∗ = M ⋅ p. This mapping is invertible if, and only
if, M is invertible. Thus, p = M−1 ⋅ p∗ , while M−1 is the inverse matrix of M. ⊕
This enormous range of potential applications explains the importance of
perspective affinities and perspective collineations. We will prove, for in-
stance, that an arbitrary planar section of a cylinder of rotation is an ellipse
(p. 164). On the other hand, the base curve of a cylinder of revolution is a
circle. Therefore, we have:

Circle and ellipse are mutual affine images.

122 Polyhedra: multi-faced and multi-sided

Fig. 3.65 on the left shows the affinity of an ellipse to its principal apex circle,
and on the right its affinity to its auxiliary apex circle.
E

g =π∩γ

E′

hc e
p

ec
pc eye point E, image plane π, base plane γ
axis of collineation g = π ∩ γ
yellow circle e ⇒ image ellipse ec
red circle p ⇒ image parabola pc
blue circle h ⇒ image hyperbola hc
Fig. 3.66 The collinear relation between a circle and the three types of conic sections, see also
the general case with a non-perpendicular image plane (Fig. 9.24).

Fig. 3.66 can be seen as an anticipation of Chapter 9: If circles (e, p, and

h) in a plane γ are projected from a center (the eye point E) to an image
plane π, we get conic sections (ec , pc , and hc ) after the intersection with the
quadratic projection cones. These can be ellipses, parabolas, or hyperbolas.

Fig. 3.67 The water basin in front of St. Charles’ Church in Vienna. Depending on whether the
plane parallel to the sensor plane through the lens center touches its boundary, intersects it, or
not, we get an ellipse (left), a parabola, or a hyperbola (right).

In every case, the central perspectivity in space appears as a perspective

collineation in the drawing plane. The image of the straight line g of γ and
3.5 Planar sections of prisms and pyramids 123

π is the axis of collineation. The image of the center E is the center of

collineation. Any intersection of the circles with their projections in π lies on
the axis of collineation.
Since π and γ commute, i.e., it does not matter which comes first, the afore-
mentioned is true not only for the circle but for all conic sections. It is, thus,
possible to “change” a hyperbola or parabola into an ellipse and even (in the
special case) into a circle by means of perspective collineation. Fig. 3.67 is
an example of the perspectively collinear mapping of an ellipse – after all, a
photograph produces a kind of collinear mapping!

Fig. 3.68 “Heidelberg horse”: How many conic sections (or parts of which) are visible on both
photos? Hint: All boundary curves on the horse, and on its neck in particular, are circular arcs.
You may stop counting once you have found 3 hyperbolas and 20 ellipses . . .

Fig. 3.68 shows two photos containing a multitude of conic sections in the
form of collinear images of circles and ellipses: The left mainly contains im-
ages of circles (the circle-shaped boundaries of the horse’s neck are depicted
as parts of hyperbolas), but also elliptic shadows on the cylinders of revolu-
tion contained in the building. On the right, there are elliptic conic sections
that also appear in the photo as ellipses.
▸▸▸ A flexible interactive installation from an Archimedean solid

Fig. 3.69 Kinetic Architecture: Guvenc Ozel (with Alexandr Karaivanov, Jona Hoier, Peter In-
nerhofer). The inside has a flexible triangular mesh (http://ozeloffice.com/).
124 Polyhedra: multi-faced and multi-sided

The final example in this section does not relate to the truncated octahedron
but to another Archimedean solid (Fig. 3.44 on the right): “Cerebral Hut”
explores the relationship between architecture, motion, and human thought.
We traditionally assume that an artificial environment influences our psyche.
What if a kinetic piece of architecture could establish a direct connection
between the thoughts of its user and itself in order to reconfigure its physical
boundaries accordingly?

Fig. 3.70 In order to create a space that is reactive to brain activity, the team hacked and
reprogrammed a commercially available device that can measure concentration levels and blinking,
interpreted these data thresholds, and wrote scripts that would translate them into motion.

Research on different folding patterns and geometric structures has led to the
creation of an environment that can act as a vessel for motion. Cerebral Hut
plugs into a vein of contemporary research that explores kinetic environments
and the relationship between technology, motion, and space, but it is the first
of its kind that creates a moving piece of architecture that directly responds
to human thought.

Fig. 3.71 The biosphere by Buckminster Fuller (Montreal)

A remark from a geometrical point of view: The triangulation of the bio-

sphere in Montreal (Fig. 3.71) is comparable to the (flexible!) triangulation
used in the installation. Cerebral Hut has no final or ideal design form. Its
interior and exterior are in constant transformation, which is triggered by
user participation. ◂◂◂
4 Curved but simple

In this chapter, we shall discuss the theory of curvature of planar and space
curves as well as basic curved surfaces that are easy to imagine, such as
spheres and cylindric surfaces. The latter differ substantially in one respect:
The sphere is “doubly curved” while cylinders are just “single curved.”
We will first talk about the sphere, despite its double curvature, because
it is very easy to define and imagine. The sphere plays a central role in
further geometric considerations. Every planar section of a sphere and also
the intersection of two spheres is a circle.
We first need to understand the theory of planar and space curves. In space,
we have two kinds of curvature, namely curvature and torsion. By using
curves, we can create surfaces. The sphere may be generated by rotating a
circle about one of its diameters. Cylindric surfaces can be generated from
straight lines that are moved parallel to a certain fixed direction through
space. The images of all possible curves on a surface envelop the contours
of the surface. If this contour only consists of straight lines for every projec-
tion, the surface is developable. It is only such surfaces that can be unfolded
(developed) into the plane without distortion.
The planar section of a cylinder of revolution is an ellipse (in general), and its
development is a sine curve. The intersection of two cylinders of revolution
is usually a space curve, but it may in special cases split into two ellipses.

© Springer Nature Switzerland AG 2020 125

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_4
126 Curved but simple

4.1 Planar and space curves

Every branch of science has its own language by means of which its ideas can
be expressed clearly and concisely. Before discussing curved surfaces, let us,
therefore, define certain terms that are particularly important in differential
geometry.
Basics from planar curve theory
Differential geometry deals with properties of curves and surfaces that de-
scribe them independently of parametrizations and coordinate systems such
as tangents, curvatures, etc., which can be calculated by means of mathe-
matical differentiation. For instance, if one considers two points P and Q of
a planar curve, then its connection line forms a secant (lat. secare = to inter-
sect). If Q now moves closer to P , then there exists an unambiguous limiting
position which is called tangent to the curve (lat. tangere = to touch). One
simply says: At the point of contact P , the tangent of a curve and the curve
share two points which are infinitely close together. The normal to the tan-
gent at P is called the curve normal. If the curve is a circle, all curve normals
pass through the center. However, in general, the totality of normals to the
curve envelops a curve – the so-called evolute.

Fig. 4.1 Tangent, normal, evolute of different curves

We can also consider two neighboring points Q and R to the curve point P .
If the three points do not lie on a straight line, they define a circle. We now
let Q and R move closer to P . This eventually leads to a well-defined limiting
position of the circle determined by three points – the osculating circle at
the point P . (A circle can also degenerate into a straight line or a point.)
The radius of the osculating circle is called the radius of curvature. In the
vicinity of the curve point, it approximates the curve much better than the
tangent. The osculating circle at P is said to have three points in common
with the curve.
4.1 Planar and space curves 127

The locus of the midpoints of all osculating circles is the evolute enveloped by
the curve normals.

Proof : The osculating circle is determined by three neighboring points P , Q, and

R. So, its center is located at the intersection of the perpendicular bisectors of P Q
and P R. Let Q∗ be the midpoint of P Q and R∗ that of P R. The perpendicular
bisectors, therefore, converge towards the curve normal in Q∗ R∗ . The circumcenter
of P QR thus converges towards the intersection of two adjacent curve normals. ⊙
▸▸▸ A circle as an evolute
Fig. 4.1 shows curves with their evolutes. One can see that evolutes have
cusps whenever the curve has an apex – a point whose radius of curvature is
stationary. The right curve is especially notable. It is constructed in such a
way that their evolute is a circle. It is, therefore, called a circle’s involute. ◂◂◂

▸▸▸ Osculating circles of the ellipse

Fig. 4.2 Evolute and osculating circles of an ellipse

In Fig. 4.2, the evolute of an ellipse is visible – a diamond-shaped curve

with four cusps at the centers of the osculating circles at the apices of the
ellipse. The figure also illustrates how to construct the osculating circles of
the apices, which are very useful when we draw an ellipse. ◂◂◂

▸▸▸ A curve can be its own evolute

In Fig. 4.3, one can see logarithmic spirals with their evolutes. These are
curves that traverse a pencil of rays under a constant angle ψ. Their equation
in polar coordinates (r, ϕ) is r = ep⋅ϕ . The constant parameter p = cot ψ
determines how fast the logarithmic spiral wraps around its asymptotic point.
The evolute of such a curve is another logarithmic spiral that is congruent
to the initial curve if p is chosen appropriately. In special cases, it can even
be equal to the initial curve (right image).
Logarithmic spirals play an important role in nature because they are related
to exponential growth. They also have their place in technology: The photo
in Fig. 4.4 shows a paving slab lifter. Due to the weight of the object being
128 Curved but simple

Fig. 4.3 Logarithmic spirals with their evolutes

Fig. 4.4 Logarithmic spirals in technology

lifted, the spirals are pressed towards the bottom, transmitting the pressure
back to the slab. ◂◂◂

▸▸▸ We steadily rotate the steering wheel

Fig. 4.5 Clothoid with winding points: The radius of curvature decreases propor-
tionally to the arc length which makes the image look so balanced.
4.1 Planar and space curves 129

The reciprocal value of the radius of curvature is called the curvature of

the curve. The distance of two points, measured along the curve, is called
arc length. Fig. 4.5 shows a so-called clothoid. It is characterized by the fact
that the curvature increases linearly with the arc length. Thus, the radius of
curvature decreases proportionally to the arc length. Clothoids are applied
in road and railway construction, where a constant rotation of the steering
wheel makes it possible to change a straight path into a circular one with
curvature continuity. The equation of the curve (it requires the numerical
solution of integrals) is given in [9]. The curve wraps around two points
(asymptotic points) in an infinite number of loops. ◂◂◂

▸▸▸ Envelopes of circles

c
E E
c

Fig. 4.6 Envelopes of circles centered on a curve c that contain a fixed point E

c
c∗ E∗
n
c∗ P
t
t
E E
c
n
E∗

Fig. 4.7 The envelopes c∗ are orthonomic curves of c with respect to E.

Motivated by Fig. 4.5, let us define a transformation as follows: Take a planar

curve c and consider the set of circles in the plane centered on c that contain
a fixed point E. What is the envelope of all circles? Fig. 4.6 shows two
130 Curved but simple

“educated guesses”: When we take a circle or an equilateral hyperbola as c,

the results look familiar (a limaçon of Pascal and a “lemniscate of Bernoulli”
(Fig. 4.6).
How can we find the envelope of the circles in a pointwise manner? For this
purpose we have to intersect two neighboring circles. Since, by definition, one
of the intersection points is always E and the other one lies on the normal n
of the connection line of the two neighboring centers of the circles – and
this connection line is the tangent t of c (Fig. 4.7). So, the points E ∗ of our
desired curve c∗ are symmetric to E with respect to the tangents of c. Such
curves are called “orthonomic curves”. The variety of orthonomic curves is
great, since we can choose both the curve c and the position of E. Clearly,
the less the degree of c, and the more special the point E that is chosen, the
more specific the curves c∗ . ◂◂◂

▸▸▸ Polarizing curves

In Chapter 1 we introduced polarity at a circle (Fig. 1.37). Points correspond
to straight lines and vice versa. Polarization of a curve c at a circle k was
done as follows: The tangents t of c are transformed into the points T ∗ of
the resulting curve c∗ . The points P of c result in the tangents p∗ of c∗
(Fig. 1.41). Conics, for instance, are transformed into conics again.

c c∗

c
c ∗
k T∗
P
p∗
t
k t P
p∗
T∗

Fig. 4.8 Different spirals obtained by polarization at a circle: The left side shows a hyperbolic
spiral c (polar equation r = a/u) which is transformed into an involute of a circle c∗ , on the right
we see another logarithmic spiral c which remarkably results in a reversely congruent logarithmic
spiral c∗ .

Fig. 4.8 shows two more special cases: If we polarize a hyperbolic spiral at
a circle around the spiral center, we get an involute of a circle (see also
Fig. 4.11). The second example is especially noteworthy, since it illustrates
that the logarithmic spiral does not change its shape when it is polarized at
a circle around its center. ◂◂◂
4.1 Planar and space curves 131

▸▸▸ Offset curves

The offset curve c∗ of a curve c at distance d is a curve that always has the
same normal distance d from c (Fig. 4.9). The curves c and c∗ can then also
be called “parallel curves”. All parallel curves have the same evolute. If an
offset curve hits the evolute, is must have a cusp since then the osculating
circle has radius zero.

Fig. 4.9 Offset curves of a given curve c.

Fig. 4.10 Involutes of a circle c and their congruent (!) offset curves

Usually offset curves with signed distance d differ for positive and negative
distance. This does not apply any more if the curve c has a cusp. As an
example take the involute of a circle (Fig. 4.11). This curve is the only one
out of all curves whose offset curves are of the same type. They even are
congruent!
◂◂◂
132 Curved but simple

Fig. 4.11 Offset curves in practice. Left image: tracks of a marine snail feeding on jelly fish on
the beach. Right image: a basket, where the curve in the middle closely resembles an ellipse.

Fig. 4.12 Planar curves provide a source of inspiration to artists (Greg Lynn, “St.Gallen”).

▸▸▸ Architectural design using curves

Fig. 4.12 shows a virtual architectural design (“Intricate Surface”) with a
number of planar intersection curves that obey certain rules that the artist
has developed (“mutations”). ◂◂◂
4.1 Planar and space curves 133

The theory of space curves

Fig. 4.13 Almost two-dimensional curves . . .

If the curve does not lie in a plane, then the whole issue becomes more
challenging. The terms secant and tangent do not change their meanings.
The curve normal, however, is no longer uniquely defined: Each normal to
the tangent through the point of contact is a curve normal and lies in the
normal plane ν of the curve tangent.

Fig. 4.14 Osculating circles of a space curve

In space, we can also consider a curve point P and two neighboring points
Q and R. If the three points do not lie on a straight line, they define a circle
in space. If we now let Q and R move towards P , the osculating circle of the
134 Curved but simple

curve at the point P converges to a limiting position. Its carrier plane is called
the osculating plane. We say: The osculating plane shares three (infinitely
close) points with the curve at the point of contact.

Fig. 4.15 Accompanying tripod (also called Frenet frame) of a space curve

Among all normals of the curve at the point P , we can now highlight one
normal in particular – the one that lies in the osculating plane, also called
the principal normal. If we now consider the normal that lies perpendicular
to the osculating plane (the so called binormal ), then we have finished the
curve’s accompanying tripod.
⊕ Remark : All osculating planes σ envelop the tangent surfaces of the curve, which
we will discuss in detail later. All normal planes ν of a curve envelop a surface
called normal developable. In Fig. 4.37 on the right side, we see an example of such
a surface. More interesting for us is a surface that is enveloped by the so-called
rectifying planes. The rectifying plane ̺ is spanned by the curve tangent t and the
binormal b (Fig. 4.15 on the right). The thus resulting surface is called the rectifying
developable. It occurs, for example, in twisted paper strips (Fig. 5.60).

Fig. 4.16 Special projections of a curve

⊕
The normal projection of a curve to the osculating plane σ in P is a planar
curve that has the same osculating circle in P ′ = P as the space curve it-
self (Fig. 4.16, left). When projected orthogonally to the rectifying plane ̺,
4.1 Planar and space curves 135

the curve has (as a rule) an inflection point P ′′ = P . (The osculating circle
appears as a straight line; Fig. 4.16, middle.) In the case of so-called handle
points – points which may be found in a symmetry plane of a curve – a flat
point occurs. When we finally project orthogonally to the normal plane ν
(Fig. 4.16, right), a cusp P ′′′ = P occurs because the curve tangent appears
as a point.
In the application on p. 142, we will discuss the shortest curves (geodesic
lines) on curved surfaces as applications of the aforementioned topics.

Fig. 4.17 Torsion of a space curve and something akin to an accompanying tripod

Apart from the curvature, geometry also employs the notion of the torsion
of a curve (Fig. 4.17).
Circles (seen as space curves) have constant curvature but zero torsion. He-
lices as “generalized circles” also have constant curvature, but they addition-
ally have constant torsion.
The torsion at a curve point is a measure of the rate at which the osculating
plane rotates about the tangent (by analogy with the curvature of a planar
curve, which is a measure of the rate at which the tangent rotates about the
center of curvature). A curve with zero torsion lies in a fixed plane, which is
the osculating plane (by analogy, a planar curve without curvature remains
in the direction of a straight line, which is the tangent).

▸▸▸ The simplest space curve

Perhaps, the one example of a space curve that is the simplest to understand,
is the helix, which will be discussed in greater detail in Chapter 7.
In a helical motion (a composition of a rotation and a proportional translation
along the rotation axis), the helix constantly winds along the axis of a cylinder
of revolution. Therefore, curvature and torsion are the same in each point of
the curve, and all the tangents and osculating planes enclose the same angle
136 Curved but simple

Fig. 4.18 A photo of a helix inside of a LCD-lamp

to the axis of the cylinder. Projected in direction of its axis, the helix appears
as a circle. Any projection perpendicular to the axis is a sine curve. Fig. 4.18
shows a photo of a helix, i. e., a perspective image. Fig. 4.19 illustrates that
such curves also occur in nature.

Fig. 4.19 A helix-like space curve in nature

Fig. 4.20 A helix (gray/green) is polarized at a sphere centered on its axis. In this special projec-
tion perpendicular to the axis, the helix appears as a sine curve. The polarized curve (red/yellow)
lies on a cone of revolution.
4.1 Planar and space curves 137

In Fig. 4.20 (and also in a general view in Fig. 1.42) such a helix c is polarized
at a sphere centered on the axis. Its osculating planes σ will result in the
points S ∗ of the polarized curve c∗ . The radial rays leading to the points S ∗
are perpendicular to σ and thus also have a constant angle to the axis.
Consequently, c∗ lies on a cone of revolution. Furthermore, the distance of σ
increases linearly with the rotation angle u during the helical motion. Since
in the polarity transformation the product of the distances of the plane σ and
its pole S ∗ is constant, the curve c∗ appears as a curve with polar equation
r = a/u (a is a constant), which is a hyperbolic spiral (Fig. 1.42 on the
right). ◂◂◂

Space curves can create surfaces

If a curve moves through space (during which its form may change), it usually
“sweeps” a surface. The different positions of the curve are called u-lines. The
orbits of the curve’s points form a second set of curves on the surface, the v-
lines (Fig. 4.21). In general, only one u-line and one v-line pass through each
surface point P . The tangents tu and tv at P to both lines span a plane τ
which touches the surface at P and is called the tangent plane. The normal
n of the tangent plane P is called the surface normal.

Fig. 4.21 Tangent plane, surface normal, and u- and v-lines of a surface

Contour and outline of a surface

Let us view a curved surface from a point E (in Fig. 4.22, it is a point E∞ at
infinity). Then the contour of the surface is usually seen immediately. It is
the locus of certain points on the surface that form the contour line k. Let K
be a contour point. It is defined such that the viewing ray EK touches the
surface, and therefore, lies in the tangent plane τ of K. This means that
it is perpendicular to the surface normal n. Let us formulate the following
important theorem:

In a contour point, the viewing ray is orthogonal to the surface normal. The
tangent plane at the contour point goes through the projection center.
138 Curved but simple

Fig. 4.22 Contour point, contour on the surface, and outline of the surface in a projection plane

Fig. 4.23 Rather tricky contour of a snail shell: The surface outline (black and red) is enveloped
by the images of arbitrary surface curves. Note that boundary lines are not part of the outline.

In the projection to an image plane π, the surface – apart from potentially

occurring boundary curves – is bounded by the projection k p of the contour
line (the outline).
The outline is also enveloped by the projections of all possible surface curves.
The following theorem is especially important for geometrically correct free-
hand drawing:

If a curve on a carrier surface intersects the contour of the surface, then, in

general, its projection touches the outline of the surface.

Only in exceptional cases, the projection cp of a surface curve c has a cusp

on the outline – namely, when the projection center E lies on the tangent of
c at the intersection with the contour (Fig. 6.93).
4.1 Planar and space curves 139

▸▸▸ Why do highlights occur on curved surfaces?

The basic geometric assumption is as follows (Fig. 4.24): A point R of

Fig. 4.24 Highlights in different surfaces. Left: one light source (the Sun). Right: reflections of
different non-point light sources (flashlights).

an illuminated surface is a highlight when the projection center Z and the

light surface L lie in the same plane through the surface normal n in R. An
additional restricting condition says that n must be the angular bisector of
LR and ZR. Fig. 4.24 shows highlights on spherical surfaces. The highlights
lie in planes that include the respective centers of the spheres, the position
of the light source, and the lens center of the camera. ◂◂◂

How can the curvature of a surface be measured?

The term surface curvature is much more complicated than the simple cur-
vature of a curve: By merely considering the normal sections of the surface
at a surface point P as in Fig. 4.21 on the left (these are the surface curves
through P in a plane through the surface normal n), one deals with an infinite
number of curvatures of curves through P .

Fig. 4.25 Elliptic, parabolic, and hyperbolic surface points, flat point (bottom right)

Even with the diversity of curved surfaces, three significantly different types
of surface points can be distinguished. For this purpose, let us consider the
surface in such a way that P is the contour point (the projection center lies
140 Curved but simple

in the tangent plane of P ). If the surface now lies in a sufficiently small

neighborhood of P on both sides of the tangent plane, then we speak of a
hyperbolic point (Fig. 4.25 bottom left). Each saddle surface possesses points
which are exclusively of this type. If the surface touches the tangent plane
at one side, then we speak of an elliptic point (top left; a typical example
is the sphere). If there exists a certain surface curve (or tangent) through
P along which the surface, only considered locally, remains completely in
the tangent plane, we speak of a parabolic point (in Fig. 4.25, top right,
we see how a tangential section “forms a cusp” at the point). Cylinders and
cones represent this type particularly well. A fourth type can only happen
singularly or alongside a curve on the surface: Fig. 4.25, bottom right, shows
a so-called flat point. This differentiation has great practical implications.
Surfaces which consist only of parabolic points (including some flat points)
are the easiest to modify. What is more, they can typically be unfolded
(“developed”) into a plane. Hyperbolically curved surfaces are much harder
to modify than even elliptically curved ones.
Both elliptically and hyperbolically curved surfaces cannot be developed, which
is especially impractical in the case of the sphere: There is no way of mapping
the surface of the Earth into the plane without distortions.
Polygon mesh on a sphere
While spheres can be approximated more or less well by many quadrangles
or triangles (the facets of insect eyes are also congruent, as can be seen in
Fig. 4.26 on the right), not all faces will be mutually congruent. Fig. 4.27
shows a not-quite-perfect distribution of triangles. This triangulation can be
derived by the “refinement” of an icosahedron.

Fig. 4.26 Left: simulation of a compound eye, derived from a refined icosahedron as triangle mesh
(Fig. 4.27). Spheres with a suitable radius are placed in the triangle’s centroid. These spheres
intersect each other and form polygon patterns that usually consist of hexagons and sometimes
also pentagons. Right: for comparison, the eye of a robber fly.

▸▸▸ A moderately equal distribution of points on the sphere

The problem of evenly distributing a given number n of points on the surface
4.1 Planar and space curves 141

Fig. 4.27 Refinement of an icosahedron

of a sphere is only exactly solvable if we have 4, 6, 8, 12, and 20 points,

because in these cases, Platonic solids are formed in the following sequence:
tetrahedron, octahedron, cube, icosahedron, dodecahedron. Based on these
solids, the pattern may be “refined” as in Fig. 4.27, thus allowing for multiples
of the aforementioned numbers.

Fig. 4.28 10, 15, 64, and 250 points distributed uniformly on the sphere

Fig. 4.28 illustrates a possible solution for other numbers of points. The
solutions were calculated by using a dynamic approach that employs quasi-
physical repulsion for the points. The shortest distances between the points
are parts of great circles on the sphere (see p. 143), and the repulsive forces
increase and decrease by the square of the distance. The translation of the
points on the surface continues until a state of equilibrium is reached. The
algorithm was developed by Franz Gruber based on the ingeniously simple
algorithm by Fruchterman/Reingold [2]. This problem is also known as the
Thompson problem, which is concerned with the minimization of energy of n
equal point charges on the sphere. ◂◂◂

▸▸▸ Triangulation as a powerful tool

Triangulation is of high importance in the computer age: It is not only use-
ful for the approximation of surfaces but also for the simplification of com-
plex geometric volumes. Many particularly difficult physical problems can be
solved with adequate accuracy by breaking down the situation into partial
problems.
142 Curved but simple

Fig. 4.29 A tilted brandy glass and a weaving ashtray

Fig. 4.29 on the left shows how a body is first cut into sections – in this case,
it is a fictitious brandy glass. The emerging intersection curves can be highly
complicated (closed) line strips that may also have an arbitrary number of
holes. In the second step, these sections are triangulated (subdivided into
triangles) and can thus be interpreted as volumes (three-sided prisms with
the thickness of a disc). For each prism, the centroid, the statical momen-
tum, and the rotational momentum regarding arbitrary axes (see [9]) can be
determined by using simple formulas. The sum of all centroids is called the
total centroid (in the picture, it is marked red). The sum of all rotational
moments is called the total rotational momentum.

Fig. 4.30 Some complications arise with the introduction of fluids . . .

Realistic physical simulations can be accomplished in this way. For instance,

it is conceivable, given the right impulse, that the ashtray in Fig. 4.29, right,
can set itself upright from its seemingly hopeless situation. When simulating
fluids (Fig. 4.30), however, simple geometry is no longer sufficient due to the
complexity of the situation. ◂◂◂

▸▸▸ Geodesic lines – the shortest connection

If two points on a curved surface need to be connected by a surface curve of
minimal length, this will (in general) not be a straight line. We shall discuss
this problem particularly in relation to the sphere and the shortest connecting
line in this special case (application p. 159, Fig. 4.32 on the right).
Let us first define the following:

A geodesic curve on a surface is characterized by the fact that in all of its

points, the respective osculating plane includes the surface normal.
4.1 Planar and space curves 143

Fig. 4.31 Art work with curved approximations to spheres (Kyoto): Bending straight sticks creates
great circles on the sphere (see also Fig. 4.32 on the right).

This leads to the following:

The shortest connection between two points in a surface is a geodesic curve on

the surface.

At this stage, let us not proceed with a mathematically rigorous proof but
merely make the matter more plausible: By definition, all principal normals
(Fig. 4.15) of a geodesic curve are at the same time normals of the carrier
surface. Let us now consider two neighboring points Q and R of a surface
point P . In the normal projection to the tangent plane in P , the surface
curve is approximated by the normal projection of the osculating circle – it
is a straight line because the osculating plane is its carrier plane.
In other words: Locally speaking, the curve works like a straight line – which,
as we know, is the absolutely shortest path. Every other connection from Q
to R would be a detour.
⊕ Remark : For two special cases, there exists a simple and easily applicable solution:

Geodesic lines on spheres are great circles. Geodesic lines on developable surfaces
become straight lines in the unfolding.

First of all, let us consider the sphere: The osculating plane of the surface curve has
to contain the surface normal and must, therefore, pass through the center of the
sphere. Thus, the osculating circle at every point is a great circle. If the geodesic
curve were to deviate from the great circle, then the osculating plane would “sway,”
which would lead to a contradiction.
144 Curved but simple

Fig. 4.32 Left: geodesic lines on a general surface. A “right triangle” has been drawn in – however,
the Pythagoraen theorem only applies if the surface is developable. Right: Geodesic lines on a
sphere are great circles.

Now let us take a look at developable surfaces: Unfolding preserves lengths, and in
the plane, the shortest connection between two points is a straight line. Fig. 4.88
shows geodesic curves on a cylinder of revolution that result from the bending of
straight lines. These are helices, which we shall discuss further in Chapter 7. ⊕
◂◂◂

How can we recognize a developable surface?

Fig. 4.33 Straight contour parts of developable surfaces must be visible from all directions.

If a surface is parabolic at every point P , then it also has, by definition, a

tangential direction in P along which the surface stays entirely in the tangent
plane. If one follows the surface along the respective tangential direction, then
one always stays in the same tangent plane. This plane constantly touches
the surface. It follows that parabolic surfaces are composed of straight lines
and that, along these straight lines, they are touched by the same plane.
⊕ Remark : However, the condition that a surface is composed of straight lines is not
sufficient. Quite the contrary: Most curved surfaces that are composed of straight
lines are not developable! ⊕
How, then, can we find out whether a surface is parabolic throughout? If
each tangent plane touches along a whole straight line, and if this tangent
4.1 Planar and space curves 145

Fig. 4.34 Developable remains developable. . .

plane happens to run through the projection center, then each point of the
straight line is a contour point. In other words: The contour of such a surface
is composed of nothing but straight line segments (Fig. 4.33, Fig. 4.34, and
Fig. 4.35).

The only surfaces that can be developed into the plane without distortion are
surfaces without elliptic or hyperbolic points. A developable surface can be
recognized by the fact that its contour is straight from any viewpoint.

Fig. 4.35 Straight outlines and shadow boundaries in a virtual enveloping cone

⊕ Remark : Insects such as hazel leaf rollers are capable of bending and rolling a
leaf in a peculiar way and laying their eggs in the protected “nest” that is thus
created. Sometimes, dehydration alone causes a leaf to assume other developable
forms (Fig. 4.36). Since the “shrinkage factor” is not the same for all parts, being
lower, for example, at the leaf-veins, “shrivelling” soon occurs in the leaves.
⊕
Even if the contour of an object in space is merely a “harmless” curve, tips
occur relatively often when dealing with the contours of hyperbolic surfaces:
In each case the tangent plane in a contour point contains the center of
146 Curved but simple

Fig. 4.36 Shrivelled leaves and curved contours

the projection. That is why the tangent at the contour often passes exactly
through the center.
At this point, let us make an additional statement about the osculating cir-
cles of a space curve. A picture says more than a thousand words – therefore,
simply enjoy the following (more information can be found in academic lit-
erature):

Fig. 4.37 All osculating circles of a space curve and their axes form a surface.

If a point moves along a space curve, the related osculating circle constantly
changes. The surface that is generated by all these osculating circles can be
very complicated (Fig. 4.37 left) and is always doubly curved.
However, if we are dealing with a surface containing the axes of all osculating
circles, we also have a curved surface, though it is only single curved in that
case. This kind of surface is called polar developable or evolute developable.
4.1 Planar and space curves 147

Its analogue in the plane is the locus of all centers of curvature of a planar
curve – the evolute. Fig. 4.37 on the right shows the polar developable of a
cubic circle (see also Fig. 6.41).

Analogous considerations for shadows

Fig. 4.38 Self-shadow and cast-shadow of curved surfaces

As we already know, there is actually no difference between projection and

illumination. All that we have learned so far about projection equally applies
to central and parallel illumination.
Instead of the contour, we now refer to the boundary of the self-shadow (ter-
minator), and instead of the outline of the projection, we now refer to the
cast-shadow boundary. The cast-shadow of a surface is surrounded by the
cast-shadow of the self-shadow boundary and possibly by cast-shadows of
the surface’s boundary curves. If a surface is developable, a part of the self-
shadow / cast-shadow boundary is straight (Fig. 4.38).
The line that separates lit areas and areas in self-shadow is called self-shadow
boundary or sometimes (especially in the context of planets) terminator.
Fig. 4.39 shows that the inner part of a hollow (and therefore non-convex)
cylinder carries both types of shadow, which are separated by a straight
line through T . The viewer cannot distinguish between the two types, and
the separating terminator is not clearly visible. However, someone who draws
such figures or writes computer programs to render them needs to be familiar
with this.
Just to make it clearer, let us repeat the above argument in other words:
An object always produces a cast-shadow on planes, on other objects, and
potentially – if the object is not convex – on itself. However, there are parts
of the surface of the object that are “simply dark” (Fig. 4.40) because they
are turned away from the light source. These parts are called self-shadow.
148 Curved but simple

Fig. 4.39 Self-shadow and cast shadow of curved surfaces: On the inner part of the cylinder, we
have a cast shadow of the cylinder on itself and self-shadow (“dark areas”).

Fig. 4.40 “Dark” parts on a surface: The parts of the surface that are turned away from the light
source are in “self-shadow.” If the surface is not convex (as in this example), additional cast-shadows
of the surface will potentially appear on itself (not displayed in these images), especially when the
light comes from higher above (images to the right).

Fig. 4.41 The test in practice: At sunrise, some hours later (hazy) and in the afternoon.
4.2 The sphere 149

4.2 The sphere

Despite the fact that we have talked about developable surfaces in adequate
detail, let us start with a surface that is not developable. The sphere is par-
ticularly easy to imagine, occurs frequently, and has many special properties.

Fig. 4.42 Spheres as distance surfaces

The sphere is the locus of all space points that have a constant distance (the
radius r) from a given point (the center M ).

The sphere can be defined in a different way, which is advantageous for many
geometric considerations:

Fig. 4.43 A sphere is generated by the rotation of a circle about its diameter.

A sphere is also formed by the rotation of a circle about an arbitrary diameter

of this circle.

Let us consider a point P of the circle during a rotation about an arbitrary

diameter d (Fig. 4.43): It describes the path circle with the axis d and the
150 Curved but simple

radius dP . At maximum radius, the straight line M P is orthogonal to the

axis d. Sphere circles with a maximal radius (the sphere radius) are called
great circles – all other circles are small circles. If P lies on d, then the small
circle shrinks to a “null circle”. Its carrier plane is orthogonal to the sphere’s
radius at the contact point (Fig. 4.44).
If we now consider that d can be chosen in an infinite number of ways, we
have proven the important theorem that every planar section of a sphere is
a circle (Fig. 4.46):

Each planar section of a sphere is a circle. The intersection circle is a great

circle only if the plane contains the center. All tangent planes of the sphere are
orthogonal to the radius at the contact point.

Fig. 4.44 Tangential planes of a sphere orthogonal to the radius at the contact point

Fig. 4.45 The intersection of two spheres is always a circle.

If we let two circles rotate about a common diameter, we get two spheres
(Fig. 4.45). During rotation, the common points of the circles generate the
same circle:
4.2 The sphere 151

Fig. 4.46 Circles as intersections of spheres

The intersection of two spheres is always a circle which lies in a plane orthogonal
to the straight line spanned by their centers.

⊕ Remark : An additional remark for advanced readers, who may think that the
theorem should be stated more precisely: Two spheres only share a circle if they
are close enough to each other. In fact, it can be calculated that two spheres always
share a circle, but it is not always a real one.
Its center is always real and lies on the straight line joining the spheres’ centers.
We can further say (in what may be an overstatement): Two spheres always share a
non-real circle in the plane at infinity – the so-called absolute circle (see also p. 194).
Once we know this, some things are no longer surprising: Two spheres should share
a space curve of degree four, but because the absolute circle is always part of it,
the intersection splits off, and there only remains a planar curve of degree two – a
circle which may be real or complex, or even a single point. ⊕

⊕ Remark : If we rotate a circle not about one of its diameters but about a chord, we
get a so-called spindle torus (Fig. 4.47 center and right). We will discuss it further
in the section concerning surfaces of revolution. ⊕
Parts of spheres often occur in architecture and technology. They may be
found, among others, in hanging domes, spherical caps, and Byzantine domes.
152 Curved but simple

Fig. 4.47 Left: a circle rotates about one of its chords, generating a spindle torus. Right: The
roof of the famous Cathedral of Santa Maria del Fiore in Florence (Filippo Brunelleschi, see also
Fig. 9.2) is an approximation of a spindle torus. Neighboring circular arcs are combined with
(elliptic) cylinders.

▸▸▸ The position of an airplane

The distance r of an airplane to a ground control station B can be determined
via radar. The airplane lies on the distance sphere around B at the distance r
(Fig. 4.48).
If the synchronous measurements of three ground control stations (with spec-
ified relative positions) are known, one gets three distance spheres that share
two points: Two spheres intersect along a circle, and all three circles pass
through the common points (Fig. 4.49). One of these (above the surface of
the earth) is the position of the airplane. ◂◂◂

▸▸▸ The “spatial Pythagoras”

We are looking for a cuboid where the three edges connected to the corner
P pass through three fixed points A, B, and C (Fig. 4.49, center). Due to
∢AP B = ∢BP C = ∢CP A = 90○ , the point lies on the Thales spheres above
AB, BC, and CA. For this cuboid corner, it can be shown very elegantly
by using vector calculus that the square of the area of the triangle ABC is
equal to the sum of the squares of the triangle areas AP B, BP C, and CP A.
The analytic proof can also be found in [9] and is given here as well due to
its brevity:
Proof : Let P be the point of origin of a Cartesian coordinate system with the
points A(u, 0, 0), B(0, v, 0), and C(0, 0, w). The triangles AP B, BP C, and CP A
then have the areas u⋅v
2
, v⋅w
2
, and w⋅u
2
.
The surface d of the triangle ABC results from the cross product

RR −u −u RR RR vw RR
1 Ð→ Ð→ 1 RRRR⎛ ⎞ ⎛ ⎞RRRR 1 RRRR⎛ ⎞RRRR 1 √
d= ⋅ ∣AB × AC∣ = ⋅ RRR⎜ v ⎟ × ⎜ 0 ⎟RRR = ⋅ RRR⎜wu⎟RRR = ⋅ (vw)2 + (wu)2 + (uv)2 .
2 2 RR⎝ 0 ⎠ ⎝ w ⎠RR 2 RR⎝ uv ⎠RR 2
RR RR RR RR
4.2 The sphere 153

Fig. 4.48 Intersection of three hemispheres: Two hemispheres always intersect each other along a
semicircle. All three hemispheres have a common point.

Fig. 4.49 Intersection of three spheres and cuboids through three points (center)

1
Thus, d2 =
4
[ (vw)2 + (wu)2 + (uv)2 ] = b2 + c2 + a2 . ⊙ ◂◂◂

▸▸▸ Newton’s kissing number problem

How many spheres of the same size can be arranged around a central sphere
of that same size so that they each touch the central sphere? Newton’s correct
assumption was twelve spheres, because a thirteenth sphere simply would not
fit even though there was a little space left. How can we provide a solution
to this problem?
Solution:
Let r be the sphere radius. Then, all the spheres that touch the central sphere
Σ0 (center in the coordinate origin) lie on a sphere Σ∗0 with radius 2r that is
concentric to Σ0 . We select the first sphere on the x-axis, say Σ1 . Now, all
spheres that touch Σ1 lie on a sphere Σ∗1 with radius 2r that is concentric to
Σ1 . The intersection
√ circle of Σ∗0 and Σ∗1 is a horizontal circle in the xy-plane
(with radius 3r). Now – as in the two-dimensional plane – one could find six
spheres in the xy-plane and then proceed from there. However, this does not
154 Curved but simple

Fig. 4.50 Left: Six circles fit precisely around a central (blue) circle of the same size. Center: A
sphere (represented here in blue) is given. The aim is to pack as many spheres of the same size as
possible around that blue sphere. Right: This cellar spider is faced with the same problem when
carrying its eggs around in a sac.

make for ideal packing, even though we could “stick” three (orange) spheres
on both sides and thus get exactly twelve spheres. It turns out that there is
still a bit of room for improvement.
Let us, then, select only one neighboring sphere Σ2 in the xy-plane and follow
the same steps (concentric circle Σ∗2 ). In the intersection of Σ∗0 , Σ∗1 , and Σ∗2 ,
we will get two further sphere centers that are symmetrical with regard to
the base plane, etc. Again, we have twelve spheres around Σ0 . There is some
space left, but not enough to add a thirteenth sphere. It was only in 1953(!)
that this could be proven (see http://de.wikipedia.org/wiki/Kissing_
number_problem). In nature, we might be able to add a thirteenth sphere by
squeezing and pushing (Fig. 4.50 right). ◂◂◂

Outline and shadow of a sphere

The following theorem provides the key for all further considerations:

The tangent planes of a sphere along a small circle envelop a cone of revolution,
and those along a great circle envelop a cylinder of revolution.

Proof : Let us imagine a circle k (center M ) in the plane (Fig. 4.51) and a point S,
from which two tangents t and t∗ to k exist. The straight line T T ∗ connecting the
contact points is – for reasons of symmetry – perpendicular to the diameter d = M S.
Let the carrier plane of the circle now rotate about d. The circle k becomes a sphere
Σ, t generates a cone ∆ of revolution with the apex S, and the contact point T
generates a small circle u with the axis d. The tangent of u at T and the line ST
span the tangent plane τ of ∆. During rotation τ envelopes ∆. If we now let S pass
into infinity (S∞ ), then the cone of revolution becomes a cylinder of revolution, and
the small circle u becomes a great circle. ⊙
How do we see a sphere from the center S? From S, we can consider an en-
veloping cone of revolution that touches the sphere. If S is a point at infinity,
then the cone becomes a cylinder of revolution. The contact curve of the
4.2 The sphere 155

Fig. 4.51 Tangential cone and cylinder

Fig. 4.52 Outline and shadow of a sphere

cone or cylinder of revolution is a small circle or great-circle u respectively.

However, this circle is already the contour of the sphere: According to con-
struction, the tangent plane in each point passes through S. The analogue is
true for central and parallel illumination. Let us summarize:

The contour and self-shadow boundary (terminator) of a sphere is always a circle

on the sphere. With parallel projection or parallel illumination, the contour and
self-shadow boundary is a great circle.

▸▸▸ The tilt of the slender moon crescent – a “result of temperature”?

Why does the slender moon crescent often appear to be “lying on its back”
in tropical regions while having a more upright position in northern regions?

Solution:
Whether or not the moon crescent is inclined or upright depends on the
moon’s terminator (see also application p. 79). It appears as an ellipse, and
its apparent inclination (tilt) depends on its principal axis. Therefore, it
matters from which direction the Moon is illuminated. If it is illuminated
156 Curved but simple

Fig. 4.53 Tilt of the slender crescent: a) at the poles, b) at the equator, beginning of spring or
fall, c) somewhere in-between in winter, d) somewhere in-between in summer

from the side, then the slender crescent appears to stand upright, and if the
light comes from above or below, then an inclination becomes apparent. We
must further be aware of the following: From the vantage point of Earth, the
path of the new moon follows the path of the Sun, though with a certain
delay (see also p. 488). This is also approximately true for the slender moon
crescent (but not for fuller moon phases).
Let us start with an extreme: At the North Pole (Fig. 4.53 a), the path of
the sun (if it is visible at all) appears as a circle at a fixed elevation. The
slender moon crescent should also appear at a similar elevation – and not
very far from the sun. When viewed from the pole, the Moon is illuminated
“from the side” so that the self-shadow boundary (terminator) is visible as an
upright half-ellipse. In the deeper polar winter, neither the Sun nor the moon
crescent can be seen. (However, as is apparent on p. 488, the full moon in
polar winter is very much visible. Upon closer inspection, it can be discovered
that the near-full moon looks the same as in the tropics!)

Fig. 4.54 Within several hours, our Moon, when photographed from one and the same location
on Earth, changes its inclination, provided we keep the lower edge of our camera parallel to the
ground. The images above show two cases: the picture on the left was taken two hours before
sunset, the picture on the right six hours later.
4.2 The sphere 157

Now for the other extreme: At the equator, there are two days of the year
when the sun moves at a circle through the zenith (Fig. 4.53 b). The slender
moon crescent follows this path. This time the terminator appears as an
inclined half-ellipse: From the perspective of the equator, in the days around
new moon, it is illuminated almost precisely from the top (waxing moon, in
the afternoon and evening hours) or from the bottom (waning moon, in the
morning hours).
In general, path circles of the Sun and the Moon in the days around new
moon are steeper in latitudes closer to the equator. The steeper they are, the
flatter the ellipse appearing as the images of the Moon’s terminator are. It can
further be said that the crescent moon will predominantly occur in summer,
not in winter, because the sun’s path is steeper in summer (Fig. 4.53 c and
d). Therefore, it is not at all far-fetched to say that the position of the slender
moon crescent is correlated to (even if not directly caused by) temperature
... ◂◂◂

▸▸▸ Spheres and spherical parts everywhere . . .

Spheres can be found everywhere, especially in nature: Because of gravity,
celestial bodies like suns, planets and even moons with a diameter of more
than 500 km are shaped in a way that closely resembles a sphere. Surface
tension causes water droplets to also take the shape of spheres (Fig. 4.55),
since a sphere allows a given volume to be enclosed by the smallest surface
possible.

Fig. 4.55 Tiny water droplets in a spider net and on a leaf

In technology, parts of spheres can also be found quite often. In Fig. 4.56
we can see some navigation systems which are commonly used on ships:
On the left, hemispheres are attached to a cylinder of revolution. In the
transition from one surface to the other, we can see a sharp bend in the self-
shadow boundary. In the center image, two sphere caps and a short cylinder
of revolution between them can be detected. On the right, the shape of a
“satellite dish” approximates a sphere cap very closely (theoretically, it is
158 Curved but simple

a paraboloid of revolution, see application on p. application p. 241). The

self-shadow of the cap is circular (see also Fig. 4.81).

Fig. 4.56 Navigation systems on ships

As a last example, let us analyze the shades and shadows in Fig. 4.57. The
bowl with small spherical tomatoes inside has the shape of a half sphere. The
half sphere is bordered by an equator circle which is composed out of two
half circles c1 and c2 . The left half circle c1 casts a circular shadow cs1 onto
the spherical interior of the bowl, the right half circle c2 casts a congruent
shadow cs1 onto the table plane parallel to the equator plane. The half sphere
is separated into two spherical parts by the half great circle (terminator) e =
e1 ∪ e2 . e is optically hard to detect, but the shadow it casts is the dominant
ellipse es = es1 ∪ es2 on the table.

e1 cs1
c1 c2

t es1
s
t

cs2
e2
es2

Fig. 4.57 The bowl has the shape of a half sphere.

Finally, let us consider the tiny tomatoes to be spheres. The sun rays touching
them along their terminator t form cylinders of revolution. The shadow ts of
the terminator onto the bowl is the intersection curve of the cylinder with
the sphere – a space curve that is “almost” a circle. ◂◂◂
4.2 The sphere 159

The shortest path on a sphere

In space, the shortest connection between two points is a straight line. On
the globe, we encounter a problem when dealing with larger distances: We
cannot, like Jules Vernes, travel through the center of the Earth. The shortest
connection between two points A and B on the sphere is a great circle –
a circle whose plane is defined by A, B, and the center M of the sphere.
Therefore, if a ship were to travel from a certain circle of latitude to another
point on the same latitude, the shortest path would not follow this circle
(except at the equator or a meridian)!
⊕ Remark : The circle of latitude would be more practical for navigation purposes:
It can be calculated without much trouble by measuring the angle of elevation to
the peak level of the sun (see p. 481). For this reason, it used to be common in old
times to travel along circles of latitude – better to cross some more sea miles than
to “miss” a life-saving island in the ocean: The precise determination of the circle
of longitude used to be very difficult at sea. This problem was eventually solved by
the use of “stormproof” clocks [9]. ⊕

Fig. 4.58 Left: shortest path from Vienna to Varadero. Right: Flight paths are – at least in
theory – great circles. In practice, jet streams especially, but also other factors, must be taken
into account. That is why many flights head up high to the North even when their origin and
destination airports lie on the same circle of latitude.

▸▸▸ From Vienna to Varadero

In the age of GPS (Global Positioning System, Fig. 4.56), attempts may
actually be made to come close to the great circle while crossing from A to
B. Of course, one must not forget the importance of water and air streams
(“jet streams”) when undertaking such an endeavor. Fig. 4.58 shows a flight
path between two continents – naturally displayed on a 2D map, on which
160 Curved but simple

great circles are not recognizable as such. These are photographs of a low-
resolution monitor within an airplane.

Fig. 4.59 The global distribution of GPS-satellites (flight altitude of approx. 20 000 km)

In the middle of Fig. 4.58, the 2D map was transferred onto a sphere. The
flight path should now come close to a great circle (the offset is negligible).
The flight from Vienna (48○ northern latitude, 16○ eastern longitude) to
Varadero in Cuba (20○ northern latitude, 70○ western longitude) leads into
northern latitudes at first. From the globe shadow, which deeply envelops
the North Pole, it can be concluded that the flight must have taken place
at the approximate time of the winter solstice. During the multi-hour flight,
the shadow boundaries move westwards. The image on the right shows the
ideal flight path (which is a great circle). Fig. 4.74 shows the distribution of
GPS-satellites worldwide. With their flight altitude of more than three times
the Earth radius, they orbit the Earth twice a day. ◂◂◂
4.3 Cylinder surfaces 161

4.3 Cylinder surfaces

The tried-and-true cylinder of revolution. . .

So far, we have already used the term cylinder of revolution because we have
assumed that everybody can imagine such an object. Let us now attempt an
exact definition:

A cylinder of revolution is the locus of all points in space that are equidistant
to a given straight line (the axis) at the fixed distance r (the radius).

As with a sphere, a cylinder of revolution can also be generated by a motion

in space:

A cylinder of revolution is generated by the rotation of a straight line about a

parallel axis.

We can conclude that cylinders of revolution are “refined” regular prisms. In-
cidentally, computers also display them as regular prisms with a high number
of edges. (The eye can barely distinguish the difference between a 200-sided
prism and a perfect cylinder.)
Both given definitions define a cylinder of revolution as unbounded, or “in-
finitely long.” In practice, cylinders of revolution are usually delimited by
sections which are perpendicular to the axis (circles). The development of
such a bounded cylinder of revolution is a rectangle with the width 2π ⋅ r and
the height h.

Fig. 4.60 Outline and cast-shadow of a cylinder of revolution

Cylinders of revolution carry only parabolic points (single-curved). Their

outline and their shadow are usually composed of two straight lines, including
the images or shadows of the boundary circles (Fig. 4.60).
162 Curved but simple

. . . and general cylinders

A cylinder is not necessarily a cylinder of revolution (as is the case in technical
drawings).

A general cylinder is created by moving a straight line parallel along an arbitrary

directrix.

Fig. 4.61 General cylinder: planar intersection, intersection with a straight line

General cylinders retain important properties of cylinders of revolution, ex-

cept for rotational symmetry. They are single-curved (and thus developable),
which can also be recognized from the fact that their contours or self-shadow
boundaries are composed of straight lines.

Fig. 4.62 Cylindrical boundaries of a sound box

The outline of a guitar’s cylindrical boundaries is a good example. The con-

tours are composed of straight lines, and this is true for the normal projec-
tion (the left photo in Fig. 4.62 was taken with a telephoto lens), as well as
the central projection (wide-angle lens used in Fig. 4.62 on the right). The
photographs at hand were shot with light flare (indoor shot) to avoid sharp
shadow boundaries. The shadow of the side wall is, nevertheless, recognizable
as being straight.
4.3 Cylinder surfaces 163

Fig. 4.63 General cylinders Fig. 4.64 Doubly rolled palm leaves

A model of a general cylinder can quickly be “crafted” by everyone: One only

needs to drag a piece of paper across the sharp edge of the surface of a table.
This procedure changes the structure of the squashed paper, and a paper roll
emerges. A phenomenon of this kind often occurs in nature, such as in plant
leaves (Fig. 4.64).
General cylinder surfaces also play an important role in architecture: They
can be used to shape rooftops or façades that are reasonably cheap to produce
– owing to the simple fact that they can be cut in the plane.
Geometrically speaking, we often deal with parallel projections or parallel
shadows with “circumscribed” cylinders – as in the case of the sphere, where
the circumscribed cylinder is a cylinder of revolution. Fig. 4.65 shows the
cylinders “underneath” different spatial spirals (see Chapter 7, p. 315).

Fig. 4.65 Different spirals on a cone of revolution. Their projection cylinders with generators
parallel to the cone’s axis are of interest.
164 Curved but simple

4.4 The ellipse as a planar intersection of a cylinder

of revolution

A classic proof of spatial geometry

Fig. 4.66 Oblique sections of cylinders of revolution

The ellipse – as defined by the Greeks – occurs as a planar skew section of a

cylinder of revolution.

Proof : The proof by the Frenchman Dandelin is as ingenious as it is simple: A

conventional ellipse is the locus of all points in the plane where the sum of the
distances from two fixed points – the focal points F1 and F2 – is constant. Dandelin
noted that these points can be seen as the contact points of two congruent spheres
that touch both the cylinder of revolution and the plane.

Fig. 4.67 Dandelin spheres Fig. 4.68 Pipe elbow

4.4 The ellipse as a planar intersection of a cylinder of revolution 165

Let us call the points F1 and F2 (they do not lie on the cylinder of revolution). If
we now take an arbitrary point P on the intersection curve, we notice that there
exists a cylinder generator e through the point P , which touches the spheres in the
points 1 and 2. We can say: The (identical) straight lines P 1 and P 2 touch both
spheres, and the sum P 1 + P 2 is constant – equal to the distance from 1 to 2. On
the other hand, the straight lines P F1 and P F2 are likewise tangents of the spheres
from P . All tangents of a sphere through a fixed point P have a constant tangent
distance and form a cone of revolution. Thus, P F1 = P 1 and P F2 = P 2. The sum
P F1 + P F2 is also constant (= 12), which is precisely what we wanted to show. ⊙

A multitude of applications
If the cylinder is upright, the lowest point A and the highest point B on
the intersection curve are the principal vertices of the ellipse (Fig. 4.68). The
midpoint M is the intersection of the plane with the cylinder axis. The points
C and D (which have the same height as M ) are the auxiliary apices.

Fig. 4.69 Construction of a pipe elbow: Two concentric congruent ellipses are overlapped by
means of a rotation.

Both parts of an obliquely truncated cylinder can be attached to form a pipe

elbow by rotating one of them by 180○ about the normal of the plane at the
center of the ellipse (Fig. 4.69). If the intersection plane forms an angle ϕ
with the cylinder axis, then the pipe axes and the plane of the ending circles
form the angle 2ϕ. For ϕ = 45○ , the pipe elbow is perpendicular.

Fig. 4.70 Vaults with ellipses as intersections

166 Curved but simple

Four cylinders of revolution that are cut at both sides at 45○ in the manner
of a wedge can be put together to create a beam connection. The intersection
consists of a pair of ellipses that intersect each other perpendicularly in the
neighboring apex (such a connection is visible in Fig. A.10).
Geometrically speaking, the same problem arises when two cylinder-of-revo-
lution-like vaults intersect (Fig. 4.70 shows the bottom view of a groin vault
and a barrel vault).

Fig. 4.71 Variations of a three-cylinder connection

The variations of such cylinder connections (Fig. 4.71) should illustrate the
enormous importance ascribed to planar intersections of cylinders of revolu-
tion.

Fig. 4.72 Intersection of two cylinders of revolution: Palmenhaus/Schönbrunn, Vienna

Proceeding with this game, one can connect six pointed cylinder-of-revolution
wedges, producing a spatial cross. Six ellipses are involved, and it truly chal-
lenges our spatial imagination to sketch such an object. (You can find more
on this topic in the appendix concerning freehand drawing.)
Fig. 4.73 on the left shows a beam connection with three pairwise orthogonal
axes. The “interior” (center left), as a physical model, rolls well. For the com-
puter, such intersections often lead to numerical problems (“near-identical”
planes must be intersected). This is why, in Fig. 4.73, the cylinder radii were
set to be slightly different, which immediately causes the intersection curves
to “break off” at double points.
4.4 The ellipse as a planar intersection of a cylinder of revolution 167

Fig. 4.73 Beam connection with three beams, a variation using eight-sided prisms

In Fig. 4.73 on the right, the cylinders were approximated by regular, eight-
sided prisms. The shared interior of the prisms is, then, a solid that is com-
posed of congruent squares or hexagons.

Fig. 4.74 Illustrations like these can be helpful to spatial imagination.

Multiple cylindrical ungulas can be attached “in sequence” (Fig. 4.76), thus
creating an approximation to a so-called torus, about which we will have
much more to say at a later stage.
During development, a cylinder of revolution bounded by circular sections
changes into a rectangle with width 2π ⋅ r. What kind of curve results from
an elliptical oblique section?
This time, a calculation is obligatory. We have to relate the arc length that
occurs during development to the height of a curve point. Let us consider an
ellipse point in plan view and front view. In the plan view, P ′ is a circle point
with the coordinates x = r ⋅ cos u, y = r ⋅ sin u (u is the polar angle in radians).
If the section plane is inclined towards the horizontal base plane (a plane at
the height of the neighboring apex) at an angle of α = 90 − β, then the height
of P equals z = y ⋅ tan α = k ⋅ y (k being constant). When developed, the point
has the coordinates P v (r ⋅ u/z) or P v (r ⋅ u/k ⋅ r ⋅ sin u). The developed curve
is, therefore, essentially a sine curve.
The principal vertices 4 and 8 of the ellipse (Fig. 4.75) correspond to the
apices of the sine curve. The auxiliary vertices correspond to the inflection
points of the sine curve. The inclination of the tangents at the inflection
168 Curved but simple

Fig. 4.75 Development of a cylindrical ungula: The development of the intersection curve follows
a sine function, of which the osculating circles and the tangents of inflection (angle β) should be
specified. (For example, the osculating circle at point 4 is equal in size to the osculating circle of
the ellipse that is orthogonally projected onto the tangent plane τ .)

Fig. 4.76 Development of a pipe connection

points equals the inclination of the plane which can be seen in the front
view.
⊕ Remark : When a pipe connection is developed, it is equally valid to develop a
prototype. The other parts are congruent. It is possible to proceed even more eco-
nomically: By imagining the ungula as rotated by 180○ before development, the pipe
connection is stretched into a single cylinder of revolution (Fig. 4.76). Therefore, we
only need to develop one cylinder of revolution with potentially multiple (though
essentially congruent) oblique sections. The result is a rectangular strip to which
one or more (often congruent) sine curves are inscribed. This minimizes wear, which
will delight the pragmatist! ⊕

Of course, it doesn’t work every time

It appears contradictory that the intersection of two cylinders of revolution
should yield two ellipses, since the following famous theorem holds:

Bézout’s theorem: Two algebraic surfaces of degree m and n generally share

an intersection curve of degree m ⋅ n.
4.4 The ellipse as a planar intersection of a cylinder of revolution 169

Fig. 4.77 Arbitrary intersections of cylinders of revolution

Proof : The two surfaces are intersected by an arbitrary test plane along two planar
curves of the orders m and n. It is sufficient to show that these m have n points in
common, because the intersection curve must be of the same order.
The theorem about planar algebraic curves is, as the name indicates, a classical
algebraic problem and remains to be proven. The algebraic curve of n-th order is
given by the implicit equation
i j
∑ aij ⋅ x ⋅ y = 0 with 0 ≤ i + j ≤ n.

The algebraic curve of degree m can be written in the same way:

i j
∑ bij ⋅ x ⋅ y = 0 with 0 ≤ i + j ≤ m.

We eliminate one variable, say y, from both equations by computing the resultant
of the two polynomials. This yields an algebraic equation of degree m ⋅ n. According
to the fundamental theorem of algebra, this polynomial possesses m ⋅ n solutions
(including complex and multiple solutions). ⊙

Fig. 4.78 Filleting of an intersec- Fig. 4.79 Arbitrary intersection of cylinders (on the left:
tion beaver food)

If one intersects two quadratic surfaces (to which cylinders of revolution

belong), then a space curve of degree four is to be expected.
170 Curved but simple

Fig. 4.77 shows the intersection curves of cylinders of revolution with inter-
secting axes and slightly different radii. (The curves appear as circles when
projected in the direction of the axes, and as parts of hyperbolas when pro-
jected orthogonally to the plane spanned by the axes.) Such intersections
occur very frequently in technical applications. Fig. 4.79 shows a curious ex-
ample from nature: A beaver has gnawed away at a mighty tree, which may
be blown over by the next storm.

Fig. 4.80 The space curves may split into parts under certain circumstances.

The intersection curve of two cylinders of revolution may also split into parts:
When put together, both section ellipses also form a space curve of degree
four. (After all, they share four points with an arbitrary plane.)
Fig. 4.80 illustrates how the intersection of two cylinders of revolution may
disintegrate if two criteria are given: First, the cylinders must have an equal
radius, and second, their axes must intersect. In other words: Both surfaces
must share an inscribed auxiliary sphere. In the normal projection to the
plane that is spanned by the axes, the hyperbola that occurs for different
radii splits into a pair of straight lines: A hyperbola does not possess a double
point.
Degenerating intersections are not only frequently employed in technology,
but they can also be observed in nature (in the context of shadows), which
is due to a second theorem:

The shadow of a planar boundary curve of a degree-two (quadratic) surface

onto itself is a conic section.

Proof : The planar boundary curve is a conic section c and, in the case of a sphere
or a cylinder of revolution, a circle (Fig. 4.81). The light rays through c form a
degree-two cone Λ (an oblique cylinder in the case of parallel shadows). In both
points T1 and T2 , where the self-shadow boundary of the surface (a conic section
4.4 The ellipse as a planar intersection of a cylinder of revolution 171

Fig. 4.81 Shadow of a sphere part or a cylinder of revolution on itself

or a pair of straight lines) intersects the boundary curve, the light cone and the
surface touch each other. Due to this double tangency, the intersection curve splits
into two conic sections, one of which is the boundary curve c. The other curve c∗
carries the boundary of the boundary curve’s shadow onto the surface itself. ⊙

Fig. 4.82 Self-shadow of a planar intersection on a sphere or a cylinder of revolution

▸▸▸ Shadows as intersection curves

Fig. 4.83 (left image) shows a lamp that appears to produce sunlight. The
other two photos show what happens with the light in the next 90 minutes
after the first photo was taken. The artist has covered a window with a sheet
of paper that has an elliptic hole cut into it, and through this hole sunrays
can enter the room. The shadow of the ellipse will normally be an ellipse,
but under specific circumstances it can also be a circle. In this special case,
the cylinder that is defined by the circle and the direction of the sunrays is
an oblique circular cylinder, and the corresponding ellipse on the paper is a
vertical section of it. In this way one will find a fitting ellipse for arbitrary
directions of the incoming light.
Fig. 4.84 shows that the light rays surrounding the tire form an approximately
oblique circular cylinder. (In the photograph, it is depicted as a cone with
the sun point for its apex.) An intersection with the half pipe (which is itself
172 Curved but simple

Fig. 4.83Sunlight, Earth rotation, paper, lamp: sunlight through an elliptic opening
in the window and its movement during 90 minutes (Liddy Scheffknecht: oculus,
2012).

Fig. 4.84 The shadow of a tire on a curved road is a space curve of degree four.

part of a cylinder of revolution) results in a space curve of degree four as the

shadow of the tire.

Fig. 4.85 Shadows of a radiator on a curtain

4.4 The ellipse as a planar intersection of a cylinder of revolution 173

Fig. 4.86 The radiator behind the curtain

Fig. 4.85 shows the shadows of the radiator of a central heating onto a curtain.
The radiator itself is depicted in Fig. 4.86. ◂◂◂

The process of development. . .

Fig. 4.87 on the left illustrates how a sheet of brass shaped like a cylinder
of revolution can be distorted into a rectangle. Mechanically, this is accom-
plished by continuously bending the brass by small amount at even time
intervals. The further away from the cutting position a generator lies, the
more often the bending procedure has to be repeated.

Fig. 4.87 Development of a cylinder of revolution and how it is used to roof a swimming pool

The position highlighted in red looks approximately like half of an ellip-

tic cylinder (though this resemblance is not exact). In Fig. 4.87, right, this
position was used as the roof of a small swimming pool. The reflection in
the water gives the additional appearance of a closed (and almost elliptic)
cylinder.
. . . and its inversion (rolling up the development)
In practice, instead of developing a surface, it must sometimes “envelop” it
(Fig. 4.88). Simple examples include the placement of a label onto a bottle
174 Curved but simple

Fig. 4.88 Inversion of the development (“envelopment”) on a cylinder of revolution. On the left:
rewinding straight lines (helices) and a circle. On the right: A circular aluminum foil rolled into a
cylinder of revolution is used as a “spout.”

and the circular aluminum foil that is rolled up and put into a bottleneck to
prevent its liquid content from dripping when it is poured (Fig. 4.88 on the
right). The “inclusion” of straight lines during envelopment produces geodesic
lines on the carrier cylinder (see p. 142). In the general case, these are helices
– we shall later address these important curves in great detail.
If a circle whose diameter equals the circumference of the cylinder is also
rewinded, a curve with a tacnode, i.e., a point where the curve touches itself,
is produced. Such a curve is highlighted in red in Fig. 4.88 on the left. The
tacnode is visible in the obscured region. In the background of Fig. 6.34,
circles rolled up onto a cylinder of revolution are visible on the bamboo
stools.

Fig. 4.89 rolling up Fig. 4.90 “Rolling up” of a circle

In purely geometric terms, the production of potato chips in Fig. 4.90 is a

tricky matter. Obviously, a thin layer of pulp is cut out. If it is then rolled
up onto a cylinder of revolution – as with a “rolling pin” – we would not yet
expect a doubly curved surface. It can be assumed that the pulp is immedi-
ately pressed onto a doubly curved stencil. The pulp is thick enough to stand
the necessary stretching.
Fig. 4.89 on the left shows two ancient Egyptian bracelets where planar curves
made of pure gold were enveloped onto an “elliptical cylinder” (the forearm).
5 More about conic sections
and developable surfaces

Like spheres and cylinders, cones are surfaces that we can easily imagine. Like
cylinders, they are simply curved and developable. Among all cone surfaces,
the cone of revolution and the oblique circular cone are especially important.
Their planar sections are the conic sections per se: ellipses (among them
circles), hyperbolas, and parabolas. We can derive many common properties
through a spatial interpretation of these most famous planar curves.
Cones and the cylinders are developable: they can be unrolled into the plane
without distortion. They only share this characteristic with so-called devel-
opables, sometimes also called torses or tangential surfaces of space curves.
Like cylinders and cones, these are composed of straight lines called gen-
erators or rulings. Developables are envelopes of one-parameter families of
planes where each plane touches along a certain generator. This makes them
easily recognizable: From any viewpoint, their contours are generators. The
shadow boundaries, too, always have straight components.
Doubly curved surfaces, such as the sphere, cannot be unfolded into planes
without distortion, but because this is such an important task, mathemati-
cians and geometers have spent an immeasurable amount of time on the issue
of transforming the surface of the globe into a planar map while leaving cer-
tain geometric properties unaffected by the process.

© Springer Nature Switzerland AG 2020 175

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_5
176 More about conic sections and developable surfaces

5.1 Cone surfaces

Like cylinders, cones are notable for their simplicity. We can explain cones
as follows (Fig. 5.1):

Fig. 5.1 The motion of a straight line through a fixed point (the apex); right: an approximation

A cone is generated by a straight line e that always passes through a fixed

point S. S is the apex, e is a generator and each pose of e can also be considered
a generator of the cone.

In geometry, a cone is not bounded and can thus always be interpreted as a

double cone, i.e., the surface consists of both parts on either side of the apex
(Fig. 5.2 on the left).
Among cone surfaces, the cones of revolution are the most important:

Fig. 5.2 Shadows of cones and cylinders of revolution

A cone of revolution is generated by the rotation of a straight line e about an

intersecting axis.
5.1 Cone surfaces 177

Fig. 5.3 Cone of revolution with circles of latitude and generatrices

Only parts of cones of revolution occur in practice, and these are sometimes
not even easily identifiable as such (see Fig. 5.4 and Fig. 5.5).

Fig. 5.4 Rotation of straight lines and a cone-shaped excavation

Fig. 5.5 Vulcania European Park of Vulcanism Conus by H. Hollein (parts of cones of revolution)

The development of a cone of revolution

All generatrices of a cone of revolution have the same length. Or more pre-
cisely: All sections of the generatrix between the apex and a circular section
are equally long. Therefore, if we develop a cone into the plane, it becomes
a circular sector.
According to Fig. 5.6, the central angle ω of the circular sector can be derived
by equating the respective circular arcs. With the notations from the figure,
178 More about conic sections and developable surfaces

Fig. 5.6 Development of a cone

we have 2π ⋅ r = ω ⋅ s and thus

r r
ω = 2π and also in degrees ω ○ = ⋅ 360○ .
s s
Given an equilateral cone with s = 2r, we get half a disc. A frustum of a cone
of revolution is developed into an annular sector. Curves on the cone become
curves that intersect the generatrices at the same angle as they did in space
(application p. 198).
▸▸▸ Rolling up onto a cone
As with cylinders of revolution, we can also “invert” the development. We
can inscribe an arbitrary curve k v into the developed lateral area of the cone
of revolution, such as a circle (as in Fig. 5.7).

Fig. 5.7 Rolling up a circle

The result is a space curve k on the cone that, if developed, produces the
curve k v . The curve k can be determined as follows, point by point: We pick
5.1 Cone surfaces 179

a generator ev in the development. We then determine the central angle ϕ

to the respective central angle ϕv on the base circle and thus reconstruct the
generator e on the cone. Finally, we measure the distance S v P v from the
development on e. ◂◂◂

▸▸▸ An unbalanced dumbbell rolling on a plane

If the co-axial wheels of trains were not conical, they would continually derail.
Even the steel ring on the inside of the wheel (Fig. 5.8) could not prevent
derailment.

Fig. 5.8 The components of the wheels of a train and the rails . . .

The slightly conical shape of the wheels (Fig. 5.8) is essentially responsible
for that effect: If the train drifts to the right, then the contact circle on the
right wheel apparently increases its diameter due to the conical shape of the
wheel. At the same time, the diameter of the contact circle on the left wheel
decreases (causing a tilt of the axis). This is what forces the axis to move
back into the correct and stable position. The smaller and the larger circle
can be interpreted as parallel circles of a virtual cone of revolution. In the
perfect equilibrium of a train that runs smoothly along the rails, the circles
have the same size and the connection surface is a cylinder of revolution.

Fig. 5.9 Left: A virtual cone – the hull surface of an “unbalanced dumbbell” – is rolling on a
plane. Right: The circles of contact of each wheel (red) also form such a virtual cone. The rigid
wheel system will steer around a left curve when the left circle is smaller.

When the train has to navigate a curve, the conic wheels are also of great
advantage. Imagine the left curve in Fig. 5.9: The tracks have to be inclined,
180 More about conic sections and developable surfaces

otherwise the centrifugal forces would tip the train over. However, these cen-
trifugal forces will still shift the train to the right. This causes the circle of
contact with the right track to increase, and the corresponding circle on the
left to decrease. This achieves the desirable outcome of the train automati-
cally tilting to the left (Fig. 5.9). Inclining not enough to the right will push
the train further right, whilst inclining too far to the right will push it back
to the left. The perfect equilibrium lies somewhere in between. ◂◂◂

▸▸▸ A plane rolling on a cone

Instead of a cone rolling on a plane we can let a plane roll on a cone. In
this case, the point in the plane that touches the apex will not move. Points
in the plane that are at a constant distance from the apex (and therefore
lie on a circle) will move on spheres. This movement belongs to the topic of
spherical kinematics, which we will discuss in greater detail in Chapter 11.

Fig. 5.10 A circle rolls on a cone with an apex identical to its center. Straight lines in the plane
leave “traces” on the cone. When the cone is developed, those traces are rectified to straight lines
again.

Straight lines in the planes leave a “print curve”, which is also called a geodesic
curve, on the cone. ◂◂◂

General cones are, in practice, not as common as arbitrary cylinders. They

are sometimes found in the form of connecting surfaces of two “directrices,”
which often reside in parallel planes (Fig. 5.15).
▸▸▸ Connecting cone between a rectangle and a circle
Fig. 5.11 shows how a connecting surface composed of (planar) triangles and
oblique circular cones can be pieced together in such a way that there are no
edges:
If, for instance, a circle k and a rectangle r in parallel planes are given as
boundary curves (Fig. 5.11 on the left), then the points 1, 2, 3, and 4 on k –
5.1 Cone surfaces 181

Fig. 5.11 Connecting surface between a rectangle and a circle in theory. . .

Fig. 5.12 . . . and practice

where the tangents are parallel to the rectangle sides – are to be determined.
These points are then connected to the rectangle sides by triangles. Four
oblique circular cones, which are touched by the triangles, remain.
If the carrier planes of the boundary curves are not parallel, then the points
1, . . . , 4 can be obtained by intersecting the rectangle sides with the section
line s and, from there, positioning the appropriate tangents to k. ◂◂◂

▸▸▸ The jet of a hairdryer

We want to connect a “rounded rectangle” with a circle in a parallel plane
by a smooth surface.

Fig. 5.13 Theory . . . Fig. 5.14 . . . and practice

182 More about conic sections and developable surfaces

Solution:
Fig. 5.13 shows a case where the rectangle is rounded off by circles. Fig. 5.14
shows a practical application of this case.
The following idea leads to an elegant solution: The longer sides of the rect-
angle can be “handled” – as in the previous example – by the use of two
triangles. Two circles of different sizes in parallel planes remain (if we imag-
ine the upper half-circles as complete). There now exist two oblique circular
cones with the apices S1 and S2 on which the circles lie. They form the
connecting surfaces that touch both triangles.
Fig. 5.15 shows an airport counter designed according to this method. The
advantage, apart from an aesthetic one, lies in the fact that there are no hard
edges that could cause injuries. ◂◂◂

▸▸▸ The cone according to Le Corbusier

Fig. 5.16 depicts a design by the French architect (1887–1965) who was
famous for his formally strict yet functional architecture. A more precise
analysis of the sophisticated object shows its remarkably simple method of
construction:
As in application p. 180 (connecting cone between a rectangle and a circle),
Corbusier developed a connecting surface between a base square and a circle.
He then simply and obliquely truncated the object composed of triangles and
oblique circular cones. The top surface is composed of ellipse fragments (see
the next section) and straight lines. ◂◂◂

Fig. 5.15 Connecting cone . . . Fig. 5.16 . . . of two directrices (after Le Corbusier )

A theorem with great technical applicability

Without doubt, geometry can be fascinating by itself, regardless of any prac-
tical applicability. It is, nevertheless, pleasing to find applications of geomet-
rical theorems in everyday contexts. Here is a good example:

If two surfaces of degree two touch at two points, then the intersection curve
splits into two conic sections.
5.1 Cone surfaces 183

Fig. 5.17 Symmetrical and asymmetrical trouser parts

It can be shown that a non-degenerate intersection curve of degree four can

only possess one double point (a self-intersection). On the other hand, it can
also be proven that whenever two surfaces touch each other, the intersection
curve must possess a double point precisely at the contact point.
Now, we can easily understand the following practically relevant theorem
based on the aforementioned statement:

If two degree-two surfaces share a common inscribed sphere, then the intersec-
tion curve splits off into two conic sections.

Fig. 5.18 Cylinder intersection including the explosion sketch

Fig. 5.19 Degenerate intersection of degree-two surfaces

At first, the intersection curve of both surfaces is of degree four. Like the
carrier surfaces, it will touch the sphere twice. Both surfaces possess the
same tangent plane at the contact points (which are also double points).
This leads to the splitting of the curve into two conic sections.
184 More about conic sections and developable surfaces

Fig. 5.19 at the center shows two cones of revolution that have a common
inscribed sphere. The intersection curve thus splits off into two ellipses. If only
a part of the cone of revolution materializes, then only one ellipse remains.

Fig. 5.20 Exposition in Geneva

Fig. 5.19 on the right goes one step further, performing an affine transforma-
tion to the cones of revolution. Thus, they do not remain cones of revolution.
In the course of the affine transformation, the common ellipse changes into
a new ellipse. This principle fascinated the architectural firm Coop Him-
melb(l)au to such a degree that they decided to immortalize it in multiple
famous buildings (as in Geneva, where each of the towers is 40 meters tall;
see Fig. 5.20 or Fig. 5.21).
⊕ Remark : A spatial affine transformation is very easy to calculate: The new coor-
dinates are simply arbitrary linear combinations of the old coordinates. ⊕

Fig. 5.21 UFA Cinema Center, Dresden and Paradise Cage

5.2 Conic sections 185

5.2 Conic sections

Planar sections of a cone of revolution

The term “conic section” (often simply called “conic”) is found in every book
on geometry, almost as often as “straight line” and “circle.” It is also known as
a curve of degree two. This name implies that the curves have, algebraically
speaking, two intersections with a generic straight line.

Fig. 5.22 The different conic sections

⊕ Remark : From a mathematical perspective, the curves are given by an algebraic

equation of degree two in the form of

a1 x2 + a2 y 2 + a3 xy + a4 x + a5 y + a6 = 0 (5.1)

(the first three coefficients must not all be zero – otherwise we are left with a linear
equation). If we now insert the linearly dependent term k ⋅ x + d for the variable y,
we always get a quadratic equation in x, which leads to the two intersection points.
(Multiple solutions and complex conjugate pairs of solutions have to be taken into
account.) ⊕
What is a degenerate conic section?
We distinguish between degenerate and non-degenerate (regular) conic sec-
tions. From a geometric perspective, regular conic sections are formed by the
intersection of a cone of revolution with a plane that does not contain the
apex of the cone. Ellipses, hyperbolas, or parabolas thus emerge depending
on the inclination of the plane.
A plane through the apex of the cone either intersects the cone in two gen-
erators, touches it along a generator, or does not intersect it at all (more
precisely, it intersects along two complex conjugate straight lines with the
apex as the real intersection).
⊕ Remark : Something for the mathematically knowledgeable: A cone of revolution
with aperture 2α centered at the origin of the coordinate system and with the z-axis
for its axis can be described by the equation

x 2 + y 2 − t2 z 2 = 0
186 More about conic sections and developable surfaces

with t = tan α > 0. A non-horizontal intersection with the plane

px + z = 0, p>0

through the origin (normal vector (p, 0, 1)T ) yields the condition
√
y = ±x ⋅ p2 t2 − 1,

or a pair of straight lines that is real whenever p2 t2 > 1, or p ≤ cot α is given.

Otherwise, we are dealing with a pair of complex conjugate straight lines. ⊕
▸▸▸ Various definitions of conics
Depending on whether the underlying geometry is Euclidean, affine or pro-
jective, conics can be defined by their focal properties, as planar sections
of cones of revolution, as perspective images of circles, by their quadratic
equation. Even more definitions can be found in [16].

Fig. 5.23 The standard definitions of ellipses, hyperbolas, and parabolas

The standard definition of conics (Fig. 5.23) is based on the focal proper-
ties. There is a strict difference between ellipses, hyperbolas, and parabo-
las. Ellipses are defined as the sets of points P in a plane with a constant
sum of distances P F1 + P F2 = 2a = const. Points Q of a hyperbola with
focal points F1 and F2 are defined by a constant difference of distances
∣P F1 − P F2 ∣ = 2a = const. The points R of a parabola are characterized
by equal distances to its focal point F and its directrix l: p = Rl. ◂◂◂

Focal points
If we also allow points at infinity, then all regular conic sections possess two
focal points. In the case of ellipses and hyperbolas, both are regular points,
and with parabolas, one of the focal points “slides” in the axis direction
towards infinity.
This way of thinking has its advantages: Many properties of ellipses and hy-
perbolas that refer to their focal points can be transferred to the parabola,
which, apparently, occupies a notable intermediate position. We do not al-
ways need to emphasize the difference between regular and non-regular conic
sections either. We can imagine a pair of intersecting straight lines as hyper-
5.2 Conic sections 187

bolas that are simply seen from a very large distance. (A hyperbola is nearly
straight at almost all points. It only changes direction very briefly in the
vicinity of its vertices.) A parallel or collapsing pair of straight lines could
also be imagined as an “infinitely flat ellipse.”
Let us, for example, take a theorem that refers to the so-called focal rays (the
straight lines connecting a point of a conic section with its focal points):

The tangent of a conic section bisects the angle between the focal rays.

Fig. 5.24 The Dandelin spheres

Proof : An elegant proof of this theorem is based on the proof by presented by Dan-
delin, where a conic section is explained as a planar section of a cone of revolution.
The tangent t at a curve point P can be interpreted spatially as the intersecting
straight line of the cone’s tangent plane τ with the plane ε. The plane τ touches
the first Dandelin sphere at 1 and ε in F1 . For reasons of symmetry, t forms the
same angle with the tangent segments P 1 and P F1 . The same is true for the second
Dandelin sphere, but since the carrier lines of P 1 and P 2 are identical, t forms the
same angle with the focal rays P F1 and P F2 . ⊙
This theorem is of practical applicability. The section concerning degree-two
surfaces provides more information on it, addressing concepts such as focal
mirrors. Furthermore, the following interesting statement can be derived:

Conic sections of different type with two common focal points (confocal conic
sections) intersect at right angles.

Proof : First of all: The two conic sections are of different types – two confocal
ellipses or two confocal hyperbolas do not possess real intersection points. Even
confocal parabolas can be disjoint (Fig. 5.28). Thus, it can only be a pairing between
either an ellipse and a hyperbola or between parabolas opened in opposite directions.
188 More about conic sections and developable surfaces

Fig. 5.25 The two Dadelin spheres in the case of a hyperbolic section.

For both, it is true that the tangent is the angular bisector of the respective focal
rays at the intersection. There are always two angle bisectors forming an orthogonal
pair. ⊙
Another theorem that refers to the focal points goes as follows:

A conic section is the locus of all points at a constant distance from a fixed
circle and a fixed point in its plane.

Proof : The “proof” for the parabola actually lies in its definition: The parabola is
given as the locus of all points that are equidistant to a point and a straight line. If
this director is seen as a circle around the focal point at infinity, everything appears
to fit together once again. Concerning the other conic sections, we shall soon prove
the generalization of the theorem that includes this special case. ⊙
Once again, there are a number of applications of this theorem. For instance,
let us consider the envelope of circles centered at all points of a conic section
while passing through a focal point. It is the “director circle” that passes
through the focal point.
The above-mentioned generalization of the theorem reads as follows:

Each conic section is the locus of the centers of all circles that touch two given
circles in one plane.
5.2 Conic sections 189

Fig. 5.26 A net of confocal conic sections

Fig. 5.27 Touching circles of two given circles and the locus of their centers

Proof : The parabola occurs only if one of the circles is infinitely large and thus
degenerates into a straight line. We have already studied this case.
Let k1 and k2 be two circles with the centers F1 , F2 and the radii r1 , r2 . We can
now construct the center P of a touching circle k with the radius r if, for example,
we intersect two concentric circles around F1 and F2 with the radii r1 ± r and r2 + r
(Fig. 5.27). However, for P , we have ∣P F1 ± P F2 ∣ = ∣r1 ± r2 ∣. ⊙
⊕ Remark : If one of the circles shrinks to a “zero circle,” then the aforementioned
theorem applies. ⊕

Parabolas and catenaries

It is remarkable that there apparently exists an infinite number of “signifi-
cantly different” ellipses and hyperbolas but only one prototype of a parabola.
All parabolas can be derived from the prototype by a mere enlargement or
190 More about conic sections and developable surfaces

Fig. 5.28 Parabolas and . . . Fig. 5.29 . . . catenaries with different parameters

shrinking. It shares this property with the circles, the catenaries (Fig. 5.30),
and also with “higher-degree parabolas” of the form y = a ⋅ xn .

Fig. 5.30 Catenaries are positions of equilibrium of hanging ropes with not too much additional
weight pulling on them.

Fig. 5.31Parabola (red) versus catenary (green): A heavy flexible rope or chain will
take a position of the catenary.

The catenary represents the position of equilibrium of a hanging rope; it is

further addressed in [9].
We can see from its explicit equation
x a x/a
y = a ⋅ cosh = (e + e−x/a )
a 2
that the factor a is only a factor of similarity.
5.2 Conic sections 191

Fig. 5.32 There is an undeniable geometric beauty in suspension bridges (here: Verrazzano-Nar-
rows Bridge). Due to the weight that the bridge’s ropes have to carry, the shape of the ropes comes
closer to parabolas than to catenaries.

In the regions of the apex, catenaries resemble parabolas to such an extent

(Fig. 5.31) that it is possible to replace catenaries by parabolas in practice
(though only approximately).
However, if we truncate the power series of the exponential function (which
is transcendental rather than algebraic) after the term of degree two, we
obtain the equation of a parabola. Catenaries emerge as the solutions of a
differential equation: The shape of the curve emerges from the local condition
that the tangential pulling forces of the left and right part of the rope should
be in equilibrium with the mass at every point.

Fig. 5.33 Upside down catenaries are used for the enhancement of load capacity

A catenary that is turned “upside down” yields the optimal form for an arched
bridge, where the pressure at each point is distributed evenly on both sides.
Fig. 5.33 shows a model and its practical execution in the Gaudí museum in
Barcelona. Fig. 5.34 shows a much bigger version.
192 More about conic sections and developable surfaces

Fig. 5.34 A catenary turned on its head as a sturdy girder

▸▸▸ Square wheels

Fig. 5.35 A bumpy road and a square wheel can mutually “neutralize each other”.

An interesting – though academic – question: How would a road have to be

constructed for a square wheel to be able to roll without vertical motions of
its axis (Fig. 5.35)?
The solution can, for instance, be found in [?] – it happens to be an upside
down catenary. ◂◂◂

How many elements do we need in order to determine a conic

section?

Ellipses and hyperbolas are defined by five points or five tangents, parabolas by
four.

⊕ Remark : We will refrain from providing an exact geometric proof. From an ana-
lytic standpoint, we must (in general) solve a system of linear equations with six
variables – the coefficients of the conic section’s equation formula, Formula (5.1),
for which only the ratio a1 ∶ a2 ∶ . . . ∶ a6 matters.
Given five points or five tangents, it is possible to precisely construct as many addi-
tional points and tangents as necessary. This requires the mutually dual theorems of
Pascal and Brianchon are required, which we will not prove here. Pascal ’s Theorem
is illustrated in Fig. 5.41 and in Fig. 5.36 at the right – a simple linear construction
for an arbitrary additional point of the conic section is plainly visible.
▸▸▸ The theorems of Pappus and Pascal
Pascal ’s theorem reads as follows: For every hexagon inscribed in a conic section,
the intersection points of pairs of opposite sides are collinear (Fig. 5.36 to the
right, Fig. 5.41). The converse of this theorem is true as well. Pascal hexagons, that
is, hexagons whose intersection points of opposite sides are collinear, are always
5.2 Conic sections 193

inscribed in some conic section. The existence of a Pascal line is, thus, not only
necessary, but sufficient for a hexagon to be inscribed in a conic section. The theorem

Fig. 5.36 Left: Theorem of Pappus; Right: Theorem of Pascal

does not only apply for all ordinary conic sections, but for degenerate ones as
well. Furthermore, it is particularly important in projective geometry in general.
However, from a historical perspective, Pappus’ theorem is much older. ◂◂◂

▸▸▸ The theorem of Brianchon

Charles Julien Brianchon proved a theorem that turned out to be the dual of
Pascal ’s theorem (Fig. 5.37, left): For every hexagon that is circumscribed about
a conic section, the three lines joining opposing vertices pass through a common
point. Conversely, if a hexagon possesses a Brianchon point, then it is circumscribed
about a conic section.

Fig. 5.37 Theorem of Brianchon(on the left) and an interesting configuration (on the right)

By changing the names of the points or tangents of the conic section, one generally
obtains a new Pascal line or Brianchon point. In general, one can find sixty Pascal
lines and Brianchon points (Fig. 5.37 on the right). There are four such lines through
each Brianchon point and four such points that lie on each Pascal line. ◂◂◂
194 More about conic sections and developable surfaces

Fig. 5.38 Two conic sections sharing four points. Center right: A parabola is defined by four
points. (Actually there may be two parabolas on four points.) Far right: Circles automatically
share two complex conjugate points at infinity (marked by crosses).

Only four points or tangents are required for a parabola because the straight line at
infinity is automatically a tangent and thus a fifth element. However and in general,
there exist two solutions, pictured in Fig. 5.38 on the right. ⊕
⊕ Remark : A parabola “in standard position” (the axis is either parallel to the x-
or y-axis) is defined by three elements. It is then possible to recognize not only
the tangent at infinity but also the respective contact point (in the direction of the
axis).
If one point and its respective tangent are known, we speak of a line element. In
general, a parabola is, thus, defined by two line elements. ⊕
⊕ Remark : The question as to how many points are required to define a conic section
uniquely can be answered very quickly in the algebraic sense – by making use of
analytic geometry. There, a curve of degree two is defined by six parameters, one
of which can be picked arbitrarily. This means: If the coordinates of five points are
inserted into Formula (5.1), we get a system of linear equations with five variables,
which can be solved uniquely. ⊕
⊕ Remark : As was proven analytically in [9], circles always share two complex con-
jugate points at infinity. Despite our intuition to the contrary, this explains many
apparent contradictions. It is now clear why a circle (which is a conic section) can
be defined by only three points: The two missing points are the so-called “absolute
points,” and they always belong to the circle (see also p. 151 and p. 276). ⊕

When do we get an ambiguous solution?

If five elements of the same type (e.g., five points or five tangents) are given,
then the conic section is defined uniquely. If the given elements are of dif-
ferent types, then there exist two solutions (mixing proportion 4 ∶ 1) or four
solutions (3 ∶ 2). Let us leave aside the exact reasons and simply solve a few
of the many possible combinations.
The inverse paper strip method
An interesting possibility of “constructing” an ellipse is the so-called paper
strip method (Fig. 5.39):
On a strip of paper, we mark three mutually distinct points P , X, and Y . The
paper strip is moved in such a way that X and Y move along the respective
coordinate axes. Then, the point P describes an ellipse whose half-axes are of
lengths P X and P Y . The proof is given in the chapter about kinematics. We
5.2 Conic sections 195

Fig. 5.39 The “paper strip method” and its inversion

can apply this knowledge in the following example: Given an axis, the apices,
and an arbitrary point P on the ellipse, the missing apex can immediately
be found by fitting a paper strip by means of a compass. If, for instance, half
of the major axis a is given, then P Y is known, and X can be constructed.
The distance P X = b is half of the minor axis.
Ellipse and hyperbola with axes given by a line element
The ellipse and hyperbola are symmetric with respect to both axes. Thus,
if one point is known, three other symmetric points can also be found by
mere reflection. A tangent yields three additional tangents, and thus, the
declaration appears to be over-determined. Due to symmetry, however, this
does not produce a contradiction.
Let us consider the line element in the first of four quadrants defined by
the axes. For the sake of clarity, let us define the axes as the x- and y-axis
without declaring which of these is the major axis.
The intersections of the tangent t with the axes shall be called X and Y .
Depending on where these points lie, an ellipse or a hyperbola is to be ex-
pected.
In the “elliptic case,” we can (for instance) use an affinity of the ellipse to
one of its apex circles. The x-axis should be its axis of affinity. X remains
static and the ray through the given point P will intersect in the “circle field”
at a right angle. The respective point P ∗ on the circle can thus, firstly, be
found at the Thales circle above M X and then, secondly, at the affinity ray
orthogonal to the x-axis (the affinity axis). Now, the circle around M can be
drawn through P ∗ . The apex of the ellipse can be found at the intersection of
the affinity axis. The other apices can be found by completion. The principal
apices can only be determined after that process.
In the case of a hyperbola, we should imagine a solution that may not be the
shortest (when counting lines) but represents an interesting combination of
the aforementioned considerations: We first determine an ellipse that passes
through the given point P and possesses a tangent there that is orthogonal
to the tangent t. This ellipse is confocal to the desired hyperbola. The focal
points F1 and F2 are thus given, and the hyperbola is more or less finished.
196 More about conic sections and developable surfaces

Fig. 5.41 Pascal’s theorem allows the linear

construction of further points X – which may
be located on a straight line x passing through
3 – given five points 1, . . . , 5 by means of the
Pascal line p, which is defined through the aux-
Fig. 5.40 Spatial interpretation of a conic sec- iliary points U = 24 ∩ 35 and V = x ∩ 14
tion construction (⇒ W = p ∩ 15 ⇒ X = 2W ∩ 3V ).

▸▸▸ Construction of conic sections via spatial interpretation

In some cases, when specific constellations are given, the solution can be
approached by spatial interpretation. The following example is already quite
sophisticated: Three points A, B, C and two tangents s and t of a conic
section are given. The lines s and t can then be interpreted as outline gen-
eratrices of a right cone of revolution in a front view. One of the two angle
bisectors is then the front view of the cone’s vertical axis. (The initial situa-
tion is rotated according to (Fig. 5.40).)
The base plane can be “pulled up” to an arbitrary height. Now, the top
views of the three points A, B, and C can be augmented by the respective
generators. The three points gain a spatial significance. They lie on the cone
of revolution and span a plane ε, whose intersection curve with the cone not
only contains A, B, and C, but also touches the cone’s generators s and t. In
the front view, we see these points as contacts – as is the case with all other
planar intersection curves whose carrier planes do not appear as straight
lines. However, the augmentation of the points is ambiguous. For each point,
there exist two possibilities – 8 combinations in total – of which certain pairs
possess the same front views. Thus, we can expect four different planimetric
solutions. An arbitrary constellation is chosen in Fig. 5.40.
In order to make the plane ε appear as a line, we augment the horizontal
principal line through C onto the straight line AB (auxiliary point 1). A side
view with the viewing direction 1C yields the desired appearance of ε. Not
only the principal vertex (as a point on the space curve) can immediately be
5.2 Conic sections 197

found but also the contact points with the given tangents (the outline points
in the front view, such as T ). ◂◂◂

Construction of the conic sections

Fig. 5.42 Pointwise construction of the conic sections

When conic sections are constructed, the osculating circles in the apices
should likewise be drawn. The constructions are shown in Fig. 5.42 with-
out proof. The ellipse always remains inside the larger and outside of the
smaller osculating circle. The transition of the curve from one osculating cir-
cle to another should occur “harmonically.” The plotting of arbitrary points
is rather counterproductive, however, since the increased demand for drawing
precision may disturb the overall balance of the curve.
The parabola can be seen as an ellipse with one apex and one focal point
at infinity. As it detaches from the osculating circle, the hyperbola gradually
approximates its asymptotes. If a hyperbola is to be sketched, the asymptotes
must always be considered!
Arbitrary points are constructed by applying the definition of conic sections
with the use of the focal points. In the case of ellipses, the sum of the dis-
tances from the focal points is constant. For hyperbolas, the magnitude of
the difference stays constant. For parabolas, the distance to the focal point
is equal to the distance from the director . The tangent at the constructed
point always bisects the angle of the focal rays.
▸▸▸ What does an ancient column have to do with an ellipse?
Ancient (Greek and Roman) columns are no ordinary cylinders of revolution
– they are “thicker” in the middle. In fact, instructions for the construction of
the meridian were found carved in stone: A circular arc is to be constructed,
then intersected with vertical lines, and finally, the vertical sections are to
be multiplied (by a factor of 10, for instance). A circle whose ordinates are
multiplied is, in fact, an ellipse!
⊕ Remark : Rotating an ellipse is not enough to produce an ellipsoid. Such an object
is only produced by rotating the ellipse about one of its two axes. For those who
are interested: The result of a rotation about an axis-parallel chord is a surface of
198 More about conic sections and developable surfaces

Fig. 5.43 The production of ancient columns and what sometimes remains of them . . .

revolution which differs from a spindle torus by an affine mapping (Fig. 4.47).
One could now argue that this is a mere splitting of hairs and that the exact shape
of the column should be unimportant – as long as it is slightly thicker in the middle,
who cares! But far from it! The ancient Greeks were enthusiastic “hair-splitters” in
this regard, and the production of columns was quite an art of its own. For instance,
columns at the corners of a building were slightly bigger – not for reasons of statics,
but because they usually appear in backlight and thus give the impression of being
thinner. ⊕ ◂◂◂

▸▸▸ “Angular distorted” conic sections

If a cone of revolution that includes a planar oblique section is developed into
the plane, then the section becomes a curve that, from an optical viewpoint,
bears little resemblance to a conic section.

Fig. 5.44 “Angular distorted” conic sections and a resemblance of them, which is, however, entirely
unrelated . . .

From a mathematical point of view, it is still a curve with a reasonably similar

type of construction. It is produced from the top view of a conic section if
the polar angle to the curve point is multiplied by a factor of t = g/w < 1,
where g is the length on the cone’s generator seen in the top view, and w is
its undistorted length. The radial distance must then be enlarged by a value
of 1/t = w/g > 1. ◂◂◂
5.2 Conic sections 199

Fig. 5.45 Process of development of a cone of revolution (including a planar oblique section)

▸▸▸ Shadows on a wall

Fig. 5.46 in the middle shows a situation from everyday life: A lamp casts
a curved shadow. These are obviously conic sections, but how are they pro-
duced?

Fig. 5.46 Shadows on the wall

Solution:
To be clear: The conic sections have nothing to do with the cone shape of the
lampshade! The circular edges of the lampshade are the important parts. For
example, let us look at the bottom edge. Due to the approximately point-
shaped light source (the filament wire of the lamp), there exists a projection
cone of revolution that intersects the wall along a conic section. The type of
the conic section depends on the inclination of the wall with respect to the
projection cone (the inclination of the cone-shaped lampshade is irrelevant).
This is why the bottom shadow in Fig. 5.46 on the left is not a parabola (as
could be spontaneously assumed) but a hyperbola.
The same considerations are also applicable to the top boundary circle. In
the general case, the projection cone is not the “extension” of the bottom
cone (as in Fig. 5.46). We are thus faced with a second conic section that is
independent from the bottom one. ◂◂◂
200 More about conic sections and developable surfaces

Fig. 5.47 Approximated parabolic trajectories

▸▸▸ Parabolic trajectory

Fig. 5.47 shows a game that uses approximated parabolic trajectories. If an
object is thrown in the air at an angle, then its trajectory (assuming no air
resistance) will resemble a parabola. The special view in Fig. 5.47 gives the
impression of a “jumping jet of water.” Irregularities in the parabolic arc arise
due to several factors invisible to the naked eye (the image was taken using
a very short exposure time). ◂◂◂

Fig. 5.48 The orbits of planets are ellipses.

▸▸▸ Why are the orbits of planets ellipses? (Fig. 5.48)

Kepler discovered this fact by means of very accurate measurements. It is
hard to find a geometric explanation, since we actually need to solve differen-
tial equations ([16], see also https://www.researchgate.net/publication/
228860752_SATELLITE_ORBITS). It took another 70 years until Newton pro-
vided the geometric proof. More on the orbit of our planet can be found in
Chapter 11. ◂◂◂
5.3 General developables 201

5.3 General developables

Fig. 5.49 Seaweed as a typical example of general developable surfaces in nature

The third and last class of simple curved surfaces (next to the omnipresent
cylinders and cones) is not very well-known. This is not surprising – these
seldom spectacular surfaces often occur in a very hidden form; their outline
is always straight (Fig. 5.49).
As we have already said, developables are tangent surfaces of space curves.
They are enveloped by planes – the osculating planes of the space curve.

In the course of any type of spatial motion, each plane envelops a developable,
a cylinder, or a cone. It touches the emerging surface at each position along a
straight line (generator).

Developables of constant slope

c Γ1
Γ2

Fig. 5.50 A space curve c generates two surfaces Γ1 and Γ2 with given constant slope.
202 More about conic sections and developable surfaces

Developables of constant slope are an important class among the devel-

opables. They are characterized by the fact that all their tangent planes
are equally “sloped” and thus possess the same angle of inclination to a base
plane. If we define the position of the generating plane β of a developable in
such a way that it runs through the corresponding tangent t of a space curve
c and has a given inclination to a fixed direction (e.g., the vertical coordinate
axis), the plane will envelop a developable surface of constant slope. However,
the task is ambiguous. If the tangent is not too steep, there will be two real
solutions β1 and β2 that generate two surfaces Γ1 and Γ2 (Fig. 5.50).
Surfaces like the ones describes above are not only of interest in theory, but
also in practice. For instance, they are quite often used as stable transitions
of road edges or for roof constructions. Sometimes they are deployed quite
efficiently as tooth flanks (developable helical surfaces, see also Chapter 7),
because the tangent plane along an entire generating line is constant and
always has the same slope.

Fig. 5.51 Sand dunes as developables of constant slope

Such developables are found in sand dunes, contributing to their equilib-

rium: Sand is “trickling” down until a certain angle of inclination is reached
(Fig. 5.51). At a certain angle, depending on the grain size, the friction of
the grains is large enough for a state of equilibrium to set in.

Fig. 5.52 The sand dunes in the Namib are world-famous and the oldest dunes on Earth.
5.3 General developables 203

Let us think of the ridge of a sand dune. From each of its points, sand trickles
down in all directions and forms a cone of revolution (cone of constant slope).
The dune surface is enveloped by all those cones (Fig. 5.51 left). Pairs of
adjacent cones intersect along a hyperbola that, in extreme cases, becomes a
pair of straight lines passing through the tip of the cone. The dunes that are
probably most famous around the world, and also of a very stable variety,
are found in the Namib desert (Fig. 5.52, Fig. 5.53).

Fig. 5.53 Due to the wind, desert sand seems to shift constantly. The grains of sand in the Namib,
however, have been sitting in the same place for at least a million years.

When planning roads through arbitrary terrain, the engineers rely on this
insight: For the material used for construction (sand or gravel), the critical
angles are known. The engineers now consider corresponding developables
through the roadside and solve two problems at the same time: Firstly, the
developables are stable and there will be no landslides, and secondly, the
necessary motions of material are minimized (Fig. 5.54).

Fig. 5.54 Surfaces of constant slope in road construction

204 More about conic sections and developable surfaces

The generating curve c does not need to be a space curve. It can be planar and
even lie in the base plane. In this case, the planes β1 and β2 and, therefore,
also the surfaces Γ1 and Γ2 are symmetrical with respect to the base plane.

Fig. 5.55 Left side: Simple horizontal generating lines c of torses Γ1 with constant slope (left:
circle, right: ellipse). The torses Γ2 are symmetrical with respect to the base plane. Right side:
The two symmetrical torses are defined by a limaçon c in the base plane.

A trivial example is the surface of constant slope through a horizontal circle,

which is a cone of revolution. If the generating line is a horizontal ellipse,
however, the surface immediately produces selfintersections (Fig. 5.55, left).
Fig. 5.55 to the right illustrates the case in which c is a limaçon of Pascal.
The images on the right in Fig. 5.56 show that surfaces of constant slope
can be used to solve the following problem of planar geometry: Given a
curve c (for instance, a limaçon), what is the locus m of all points that have
the same orthogonal distance from the curve? One possible solution works
with surfaces of constant slope through c: Their straight parameter lines are
equally inclined and perpendicular to the curve’s tangents, both in space
and in the plan view (due to the law of right angle). If we project the spatial
intersection line m of two neighboring surfaces onto the ground plane, we
discover points on the desired locus m′ .

Fig. 5.56 Left side: The horizontal generating line c is a limaçon. Its midline m′ is a circle. Left:
spatial solution, right: plan view. Right side: A function graph takes over the part of the otherwise
much more complicated developable surfaces.

▸▸▸ Function graphs with constant slope

In practice, it is generally hard to explicitly calculate the intersection line of
the two different developables (Fig. 5.56 on the right). Therefore, finding a
graphic solution that is easier to implement, is a better idea. The definition
goes as follows: We consider a closed planar curve c in the (x, y)-plane and
define a function graph Γ in the environment of c in such a way that each
point is assigned a z-value that is the shortest distance from c. This graph
5.3 General developables 205

contains parts of the developable surfaces of constant slope which we have

discussed in the previous section. The graph has arrises where two surfaces
intersect. Whenever a part of one developing surface overlaps with a part of
another surface, only points that are closer to the (x, y)-plane count.

Fig. 5.57 Left side: A set of closed lines and the corresponding function graph. The plan view of
the arrises can be interpreted as “skeleton” of the boundaries. Right side: Instead of a base plane,
one can also use a base sphere. Distances have to be measured on great circles, and heights have
to be measured on straights through the midpoint.

The distance can either be always positive, or a negative value can be as-
signed to it if the point lies outside of the area defined by c. In any case
the zero-manifold of the graph is the curve c. This kind of definition is very
robust, even though it is computationally expensive, and allows to graphi-
cally determine midlines of arbitrary complicated lines. The projection of the
arrises (in simple cases the “midline” or a “median axis”) of a more complex
shape can be interpreted as “skeleton” of the boundary (Fig. 5.57 to the left).
If we consider several closed loops, we can assign negative z-values for points
that are outside of any given loop. When the loops are tiny circles (“zerocir-
cles”), the plan view of the arrises results in the Voronoi diagram (Chapter
12) belonging to the scene, otherwise we can speak of a generalized Voronoi
diagram. ◂◂◂

▸▸▸ Star-shaped surfaces with locally constant slope

The definition of the function graph Γ can be extended in a way that, instead
of the (x, y)-plane, we consider a curved base surface Σ. We then have to
measure distances along geodesics of Σ and find the corresponding points on
the “generalized graph” on the normals of Σ. In the easiest case Σ is a sphere.
Then the geodesics are great-circles. The generalized graph is sometimes
called “star-shaped surface”: Associated with each normal of the sphere we
have exactly one distance value, i.e., the shortest distance to the spherical
curve. In Fig. 5.58 the outlines of a continent are chosen in order to create a
new topology of the continent. Note that lakes, for instance, can significantly
change the surface.
206 More about conic sections and developable surfaces

Fig. 5.58 Left side: A set of closed lines (continents and islands) and the corresponding star-shaped
surface. The surfaces have locally constant slope with respect to the corresponding normals. Right
side: The projection of the arrises of the star-shaped surface onto the sphere produces a “skeleton”
of the boundary curves.

The curvature of the base sphere Σ in Fig. 5.57 on the right is comparatively
small. In Fig. 5.58 on the left the starshaped surface does not consist of
parts of developable surfaces any more. If we project the arrises of the star-
shaped surface onto the sphere, we get a skeletonization of spherical curves
(Fig. 5.58 to the right). Similarly to planar curves, we can also generate
spherical Voronoi diagrams (Chapter 12). ◂◂◂

How is a developable strip created?

We know that, next to the “trivial” cylinders and cones, only a class of surfaces
called developables can be developed. Let us take a rectangular strip of paper
and attach the narrow edges to each other. The hereby generated ribbon
is, of course, developable, since we cannot stretch or compress the paper
without tearing it. If we do not twist the paper strip, then we get a cylinder.
The midline of the strip is, thus, its normal section. Twisting produces a
developable with the midline being a curve in space.
▸▸▸ Creating paper strips
Is there a “paper strip” for an arbitrary space curve c so that c is its midline
(Fig. 5.59, Fig. 5.60)?
Let us consider three developables that are connected to the space curve:
the tangential surface as the envelope of all osculating planes, the normal
developable as the envelope of all normal planes, and the rectifying developable
as the envelope of all rectifying plane (planes orthogonal to the principal
normal).
The midline c becomes a straight line when spread out in the plane and is,
thus, a geodesic line on the developable. We know of these lines that the
5.3 General developables 207

σ ̺

t
c

Fig. 5.59 The rectifying developable of a space curve can be parameterized in such a way that
the entire surface develops into a rectangular strip with the midline as the rectified curve.

Fig. 5.60 Rectifying developables of three space curves (blue)

osculating planes must be perpendicular to the tangent plane at any point

(p. 142). The tangential surface, whose osculating plane is also its tangent
plane thus does not qualify for our purpose. We can see from Fig. 4.37 on
the right that the normal developable does not even carry the curve. This
only leaves the rectifying developable, for which all the necessary conditions
are met: The curve lies on the developable and is also geodesic line on it.
Despite the fact that the development preserves angles, the generators of the
developable do not meet the midline orthogonally. Quite the opposite: The
only straight line in the rectifying plane which is orthogonal to the tangent of
c is the binormal, and the locus of all binormals is not a developable. Thus,
the method of intersecting two neighboring normal planes of the curve does
not lead to the solution. ◂◂◂
208 More about conic sections and developable surfaces

Fig. 5.61 “Development of a developable”: How is it done? Center: Two congruent parts, one
inside the other. On the right, one of these parts was developed.

▸▸▸ “Development of a developable”

Every developable can be seen as the tangent surface of a space curve g.
The curve g is known as the developable’s edge of regression. The tangents
g are the generators of the developable. The development makes the edge of
regression a planar curve g v . The generators of the developable are tangents
of g, and thus, the corresponding lines in the development are the tangents
of g v .

Fig. 5.62 Development of a developable which contains the Möbius strip. Each intermediate
position is a developable, on which the jointly transformed strip is bounded by geodesic lines.

The edge of regression is seldom relevant in practice because, out of the in-
finitely large tangent plane, only the parts that do not reach the edge of
regression are normally used. The edge of regression is meaningless with re-
gard to the windmill in Fig. 5.61. Similar conditions apply to the oloid, whose
development is depicted in Fig. 6.63, and to the development in Fig. 5.62. ◂◂◂

▸▸▸ Approximation of a curved surface through rectangular strips

A common technique (especially in shipbuilding) is to approximate moder-
ately curved surfaces by means of developable strips.
5.3 General developables 209

Fig. 5.63 A curved surface is “modeled” from planks.

Wooden planks – but also aluminum planks, which are more commonly used
under tensile loadings – are, for example, well-suited for “twisting and bend-
ing”. This system of modeling can be seen in Fig. 5.63. ◂◂◂

Let us generate arbitrary developable surfaces

Architecture often demands the use of developable surfaces, and this is not
surprising: The creation of such surfaces is much easier and cheaper than
that of doubly curved surfaces.

Fig. 5.64 How architects construct curved and reasonably priced glass façades . . .

Some good news and some bad news: While it is always possible to find a de-
velopable surface connecting two arbitrary boundary curves, developable sur-
faces are characterized as always having straight-line contours. The connect-
ing surfaces could, therefore, not have anything “curvy” or “spherical” about
them. Therefore, in many cases, the surfaces would not be curvy enough. A
program that performs direct modeling with developable surfaces – and can,
nevertheless, achieve good results – is introduced on p. 360.
▸▸▸ Developable connecting surfaces
Let c1 be an arbitrary curve in space and c2 a curve in a plane ε2 (Fig. 5.65).
We will now introduce a procedure for finding the developable connecting
surface Φ of c1 and c2 :
We determine the tangent t1 for an arbitrary number of points P1 ∈ c1 ,
intersect it with the carrier plane ε2 , and draw a tangent t2 towards c2 from
210 More about conic sections and developable surfaces

Fig. 5.65 Construction of a connecting developable (on the left: oloid)

the intersection point T1 = T2 . The respective contact point P2 determines,

along with the initial point P1 , a generator e of the surface Φ. The tangent
plane of Φ is determined along the whole generator by the connection of e
with t1 . Due to our construction, t2 also lies in the tangent plane – which
is no contradiction. The thus defined series of tangent planes envelops a
developable. This construction also applies if both c1 and c2 are planar curves.

Fig. 5.66 Centrum Bank of Vaduz (Hans Hollein) with cylinders as surfaces

If both curves c1 and c2 are space curves, the problem is more difficult but
still solvable: Instead of the carrier plane ε2 of c2 , we now consider the tangent
plane Φ2 of c2 . The tangent t1 at c1 should not be intersected with the plane
ε2 but with Φ2 – a task for which we need a computer!
An interesting example of a special connecting developable is pictured in
Fig. 5.66. Here, the directrices lie in parallel planes. In such cases, the gen-
erators of the connecting developable are the straight lines connecting the
points on both generators in which the tangents are parallel to the directri-
ces. However, the image on the right indicates that all generators are parallel
5.3 General developables 211

– and thus, a cylinder is given. The image is based on a cylinder that was
intersected with the side surfaces. ◂◂◂

▸▸▸ The connecting developable of two circles

Let us consider two circles c1 and c2 in different non-parallel planes ε1 and
ε2 (with the section line s). If the circles are not accidently circular sections
of quadratic cones, we can find the connecting developable according to our
rules: Choose a point P1 ∈ c1 , intersect the corresponding tangent t1 with S
(point T ) and find the two tangents from T to c2 . Choose one of the contact
points (T2 ). Then the line T1 T2 generates the desired developable. Fig. 5.67
shows a practical application in architecture by W. Tschapeller.

Fig. 5.67 Wolfang Tschapeller’s vision of the “Vienna Museum” (2015). The buildings consist of
developables that connect circles.

Fig. 5.68 shows how two sets of n circles can be connected using the described
method. Note that all contours are straight lines, which indicates that we
are dealing with developables. In this special case the 2n surface parts are
all congruent.

Fig. 5.68 If n larger and n smaller circles are evenly distributed in meridian planes, they can be
connected by a series of 2n congruent developables.
212 More about conic sections and developable surfaces

Fig. 5.69 shows two additional views of the object which is inscribed to an
outer torus and circumscribed to an inner torus (tori will be discussed in
greater detail in Chapter 6).

Fig. 5.69 Two more different views of a “torus-like object” (some part surfaces are removed).

Max Klammer, one of Tschapeller’s students, investigated the development

of part surfaces like that, which resemble the development of conical frustums
(Fig. 5.70). He connected them in the indicated way – first by “tooths”, later
on also by paper clips and staplers.

Fig. 5.70 Max Klammers investigations of the development

So far, the project does not appear to be extraordinary. Klammer, however,

found out that the whole object can be interpreted as a machine that is able
to revolve, as long as it stays “a little bit flexible” and only some parts of the
surfaces are materialized (Fig. 5.71).

Fig. 5.71 Theoretically impossible – and yet it moves!

5.3 General developables 213

In the presentation of his diploma thesis, Klammer made huge models move
to demonstrate that such objects can be used in “flexible architecture”. Fun-
nily enough, the German word Klammer-Maschine means stapler in English,
so we assigned his model an amusing name: the “Klammer-Torus”. ◂◂◂

Reflections on a developable surface

Fig. 5.72 A reflection of light rays on a plane

Reflections on curved surfaces are part of very sophisticated geometry. If all

light rays that emanate from a point-shaped light source are reflected on
such a surface, we get a congruence of rays which envelop a new surface:
the so-called focal surface or caustic. In general, such focal surfaces are very
complicated and full of “singularities”. In application p. 228, we shall discuss
the reflection in a sphere.
The problem is simplified significantly if the reflecting surface is only single-
curved: Cylinders, cones, and developables are touched by a fixed plane along
a whole generatrix.
Fig. 5.72 shows what happens during reflection on a plane: All light rays from
a point-shaped light source (which – like the Sun – may be infinitely far away)
through an arbitrary straight line of the plane form a plane. The reflected
light rays are restricted to the reflected plane, which is itself determined by
the straight line and the reflected light source.
If the reflecting surface is single-curved, we can replace it along each generator
by its tangent plane. As with the plane (Fig. 5.73), the following simple
situation applies:

All light rays that are reflected in a generator of a simply curved surface lie in
a plane (the focal plane) which passes through the point that is reflected in
relation to the tangent plane along the generator.

If we now let the generator move along the surface Φ, we get a pencil of focal
planes which envelop the developable. The focal surface of a developable
surface is, thus, itself developable. To give a more concrete example, the
focal surface of a cylinder is again a cylinder. In application p. 227, we will
214 More about conic sections and developable surfaces

Fig. 5.73 Reflection in a simple curved surface

find out more about the reflection in a cylinder of revolution. The locus of
all light sources reflected in the tangent planes is a curve – the focal line.
▸▸▸ Shiny generators on aluminum cylinders
Let us consider Fig. 5.73 on the right: One of the rays that are reflected along
a generator appears as a point. We can now imagine parallel light and picture
the surface from a greater distance, which causes the viewing rays through
the points of the generators to be approximately parallel. Thus, every point
on the generator “flashes up”, and the whole generator glows brightly. Such
effects are often visible on turned parts of aluminum (cylinders of revolution
or cones of revolution) and also on glass bottles in sunlight. ◂◂◂

▸▸▸ Glowing curves

We encounter the following situation with reflections in a plane: A devel-
opable surface reflects a light source (such as a spotlight or the Sun). All
focal planes intersect the plane in a straight line. The focal lines envelop
a curve – the intersection curve of the focal surface with this plane. Thus,
the following applies: The reflection of a developable surface on a plane is
a glowing curve. The curve glows because it focuses all light that interacts
with the surface (and sometimes even causes heat effects). ◂◂◂

▸▸▸ Light games in the water (Fig. 5.74):

Everybody knows the fascinating, constantly changing reflection patterns of
sunlight on the sides of boats or buildings. The focusing effect of the light
rays is evident here.
This permits us to speculate that the surface of moving water is only single-
curved. This is, in fact, true (though only approximately so) if the motion
of water is caused by wind. In such situations, the emerging wave surfaces
are (ideally) cylinders – and hence, single curved. However, we are also deal-
ing with the reflections of waves on harbor moles and boats. This produces
new cylindrical waves which overlap the original ones. Taken together, these
circumstances create the ideal conditions for a beautiful spectacle. ◂◂◂
5.3 General developables 215

Fig. 5.74 Light games above water (left) and below water (right)

▸▸▸ Reflecting compact discs – and curious effects

The curious reflections on CDs must have aroused the interests of many. Two
things are apparent: Firstly, these reflections seem to be glowing straight lines
that can swiftly change position (and essentially rotate about the midpoint
of the CD). Secondly, all colors of the rainbow seem to be visible among the
reflections. This cannot be caused by the tracks (of which there is actually
only one): The same effect is visible on blank CDs.

Fig. 5.75 A CD reflects all colors of the rainbow.

Apart from mere reflection, other properties of light are at play here. Roughly
speaking, we are dealing with interferences. CDs are essentially manufactured
from plastic polycarbonate, one side of which is vaporized with an aluminum
layer (the aluminum reflects the laser beam which reads the data). In order to
prevent chemical reactions, the disc is then coated with transparent plastic.
When a light ray passes through the plastic coating, a part of it is reflected
on the surface. What remains is refracted at the surface based on the angle
of incidence (and also splits up into its color components). On the other side
of the plastic coating, at least by the time it reaches the aluminium layer,
the light is reflected, as well as “refracted,” and the ray appears parallel
(but slightly shifted) to the ray reflected on the first surface. Diffraction of
the light rays also takes place due to the presence of an optical diffraction
grating (many thousands of openings per inch). The reflected and (following
the many consecutive reflections) slightly shifted light rays now interfere
with one another. This may lead not only to the augmentation but also to
216 More about conic sections and developable surfaces

the cancellation of entire color components. Similar effects occur on butterfly

wings and fish scales.

Fig. 5.76 This unusual photograph demands an explanation.

Fig. 5.76 definitely requires additional explanation. The photo was taken in
a room on the top floor. The sun shines through a window in a sloping wall
and almost illuminates one half of a slanted CD. One might expect only
the illuminated half of the CD to produce a reflection on the sloping wall.
In fact, the entire CD is reflected on the wall, but the reflection is divided
perfectly in two halves, which also look different. A detailed explanation of
this phenomenon falls within the remit of physics rather than geometry. ◂◂◂
5.4 Maps and “sphere developments” 217

5.4 Maps and “sphere developments”

The dilemma regarding the development of doubly curved sur-

faces

Fig. 5.77 However you may look at it, the sphere is definitely not developable!

We know at this point that it is not possible to develop doubly curved surfaces
– let alone a sphere! Every attempt to press the surface of a sphere into the
plane ends with expansions and compressions, or with the tearing apart of
the object altogether. The method of peeling tangerines pictured in Fig. 5.77
comes close to Walter Wunderlich’s approach in Fig. 5.78. The method in
Fig. 5.77 on the right exhibits certain similarities to the approximation of the
sphere by cylindrical strips in Fig. 5.78 (center). Nature occasionally solves
the problem abruptly (Fig. 5.79), artists find their individual solutions.

Fig. 5.78 The “apple peeling method” by Wunderlich and an approximation by a cylinder

However, the development of a sphere would be important for so many appli-

cations – especially in cartography! It is obvious that the maps with which
we constantly deal must have been produced by using certain tricks that
accommodate the curvature of the sphere. This must necessarily have oc-
curred at the expense of geometrical properties that are well known to us.
See, for example, Fig. 5.81, which highlights those parts of the globe that are
illuminated by the Sun at a given time.
218 More about conic sections and developable surfaces

Fig. 5.79 The “brute-force approach” sometimes works as well . . .

Fig. 5.80 Orange origami art (Gerda Tomaselli, according to http://okadas.com/promo.htm)

The situation in the left image is very clear: The shadow boundary of the
globe is a great circle. The right image shows the respective situation in a
relatively common world map, where geographic longitudes and latitudes are
simply used as Cartesian coordinates. The great circle becomes a wave-like
curve (but not a sine wave). With such transformations, it is very simple to
find a geographical position. However, the benefits of such a map already end
here: Angles, lengths, and areas appear greatly distorted. This is especially
evident in polar regions – around the equator, distortions tend to be the
smallest.
▸▸▸ Is it developable after all?
Architects are usually keen on building models of their sketches. Nowadays,
they mostly employ state-of-the-art modeling software for this purpose. If
5.4 Maps and “sphere developments” 219

Fig. 5.81 The shadow boundary in the distorting coordinate grid

Fig. 5.82 A slightly unusual “development” of a sphere . . .

they then activate the “develop” function of such a software package, the
computer will be up to the task. In the case of a sphere, a strangely frayed
object will emerge. Upon closer inspection, it turns out that this object con-
sist of ring sectors. Why is this the case?
Fig. 5.82 illustrates the spatial situation: The sphere is more or less accurately
approximated by frustums of cones of revolution. Each single frustum is
developable, producing a ring sector. The better the approximation, the more
circular rings are produced.
As original as this idea is, it is not applicable to cartography: One only needs
to imagine a world map where all the continents are “dissected” on separate
strips . . . ◂◂◂

An angle- and circle-preserving projection

We now consider the so-called stereographic projection of the sphere. It is a
central projection with the center of the projection on the sphere, and the im-
age plane is (for example) the opposite tangent plane. This kind of projection
plays an important role in cartography due to the following properties:

After stereographic projection, intersection angles and circles are preserved.

Proof : Let a point Z on the sphere be the center of the stereographic projection
from the sphere to the plane and π be the tangent plane through the antipode of Z.
220 More about conic sections and developable surfaces

Fig. 5.83 Stereographic projection of the globe from the South Pole

Without loss of generality, we may assume π to be horizontal. The straight lines of π

correspond to circles through Z that result from the intersection of their connecting
planes with the sphere. Let us take a closer look at two such straight lines a and b
with the intersection P and the enclosed angle ϕ. They correspond to two circles
ak and bk on the sphere through Pk and Z.

Fig. 5.84 Stereographic projection of the globe from the antipode of Vienna

Due to symmetry, the angle enclosed by both circles on the sphere is the same in
both common points. However, the angle enclosed by ak and bk at Z equals ϕ,
because the respective circle tangents lie in the horizontal tangent plane of Z and
are, thus, parallel to a and b. Therefore, ak and bk also intersect in the point P k at
an angle of ϕ. Whatever is true for the angle between two straight lines is also true
for the angle enclosed by two curves, since the angle between curves is defined as the
angle of the respective tangents. We have thus shown that stereographic projection
preserves angles. It is also said that stereographic projection is conformal.
We must briefly elaborate on the proof that stereographic projection also preserves
circles: Stereographic projection on π is basically no different from a central projec-
tion on π – under special consideration of the points on an auxiliary sphere, which
was placed “between” the center and the image plane (more precisely: the restric-
tion of a central projection to the sphere). It is not forbidden to project points or
5.4 Maps and “sphere developments” 221

straight lines that do not lie on the sphere. In general, however, the preservation of
angles and circles only applies to sphere elements.
Let kk be a circle on the sphere. We now observe a cone of revolution that touches
the sphere along kk . It has the apex Sk and nothing but generators that intersect
the circle on the sphere at right angles. Let us project Sk from Z towards S ∈ π.
Then, the generators correlate to straight lines through S. On each such straight
line, there exists the image of a point of kk . The preservation of angles applies there
(and only there). As on the sphere, the tangent of the projection k ∈ π of kk must
intersect the projected cone generator orthogonally. However, there exists only one
type of curve that intersects a pencil of rays orthogonally: the circle. Moreover, S
must be the midpoint of this circle!
⊕ Remark : If the sphere circle kk passes through the center Z, then the apex Sk of
the “adjacent” cone of revolution is located at the same height as Z. Its projection S
is, therefore, a point at infinity. The pencil of rays of the projecting cone generator
becomes a pencil of parallels. The projected curve k becomes a straight line that
intersects this pencil orthogonally – but also, if you like, to a circle with its “center
at infinity.” ⊕ ⊙
A practical map
If the points on the globe are projected from the center of the globe onto an
arbitrary tangent plane, we get a world map where all planes – including the
great circles – appear as straight lines through the center. However, great
circles are (as we will soon discover) the shortest connecting paths between
two points on the sphere surface. On such a map, it would thus suffice to
connect two points with a ruler in order to draw the shortest path (Fig. 5.85).

Fig. 5.85 From Tunis to Tokyo

As remarkably simple as this principle may be, it produces a very significant

map distortion, because the center of projection is relatively close to the
projection plane. If, for instance, the northern hemisphere was projected
222 More about conic sections and developable surfaces

onto the tangent plane of the North Pole (many airplanes choose routes
far up north on long-distance flights), then the entire southern hemisphere
(including the equator) would not even appear on the map. Even the latitudes
south of the 30th parallel would lie very far outside of it.
▸▸▸ From Bangalore to Buenos Aires
The pragmatic American architect Richard Buckminster Fuller came up with
his own solution to the problem of flight paths in his own way (Fig. 5.86):

Fig. 5.86 With Buckminster Fuller on the cuboctahedron from Bangalore to Buenos Aires

He projected not onto the plane but onto the side surfaces of a cuboctahe-
dron that was ingeniously designed to fit onto the Earth and consists of eight
equilateral triangles and four squares (the name derives from a combination
of a cube and an octahedron). By complete accident, the continents turned
out to fit relatively well on the surfaces of the polyhedron. If such a cuboc-
tahedron is developed in the plane, one gets a map that is not very distorted
(but not connected either). On this map, the shortest flight paths appear
as straight lines as long as one remains within the side surfaces. The kink
happens upon transition to the next surface. However, within continents, the
flight routes are very easy to find. ◂◂◂

Stereographic projection offers a theoretically interesting solution to the flight

path problem. On a map constructed by such a projection of the globe, the
longitudes and latitudes appear as circles due to the preservation of angles
and circles – with each circle of longitude intersecting each circle of latitude
perpendicularly. The shortest paths between two points are, thus, also circles
that must fulfil a certain condition that we will determine shortly. By defining
a circle, one also knows the course angle at every point of the flight or ship
path.
5.4 Maps and “sphere developments” 223

▸▸▸ The shortest connection

In a stereographic projection, the shortest path between two distant globe
points A and B is to be determined.

Fig. 5.87 The shortest path in a stereographic projection

Solution:
The shortest path from A to B on the sphere is a great circle (Fig. 5.87). It is
unambiguously defined by the carrier plane ABM (M being the midpoint of
the Earth). If two arbitrary great circles are intersected, then the intersection
points lie antipodal (relative to M ). In stereographic projections, great circles
are, thus, circles that pass through two opposite points on the inversion circle
(which is the image of the equator). According to the theorem concerning the
constant power of a point regarding a circle (p. 16), we thus know the product
of the distances on the secants through M (through A as well as B). This
way, two additional points A and B on the great circle in question can be
determined (though only one is sufficient for construction). ◂◂◂

The inversion at a circle from a “higher perspective”

Let us once again return to the inversion in the plane. As interesting as it
might have been theoretically, it still seemed a little bit arbitrary. Does the
transformation M P ⋅ M P ∗ = r2 actually have a “higher” significance?
Let the equator plane π be the image plane of the stereographic projection
and the North Pole Z be the center of projection (Fig. 5.89). Furthermore,
let S denote the South Pole. Pk and Pk∗ are two points on the sphere with
the same orthogonal projection (top view) in π. The points Z, Pk , and Pk∗
now lie on the same great circle, which we shall, from now on, imagine as the
sphere outline.
Let P and P ∗ be the respective stereographic images of P and P ∗ . The points
Pk , P ∗ , and S thus lie on a straight line (reflected in π) which, according to
Thales’s theorem, is perpendicular to P Z.
For the following simple calculation, let us imagine a two-dimensional coor-
dinate system with the center M of the sphere as its origin, the x-axis in
the direction of P , and the y-axis in the direction of Z. If x0 denotes the
x-coordinate of P , then the straight line P Z has a slope of − xr0 (r = sphere
224 More about conic sections and developable surfaces

Fig. 5.88 A beautiful spatial interpretation of the inversion

Fig. 5.89 Inversion and stereographic projection

radius). The slope of the straight line Pk S is, thus, xr0 and its equation is,
therefore, y = xr0 ⋅ x − r. In order to preserve P ∗ , we only need to intersect it
with the x-axis:
x0
⋅ x − r = 0 ⇒ x0 ⋅ x = r 2 .
r
This already shows that P and P ∗ are inverted in the circle around M with
the radius r.
If the tangent plane at S is chosen instead of π as the image plane of the
stereographic projection, then all image lengths are doubled, and we arrive
at the following theorem:

Let π be the horizontal drawing plane in whose origin a sphere with diameter 1
is centered. If overlapping points on the sphere are projected from the topmost
point of the sphere onto π, we get points which form an inverse pair with
respect to the unit circle in π.

⊕ Remark : Strictly speaking, the theorem would later have to be formulated as

follows: Let Z be a point on the sphere, π be the tangent plane of the sphere in the
opposite point, and k0 be the great circle of the sphere parallel to this plane. Then,
5.4 Maps and “sphere developments” 225

both hemispheres (bounded by k0 ) are mapped via the central projection from Z
to π onto point fields which are related by an inversion with respect to k0 . ⊕
This admittedly far-fetched inversion of the plane is considerably easier for
a three-dimensionally thinking being – it is merely the reflection of a sphere
half in a plane. The “missing link” is a stereographic projection. Of course, we
could now think about whether or not a four-dimensionally thinking being
would find it equally easy to invert three-dimensional space in a sphere –
such as using the generalized stereographic projection of space onto a “hy-
persphere”. It is, indeed, possible to extend this line of reasoning, and from
a mathematical standpoint, it works! The only catch: Calculating such a
phenomenon is perfectly feasible, but imagining it is no longer possible.
From a higher perspective, the most important properties of inversion can
easily be proven:

Inversions map circles to circles and preserve angles. Straight lines correspond
to circles through the midpoint of the inversion circle.

Proof : Preservation of angles (conformality) and preservation of circles alike are

deducible from the respective properties of the stereographic projection, which is
applied twice (in opposite directions).
Likewise, the reflection in the sphere does not affect the section angles or the circle
property. In the case of stereographic projection, straight lines correlate with circles
through the projection center (the intersection circle of the image plane with the
sphere). They become circles through M during reflection and circles through M in
an inverted stereographic projection. ⊙
⊕ Remark : Another proof for the preservation of circles can be found in [9], where it
is conducted in a rather elegant way by the use of complex numbers. The advantage
of the geometric proof is, once again, the simple fact that its “residual products” pro-
mote geometrical comprehension. One piece of insight somehow always produces the
other, eventually leading to a rounded geometrical “worldview,” which yields many
conclusions whose proofs would otherwise have to be “derived” mathematically. ⊕
226 More about conic sections and developable surfaces

5.5 Reflection in a circle, a sphere, and a cylinder of

revolution
So far, we have often mentioned the reflection in a circle (for instance, with
inversion), even though references to physical reflection were not always in-
tended. The reflection in a surface which we know from nature follows two
rules:
Firstly, the incoming and the outgoing light rays lie in a plane which also
contains the surface normal. Secondly, the angle enclosed by the normal and
the incoming ray equals the angle enclosed by the normal and the outgoing
ray (“angle of incidence = angle of reflection”).
▸▸▸ Reflection in a plane
Let E be the eye point and S the point that emits light (Fig. 5.90 on the
left). If we look into the reflection plane τ , we see a reflecting point R. We
can determine R geometrically by reflecting S in τ and connecting this point
S ∗ with E. The eye can no longer distinguish whether it is looking at the
reflecting point R or the reflection point S ∗ .
In fact, the two aforementioned rules are observed: SR and RE lie in a plane
which includes the normal n, just as the angle of incidence ζ1 and the angle
of reflection ζ2 are identical. The reflection in a plane is, therefore, a simple
(linear) problem. ◂◂◂

Fig. 5.90 Reflection in curved surfaces

▸▸▸ Reflection in a curved surface

Generally speaking, the reflection in a curved surface is, of course, a remark-
ably difficult undertaking in the general case. Let τ be a tangent plane in
an arbitrary surface point R (Fig. 5.90 on the left). The point S in space
can thus be reflected on τ . If, by coincidence, the straight lines ER and ES ∗
happen to be identical, then we have found the reflecting point of S in R
(complicated surfaces may contain multiple “reflecting points”).
5.5 Reflection in a circle, a sphere, and a cylinder of revolution 227

Fig. 5.91 Reflection in a doubly-curved surface (Cloud Gate Chicago, Anish Kapoor )

In practice, we can find all reflecting points by conducting the aforementioned

test for a high number of surface points. In cases where such reflection points
are “almost” found, the immediate vicinities can be further scanned using
approximation techniques. ◂◂◂

This generalized search method is, of course, computationally expensive –

but we will not discuss it further here. In his diploma thesis, Franz Gruber
developed a method for determining reflecting points as in Fig. 5.91 in “real
time” (multiple images per second).
Two special and somewhat frequent cases are interesting for us.
▸▸▸ Reflection in a cylinder of revolution
Let Φ be a cylinder of revolution (Fig. 5.90 at the center), which, for the sake
of simplicity, we shall imagine as being positioned vertically. If we project all
points normally into the base plane, the problem of reflection is reduced to
the two-dimensional reflection in the base circle. We now just have to find
a point R′ , which, if seen from E ′ , is the reflecting point of S ′ in the base
circle: The same angles in the base plane cause an equality of angles in the
plane that is formed by E, R, and the cylinder normal n = M R.

Fig. 5.92 Anamorphoses: Reflection in the cylinder of revolution

228 More about conic sections and developable surfaces

⊕ Remark : The reflection in the cylinder of revolution, which is familiar to us from

many halls with reflecting columns, was historically used to create so-called anamor-
phoses. The observer sees a bizarrely distorted image that appears “normal” when
seen in a cylindrical mirror (Fig. 5.92). ⊕ ◂◂◂

▸▸▸ Reflection in a sphere

Fig. 5.93 Reflections in spheres

The reflection in a sphere can likewise be traced back to the reflection in a

great circle: The sphere normal always contains the center M , and the light
rays in question, ER and RS, lie in the great-circle plane M ES. ◂◂◂

Thus, the problem of trying to get a grip on the physical reflection in a circle
is becoming more urgent. Let us say it upfront: The problem is not easily
solvable because it can be shown that it would require the solution of an
algebraic equation of degree four. This is not a problem for the computer
(there are formulas for solving such equations), but it is generally unsolvable
with just a ruler and a compass alone.

Fig. 5.94 The physical reflection in a sphere

A simple special case can be solved relatively quickly: If the point S (which
is to be reflected) happens to be at the same distance from the circle (in
practice, it would be from the sphere or the cylinder of revolution) as the
5.5 Reflection in a circle, a sphere, and a cylinder of revolution 229

eye point E, then the reflecting points Ri are located precisely on the angle
bisector of M S and M E. Armed with this knowledge, we can easily locate
the specular points in the eyes (Fig. 5.95) emanating from a light source. The
location of the light source, on the other hand, is equally easy to reconstruct
once we know the location of the specular points.

Fig. 5.95 Specular points on sphere caps

Even in the general case, we should not immediately give up. Let us shed
some light on its “trappings”:
▸▸▸ Conic sections with given focal points that touch a given circle
We know that in the case of conic sections, the curve tangents and curve
normals bisect the angle of the focal rays. A light ray through the focal point
E is thus reflected in another focal point S.
If such a conic section touches a given circle k at the curve point R, then
the tangent to the circle equals the tangent to the conic section at R, and
we have thus reduced the problem to the reflection in a circle.

Fig. 5.96 Four confocal conic sections which touch a given circle

In order to find an additional point on the conic section in question, we now

search for the point R on k, which is the reflecting point of S on the circle
k with respect to E. In the case of the outer reflection, the solution seems
230 More about conic sections and developable surfaces

to be unambiguous, but it still cannot be done without the solution of a

degree-four equation. Not all of the four possible solutions have to be real.
We are dealing with four confocal conic sections (Fig. 5.96) which intersect
each other orthogonally, as we have already proven. ◂◂◂

Fig. 5.97 The dragonfly dents the water surface in three points, which are immediately visible
through the specular highlights.

The question still remains as to how the four “reflecting points” of S on the
circle k can be geometrically determined for the given eye position (Fig. 5.98
on the left). The answer can only be: by trial and error. By moving R along
the circle, the light ray ER is varied. The respective incoming light ray
is always the reflection in relation to the circle normal and can easily be
constructed. At some moment, S will change sides relative to this reflection
– and the solution lies precisely there, with graphical accuracy.

Fig. 5.98 The reflected rays envelop a curve with cusps.

5.5 Reflection in a circle, a sphere, and a cylinder of revolution 231

▸▸▸ Focal lines of a circle

The aforementioned method of “trial and error” produces a family of reflected
rays which envelops a curve. The form of this so-called focal line (caustic)
may vary. For points outside of the circle, it will look approximately as in
Fig. 5.98 and Fig. 5.99.
The term “focal line” is derived from the fact that instead of the eye point E, a
line-shaped light source can be imagined, and then such brightly illuminated
curves appear on cylindrically reflecting walls. A point E inside the circle k
produces focal lines of the same type as in Fig. 5.100.

Fig. 5.99 Caustic with snout beetle. Note that the animal is directly in sunlight.

In each case, there exist, emanating from E, multiple (at most four) tangents
of the focal line which lead to the desired reflecting points.

Fig. 5.100 The point may also be located inside the circle.
232 More about conic sections and developable surfaces

In reflections on the outer wall of the circle, only one of the four solutions is
practically relevant. It can also be found by trial and error. If E lies inside
the circle, then all solutions may actually be visible.
Fig. 5.101 serves to illustrate why spatial inversion is often referred to as
reflection in a sphere despite the fact that a real physical reflection in a
sphere has nothing to do with it: Inversion gives us a well-defined spatial
object, which is generated regardless of the observer (in this particular case,
it is a Dupin cyclide, which is further discussed on p. 287). The reflected
image which appears on the sphere changes its appearance with the location
of the viewer.

Fig. 5.101 Inversion and physical reflection in a sphere

When put this way, the optical similarity of both images in Fig. 5.101 is
only striking because the inverted object is observed from the right vantage
point. By the way: M.C. Escher has approximated his spherical reflections
by means of inversion. ◂◂◂
6 Prototypes

The majority of all curved surfaces cannot be developed into a plane. “In
the neighborhood of a surface point,” doubly curved surfaces can bend in
one direction, lying on one side of the tangent, as they do with spheres, or
being bent in two directions and lying locally on both sides of the tangent
plane, as is the case with saddle surfaces. By examining the simplest cases
of doubly curved surfaces, i.e., degree-two surfaces, from the perspective of
their behavior regarding the tangent plane, far-reaching statements about
local curvature characteristics of all surfaces can be made.
Surfaces that consist locally only of elliptic or hyperbolic points cannot
be flattened out in the plane without deformation; they would need to be
parabolic in all points.
However, degree-two surfaces are not only interesting as prototypes. They
possess a multitude of special properties with technical applicat ions. A clas-
sical example is the parabolic headlamp.
Surfaces of revolution are the first type of more general surfaces to be dis-
cussed in the following sections. Aside from degree-two surfaces of revolution,
the torus also plays an important role. This degree-four surface has numer-
ous applications; it is not only generated by the rotation of a circle about a
coplanar axis but also by the rotation of a sphere.

© Springer Nature Switzerland AG 2020 233

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_6
234 Prototypes

6.1 Surfaces of degree two

The simplest type of surface is a plane, which possesses one intersection point
with every (general) straight line in space. It is therefore said that planes are
degree-one surfaces (and there exists no other degree-one surface).
Each surface that possesses two intersection points with a (general) straight
line (these need not always be real and may also coincide) is called a quadratic
(degree-two) surface. We are already familiar with several types of them: the
sphere, the cylinder of revolution, and the cone of revolution.
By applying affine mappings (which do not change the aforementioned num-
ber of intersections) to such surfaces, ellipsoids, elliptic cylinders, and quadratic
cones can be produced.
Through collinear mappings (which are also linear mappings that do not
change the degree of the surface), paraboloids and hyperboloids, as well as
parabolic and hyperbolic cylinders, can be produced. As we shall soon discover,
it is impossible to distinguish between different types of cones produced by
collineation, which is why they are generally referred to as quadratic cones.

Doubly curved degree-two surfaces – ellipsoids, paraboloids, and hyperboloids

(two-sheeted) – differ with respect to affine mappings, but there are collinear
mappings that can transform them into a sphere. Single curved degree-two
surfaces – elliptic, parabolic, hyperbolic cylinders, and quadratic cones – are
affine and collinear with respect to the cylinder or the cone of revolution.

Fig. 6.1 Elliptic cylinders carry circular intersections.

Fig. 6.1 shows an interesting “experiment” with elliptic cylinders that are
actually oblique circular cylinders: in contrast to the ellipse-shaped sections
of cones of revolution, the single “hooves” can be twisted steplessly due to
their circular shape.
▸▸▸ A new arm for Eva
Inspired by geometric models like in Fig. 6.1, Philomena Strack, in her
diploma thesis, designed a “new arm for Eva” (Fig. 6.2 and Fig. 6.3).
6.1 Surfaces of degree two 235

Fig. 6.2 Elliptic cylinders with a common circular section can successfully be used for prosthetic
arms.

While the human elbow joint is a hinge joint that only has a single degree of
freedom, most natural movements of the arm consist not only of bending the
elbow joint, but also include a rotation of the lower arm. By recreating the
elbow joint not as a hinge joint, but as a swivel joint, this single joint can
replicate the combined movement and enables natural-looking movements of
the prosthetic arm.

Fig. 6.3 Eva, the trial participant, confirmed that she felt very comfortable with the new arm.

The shapes of both the upper and the lower arm resemble that of an elliptical
cylinder, which is more easily interpreted as a circular cylinder with approx-
imately 45○ obliqueness (then, the elliptic cross section has the proportion
≈ 1.4 ∶ 1). The swivel joint comprises two congruent circular cross-sections of
the upper and lower arm, ensuring the structural integrity of the prosthetic’s
surface at every angle of the joint. In comparison, a hinge joint would open
the surface while bending, and thus interfere with the design of the prosthe-
sis. ◂◂◂
236 Prototypes

Ellipsoids
As we already know, all cylinders and cones are developable because they
are single curved. Ellipsoids, on the other hand, are doubly curved – just like
spheres. In general, ellipsoids possess “three axes” but are not rotationally
symmetric – they possess three pairwise-perpendicular planes of symmetry.
By examining the ellipsoid that is produced by the rotation of an ellipse,
one can distinguish between an oblate and a prolate ellipsoid (egg-shaped)
of revolution.

Fig. 6.4 On the left: triaxial ellipsoid; on the right: ellipsoid of revolution, even though the
parametrization suggests three axes.

▸▸▸ The shape of the Earth

As is well-known, the globe is not precisely a sphere; it rather approximates
an oblate ellipsoid of revolution, also called a geoid. The exact shape of the
Earth corresponds to an ellipsoid with many tiny indentations and protru-
sions. These “blemishes,” including the oblateness at the poles, are highly
relevant for physicists and geodesists. From a purely optical perspective, the
Earth is practically indistinguishable from a sphere. ◂◂◂

Due to their affinity, ellipsoids “inherit” important properties from the sphere:

Each planar section of an ellipsoid is an ellipse. The outline of an ellipsoid is

an ellipse in cases of normal projection and can by any conic section in cases
of central projection.

▸▸▸ Ellipsoids in dome construction

Fig. 6.5 on the left shows the dome of the Reichstag in Berlin, which has
the exact shape of a half-ellipsoid of revolution. Fig. 6.6 shows the normal
projections as well as one central projection (perspective) of the dome.
The cathedral in Florence is duly famous, not least for its dome (built by
Brunelleschi , Fig. 9.2). However, the dome is not an exact ellipsoid – not
6.1 Surfaces of degree two 237

Fig. 6.5 An exact elliptical dome and “something similar” (Karlskirche, Vienna)

Fig. 6.6 The Reichstag dome in Berlin is a half-ellipsoid of revolution.

even approximately! In fact, it is approximated by cylinders that intersect

along ellipses. ◂◂◂

Fig. 6.7 Aesthetic lamps “Queen Titania”, designed by Alberto Meda and Paolo Rizzatto. The
lamps are perfect ellipsoids with three axes. Many artists are inspired by the aesthetics of ellipsoids.
238 Prototypes

▸▸▸ Ellipsoids in industrial design

Fig. 6.7 and Fig. 6.8 show two examples of triaxial ellipsoids in Industrial
Design. All the planar sections are ellipses.

Fig. 6.8 Elliptic fruit bowl (half of a triaxial ellipsoid) with parallel planar sections

Fig. 6.9 Spherical and ellipsoid-shaped teapot

Fig. 6.9 on the right shows Boris Sipek’s teapot in the form of an ellipsoid.
Strictly speaking, the teapot resembles a “magic egg” given the various ways
in which its individual pieces can be assembled together. Sipek deliberately
uses affine transformations in much of his work. A spherical counterpart to
his teapot is also available (on the left). ◂◂◂

⊕ Remark : Mathematically speaking, degree-two surfaces are expressed by a so-

called implicit equation of the form

a1 x2 + a2 y 2 + a3 z 2 + a4 xy + a5 yz + a6 xz + a7 x + a8 y + a9 z + a0 = 0.
6.1 Surfaces of degree two 239

If a straight line with the parametrization

x = u1 + t v 1 , y = u2 + t v 2 , z = u3 + t v 3

is intersected with the surface, an algebraic equation of degree two is found for the
parameter t. Each of the two solutions corresponds to an intersection point.

Fig. 6.10 Convex surfaces but not of degree two

When counting intersection points, it would be unwise to rely on mere visual ap-
pearance. The convex surfaces in Fig. 6.10 (which possess neither indentations nor
constrictions) seem to have at most two intersection points with any straight line.
However, not every surface of a convex solid is of degree two. The surface on the right
in Fig. 6.10 is, in fact, a supersphere with the implicit equation x4 + y 4 + z 4 = r4 .
Therefore, it is a degree-four surface. The surface tangents in the six flat points
touch the surface of a “higher degree”.
In this case, a degree-two surface has enormous advantages for our deliberations:
For instance, we then know that each planar section with this surface is a degree-
two curve – and therefore, a real or complex conic section (in general). This conic
section may also degenerate to a pair of straight lines (real or conjugate complex)
or a single straight line (always real) that should be counted twice. ⊕
Listening and focusing . . .
Due to their elegant form, egg-shaped ellipsoids of revolution are often found
in design. Using the ellipse’s focal point property, a focusing lamp with a
reflecting interior that resembles an egg-shaped ellipsoid of revolution can be
created. It focuses all light that is created at the focal point of the lamp onto
the second, non-materialized focal point.

Fig. 6.11 Wavefronts emanating from points inside the ellipse

Aside from light, other types of waves may also be focused this way, such as
thermal radiation (burning mirror) and sound waves (Fig. 6.11). The focusing
240 Prototypes

works best if the source is located exactly at one focal point (left to center.)
However, the phenomenon is not spoiled by slight deviations (right images).
The figure shows the progression of wavefronts and enables us to observe
interferences.

Fig. 6.12 Visible wavefronts

▸▸▸ Whispering bowls and odd acoustic phenomena

Fig. 6.13 shows two opposing bowls in the library of Vorau Abbey in Styria
(Austria). These two bowls resemble the cut-off caps of ellipsoids of revo-
lution. Without having measured them precisely, one may assume that the
bowls are spherical: The difference between the ellipsoid cap and the respec-
tive osculating sphere at the apex is practically negligible.

Fig. 6.13 Whispering bowls . . . Fig. 6.14 . . . and soundscapes

By placing one ear close to a focal point, it is possible to hear words which
have been whispered over 20 meters away near the other focal point.
Fig. 6.14 shows an artistic version of the acoustic effect created by Bernhard
Leitner. Paraboloids were used in that example – these, as we shall soon see,
can be regarded as extreme cases of ellipsoids. Sound is projected towards the
bowl from the focal points and is then focused parallel and directed towards
a wall, where, after being reflected, the acoustic sensation appears at the
angle of reflection. ◂◂◂
6.1 Surfaces of degree two 241

Paraboloids

Fig. 6.15 Wavefronts of a parabola

Fig. 6.15 illustrates the reflection of wavefronts when using parabolas instead
of ellipses. It shows that wavefronts are generated by plains when the starting
point coincides with the focal point.
A paraboloid of revolution can be constructed by rotating a parabola about
its axis – Fig. 6.15 further illustrates the many technical applications of such
surfaces as reflectors (radar screens, satellite dishes, radio telescopes, etc.).
▸▸▸ “Receiving dishes” (Fig. 6.16 left)
In theory, a satellite dish has to be an exact paraboloid of revolution, and
the receiving device has to be located exactly at its focal point. As with the
whispering bowl previously mentioned, however, it is possible to approximate
it quite well by means of an osculating sphere.

Fig. 6.16 Paraboloids of revolution as receivers or effective search lights

The only important requirement is that the receiver is not positioned at the
center of the sphere but halfway between the center and the surface: The
radius of the parabola’s osculating circle at the apex is twice the distance of
the focal point to the apex. Since the receiver is not point-shaped to begin
with, the devices allow for the axis of the bowl to be slightly different from the
242 Prototypes

signal direction. In such cases, it is required for the sensor to be positioned

off-center (compare it to Fig. 4.56)! ◂◂◂

▸▸▸ Parabolic headlights and sun collectors

By positioning a point light source at the focal point of a reflecting paraboloid
of revolution, it can be ensured that all light rays that hit the mirror will
be reflected parallel to the axis. Such lights are called “long distance beams”
(see Fig. 6.16 on the right).

Fig. 6.17 Parabolic headlights Fig. 6.18 . . . and their inversion

Rod-shaped light sources are often used in practice. Parabolic cylinders are
utilized in such cases (Fig. 6.17); they are also useful for focusing sunlight on
a pipe. This pipe may contain liquid (e.g., thermo oil) which is first heated
and then diverted. The screen is oriented in an east-west direction and is
constantly moved to ensure that the Sun stays in its symmetry plane. ◂◂◂

Parabolic translation surfaces

Paraboloids of revolution which have undergone affine transformations – in
other words, elliptic paraboloids – are also degree-two surfaces. It is possible
to get hyperbolic paraboloids by means of certain collineations. We prefer
another definition:

When a parabola is moved along a second, fixed parabola with a parallel axis
and a different carrier plane, the moving parabola sweeps a paraboloid.

If both parabolas open to the same side, we get an elliptic paraboloid, and
otherwise, a hyperbolic paraboloid (Fig. 6.19). If one of the parabolas is
not curved at all (and is thus a straight line), we get a parabolic cylinder.
The two parabolas are interchangeable in terms of their function. Thus, the
paraboloids carry at least two pencils of parabolas of equal importance (as we
shall soon show, there exists an infinite number of pencils).
6.1 Surfaces of degree two 243

Fig. 6.19 Paraboloids as translation surfaces

⊕ Remark : A paraboloid in its standard position (containing the origin – the axis
is the z-axis) can be described analytically by the equation

z = A ⋅ x2 + B ⋅ y 2 .

If A and B are of equal sign, then the paraboloid is elliptic – otherwise, it is hyper-
bolic. If A or B disappears, then we are dealing with a parabolic cylinder.
If a paraboloid is intersected with an arbitrary plane y = k ⋅ x + d parallel to its axis,
then the equation of the intersection can be given by inserting the plane’s linear
equation into that of the paraboloid. This yields the equation of a parabola

z = A ⋅ x2 + B ⋅ (k ⋅ x + d)2 = (A + B k 2 ) ⋅ x2 + 2Bkd ⋅ x + Bd2 .

This means that each intersection with a plane parallel to the axis yields a parabola.
Its curvature depends only on k, not on d. Thus, on this paraboloid, there exist
infinitely many congruent parabolas that can be translated into each other for each
position of the planes (as expressed by k).

Fig. 6.20 Hyperbolic paraboloids (HP bowl)

In the case of hyperbolic

√ paraboloids, A and B have different signs, and A/B is neg-
ative. With k = ± −A/B, the coefficient of the quadratic term disappears and the
section parabola becomes a straight line. This means that a hyperbolic paraboloid
244 Prototypes

Fig. 6.21 A hyperbolic paraboloid carries two families of straight lines, hyperbolas, and parabolas.

carries two families of straight lines, which appear as a grid of parallel straight lines
if seen from the top. Let us summarize: ⊕

Each planar section of a paraboloid that is parallel to the axis is, generally
speaking, a parabola. With hyperbolic paraboloids, the parabolas degenerate
into lines for two particular positions of planes. Hyperbolic paraboloids, thus,
carry two different families of straight lines, which appear parallel when pro-
jected in the axis direction.

Fig. 6.22 Hyperbolic paraboloid as the result of congruent rows of points

The boundaries of the surface in Fig. 6.22 are the generators that form a skew
quadrangle. The rows of points formed by correlated points, i.e., points that
are joined by generators, are congruent if the boundary generators appear
6.1 Surfaces of degree two 245

symmetric when viewed from above – otherwise, they are only similar. The
following can, thus, be said:

A hyperbolic paraboloid is also generated by congruent or similar rows of points.

Classically beautiful and very useful

Paraboloids play an important part in surface theory, as all curved surfaces
behave like paraboloids on a local level. We will discuss this in greater detail
in the subsequent section.
Hyperbolic paraboloids are also important in architecture, where they are
simply called hyperbolic paraboloid s. They can, for instance, be used to roof
rectangular areas in an elegant and very stable manner (Fig. 6.20). Their
construction is particularly simple, as the wireframe can be created from
straight rods.

Fig. 6.23 Skyscrapers with hyperbolic paraboloids as interior walls. (The picture in the middle
shows the top view.)

Fig. 6.23 shows a sketch of twin towers designed by Paolo Piva where the
lateral surfaces are derived from hyperbolic paraboloids.
Hyperboloids
HP surfaces share the property of being able to carry two different families of
straight lines with only one other type of surface: one-sheeted hyperboloids,
which are also degree-two surfaces.
A hyperboloid of revolution is formed by rotating a hyperbola about one of
its axes. Depending on the axis of rotation, the result is either one-sheeted
or two-sheeted.
The two-sheeted hyperboloid of revolution (Fig. 6.25) is not as important
in practice as the one-sheeted variant. It can be used as a lampshade for a
246 Prototypes

Fig. 6.24 Hyperbolic paraboloid at the Polytecnico di Milano. Note that, due to the thickness of
the surface, the contour is not a conic.

precisely point-shaped light source (Fig. 2.5) because, due to the fundamental
properties of conic sections, light that shines at a focal point is reflected in
such a way that it appears to emanate from the second focal point (and, in
the case of hyperbolas, from its extension). However, direct incidence of light
would have to be prevented by local coverage, which ensures that no lighting
occurs near the light axis.

Fig. 6.25 Two-sheeted hyperboloid of revolution from different perspectives

The one-sheeted hyperboloid of revolution is much more important in prac-

tice. It is generated not only by the rotation of a hyperbola about its conju-
gate axis but also by the rotation of a straight line about a skew axis. This
not-at-all-trivial insight is attributable to Christopher Wren, the constructor
of St. Paul’s Cathedral in London.
⊕ Remark : A hyperboloid of revolution with the axis z can be described by the
equation
x2 + y 2 z 2
− 2 = 1.
a2 c
6.1 Surfaces of degree two 247

Fig. 6.26 Generation of a hyperboloid by means of a hyperbola

If it is intersected with a tangent plane x = a (parallel to the z-axis), then the

equation of two straight lines can be found as

y2 z2 c
− = 0 ⇒ z = ± y.
a2 c2 a
A rotation gives us two families of straight lines lying in the plane.

Fig. 6.27 Double generation of a hyperboloid of revolution by means of rotating straight lines

Let us now consider an arbitrary point P in the plane. One straight line e from the
first family and another line f from the second family pass through it. The straight
lines e and f span a plane τ . The section of a plane with a degree-two surface is al-
ways a degree-two curve. This curve degenerates into two intersecting straight lines.
Therefore, it could be said that e together with f forms a “degenerate” hyperbola.
We now know two curves e and f through the surface point P (straight lines are
degree-one curves which can, for instance, be deduced by their linear parameter
representation). By definition, their tangents in P – the straight lines themselves –
span the tangent plane, and thus, τ = ef is the tangent plane at P .
Nothing changes in this statement even if we apply an arbitrary spatial affine map-
ping to the hyperboloid of revolution. We, therefore, arrive at the following:

Every tangent plane τ of a one-sheeted hyperboloid intersects at a pair of straight

lines e and f . Within each of the four sectors defined by e and f , the surface always
stays on the same side of τ .
248 Prototypes

Following a similar train of thought, it can be shown that every non-degenerate

degree-two surface that carries one family of straight lines {e} must also carry a
second family {f }: Each planar section through e yields a degree-two curve which,
nevertheless, possesses a “straight-line component” through e and must, therefore,
degenerate into a pair of straight lines.
Thus, it is clear that two pencils of straight lines must exist on HP surfaces as on
hyperboloids. But what about cylinders and quadratic cones? The tangent plane τ
touches not only at a single point, but along a whole generator. Thus, two “neighbor-
ing generators” (infinitely close together) lie on τ : The degree-two curve degenerates
into a pair of identical straight lines. ⊕

Fig. 6.28 Barcelona and its affinity to hyperboloids: When arriving at the airport in Barcelona,
the control tower (designed by Ricardo Bofill Levi) is the first hyperboloid you see. On the right:
Antoni Gaudí’s model for the hyperboloids in the roof of the Sagrada Família (compare Fig. 6.29).

▸▸▸ The envelope of a rotating cube (Fig. 6.30)

What happens when a cube rotates around a spatial diagonal (e.g. AG)?
Solution:
Of the twelve edges of a cube, three pass through A and G, intersecting the
axis of rotation at the same angle (the angle occurs in congruent right trian-
gles formed by the hypotenuse AG and the edge length or square diagonal).
The three edges thus act as generators of one and the same cone of revolu-
tion, which is traced three times during their rotation. The three points B,
D, and E or C, F , and H respectively form an equilateral triangle orthogonal
to the axis of rotation, tracing one and the same hyperboloid of revolution
six times. Three of these six edges belong to one family of straight lines and
the other three edges to the other family of straight lines of the hyperboloid
(Fig. 6.30 middle and right). ◂◂◂
6.1 Surfaces of degree two 249

Fig. 6.29 Roof of the Sagrada Família in theory (middle) and practice

D F Φ

B
E

Fig. 6.30 A cube rotates around a spatial diagonal. Three points move on a circle. The edges
that intersect the axis of rotation trace two cones of revolution. The remaining six edges, which
are skew in relation to the axis of rotation, trace a hyperboloid of revolution.

▸▸▸ Result of rows of points on circles in parallel planes

If one point moves on a circle and the second point moves on a circle in
a parallel plane at the same angular velocity, then the connecting straight
line generates a one-sheeted hyperboloid of revolution. If the respective circle
tangents are parallel, a cylinder is produced (this also works if the circles are
not positioned directly above each other – unlike in Fig. 6.31).
It is also possible to consider a one-sheeted hyperboloid as the product of
congruent rows of points on parallel circles. If the points move through their
carrier circles in opposite directions, a new surface that is no longer of degree
two is created. In the special case pictured in Fig. 6.31 on the right, a so-
called circular conoid is generated, which we shall discuss at a later stage
(Fig. 7.3). ◂◂◂
250 Prototypes

Fig. 6.31 Generating rows of points on circles in parallel planes

The fact that two independent families of straight lines lie on a one-sheeted
hyperboloid of revolution has implications for practical applications. Stools
of this shape (Fig. 6.34) have an enormous carrying capacity. The gargantuan
cooling towers of power plants are also shaped like hyperboloids of revolution.
This allows them to be stable despite having relatively thin walls.
▸▸▸ Stable and light-weight
The cooling towers of many power plants have the shape of hyperboloids

Fig. 6.32 Cooling towers of a nuclear power plant. Their geometry would be perfect – events in
the past, however, have proven that uncontrollable reactions occasionally occur in them. . .

of revolution. The ones depicted here (Temelin) are 155 m high and have
a maximum diameter of 130 m. The diminution towards the top causes an
increase in the speed of the hot water vapor. This leads – according to the
aerodynamic paradox – to a diminution of pressure, which chills the steam.
◂◂◂

▸▸▸ Generalization of the generation

It may stand to reason that the double generation of a hyperboloid of revo-
lution can be generalized by two rotating straight lines. In fact, the double
generation works as long as the generating circles lie in parallel planes. The
6.1 Surfaces of degree two 251

Fig. 6.33 Generalization does not always work.

result is then a general hyperboloid (Fig. 6.33 left) that carries two families
of straight lines. By “tilting” the circles, the two families of straight lines
produce two different surfaces that have nothing to do with a hyperboloid
(Fig. 6.33 center and on the right). ◂◂◂

Two congruent hyperboloids of revolution can, under some circumstances, fit

together (Fig. 6.35). The surfaces touch along a generatrix. If one imagines
both axes as fixed and allows one of the hyperboloids to rotate about its axis,
then the other one would also have to rotate about its axis – given “glide-free
friction”.

Fig. 6.34 Hyperboloids of revolution and – in the background on the bamboo stools – interesting
“envelopments” on cylinders of revolution (see also p. 173).

In this way, the rotation about this axis can be transferred consistently onto
a rotation about a second skew axis. In practice, this is used for the design
252 Prototypes

of so-called hypoid gears, despite the fact that such gears are not optimally
suited for transferring rotations onto skew axes. This is due to the fact that
hyperboloids do not roll but rather glide simultaneously along each other in
so-called grinding motions. Involute gears, which we shall discuss in a later
section, are found much more frequently. Their gear flanks are developables,
allowing “instantaneous rolling” without gliding. Another highly original so-
lution to this problem can be found in Fig. 11.18.

Fig. 6.35 Grinding hyperboloids of revolution

We already know that every degree-two surface intersects an arbitrary plane

along a degree-two curve. The following can also be shown:

The contour of a degree-two surface lies in a plane and is, therefore, a degree-
two curve. In the case of doubly curved surfaces, it is a conic section, and in
the case of single curved surfaces, it is a pair of straight lines (as a rule).

Proof : A doubly curved degree-two surface arises from a sphere by means of

collineation. The inverse of the collineation allows us to transform such a surface
back into a sphere. The center of the projection transforms into a point. At this
point, the cone (of revolution) created by projecting surface tangents is centered. In
the collineation, the circle of contact corresponds to the contour of the degree-two
surface, which is, therefore, a conic section.
By analogy, it is possible (using collineation) to transform a single curved degree-two
surface back into a cylinder or cone of revolution. ⊙

⊕ Remark : The contour of the hyperboloid of revolution in Fig. 6.35, a hyperbola

appears five times, and an ellipse appears once.
Fig. 6.25 illustrates that the contour of a degree-two surface does not always have
to be real: In the image, no outline seems to exist for the two-sheeted hyperboloid
of revolution. Mathematically speaking, a contour does, indeed, exist – but merely
a complex one. Complex contours also occur if the center of the projection is inside
the degree-two surface. ⊕
6.1 Surfaces of degree two 253

Whatever holds true for the contour also holds for the self-shadow boundary.
It can be shown that any self-shadows of the surface’s boundary curves must
also be conic sections:

Self-shadow boundaries and shadows of a planar section of a degree-two surface

onto itself are degree-two curves.

Fig. 6.36 The ellipse as a shadow on the inner side of a surface of degree two

⊕ Remark : The proof for this statement shall now be sketched: Let L be a light
source and c the shadow-casting planar section of the surface Φ (a degree-two curve).
Then, the projection cone Λ from L is a degree-two cone (in the case of parallel
light rays, it is a degree-two cylinder). Λ ∩ Φ is to be determined. Firstly, according
to Bézout’s theorem (p.169), a spatial degree-four curve is to be expected. However,
this curve must already contain the planar section c – a degree-two curve. Thus,
the remaining section, which is the shadow of c, may only be of degree two. ⊕
Fig. 6.35 illustrates this remarkable theorem. Shadows of hyperboloids on
other hyperboloids are also visible, which are, in fact, more complicated space
curves (of degree four).
▸▸▸ Geodesics on ellipsoids
The Earth is almost a sphere, and therefore, geodesics on the Earth are
almost great circles.
Say, we want to fly along the shortest route from A to B. On a sphere, we
somehow connect the Earth’s center M with A and B and get the great
circle’s plane. Since we cannot see M directly, it is a good idea to take a
third point C on the normal through A. The plane ε = ABC has to be
intersected with the sphere. The angle of ε with the corresponding meridian
plane indicates the starting angle of our airplane. If we want to fly back
from B to A, we can take the third point C on the sphere’s normal through
B, and of course, we get the same result.
If we are overly precise , we have to say that the Earth is, in a better approx-
imation, an oblate ellipsoid. In Fig. 6.37 we exaggerate the ellipsoid a little
bit. Clearly, we can see that neither the method with the normal in A nor
the one with the normal in B works. This results in to two different ellipses
on the ellipsoid. The truth does not even necessarily lie in the middle, as we
254 Prototypes

⃗A
n
⃗A
n
A

B
B
⃗B
n ⃗B
n

Fig. 6.37 Shortest connection: The normals nA and nB to the ellipsoid at the points A and B are
skew. The red geodesic connecting A and B lies either between the blue and green planar sections
through A and B, or even somewhere else.

can see in Fig. 6.37 (right). In general, a geodesic on an ellipsoid is not an

ellipse. ◂◂◂

▸▸▸ Saturn’s shape and its rings

On Saturn (Fig. 6.38), which has an equator radius ten times that of the
Earth and much more flattened polar regions (10% less circumference of the
meridians!), this could make much more of a difference than on Earth when
we look for geodesic lines as we did in the previous application.

reflections?

Sun
Fig. 6.38 NASA photo of Saturn. The Sun is glancing through the atmosphere. File: Saturn HST
2004-03-22.jpg

Fig. 6.38 looks strange and definitely requires some more contemplation.
Luckily, there exist comparable photos from other positions, like the ones in
Fig. 6.39. Firstly, Saturn is an oblate ellipsoid and both the contour and the
“terminator” (self-shadow border) are ellipses. The fabulous rings lie in the
equator plane and can be considered to be completely flat. The shadow of
Saturn onto its rings is, therefore, also an ellipse. The rings themselves cast
shadow curves onto Saturn which are the intersection lines of oblique elliptic
cylinders with the ellipsoid, i.e., they are space curves of degree four.
6.1 Surfaces of degree two 255

Fig. 6.39 Two more NASA photos of Saturn from Wikimedia Commons

Fig. 6.40 Computer simulation

Let us now look at the dark side of Saturn (Fig. 6.38), where there should be
complete darkness. We are above the rings. The outer rings, however, are still
partly visible, since they are illuminated (the sunrays are slightly refracted
through the atmosphere). ◂◂◂

Cubic circle:
When a degree-two cylinder is intersected with a degree-two cone, a spatial
degree-four intersection curve can be predicted for the general case. However,
if the cylinder and the cone share a generatrix, then this straight line is part
of the curve. It splits off from the intersection. If both surfaces touch along
this generator, then the remaining section is a conic section – otherwise, it
is a spatial degree-three curve. Such a curve is visible in Fig. 6.41, where a
cylinder of revolution and an oblique circular cone have been chosen. The
result is called an upright cubic circle. When viewed from above, the curve
appears as a circle, and in other projections as a degree-three curve. A cu-
bic circle belongs to the simplest non-planar algebraic space curves and is,
therefore, frequently employed for purposes of illustration.
For interested readers: A possible parameter representation of the curve is
given by

x = r ⋅ cos u, y = r ⋅ sin u, z = h ⋅ tan u/2 (u ∈ [−π/2, π/2]),

256 Prototypes

Fig. 6.41 Cubic circle as an intersection curve

where r is the cylinder radius and h is a scaling factor for the height.
A cubic space curve with an even simpler parameter representation is the
cubic parabola
x = t, y = t2 , z = t3 .

▸▸▸ Movements on and around triaxial ellipsoids

Ellipsoids – and more generally degree-two surfaces – have many remarkable
properties that are subject of numerous books ([22]). Two of them will be
investigated in detail in the following paragraphs. The first consideration was
only a conjecture until recently:
For any given ellipsoid Φ, there is a three-parameter family of planes which
intersect Φ along ellipses. However, the size of an ellipse depends only on
its two semiaxes. Consequently, on each ellipsoid there exist ellipses with a
one-parameter family of congruent copies on Φ. One can expect that some
conics can even be moved on the ellipsoid.

Fig. 6.42 Congruent ellipses can be moved on a triaxial ellipsoid.

6.1 Surfaces of degree two 257

The proof for this statement, and also generalization to the other degree-
two surfaces, is given in [22]. Fig. 6.42 and Fig. 6.43 illustrate this non-trivial
fact for two different triaxial ellipsoids. In every image all ellipses of the same
color have the same ratio of semiaxes.

Fig. 6.43 With larger semiaxes of the congruent ellipses, larger parts of the ellipsoids are covered.

When an ellipse of a given size is moved along on the ellipsoid, one can
observe the contact point of the accompanying parallel tangent plane. Its
orbit stays the same for all ellipses with constant ratio of the semiaxes. If
the ratio is 1 and the moving ellipse is a circle, the orbit degenerates to the
umbilical points of the ellipsoid. Circular intersections can, therefore, not be
moved.
Another astonishing theorem, proofed in [22], states that all possible “bound-
ing boxes” (circumscribed cuboids) of a triaxial ellipsoid have their vertices
on a concentric sphere (Fig. 6.44 on the right).

Fig. 6.44 Ellipse with circumscribed rectangles (on the left) and ellipsoid with circumscribed
cuboids.

This is the 3-dimensional version of the better known fact, that all rectangles
that are circumscribed to an ellipse have their vertices on a concentric circle
(Fig. 6.44 on the left, proof see [16]). ◂◂◂

▸▸▸ Moving along the generator of a degree-two surface

Without going into detail (please, again refer to [22]): If we move along a
258 Prototypes

generator of a degree-two surface (hyperboloid or hyperbolic paraboloid),

the corresponding tangent plane will rotate about 180○ (Fig. 6.45).

Fig. 6.45 Moving along a generating line of a hyperboloid.

The rotation, however, is not proportional to the distance travelled. If we

have a hyperboloid of rotation, the rotation angle of the tangent plane is
proportional to the tangent of the distance, measured from the point with
shortest distance to the axis of the surface. ◂◂◂
6.2 Three types of surface points 259

6.2 Three types of surface points

In the previous section, we discussed the topic of degree-two surfaces ex-
tensively. This was for two reasons: Firstly, as we have seen, these surfaces
have numerous practical applications. The second reason should soon become
clear:
Curvature of a planar curve

Fig. 6.46 Curves with intricacies

Let us start with a two-dimensional comparison and imagine an arbitrary

curve in the drawing plane (without intricacies such as tips, kinks, or jumps
as in Fig. 6.46).

Fig. 6.47 Osculating circles of a bent planar curve

At each point P of this curve, there exists a tangent. It is the “linear ap-
proximation” of the curve at the point P . At each point, there also exists an
osculating circle, which does a much better job at approximating the curve.
It could be said that it represents a quadratic approximation. Depending on
260 Prototypes

the side on which the osculating circle is located, it is already possible to dis-
tinguish between positive and negative curvature. The smaller the osculating
circle, the larger the curvature (Fig. 6.47).

Fig. 6.48 Osculating parabolas of a curve

Fig. 6.48 shows another kind of quadratic approximation of a curve. This

time, osculating parabolas were used instead of osculating circles. Like oscu-
lating circles, they are each defined by three neighboring points as well as a
fixed axis direction.

Fig. 6.49 Osculating circles and osculating parabolas (seals in the harbour of Capetown)

In space, curvature is not at all easy to define!

Let us now try to extrapolate the procedure into space: If P is a surface
point, then the corresponding tangent plane τ ∋ P is a linear approximation
of the surface. However, what corresponds to the osculating circle? Let us
consider an arbitrary surface tangent t ∋ P . There exist infinitely many space
curves c in the surface with t as their tangent. The osculating circles may
be different for various curves. Meusnier has shown that all the osculating
circles of the surface curves defined by the line element (P, t) form a sphere,
6.2 Three types of surface points 261

the so-called Meusnier sphere (Fig. 6.50). The center of this sphere is also
the center of curvature for the normal section of the surface through t.

Fig. 6.50 Meusnier spheres for different tangent directions

Principal curvatures
The existence of the Meusnier sphere is remarkable in and of itself. Yet, what
happens if we change t? Of course, this problem has already been investigated
in great detail. However, the issue is very elaborate and does not yield any
impressive visuals – it, thus, falls outside the scope of this book.

Fig. 6.51 Planar intersections parallel to the tangent plane, Dupin’s indicatrix

The following shall be presented without proof: At each surface point, there
exists a greatest and a smallest Meusnier sphere. They belong to a pair of
orthogonal surface tangents, so-called curvature tangents. Let us intersect
the surface with a plane which is parallel to the tangent plane but only “in-
finitesimally” distant to it (Fig. 6.51). In an infinitesimal neighborhood of
the surface point, the intersection curve with the surface can locally hardly
be distinguished from an ellipse, a hyperbola, or a pair of parallels – known
henceforth as indicatrices. The axes of the indicatrix are the principal curva-
ture tangents. This also means that the curvature of all normal sections is
known once the indicatrix is defined – more specifically, knowing the curva-
ture along the principal curvature tangents is enough. This insight goes back
to Dupin.
Principal curvature lines are the surface curves where all tangents are curva-
ture tangents and which, therefore, follow the direction of the strongest and
weakest curvature.
262 Prototypes

Fig. 6.52 Principal curvature lines s on an ellipsoid with three axes.

There are two families of principal curvature lines. The curves from one family
intersect all curves from the other family orthogonally (except at umbilics).
If we consider a surface point and intersect the surface orthogonally to its
tangent plane τ , we get ellipses or hyperbolas whose axes are parallel to the
tangents of the curvature lines in τ . Fig. 6.52 in the middle shows curvature
lines on a triaxial ellipsoid. It can be shown that the curvature lines are
intersection curves with “confocal” hyperboloids (Fig. 6.52 on the right).

Fig. 6.53 Triaxial ellipsoid with a clever triangulation

Fig. 6.53 in the middle and on the right shows the design and the construction
phase for the project “hangar8” (Volkmar Burgstaller). A triaxial ellipsoid is
triangulated by being first cut into slices, after which the points of an in-
tersection line are connected to the points of the next intersection line in
a way that produces approximately equilateral triangles. Emphasis must be
put on the word “approximately”, as unless we are dealing with very spe-
cial polyhedra, no surface can be approximated only by congruent triangles
Fig. 10.26 shows the same surface but from the inside. Fig. 6.53 on the left
shows a perfume bottle by Guerlain (an ellipsoid of revolution), where a sim-
ilar method of triangulation has been applied, with groups of six triangles
being combined into hexagons.
Osculating parabolas lead to an important theorem
The other approach would mean using osculating parabolas instead of os-
culating circles. They agree in curvature with any intersection of a plane
through the surface normal. Like the osculating circles of the normal sec-
tions, they should be defined by three neighboring points on the normal
sections. This time, the result – the surface formed by all possible parabolas
6.2 Three types of surface points 263

– is simple: It turns out to be either the tried and true paraboloid (elliptic
or hyperbolic) or a parabolic cylinder!
If we attempt to approximate the surface quadratically by using parabolas,
we always get the same previously noted three surfaces. The approximation
quality will improve, the smaller the chosen area where the comparison takes
place. Thus, from a local perspective, there exist only three types of surfaces:

Essentially, every surface of any conceivable complexity is locally, i.e., in a

sufficiently small neighborhood of a point, curved like an elliptic or hyperbolic
paraboloid or like a parabolic cylinder (Fig. 6.54). The surface point is, therefore,
called either elliptic, hyperbolic, or parabolic.

Fig. 6.54 The three different types of surface points

⊕ Remark : Paradoxically, the two-sheeted hyperboloid consists only of elliptic points

even though it carries a multitude of hyperbolas (Fig. 6.25). Even the contours (if
real) appear as hyperbolas! ⊕
If we consider not only a “microscopic fragment” but the whole surface, then
the issue becomes much more complicated. At the level of the whole surface,
there are no limits to the possible variations. Nevertheless, it is still possible
to speak of elliptic, hyperbolic, or parabolic curved parts of a surface. In
general, the local properties do not change arbitrarily but continuously.

Fig. 6.55 The “saddle point” is always of a different type.

Fig. 6.55 shows three surfaces which, at first glance, resemble an hyperbolic
paraboloid. At the supposed saddle point, the surfaces are differently curved
264 Prototypes

(elliptic on the left, parabolic in the middle, and hyperbolic on the right). In
the neighborhood of the point in question, only the right surface is actually
related to the hyperbolic paraboloid. This can also be seen due to the fact that
the tangent plane “penetrates” the surface in more than just the contact point
(just locally). In the case of the middle surface, there is also an intersection
with the tangent plane, but this intersection does not occur in the immediate
neighborhood of the point.
The exact hyperbolic paraboloid and the one-sheeted hyperboloid of revo-
lution carry only hyperbolic points, while convex surfaces (surfaces which
envelop a manifold without indentations or neckings) are always either ellip-
tically or parabolically curved.
▸▸▸ An exciting experiment
Can a two-dimensional being on a surface find out to what extent its “car-
rier surface” is curved without leaving the surface and thus entering three-
dimensional space?

Fig. 6.56 The test using a circle

The mathematician’s answer is: Yes! The being would merely have to draw
a circle with the radius r (Fig. 6.56) and measure its circumference. If this
circumference is 2π ⋅ r, then the being lives on a parabolic surface – if it is
greater, then it lives on a hyperbolic surface. A being on an elliptic surface –
just like us humans on the globe – would see that the circle is smaller than
expected.
Let us check this by way of a specific example: We draw a circle around
the North Pole with a radius of r = 10 000 km (a quarter of the Earth’s
circumference). Thus, all points of the circle lie on the equator, which has
a circumference of 40 000 km. It is, in fact, 2π ⋅ r ≈ 62 800 km, which means
that our carrier surface must be elliptically curved – and quite strongly so.
This mundane fact of contemporary science was assumed to be true for many
centuries (and was prefigured by several great minds of the ancient world,
such as Eratosthenes, who worked out the circumference of the Earth with
remarkable accuracy 2200 years ago). The “rigorous proof” was not provided
until the age of rocket science.
⊕ Remark : This experiment has an additional, particularly exciting extrapolation:
Mathematicians can prove that it also works in any dimension. We can determine
whether space is curved without entering the fourth dimension. However, it is ac-
tually not that simple. Globally, our space does not seem to be curved on a large
6.2 Three types of surface points 265

scale. Locally, however, there are lots of noticeable indentations in the neighborhood
of large masses (such as black holes). ⊕ ◂◂◂

▸▸▸ Light edges

Fig. 6.57 shows two surfaces of translation which differ only slightly: In
the right image, a cubic parabola is being translated along a second cu-
bic parabola. The surface is continuously differentiable to the second degree
– even at the critical “midline” where the color changes. In the left image,
a translated cubic parabola is being approximated by two regular parabolas
which do not have a transition of constant curvature at their apex. A layman
has to examine the images many times to notice that in the left, there are
bends at the midlines next to the shadowing. Nevertheless, such subtleties
play a great role in practice: In the construction of car bodies , light edges
are mostly undesirable. Due to the reflective paint, car exteriors are visually
very sensitive to such details. Looking for such defects can be compared to
detecting indentations on a fly’s eye (Fig. 6.59).

Fig. 6.57 Curvature jump . . . Fig. 6.58 . . . or not?

Fig. 6.58 shows two surfaces of revolution where two differently curved merid-
ian circles are combined. The transition occurs on the left at an arbitrary
point and on the right at the exact location where the tangents of the circle
are horizontal. On the left surface, the change from elliptic to hyperbolic
points comes abruptly. On the right, the horizontal tangential plane touches
both partial surfaces along the common circle. Its points are parabolic for
both partial surfaces.
This is why the curvature jump in the right image, as opposed to the one in
the left image, is not noticeable as a “light edge.” ◂◂◂

If a surface is composed exclusively of parabolic points, then the surface is

only single and not doubly curved. There only exist a few classes of surfaces
with this property, and they play an important role in practice: There are de-
velopable surfaces, which include the cylinder and the cone (see also chapter
5). They can be manufactured relatively easily by bending the “blank cut”
without elongation or compression (Fig. 6.61).
266 Prototypes

Fig. 6.59 The light reflection shows an indentation on the fly’s eye.

Fig. 6.60The development of a Möbius strip. Each surface in between is, of course,
also developable.

Developable surfaces can be visually recognized by their contours, which are

composed entirely out of straight-line segments independent of the viewing
direction. Here, one ought to be reminded of the contour generators of cones
and cylinders (boundary curves are not counted as part of the contour –
only the surface curves along which the tangent planes appear as a line
segment). Some lesser-known examples include the developables of constant
slope, which are used during street construction; the oloid (Fig. 6.63); and
the Möbius strip (Fig. 6.61).
6.2 Three types of surface points 267

Fig. 6.61 Three different views of the Möbius strip

Fig. 6.62 Developable surfaces from different perspectives

▸▸▸ Revolvability with practical consequences

The oloid (cf. Fig. 5.65 and 6.103), invented by the Swiss engineer Paul
Schatz, is one of the developable surfaces with practical applications. It can
be rolled along a table surface in a wobbling motion, which must be the
reason for its common English name, the “wobbler” (Fig. 6.63 on the right).

Fig. 6.63 Left, center: Revolution of the oloid according to Paul Schatz (www.oloid.ch). Right:
Development according to Stachel/Dirnböck (www.geometrie.tuwien.ac.at/stachel).

As a consequence, the oloid can be revolved in a rather unusual way (Fig. 6.63).
If this revolution is conducted in water, it produces radiating currents that
display a remarkably wide range. These can, for instance, be used to treat
sewage water. ◂◂◂
268 Prototypes

6.3 Surfaces of revolution

Rotationally symmetrical surfaces play an important role in nature and tech-
nology.

A surface of revolution is created by rotation of an arbitrary curve about a fixed

axis. If the generating curve lies in a plane through the axis (meridian plane),
it is called the meridian of the surface.

Fig. 6.64 Surface of revolution in theory and nature

If rotated, each point of the generating curve creates a circle of latitude

(sometimes also called a parallel circle) of the surface. If the curve does not
lie in the meridian plane, the axially symmetric meridian can be determined
by intersecting all circles of latitude with a fixed meridian plane.

Fig. 6.65 Surfaces of revolution with meridians and circles of latitude (oblique axis)

If the meridian has a tangent parallel to the axis of rotation, then a rotation
of the associated point produces a circle with a locally minimal or maximal
6.3 Surfaces of revolution 269

radius (e.g., equator circles, gorge circles, waist lines). In general, boundary
circles do not belong to this group of circles. If the meridian tangent is
orthogonal to the axis, the circle’s plane touches the surface along the entire
circle.
⊕ Remark : During rotation, the tangent of the meridian generally traces a cone
of revolution with its apex on the axis of rotation. Alternatively, it can be said
that the surface is enveloped by a family of tangent cones. The surface can also be
interpreted as the envelope of all cones whose apices lie on the axis and which also
touch the meridian (Fig. 6.64 on the right). ⊕

Fig. 6.66 Two families of curves on a surface of revolution

Surfaces of revolution are invariant with respect to any rotation about the
axis and reflections in an arbitrary meridian plane. If the generating curve is
not a planar curve (Fig. 6.66), then, in addition to the curve itself, any curve
reflected in an arbitrary meridian plane is a “generating” one. (Any curve on
the surface of revolution generates at least a part of this surface.)

Fig. 6.67 Along each parallel circle of a surface of revolution, one can inscribe a sphere (left and
middle) and a cone of revolution (right).

▸▸▸ Vase with an ellipse as its finishing part

The lower part of the vase pictured in Fig. 6.68 is created by the rotation of
270 Prototypes

a sine wave. In the upper part, it is designed in such a way that its finishing
part is an ellipse inclined towards the axis.

Fig. 6.68 Vase with an ellipse as an oblique finishing part

Now the question arises which surface of revolution allows this simple oblique
intersection. Locally, it must be a degree-two surface of revolution! In this
particular case, the surface is hyperbolically curved. In its upper part, the
surface is a hyperboloid of revolution, and in the part near the oblique inter-
section, it is almost a cone of revolution. ◂◂◂

▸▸▸ Distorted wine barrel

A wine barrel is a well-known surface of revolution and used to function
as the starting point for a number of important calculations. Kepler , for
instance, used it to develop a simple formula which is of high significance in
mathematics (“Simpson’s rule”).

Fig. 6.69 Affinely distorted surfaces of revolution (original on the left)

Fig. 6.69 shows a “poster” of a visitor’s lounge: The wine barrel is affinely
distorted. The visitor would only notice this when moving close to the object
and viewing it from its side. On the right side, the same photo has simply
been stretched planimetrically in its horizontal direction, which re-establishes
the appearance of a surface of revolution. ◂◂◂
6.3 Surfaces of revolution 271

▸▸▸ When geometers take a walk in nature . . .

. . . it is inevitable that they are attracted to the shapes of blossoms (Fig. 6.70,
Fig. 6.71, and Fig. 6.72 on the left). Also inevitable are associations that come
to mind concerning surfaces in technology (Fig. 6.70).

Fig. 6.70 Contour of a surface of revolution in nature and in technology

Fig. 6.71 Blossom of a bluebell: The bordering lines are the intersection lines between a surface
of revolution and cylinders of revolution (compare computer graphic in Fig. 6.72 on the right).

Fig. 6.72 Broken surface contours in nature (fire lily) and in computer graphics.

In the early 1980s, the author was inspired by bluebells and consequently
created one of his first computer drawings (Fig. 6.72 on the right). The
problem at that time was that only raster graphics were available. This forced
the programmer to develop several algorithms, e.g. a hidden surface algorithm
and an algorithm for the detection of contours ([3]). ◂◂◂
272 Prototypes

The torus is a frequent building block of higher degree

A sphere is constructed by rotating a circle about an arbitrary diameter. If
the axis of rotation is translated within the circle plane into an arbitrary
position, then a much more complicated surface is produced by subsequent
rotation. It is called a torus and it is, as can be calculated, an algebraic
surface of degree four. Geometrically speaking, it can be reasoned that a
pair of circles lies in each meridian plane (Fig. 6.73) and is intersected by an
arbitrary straight line in no more than four points.

Fig. 6.73 The torus as a surface of revolution: A circle rotates about a straight line in its plane.

The same surface is also enveloped if a sphere for which the circle in the
meridian plane is a great circle rotates. It is, thus, possible to “roll around”
a sphere inside a torus – this has many implications for practice.

Fig. 6.74 Torus as an envelope of spheres

6.3 Surfaces of revolution 273

A torus is a degree-four surface of revolution that is created by the rotation of

a sphere or a circle about an axis located in its plane which does not contain
the center.

The outer ring of a torus is composed of elliptic points and the inner ring of
hyperbolic points. The circles, on which the torus can be placed in a plane,
carry parabolic points. Each circle of this kind is entirely contained in the
tangent plane of each such point.

Fig. 6.75 Torus chains: Each link is made of half-ring surfaces which are joined cylindrically by
cylinders.

A standard steel chain (Fig. 6.75) is composed of links made of two half-ring
surfaces connected by cylinders of revolution. The radius of the circle which
generates the torus is slightly smaller than the radius of the inner equator
circle.

Fig. 6.76 Torus part and affinely distorted tori

Rotating only parts of the generating circle (meridian circle) produces open
surfaces of revolution which are, geometrically speaking, still labeled as tori.
The same is true if the rotation is applied to the whole circle but does not
reach 360○ (Fig. 6.76 on the left).
Fig. 6.77 shows the interior view of a torus – a challenge to the light sensitivity
of the photo film and to the nerves of the “participants.”
Fig. 6.78 shows two surfaces which are closely related to the torus. Put more
precisely: The left surface is a half-torus, or it is bounded by surface parts
274 Prototypes

Fig. 6.77 A light at the end of the torus-shaped tunnel

Fig. 6.78 One of the two surfaces is not a torus.

which belong to a half-torus. The right surface is constructed in a way that

resembles a torus but the meridian circle grows continuously smaller. Thus,
this surface is no longer a surface of revolution.

Fig. 6.79 Other variants . . .

Fig. 6.79 shows further variations. The left image is a computer graphic. If
a doubly curved torus was to be modeled using a paper strip, it would only
result in an approximation because the surface thus generated is only single
curved. The model on the right dates back to M.C. Escher and is often called
an “Escher torus,” even though this is not quite correct.
6.3 Surfaces of revolution 275

Non-trivial sections of the torus

Fig. 6.80 Intersections of a torus with planes parallel to the axis

Algebraically speaking, the torus is a degree-four surface. Therefore, each

planar intersection curve is, generally speaking, a degree-four curve. Among
these are the so-called Cassini ovals, which are produced if the plane that
intersects the torus is parallel to the axis of the torus. These curves have
already caught our attention (Fig. 1.33, Fig. 6.80 on the right). If the inter-
section plane includes the axis, then the curve splits off into two circles.
⊕ Remark : There exist two other positions of the intersection plane where the inter-
section curve splits off into two circles: Each plane orthogonal to the axis intersects
along two concentric circles. These two circles coincide if the plane (orthogonal to
the axis) touches the torus. ⊕

Fig. 6.81 Villarceau circles in theory, as a wooden model, and in art (Benno Artmann)

⊕ Remark : There even exists a non-trivial case where the section curve splits off
into two circles: The plane has to touch a pair of axis-symmetric meridian circles
in hyperbolic surface points. We now have the case that two surfaces (the torus
and the plane) share a degree-four curve, which – due to double contact – has to
possess two double points. However, this is only possible if the intersection curve
degenerates. In this case, the curve would have to split off into two circles, but in
the case of a different degree-four surface, they might be arbitrary conic sections.
Both conic sections are envelopes of the family of circles that is cut out from the
torus-generating spheres (Fig. 6.82). The envelopes are circular (i.e., they are pass-
276 Prototypes

Fig. 6.82 Villarceau circles as envelopes

ing through the absolute points of the intersecting plane – see p. 194). This remark-
able discovery goes back to the French geometrician Villarceau. ⊕

Fig. 6.83 Rhumb line on ring surfaces

⊕ Remark : Villarceau made even more discoveries: The circles named after him
intersect all meridian circles – and thus all parallel circles – at the same angle. Such
a curve is called a surface’s rhumb line or loxodrome.

Fig. 6.84 Sphere rhumb lines (loxodromes) with different course angles

The course angle of this rhumb line is, of course, extremely special. If the angle is
changed even slightly, the curve no longer stays in a plane. In most cases, it does
not even remain a closed curve but will wind around the torus “for all infinity”
without ever returning to the starting point. For special angles, however, these
6.3 Surfaces of revolution 277

curves are closed (Fig. 6.83), and they are also algebraic, which means that they
can be described by algebraic equations.
The sphere, if you like, is also a special torus where the spine circle is shrunk to
a single point. Interestingly enough, rhumb lines on spheres are never algebraic
(except for course angles 0○ and 90○ ). Sphere rhumb lines are spiral-shaped curves
that wrap around the poles. A ship traveling at a constant course angle relative to
the north direction travels along such a rhumb line! This is actually a productive
strategy in tropical regions for traveling from A to B without great navigational
effort. Upon reaching polar regions, however, the captain would have to change his
strategy. ⊕

Fig. 6.85 Closed rhumb lines on a ring torus (compare Fig. 6.107), among them a Villarceau
circle.

Fig. 6.86 Very special triangulation of the surface of a torus with the help of rhumb lines (in the
left image with Villarceau circles.) In each knot, the angles of the intersection angles are the same.
278 Prototypes

6.4 The torus as a prototype for all other surfaces

of revolution
The importance of the torus in geometry results, among other things, from
the fact that each surface of revolution can be approximated by a sequence of
torus segments. This directly follows from the insight that each planar curve
can be approximated by a sequence of osculating circles. (Curves that are
produced by chaining circles are called basket handle arches.) It can, thus,
be said that:

From a local perspective, each arbitrary surface of revolution is curved like its
“osculating torus”.

Some surfaces of revolution found in architecture are parts of a torus even

if this is not immediately visible. The dome of the Mosque of Ibn Tulun
in Cairo (built in the 8th century AD) is part of a so-called spindle torus
because the meridian circle intersects the axis of rotation (Fig. 6.87 on the
left, Fig. 6.88).

Fig. 6.87 Spindle torus in architecture and nature

The inner parts of a torus are often found in nature (Fig. 6.87 on the right),
though we have to be generous in this assessment and speak only of the “basic
form.” Seen this way, even the pressurized stream of water escaping from a
valve in Fig. 6.89 could be likened to a spindle torus in its incipient phase.
▸▸▸ Contact lenses and astigmatism
Wearers of contact lenses (or glasses) often hear the words “spherical” or
“toric” from their ophthalmologist or optician. The surfaces of spherical lenses
(spectacle lenses) are simply sphere caps with a slightly different radius.
This causes the focal point of the lens system to be offset (when “focused
6.4 The torus as a prototype for all other surfaces of revolution 279

Fig. 6.88 Spindle torus as an ornament on an ancient wall

Fig. 6.89 A stream of water escaping from a valve

on infinity”, the focal point should lie precisely on the retina). This does not
matter in spherical lenses, which are attached to the eye by adhesion: The
sphere has the same curvature regardless of its tangential direction.
Not only is the exact working of the eye’s optical system relatively complex, it
is also different from what most people believe it to be. The actual refraction
happens on the cornea while the lens inside the eye is only responsible for
fine tuning. More information can be found on page 409.

Fig. 6.90 Contact lenses

The issue is further complicated if one additionally suffers from astigmatism.

In such cases, the principal curvatures of the cornea – and sometimes also of
the lens inside the eye – are different. This causes points on the retina to be
280 Prototypes

rendered in a “rod-like” manner. The corrective lens then has surfaces in the
form of torus caps. On top of that, the axis of the torus can be twisted to
varying degrees. As a result, toric lenses exist in a great variety of shapes,
while the number of spherical lenses on offer (at granularities of quarter-
diopters) is relatively limited and usually available in their full range in any
optician’s store.
Hard spherical contact lenses are often capable of correcting astigmatism by
the mere fact that they allow smaller cavities between the lens and the eye
to be filled with lacrimal fluid, which possesses nearly the same refractive
index as the cornea itself. The liquid, thus, complements the cornea into a
sphere (Fig. 6.90 on the left). Soft lenses, on the other hand, adapt to the
toric shape of the cornea. ◂◂◂

Cusps of contours
Torus parts – and by extension, many surfaces of revolution – tend to de-
velop contour cusps during projection. Fig. 6.91 shows different perspectives
– most of them exhibit such cusps. During a normal projection orthogonal
to the axis, the contour degenerates into two circles and into the lowest and
uppermost circle that appear as one segment (Fig. 6.91 top left, also to some
extent the picture of the anemone 6.92). Equator circles form the contour in
axis direction (Fig. 6.91 on the right).

Fig. 6.91 Cusps of the contour of a torus

Fig. 6.93 How contour cusps are formed: In

each point K of the contour line k, the associ-
ated tangent plane τ passes through the center
Z. A cusp S occurs when the tangent s of the
Fig. 6.92 Quarter-torus on an anemone contour k passes through Z.
6.4 The torus as a prototype for all other surfaces of revolution 281

The respective contour in space is an arbitrary space curve k (composed of

two branches, as can be seen in Fig. 6.93) that does not have any cusps.
In each point K of k, the associated tangent plane τ – spanned by the
tangent t of k and the respective projection ray – passes through the center
of projection.

Fig. 6.94 Left: The contour of a torus in central projection (consisting of two curves k1 and k2 )
has four cusps in the projection. Right: Viewed from another projection center, k1 and k2 appear
as space curves without cusps.

The probability that a tangent of k passes directly through the center Z of the
projection at some point S is greater than one might expect. In those specific
cases, the curve appears, locally speaking, as a point, and its orthogonal view
possesses a cusp. The contour lines k in Fig. 6.93 and Fig. 6.94 are “jagged”
in some parts, which results from the fact that the torus is triangulated. This
causes the contour edges of the approximate polyhedron to become visible.
In the projection from Z, these jagged edges cannot be seen.

Fig. 6.95 Contour of a torus, hyperboloid, and catenoid: Cusps on the contour never occur with
hyperboloids but they do occur very often with the interior parts of tori.

Fig. 6.95 illustrates how the normal views of three visually similar surfaces
of revolution may or may not have cusps despite the fact that the direction
of the projection is the same. The catenoid is formed by the rotation of
a catenary. The contour of a hyperboloid is always a hyperbola (Fig. 6.95
center), which is why cusps on the contour never occur with this type of
surface.
282 Prototypes

Fig. 6.96 Coop Himmelb(l)au: Archaeological Museum Egypt in Cairo

⊕ Remark : The question of which surface is shown in Fig. 6.96 can thus be ap-
proached from this point of view. It is actually a hyperboloid which changes its
shape into a torus towards the roof.
Fig. 6.97 shows the cusps on the contour of general surfaces of revolution. The
meridian is easily discernible on the right – no cusps occur there. A cusp is visible
in the other two pictures (see also the appendix about freehand drawing). ⊕

Fig. 6.97 Cusps on the contour of surfaces of revolution (left and center)

Intersection of surfaces of revolution

The flower petals of the bellflower in Fig. 6.97 form boundary curves which,
when projected in axis direction, resemble circular arcs with great accuracy.
The boundary curves of the petals can, thus, be said to emerge as the inter-
section curves of the surface of revolution with cylinders of revolution. This
choice of phrase is in no way exaggerated: A designer at a computer is easily
and quickly capable of modeling the curve in this way!
In mechanical engineering, intersection curves of surfaces of revolution hap-
pen particularly frequently. The axes of these surfaces often intersect, and for
this special case, Gaspard Monge has already specified a quick and efficient
procedure.
6.4 The torus as a prototype for all other surfaces of revolution 283

Fig. 6.98 Principle: Variation of Monge’s auxiliary sphere . . .

If an auxiliary sphere is put around the intersection of the axes, it will inter-
sect both surfaces along circles. The points of the intersection curve of both
surfaces of revolution are then the section points of the circles (Fig. 6.98).

Fig. 6.99 . . . during normal projection on the connecting plane of the axes

If the surfaces are projected orthogonally to the plane spanned by both axes
of rotation, then any common circles with Monge’s auxiliary sphere appear
as line segments. However, this immediately provides their section points. If
the size of the auxiliary sphere is now varied, an arbitrary number of points
of the intersection can be found.
⊕ Remark : A note for advanced readers: Even if the circles on Monge’s auxiliary
sphere no longer intersect in real space, they still share complex conjugate inter-
section points. The real straight line connecting these points remains the straight
intersection of both carrier planes of the circles on the sphere. It can be shown that
the normal projection of the intersection curve, if “extended”, passes through the
orthogonal view of each such straight intersection. These points are called “parasitic
points.” ⊕

Reverse engineering
Let us, once again, consider the bellflower from before: The difficulty seems to
lie in finding the surface parts of seemingly arbitrary surfaces that belong to
a known surface class, such as the surfaces of revolution. In computer geom-
etry, the term for this procedure is reverse engineering. Special programs are
capable of analyzing objects digitized by 3D scanners and try to subdivide
284 Prototypes

them into parts which are, for instance, surfaces of revolution or developable
surfaces.

Fig. 6.100 Surface of revolution (an ellipsoid of revolution in its upper part)

This can be greatly advantageous in practice. Consider a brilliant architec-

tural model with many curved surfaces. Once this project is due to be put
into practice, the construction costs can be greatly reduced if the surfaces
involved can be reduced to certain standard surfaces without substantial
changes to the design.

Fig. 6.101 Interior of the surface of revolution above (part of an affinely distorted torus)

Fig. 6.100 shows the limestone shell of a sea urchin – a general surface of
revolution. Upon closer inspection, the entire upper part appears as an ellip-
soid of revolution – an assumption that is further supported by its shadow.
The lower part appears to be an affinely distorted torus, which, after a mere
glance at Fig. 6.101, may not be immediately obvious to the layman.
With some practice, it is possible to recognize planar intersections of a torus
in Fig. 6.102 on the left and torus-shaped grooves on the right.
Reverse engineering attempts to develop methods enabling computers, which
are “stupid but fast,” to detect and prove circumstances which might not be
easily seen with the naked eye. Much geometric theory is applied for this
purpose:
6.4 The torus as a prototype for all other surfaces of revolution 285

Fig. 6.102 Planar torus intersections and torus-shaped grooves in a half-sphere

▸▸▸ Gauss map

Points on a general surface can be associated with points on a sphere. We
translate the associated surface normal to the center of the unit sphere and
intersect it with the sphere. In general, the associated points will be dis-
tributed over the entire surface of the unit sphere.

Fig. 6.103 Spherical tangent, normal, and binormal image of the oloid

If the initial surface is developable, it will be touched along each generator by

one and the same tangent plane with a fixed normal direction. The Gaussian
image of a curve on the sphere equals the spherical binormal image of the
surface’s edge of regression. Fig. 6.103 shows the spherical normal image of
an oloid (blue). The curve could be called a “tennis ball curve.” It divides the
sphere into two congruent halves and possesses a congruent top and front
view in relation to its symmetry plane.
If all surface generators are translated through the sphere midpoint, we get
the spherical generator image. In the case of the oloid, this happens to be
a curve with four cusps (in red), which also possesses a high degree of sym-
metry. This curve is the spherical tangent image of the surface’s edge of
regression.
286 Prototypes

Finally, the spherical primary normal image of the curve of regression (which
appears in green) may be considered. ◂◂◂

▸▸▸ Automatic detection of developables

The spherical image of a surface is not only interesting from a theoretical
perspective; it is also utilized during reverse engineering: In order to detect
whether or not a surface is developable, it can be scanned using a 3D scanner.
This results in a “poiint cloud,” which exhibits “noise” due to measuring
inaccuracy. The cloud’s surface normal is now interpolated and translated
through the center of the test sphere. The more accurate this spherical image
approximates the surface in the neighborhood of the curve, the more likely
it is that the surface is developable.

Fig. 6.104 Reconstruction of a snail shell according to Boris Odehnal et al: The shape is essentially
a spiral surface (see the next chapter), which can be “idealized” by a great number of measured
points. If one compares the perfect shape to the real object, one notices that the snail changes its
building strategy in the last phase.

This makes it very easy for the computer to detect general cylinders: Their
spherical normal image is a great-circle of the sphere whose axis direction is
the generator direction of the cylinder.
The spherical normal image of a cylinder of revolution is a small-circle of the
test sphere. ◂◂◂
6.5 Pipe and channel surfaces 287

6.5 Pipe and channel surfaces

How to invent a new class of surfaces . . .

A remarkable surface appears when a torus is inverted in a sphere (Fig. 6.105).

Fig. 6.105 Inversion of a torus in a sphere

The inversion in space preserves circles and angles like the planar version.
The emerging surface, called the Dupin cyclide, was examined by the French
geometer Charles Dupin and inherits a multitude of beautiful properties from
the torus. Algebraically speaking, it is also – quite remarkably – of degree
four.

Fig. 6.106 Dupin cyclide (Ring cyclide)

Dupin cyclides such as the torus possess two planes that touch the surface
along two entire circles and carry two families of circles which correspond to
the circles of latitude and the meridian circles. Furthermore, in the case of
tori, two Villarceau circles lie in a double tangent plane. During inversion,
this plane transforms into a sphere. For their part, the Villarceau circles pass
over into circles on the cyclide which have to lie on this sphere and are,
therefore, no longer coplanar (E. Hartmann).
Of course, such surfaces are no longer surfaces of revolution. Nevertheless,
they share some properties with them. This is due to the fact that they can
288 Prototypes

Fig. 6.107 Closed rhumb lines on a Dupin cyclide: the images were obtained by inversion of the
closed rhumb lines on a torus (Fig. 6.85).

be generated in two ways as envelopes of spheres with constantly varying

radii. With surfaces of revolution, the spheres are centered on the axis of
rotation (Fig. 6.108), and with Dupin cyclides, they are centered on the spine
ellipse and spine hyperbola, or on both spine parabolas. Surfaces enveloped
by spheres with varying radii are called channel surfaces.
Tori, in particular, can also be generated by spheres with a fixed radius. Such
surfaces are called pipe surfaces.
Of course, every pipe surface is automatically a channel surface. Thus, the
torus is a channel surface in two different ways: firstly, as a surface of revo-
lution (spheres centered on the axis), and secondly, as a pipe surface (spheres
centered on the spine circle).
We can immediately deduce why the spine curves of a Dupin cyclide are conic
sections: An ellipse or hyperbola is the locus of the centers of all circles that
touch two given circles (see page 188). It is thus possible to define each Dupin
cyclide as the envelope of spheres in two ways: The centers of the spheres lie
in a plane; in it, the spheres touch two circles.
If we allow the given circles to mutually touch or intersect in this situation,
three types can be differentiated: the ring-shaped (Fig. 6.106), the “single-
horn”, and the “double-horn” type (Fig. 6.110).
6.5 Pipe and channel surfaces 289

Fig. 6.108 Each surface of revolution is, in a trivial way, also a channel surface (see also Fig. 6.67).

Fig. 6.109 Dupin cyclides as envelopes of a family of spheres whose equator circles touch two
given circles in a plane.

Pipe surfaces and blending surfaces

Pipe surfaces are created by the motion of a sphere along a spine curve.
Fig. 6.112 left shows in what way such surfaces depend on the form of the
spine curve and the radius of the moved sphere. Two closely adjacent po-
sitions of the sphere share a circle – it lies in the radical plane of the two
corresponding sphere centers. If we let both sphere positions move together,
the straight line connecting the centers becomes the tangent of the spine
curve, and the circle becomes a great circle of the sphere.

Fig. 6.110 Dupin cyclides (“single-horn” and two “double-horns”)

290 Prototypes

Fig. 6.111 Michael Schrott has strung together parts of such double-horns into a chain (Fig. 6.111)
which is located inside a torus.

Fig. 6.112 The pipe surface as the envelope of a moved sphere (left). Pipe surfaces and art:
Soundscapes by Bernhard Leitner

Fig. 6.113 A helical pipe surface

6.5 Pipe and channel surfaces 291

If a circle is moved along a curve in such a way that its axis is the tangent of
the curve, it sweeps the same pipe surface as the family of concentric spheres
centered at the spine curve.

Pipe surfaces are often found in nature as well as in technology. They can be
rendered flexible by keeping the radius of the generating circle small.
⊕ Remark : Fig. 6.112 (to the right) shows an organically conducted tube through
which sound is directed. Openings that serve as sound sources were made in 40
places. Depending on the programming, curvatures are emphasized, counter-punctually
combined, or transformed into fugue-like sequences. ⊕

▸▸▸ Blending surfaces

Pipe surfaces are well-suited for filleting (blending) planar or cylindrical side
surfaces against a plane: A sphere of given diameter should be rolled along
the plane so that it touches the work piece. The remaining part of the sphere’s
envelope is the sought-after blending surface.

Fig. 6.114 Blending surfaces in theory and practice

Fig. 6.114 shows the case that is by far the most common: A blend of two
planes forms part of a cylinder of revolution, whereas a blend of a cylinder
of revolution and a plane consists of parts of ring tori. In the right image, a
number of tori are glued together. In the case of the anemone in Fig. 6.92,
the shaft shaped like a cylinder of revolution transforms into a quarter-torus
– just as it would during a classical blending.

Fig. 6.115 Blending surfaces in architecture: BMW world, Munich (Coop Himmel(b)lau)
292 Prototypes

In mechanical engineering, such blendings are important for increased stabil-

ity but are sometimes also employed in order to avoid sharp edges (Fig. 4.78)
or simply for design purposes. Even “in the grand style”, blending surfaces
(Fig. 6.96, Fig. 6.115) are required to make the dreams of architects become
reality.

Fig. 6.116 Blending surfaces in architecture: Niterói Contemporary Art Museum, Brasilia (Oscar
Niemeyer). The blending surface consists of two different torus sections connected by a cone of
revolution.
◂◂◂

▸▸▸ The altitudes of a tetrahedron lead to an aesthetic lamp

Fig. 6.117 The axes of the four cylinders of revolution are altitudes of a regular tetrahedron. The
blendings are comparatively complicated.

The lamp in Fig. 6.117 reveals a lot of geometric insight on the part of its
designer: Four cylinders of revolution with equal diameters and intersecting
axes are arranged in a well-balanced way. This requires the cylinders to have
the altitudes of a regular tetrahedron (Fig. 6.117, middle). Therefore, the
intersections without blending consist of six partial ellipses in their respective
symmetry planes.
When we try to blend the cylinders, we have to guide a sphere along a curve
in the symmetry plane in such a way that it touches those ellipses. Thus, the
path of the sphere’s center consists of parallel curves of ellipses. ◂◂◂
7 Further remarkable classes
of surfaces

In the ideal case, surfaces can be defined by motion. Surfaces of revolution

are produced by a rotation of a curve about an axis. In the case of helical
surfaces, the generators additionally experience a translation along the axis
of rotation. Surfaces of translation are generated by the translation of a
curve along another curve. Ruled surfaces are generated by straight lines
moving along arbitrary curves. Different surfaces (such as spiral surfaces)
can be generated by moving space curves along other such curves and by
changing the former according to certain rules. All noted classes of surfaces
play significant roles in nature and technology. Helical surfaces, for instance,
may be employed in order to “transform” rotation into translation, which
makes them useful as propeller surfaces or drills. Several types of spirals can
be found in nature, for example, in snail shells, animal horns, or even in the
shapes of galaxies in outer space.
Translation surfaces are particularly noteworthy. They can also be inter-
preted as midsurfaces of space curves. A special difficulty arises when a
surface cannot be put into any of these categories. Minimal surfaces, for
instance, are defined by a very characteristic curvature property and play a
great role in physics, architecture, and nature.

© Springer Nature Switzerland AG 2020 293

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_7
294 Further remarkable classes of surfaces

7.1 Ruled surfaces

Next to surfaces of revolution, ruled surfaces are particularly significant.

Ruled surfaces are generated by a moving straight line, and despite carrying
only straight lines, they are not developable in the general case.

Fig. 7.1 Motion of a straight line

Fig. 7.1 illustrates this situation: A straight line is moved through space
and generates a surface which is, generally speaking, doubly curved (this is
immediately evident from the curved contours).
Let us begin with the developable exceptions: cylinders are formed by the
motion of a straight line which is parallel to a fixed direction during the entire
motion, and they are ruled surfaces by definition. They are developable, and
therefore, their contours consist only of straight lines.
Cones are developable ruled surfaces, too: They are generated by moving a
straight line through a fixed apex in an arbitrary way.

Fig. 7.2 Tangent surface of a space curve. A different section of the same surface (right); the edge
of regression proceeds on an ellipsoid, and its top view is a “nephroid (left).”

Finally, developables – the most general single curved surfaces – are also
ruled surfaces because they are swept by the tangents of a space curve. The
7.1 Ruled surfaces 295

associated motion of the straight line can, without proof, be interpreted as

follows: A developable may be created by following a point K along a direc-
tor k and considering the straight line as the tangent of k. In such cases, the
director is called the curve of regression or edge of regression, because it can
be shown that along its trajectory, the generated tangent surface possesses
a sharp ridge (Fig. 7.2).
Thus, we can say:

All developable surfaces (cylinders, cones, and developables) are ruled surfaces.

Now, let us move on to general ruled surfaces, whose shapes permit many
(though not quite arbitrary) variations. In each case, an arbitrary tangential
section through a surface point P will retain the direction of the generator
which passes through it. Therefore, we can say:

All non-developable ruled surfaces are hyperbolically curved and, thus, con-
stantly “saddle-shaped”.

Conoids

Fig. 7.3 Conoidal ruled surfaces as the shapes of roofs (circle conoid)

So-called conoids represent an interesting category of non-developable ruled

surfaces. The motion of their generating straight line is defined in such a
way that the straight line always intersects a straight director and another
arbitrary given curve. As an additional condition, the surface must possess
a directing plane which is parallel to all generators. Fig. 7.3 shows a circle
conoid where the director is a half-circle.
Fig. 7.4 shows a conoidal roof shape, which is basically defined in such a way
that a horizontal director is connected to a wave-shaped curve by straight
lines which appear parallel in a top view (compare also Fig. 7.7). The question
is whether or not the resulting ruled surface is developable, which would
imply advantages for its practical construction.
296 Further remarkable classes of surfaces

Fig. 7.4 Arbitrary conoidal ruled surfaces as roof shapes: Pictured on the right is a model by
Antoni Gaudí for practical construction.

This question cannot be answered merely by evaluating the visual image,

since the recognizable contours are too short to discern any curvature.
On the other hand, it is theoretically possible to negate the question outright,
considering the following:

A ruled surface that contains a straight transversal of all generators is not

developable. A conoid, in particular, is not developable.

If P is a point on the director, then the tangential section in P is composed

of the generating straight line and the director. Thus, the point is hyperbolic.

Fig. 7.5 Conoidal ruled surfaces used in design (Santiago Calatrava).

Fig. 7.5 shows a similar problem. It again concerns two conoids with a hor-
izontal director which share a space curve (from a certain perspective, it
appears as a parabola, as in Fig. 7.5 on the left). Due to a twisting of the
surface, a slight curvature in the contours can be recognized, from which it
can be deduced that the surfaces are “almost” but not “wholly” developable.
In practice, this means that a slightly ductile surface – such as the fabric of
an elastic tent – should be able to cope with the curvature.
Let us examine this example further: Both surfaces in Fig. 7.5 can be “an-
imated” as in Fig. 7.6. Proceeding from a parabola, which determines the
length of the rods in the initial surface, the lower “crossbar is raised to a
7.1 Ruled surfaces 297

Fig. 7.6 . . . or as a fancy sliding gate

Fig. 7.7 Model of a conoidal surface

higher level”. Appropriate hinges, which prevent the rods from tilting side-
ways, have to be placed at the buckling spots.
Two important and two beautiful applications
There exist two conoids which are known under a special name in academic
literature.

Fig. 7.8 The hyperbolic paraboloid is a conoid, a translation surface, and a surface of degree two.

The hyperbolic paraboloid (Fig. 7.8) is a particularly important conoid, and

its second director is also a straight line. From this it follows that, as we
have already seen, the surface possesses an infinite number of directors. Any
two straight lines from one family together with any two straight lines from
the other family form a skew generator quadrangle. From a special perspec-
298 Further remarkable classes of surfaces

tive, both pencils of generators appear as pencils of parallels; otherwise, the

contour is always a parabola.

Fig. 7.9 Each ruled surface is touched by an hyperbolic paraboloid along a whole straight line.

Hyperbolic paraboloids are not just extremely interesting surfaces. They are
also very practical, because locally, they can be considered as the prototype
for any arbitrary ruled surface. Fig. 7.9 illustrates how a ruled surface is
touched by an HP shell along a whole straight line.

Fig. 7.10 Degree three conoid with a parabola as a director

A very visually appealing surface can be created by defining the director as

a parabola.
⊕ Remark : Such a surface is of degree three, and its prototype can be given by the
simple equation z = x2 y. The prototype of an hyperbolic paraboloid can be given
by the almost identical equation z = xy. Furthermore, arbitrary affine mappings can
be applied to these prototypes. Due to the fact that affine mappings preserve par-
allelism, the surfaces stay conoidal, and their degree likewise remains unaffected. ⊕

Besides the HP shell, the right helicoid is also of great significance. It is

created by rotating a generator about an orthogonally intersecting axis and
translating it in proportion to the angle of rotation along this axis. The axis
is the director, and all generators are parallel to the axis’ normal plane. Each
path curve of a point on the generator could serve as the second director.
7.1 Ruled surfaces 299

Fig. 7.11 Tangential intersection of degree-two and degree-three surfaces

Fig. 7.12 The right helicoid is a conoid and a helical surface. Left image: normal projection,
winding to the left. Right image: central projection, winding to the right.

These path curves are helices. The right helicoid can also be categorized as
a helical surface, which we shall examine more closely in a separate chapter.
⊕ Remark : The right helicoid does not possess a “degree” in the algebraic sense –
rather, it is called transcendental. It is, therefore, impossible to correlate the coor-
dinates x, y, and z of its points in an algebraic relationship. However, it is possible
to formulate a parameter representation or implicit representation as follows:
x z
x = v ⋅ cos u, y = v ⋅ sin u, z = c ⋅ u (u ∈ R, v ∈ R) ⇒ = tan ,
y c
where u is the angle of rotation about the z-axis and v denotes the distance from
this axis. Depending on the choice of parameter intervals, we get different parts of
the surface. ⊕
Our fourth and last conoidal example is Plücker’s conoid. It can be generated
in a similar way as the right helicoid, the only difference being that the
300 Further remarkable classes of surfaces

Fig. 7.13 Plücker’s conoid

translatory motion of the generator in the direction of the director proceeds

not proportionally to the angle of rotation but to its sine. This produces the
straight line’s harmonic oscillation.
The path curves of generator points are oscillation curves. Despite its sim-
ilarity to the right helicoid, Plücker’s conoid is remarkably algebraic – of
degree three, to be precise. This means that each test straight line yields
three exactly calculable intersection points, two of which may be complex
conjugate.
⊕ Remark : Due to its generation, Plücker ’s conoid possesses the parameter repre-
sentation
x = v ⋅ cos u, y = v ⋅ sin u, z = c ⋅ sin 2u (u ∈ R, v ∈ R).
It is now possible to eliminate the parameters due to sin 2u = 2 sin u ⋅ cos u and
sin2 u + cos2 u = 1, which results in the following implicit equation of the surface:

2c ⋅ x ⋅ y = (x2 + y 2 ) ⋅ z.

From this, it can be seen that Plücker’s conoid is of degree three. ⊕

Fig. 7.14 Ruled surfaces during the construction of bridges

Architecture often employs “wireframe models” of ruled surfaces. Fig. 7.14

shows a pedestrian bridge by Calatrava in Bilbao.

▸▸▸ A spatial version of the theorem of the angle of circumference

According to the theorem of the angle of circumference (p. 9), a straight line
segment EF in 2-space is seen at a constant angle α from any point of a
7.1 Ruled surfaces 301

pair of circular arcs passing through E and F . For α = 90○ , the pair of circles
becomes one circle (the Thales circle) with diameter EF .

Fig. 7.15 Two simple generalizations of the theorem of the angle of circumference.

By looking for all points in 3-space that see a straight line segment EF
under a constant angle α, the locus is a torus, obtained by rotation of the
situation in 2-space about the axis EF (Fig. 7.15, left). Another possible
generalization could look for all points P with ∢(P e, P f ) = α, where e and
f are given lines in space. When e and f are parallel, the locus is a pair of
cylinders of revolution (Fig. 7.15, right).

Fig. 7.16 Non-trivial spatial version of the theorem of the angle of circumference. Left α ≠ 90○ ,
right α = 90○ .

If e and f are skew, then things become more complicated: The result is
a ruled surface of degree four in the general case α ≠ 90○ . For α = 90○ , the
surface degenerates to an elliptic hyperboloid. The proof can be found in [17].
◂◂◂
302 Further remarkable classes of surfaces

7.2 Helical surfaces

The fascination behind the helix

In planar geometry, there exists exactly one curve that is invariant under
an infinite number of reflections and rotations – the circle. This undoubt-
edly accounts in no small part for the circle’s beauty as well as its practical
applicability.
In space, there exists a class of curves that may allow self-motions – the
helices.

A helical motion is defined as the combination of a rotation about a fixed axis

and a simultaneous proportional translation along this axis. The pitch belongs
to a full revolution as the distance of translation.

Fig. 7.17 Helices with an “accompanying tripod”

Helices are the path curves of points undergoing a helical motion. During a
helical motion, they are continuously translated and screwed into each other.
Additionally, they are invariant under an infinite number of point reflections
as long as the center lies on the axis, and translations in the direction of the
axis if the distance of translation is an integer multiple of the pitch.
The factor of proportion is called the screw parameter and is henceforth
denoted by p. If a point is rotated by an angle ϕ about the screw axis, then
it is simultaneously translated by an offset p ⋅ ϕ along this axis. If h is the
h
pitch, then p = .
2π
If the screw parameter is positive, one speaks of a right helical motion. If it
is negative, we speak of a left helical motion. The case p = 0 is special and
yields a pure rotation, whereas p = ∞ yields a pure translation.
⊕ Remark : The rotation and the associated translation need not necessarily be uni-
form, but what matters in any case is the proportionality. As a plant coils itself
around a pole (Fig. 7.18), its speed of growth may vary, but the result is, neverthe-
7.2 Helical surfaces 303

less, the same. A mathematician would say that the helix need not be parametrized
by time. ⊕

Fig. 7.18 Normal projections (orthogonal to the axis) of helices

The following properties can hardly be improved in terms of elegance:

Helices always lie on cylinders of revolution about the screw axis a. They have
constant slope with respect to a base plane π ⊥ a. In an orthogonal projection
on π, a helix appears as a circle. In an orthogonal projection on a plane ν ⊥ π
(ν ∥ a), a helix appears as a sine curve. The development of the carrier cylinder
shows a straight line as the development of the helix.

Proof : Neither a translation in the direction of the axis nor a rotation about the
axis change the distance from the axis. Thus, helices lie on cylinders of revolution,
which appear as circles in a normal projection in the direction of the axis.
By definition, a helix intersects each generator of the carrier cylinder at the same
angle. The ratio between the translation component and the rotation component
is likewise constant – even if only an infinitesimal segment (the tangent t) of the
curve is considered. The tangent is contained in the tangent plane τ of the cylinder,
where the angles to the generator and the base plane π can be measured. This angle
α = ∢tπ has the value tan α = p/r, where r denotes the distance of P to the screw
axis.
If t is projected orthogonally to a plane ν ⊥ π, then the translation component
remains undistorted while the rotation component is shortened depending on the
angle ψ = ∢τ ν by a factor of cos ψ. The projection curve is, thus, a cosine – and,
therefore, also a sine. Both curves do not differ in their shape but are merely phase-
shifted. Since the helix intersects all generators of the carrier cylinder at the same
angle, the development also intersects the parallel straight developments of the
cylinder’s generators at a constant angle and is, therefore, also a straight line.
⊕ Remark : Let us now use analytic methods in order to capture the path of a point
P at a distance of r to the axis as a way of proving the above theorem.
For this purpose, we shall adapt a Cartesian coordinate system to the situation
as simply as possible: Let the axis be the z-axis and P (r, 0, 0) lie on the x-axis.
After the rotation through ϕ and the simultaneous translation about p ⋅ ϕ, P has
the coordinates
x = r ⋅ cos ϕ, y = r ⋅ sin ϕ, z = p ⋅ ϕ.
The front view of the path curve can be represented by y = r⋅sin ϕ, z = p⋅ϕ, and it is,
thus, a sine curve with the equation y = r ⋅ sin(z/p). Every other horizontal normal
304 Further remarkable classes of surfaces

projection with viewing angle α also yields a congruent (and merely phase-shifted)
curve in a (u, z) coordinate system:

u = x ⋅ cos α + y ⋅ sin α = r ⋅ cos ϕ ⋅ cos α + r ⋅ sin ϕ ⋅ sin α = r ⋅ cos(ϕ − α), z = p ⋅ ϕ.

The development of the curve is given in a (v, z)-system by the linear equation

v = r ⋅ ϕ, z = p ⋅ ϕ ⇒ z = p/r ⋅ v,

which implies that the curve is a straight line. ⊕ ⊙

Fig. 7.19 Helix with a shadow

Arbitrary orthogonal views of helices (Fig. 7.19) stand – as we shall soon dis-
cover – in an affine relation to so-called cycloids. Central views as in Fig. 7.19
on the right have an appropriately complicated shape and pose a challenge
to freehand drawing.

Fig. 7.20 “Glowing” helix (LED lamp) with reflections in glass

7.2 Helical surfaces 305

When straight lines or circles undergo a helical motion . . .

All path curves during a helical motion are helices, except the paths of points
on the axis. What kind of surfaces are produced if, instead of points, we let
other objects – such as straight lines – undergo the same motion?

Fig. 7.21 On the left: The right helicoid alone may help to overcome the tedium of this car park.
Center: Right helicoid in the Sagrada Familia (Antoni Gaudí). On the right: Intercity Bank in
Lima (Hans Hollein).

When discussing ruled surfaces, we have already touched upon an equally

important helical surface – the right helicoid (Fig. 7.12). It is produced when
a straight line which intersects the screw axis at a right angle undergoes a
helical motion.
Right helicoids are primarily found in technology. For instance, the smoothly
cleaned wall on the underside as well as the path surface (upper) on the
upwards-leading driving path the car park in Fig. 7.21 both belong to this
class. Right helicoids are similarly employed for the surfaces of propellers or
cooling fans.

Fig. 7.22 Ship’s propeller, Archimedean screw

306 Further remarkable classes of surfaces

With the help of helicoids, it is possible to induce propulsion by means of

mere rotation – the screw motion is, thus, decomposed into its components
of rotation and translation. Due to the inertia of the medium through which
the motion should proceed (air or water), the surface “supports” itself on the
medium and propels the airplane or ship forwards (or backwards, as in the
case of ships’ propellers – see Fig. 7.22). Right helicoids can be employed to
lift liquids – this property was already recognized and utilized by Archimedes
(see Fig. 7.22 center and on the right).

Fig. 7.23 Impeller of a snow blower

Another interesting application of right helicoids can be found in so-called

impellers (Fig. 7.23), which are, for instance, built into excavator shovels (and
turned electrically) and snow shovels. By a mere rotation of the apparatus,
the mixture of sand and cement in one direction by the left part of the
left-handed screw and in the opposite direction by the right-handed screw.
Variations exist with a continuous outer screw and a counter-rotating inner
screw.

Fig. 7.24 Theoretical version of an impeller

Fig. 7.24 shows a “purely geometric” variation of a double screw, where the
transition from one helix to the other proceeds “smoothly” (or differentiably).
7.2 Helical surfaces 307

▸▸▸ The DNA double helix

The double helix is a popular symbol of molecular biology. Hereditary mate-
rial is preserved in the form of deoxyribonucleic acid (DNA), which is made
up of two strands (or a double helix) of molecules in the chromosomes of the
cell nucleus. These DNA stands wrap themselves about an imagined axis –
similarly to opposite helices on the right helicoid. More on this matter can
be found in application p. 314. ◂◂◂

Fig. 7.25 Vises

The conversion of rotation into translation does not necessarily have to pro-
duce propulsion. All olive presses, vises, and corkscrews (Fig. 7.27, Fig. 7.31)
take advantage of the same principle. On a miniature scale, the conversion
of rotation into translation is done with all ordinary screws!

Fig. 7.26 Bolts and nuts

Fig. 7.27 Classic corkscrew

The following question arises: What kind of helical surfaces are employed
in screw threads? They do not seem to be right helicoids – as, for instance,
308 Further remarkable classes of surfaces

those in vises. The surfaces are created by straight lines skew to the screw
axis. Let us consider the matter from a geometric standpoint.
In machine drawing, screws and bolts are displayed in a simplified manner
(Fig. 7.28).

Fig. 7.28 Blind holes and clearance holes

Different types through different directions of the straight line

Fig. 7.29 Left-handed and right-handed straight ruled helical surfaces

If the generating straight line of the right helicoid is translated so that it no

longer intersects the screw axis, so-called straight open ruled helical surfaces
are obtained. Fig. 7.29 shows two such surfaces, where the left surface pos-
sesses a negative screw parameter and the surface at the center possesses a
positive one. A helix with none-zero axis distance can be seen in the corre-
sponding top view.
The general case is given if the generating straight line is skew with respect to
the screw axis. Both rightmost skew open ruled helical surfaces in Fig. 7.30
serve as illustrations. If the skew generator meets the screw axis, then we
speak of an skew closed ruled helical surface.
Skew ruled surfaces possess a self-intersection, which is naturally also a helix.
7.2 Helical surfaces 309

Fig. 7.30 Different skew ruled helical surfaces

Fig. 7.31 Another corkscrew (“in action”). One part of the surface is a ruled helical surface.

There exists only one type of developable helical surface

None of the aforementioned ruled helical surfaces are developable in the gen-
eral case, and this is recognizable from their curved contours. The surface in
Fig. 7.30 on the right, however, seems to possess straight contour parts. It
is, in fact, possible for an skew open ruled helical surface to be developable
under a single condition: The generators have to be tangents of a helix g.
The whole surface is then a tangent surface (developable) of g, and g is a
310 Further remarkable classes of surfaces

Fig. 7.32 Helical developable with a helix for its edge of regression

sharp ridge (curve of regression) on this developable. Fig. 7.32 shows another
helical developable – in the right image, another part of the surface is visible
(every ruled surface is, in principle, unbounded).

Fig. 7.33 A developable special case of a ruled helical surface (cylinder of revolution)

Fig. 7.33 illustrates one further exception for ruled helical surfaces: If the
screwed straight line is parallel to the screw axis, a cylinder of revolution is
created. Following this interpretation, a cylinder of revolution can be seen as
a surface of revolution, a helical surface, and a translation surface.

Fig. 7.34 Cone wheels with efficient gear flanks

Helical developables are often used in engineering practice. They occur in

mechanical engineering as gear flanks for so-called involute gears or mills
(Fig. 7.35), which serve to transfer rotations onto skew axes. During road con-
7.2 Helical surfaces 311

Fig. 7.35 Mill flanks Fig. 7.36 Highway entrance in theory . . .

struction, helical developables form the embankment for highway entrances

(Fig. 7.36 and 7.37).

Fig. 7.37 . . . and practice (Euler spirals occur in top view)

Fig. 7.38 Masonry drillmasonry drill with helical developables as flanks

It is possible to apply screw motions to circles or spheres

By screwing curves other than straight lines, complicated surfaces may be
produced. The most common case that occurs in nature is the application of
screw motion to a circle. Even by restricting oneself to these circular helical
surfaces, it is possible to distinguish between several types that differ in the
initial position of the circle.
⊕ Remark : Fig. 7.39, for instance, shows a nifty keychain, which approximates a
circular helical surface. The twisted circle is the “profile” of the surface in a cross-
section perpendicular to the screw axis. ⊕
By far the most frequent type of circular helical surface is the helical pipe
surface.
312 Further remarkable classes of surfaces

Fig. 7.39 Special circular helical surface: The screwed circle is situated in a cross-section plane
perpendicular to the screw axis, which can be deduced from its shadow. The colored circle sections
form areas on the surface that are bounded by a helix.

A helical pipe surface is formed if a circle undergoes a helical motion such that
its axis is tangent to the helix that is the path of the circle’s center. It can
also be created as the envelope of a sphere, which counts the circle among the
many possible great circles.

Fig. 7.40 Helical pipe surface in different normal projections

⊕ Remark : This assertion is easy to verify: The circle undergoes a motion which, by
definition, is orthogonal to its plane. Thus, the surface is touched by a cylinder of
revolution along the circle, and a sphere fits into this cylinder, which is in contact
along the original circle. What makes this surface significant is the generation of
the helical surface by means of a sphere. ⊕
Therefore, it is possible for a sphere to roll through the surface while touch-
ing the surface along a great circle in each moment. Many water slides in
swimming pools are comprised of parts of helical pipe surfaces (Fig. 7.41 on
the left). There is also another variation, where the circle in a meridian plane
is twisted (visible in Fig. 7.41 at the center). The thus generated surface is
called the meridian circular helical surface.
⊕ Remark :
7.2 Helical surfaces 313

Fig. 7.41 Helical pipe surface and meridian circular helical surface as slides

Fig. 7.42 Helical pipe surface, this time generated by the cross section

Fig. 7.40 shows different normal views of the helical pipe surface (on the left, the
projection is perpendicular to the axis; on the right, we have a projection in the
direction of the axis). It is clearly visible that the generating circle is not vertical.
Instead, it has the respective tangent of the mid helix as its axis. The contours often
exhibit cusps, depending on the radius of the twisted circle.
Fig. 7.42 shows the same helical pipe surface but generated by a kidney-shaped
cross-section instead of a circle. Points of this cross-section can be created by pro-
ceeding from the originally twisted circle and applying the helical motion to its
points P until they “reach” a cross-section plane π that is normal to the axis.
The height difference ∆z = P π corresponds to a proportional angle of rotation
ϕ = c ⋅ ∆z. ⊕

Fig. 7.43 Steel cable (two helical pipe surfaces) under a small workload
314 Further remarkable classes of surfaces

Helical pipe surfaces are among the most frequently found surfaces in na-
ture and technology. The “helices” in Fig. 7.18 and Fig. 7.19 should, strictly
speaking, also be counted among them.
▸▸▸ The secret of DNA

Fig. 7.44 Deoxyribonucleic acid encodes hereditary information and passes it on to the following
generation via germ cells. The strands may be visualized from a geometric standpoint by two
interlocked helical pipe surfaces.

It would be presumptuous even to attempt to describe the manifold secrets

of DNA at this point, but as for its geometry, this much can be said: As
described in application p. 306, DNA molecules take the form of the famous
double helix. Each strand represents a sequence of molecular building blocks
(phosphoric acid, deoxyribose, and one of the four bases – adenine, guanine,
cytosine, and thymine). The order of these bases determines the genetic code.
◂◂◂

Fig. 7.45 Left: a strange kind of “double helix”; right: helical stairs, looking downwards
7.3 Different types of spiral surfaces 315

7.3 Different types of spiral surfaces

Fig. 7.46 Helix, helispiral, and the “classical” spiral all (right-handed)

Helispiral motion

Fig. 7.47 A typical helispiral surface

An interesting variant of helical motion is often found in nature – the helispi-

ral motion. While the regular helical motion leaves the diameters unaffected,
helispiral motion makes them smaller or larger in linear proportion to the
height.
Fig. 7.46 juxtaposes the helical motion (left) with the helispiral motion (cen-
ter). Fig. 7.48 pictures the surface that is created if this kind of transforma-
tion is applied to an equilateral triangle that intersects the axis orthogonally
at its midpoint.
The principle is as follows: Perform a rotation about the axis by an arbitrary
angle ϕ while simultaneously changing the distance to the center by the
316 Further remarkable classes of surfaces

Fig. 7.48 Helispiral motion of a horizontal equilateral triangle (left-handed)

Fig. 7.49 Helispiral motion: Pieter Bruegel the Elder, The Tower of Babel

factor p ⋅ ϕ. The value p is called the parameter of the helispiral motion.

Thus, the increase in distance is linear, as the double angle produces the
double distance.

A helispiral motion is given if the points in space are rotated about an axis and
simultaneously scaled proportionally in relation to a fixed point on the axis.

This is analogous to the following “growth recipe” of nature: “Grow larger

from a center at a linear speed and simultaneously rotate about an axis
through the center.”
This principle seems to be applied in the growth of animal horns, plant
sprouts, and even certain trees.
Geometrically, this instruction indicates that, at every turn, all points change
their distance to the axis by the same amount.
An exceptional point exists on the axis – the origin of growth – from which all
path curves (called helispirals) make their way. Their top views are Archimedean
spirals. These planar curves are created by letting a point move at equal ve-
locity on a constantly rotating rod.
From a spatial perspective, a helispiral may also be created by letting a point
move at equal velocity on a straight line, which, in turn, rotates about an
non-perpendicularily intersecting axis. Another variation may be described
as follows: If a point undergoing a helical motion is moved at equal velocity
7.3 Different types of spiral surfaces 317

Fig. 7.50 Variation of circular helispiral surfaces in theory . . .

Fig. 7.51 . . . and practice

on a normal to the screw axis, its path curve is a helispiral. During this type
of motion, the point moves on a right helicoid.

The classical spiral motion

In addition to helispiral motion, there exist numerous other types of spatial
spiral motions which share the same principle that always involves rotating
about an axis and a change of distance that is somehow depending on the
angle of rotation. The ensuing planar spiral motion proceeds by complete
analogy.

Fig. 7.52 Helispiral-shaped recess – design by Anita Aigner and Franz Lesák
318 Further remarkable classes of surfaces

Fig. 7.53 The tangent surface of a helispiral (on the left and right) and a fascinating parallel in
nature (feather dust worm, Dalmatia).

Fig. 7.55 In this corkscrew, the helical pipe surface trans-

Fig. 7.54 This spiral is “merely” forms into a helispiral surface, which is no longer a pipe
a central projection of a helix. surface but a channel surface (S. 287).

The classical (so-called cylindro-conical ) spiral motion works as follows: Per-

form a rotation about an axis by a very small angle ϕ and simultaneously
increase the distance from the center by a value that is proportional to the
size and rotation angle, which also equals a multiplication by (1 + p ⋅ ϕ). The
value p is called the parameter of the spiral motion. This process could be
compared to a “compound computation of interest”, as the “initial capital”
increases with each rotation. Thus, the growth is exponential.
If we allow for arbitrary rotation angles, we arrive at the following definition:

Fig. 7.56 Spiral right helicoid (on the left) and spiral developables
7.3 Different types of spiral surfaces 319

Fig. 7.57 Computer simulations of “real” shellfish and snail shells

A classical spiral motion is given if the points in space are rotated about an
axis and simultaneously scaled from a fixed center on the axis in an exponential
manner.

This definition correlates with the following “growth recipe” in nature: “Ex-
pand by a certain percentage in relation to a center and simultaneously rotate
in proportion to the percentage about an axis through this center.” This kind
of growth can be observed in the calcite shells of snails and shellfish, which
bear a remarkable resemblance to their computer simulations (Fig. 7.57).
However, studies by Boris Odehnal have shown that specimens of these shells
are often subject to irregularities (Fig. 6.104).
The contour of a spiral surface

Q
Fig. 7.58 Left: The interior of a snail’s shell can be quite complicated. Right: the corresponding
contour with cusps.

Fig. 7.58 shows the inner part of a shell. On the right, one can see that the
contour starting at P is first “responsible” for the outer part and then changes
to the “inner contour”. Fig. 7.59 shows a shell under three only slightly dif-
ferent viewing angles.
320 Further remarkable classes of surfaces

Fig. 7.59 A computer-generated shell under slightly different projection angles

Fig. 7.60 A living snail’s shell under slightly different projection angles, . . .

Fig. 7.61 . . . and its petrified ancestors.

The relation to the helix

Let us once again juxtapose the generation of the three aforementioned path
curves (helix, helispiral, classical spiral), assuming the most equal initial con-
ditions.
7.3 Different types of spiral surfaces 321

Let a fixed axis a (e.g., the z-axis), a fixed angle of rotation ∆ϕ, and a fixed
distance of translation ∆z (in axis direction) be given.
Let P be an arbitrary point to which we shall repeatedly apply the following
instructions: First, P is to be rotated through ∆ϕ about a and then translated
along a by ∆z.

Fig. 7.62 Screw motion, helispiral motion, and classical spiral motion in a top view

This already yields the points of a helix. For a helispiral motion, P is addition-
ally translated orthogonally to the axis direction by a distance proportional
to the angle of rotation s = p ⋅ ∆ϕ, while for classical spiral motion, P is
changed in relation to the origin by a factor depending linearly on the angle
of rotation k = (1 + p ⋅ ∆ϕ). This, in turn, yields points on the helispiral and
on the classical spiral.
If the objective is to calculate n times as many points, then the n-th part
of ∆z, ∆ϕ, and the translation distance s are to be considered. The scaling
factor of the classical spiral is to be replaced by its n-th root.
Helices and classical spirals are also slope lines (lines with a constant angle
to the axis or to the base plane). They lie on cylinders with axis-parallel
generators: In the case of helices, it is a cylinder of revolution. In the case of
classical spirals, it is a logarithmic cylinder. Helispirals, however, are no slop
lines.
322 Further remarkable classes of surfaces

7.4 Translation surfaces

Translation surfaces as the loci of chord midpoints

Remark: The proofs for the theorems shown in this section can be elegantly
realized by means of vector calculus, and they can be found at www1.uni-ak.
ac.at/geom/dld/canada-lines.pdf.

If we connect all points P1 of an arbitrary curve c1 with all points P2 of another

arbitrary line c2 , the locus of all midpoints M is a surface of translation. The
surface contains two families of congruent curves. The prototypes c∗1 and c∗2 of
these profile curves are similar to c1 and c2 (factor 1/2).

c2 c∗1
c∗2 c1
P2
M
P1

Fig. 7.63 Given two lines c1 and c2 : We consider the locus of all midpoints of P1 ∈ c1 and P2 ∈ c2 .

Paraboloids as translation surfaces

Apart from the trivial case where both profile curves are straight lines and
the surface is thus a plane, Fig. 7.63 shows the famous, simplest, and non-
trivial case: If c1 and c2 are parabolas with parallel axis direction and non-
parallel carrier planes, the corresponding translation surface is a paraboloid
(degree two). When one of the generating lines is straight, the result is a
parabolic cylinder. Trivially, if both generating lines are straight, the trans-
lation surface is a plane.
The above theorem holds for “genuine space curves” if c1 = c2 , which means
that one single curve can already define a translation surface. Henceforth, we
shall refer to such a surface as midsurface.
The helicoid as a translation surface
Besides the classical example of a midsurface (the paraboloid), there is also
the following non-trivial example:
7.4 Translation surfaces 323

The midsurface of a helix is a helicoid. Therefore, a helicoid is not only a helical

surface but also a surface of translation.

P2
M
c∗1
P1
c∗2 c1

c2

Fig. 7.64 If c1 = c2 is a helix, the midsurface is a helicoid which is, therefore, also a surface of
translation with two non-trivial families of helices on it.

a a

Φ c∗1 c∗2

c∗2
c∗1

Fig. 7.65 Left: Two congruent non-trivial helices with arbitrarily chosen equal radius on a helicoid
are displayed. Right: With an adequate change of the projection in the direction of a tangent of
the green helix, the red helix becomes a contour line.

Since c1 = c2 , the prototypes c∗1 and c∗2 of the pencils of parameter lines are
congruent. All parameter lines intersect the screw axis; they are not helical
orbits of the screw motion that generates the helicoid.
The generating helix c = c1 = c2 lies on the helicoid Φ that is thus uniquely
defined. If we vary the radius r of c, this will generate the same surface. This
324 Further remarkable classes of surfaces

will result, once again, in two families of non-trivial helics on Φ with radius
r
2 (Fig. 7.65).

A helicoid (parameter p) carries a two-dimensional manifold of congruent he-

lices (parameter p2 ). Their projections in the direction of the axis yields circles
through the image of the axis.

There is a well-known theorem about helicoids in relation to “non-trivial”

helices:1

The contour of a helicoid for any parallel projection is a helix.

Other surfaces of revolution that are also translation surfaces

The cylinder of revolution is a trivial example of surfaces of revolution
that are at the same time translation surfaces. The paraboloid of revolu-
tion (Fig. 6.19 on the left) is less trivial – like all paraboloids, it is a trans-
lation surface. Yet, there is more (the proof can, once again, be found at
www1.uni-ak.ac.at/geom/dld/canada-lines.pdf):

P1
M

P2
c∗1
c∗2
c2

c1
Fig. 7.66 If two coaxial helices c1 and c2 are reversely congruent, the midsurface is a surface of
rotation.

When we rotate a sine curve (amplitude r) about its axis, the generated surface
of rotation can be interpreted as a surface of translation with two reversely
congruent helices that lie on cylinders of revolution with diameter r and contain
the screw axis.

1 Thistheorem competes in beauty with the theorem that every contour of a quadric for any
projection is a conic section.
7.4 Translation surfaces 325

▸▸▸ Regarding rhombuses and huge towers

Fig. 7.67 Applications of the remarkable surface of revolution and translation. On the right: the
Gherkin tower in London (Norman Foster ).

The aforementioned surface of revolution, which is at the same time a surface

of translation, has several applications in practice.
Fig. 7.67 on the left shows how polyhedra, entirely made up of rhombuses
with constant side length but different interior angles, converge to a remark-
able surface with a sine curve as its meridian. The corresponding adjacent
sides of the rhombuses converge to helices. See also application p. 113. ◂◂◂
326 Further remarkable classes of surfaces

7.5 Minimal surfaces

Fig. 7.68 Under certain circumstances, surface tension can carry a coin, even if it has a greater
density – the coin just needs to be small enough!

Due to surface tension, each liquid or elastic surface (such as a rubber mem-
brane) strives to minimize its surface area. Mercury, for instance, accumulates
into tiny round droplets if it is poured onto a planar glass plate. The spher-
ical soap bubble similarly owes its shape to surface tension: A sphere simply
covers the smallest possible surface for a given volume (Fig. 7.83, right).

helicoid

Scherk surface

catenoid

Fig. 7.69 Special minimal surfaces emerge when the wire frame forms a cube, a helix, a pair of
parallel circles, or a cube with only 8 edges (on the right).

Due to the surface tension of water, a small coin (Fig. 7.83) stays afloat, even
though its much larger density would otherwise make it sink.
7.5 Minimal surfaces 327

If a soap film is bounded by a wire, it can arrange itself into many different
forms. Fig. 7.69 shows some examples of shapes that the soap film can take
when the wire frame fulfils certain conditions. In some cases, figures such as
helicoids or catenoids can be generated.
Open umbrellas are under a similar kind of pressure, as are tents and roofs.

Fig. 7.70 Surfaces exhibiting tension in theory and in Arts: Pictured on the left is the famous
“chessboard minimal surface”, also known as Scherk surface; on the right, an installation by Maria
Wambacher.

The principal curvatures must be of equal magnitude at each

point
Aside from the sphere, where all normal sections have the same curvature,
the membranes of minimal surfaces can be geometrically distinguished by
the fact that the surface’s principal curvature (see p. 261) are of equal size in
each of their points. In every surface point, the Dupin indicatrix (Fig. 6.51)
is, therefore, a pair of equilateral hyperbolas. The arithmetic mean of the
two principal curvatures is the mean curvature.

A non-spherical minimal surface has a mean curvature of zero everywhere. In

every point, its Dupin indicatrix is a pair of equilateral hyperbolas.

This is certainly the case for two minimal surfaces that we already know: the
catenoid (Fig. 6.95 on the right and Fig. 7.83 on the left) and the helicoid
(Fig. 7.12 and Fig. 7.69 on the right). Fig. 7.70 on the left shows a minimal
surface named after Scherk in top and front view, as well as from a general
perspective.
▸▸▸ Hyperbolic paraboloids are “almost” minimal surfaces.
A computer test will show that the hyperbolic paraboloid comes close to
a Schwarz minimal surface in certain areas. In Fig. 7.71 on the left, these
areas are marked black (a very good approximation) and red (still a good
approximation) – the top view is visible in the middle. The extent of the area
that is colored black by the computer depends on the choice of a tolerance
interval.
328 Further remarkable classes of surfaces

Fig. 7.71 The hyperbolic paraboloid resembles the so-called Schwarz minimal surface, and it does
not significantly deviate from the “minimal condition”. The right-hand side shows a function graph
that has the sum of the two principal curvatures as its function value. In order to be a “genuine
minimal surface,” the graph would need to have a constant z value of zero.

Fig. 7.72 A minimal surface (gray) can be approximated quite well by a hyperbolic paraboloid
(yellow). Right: Gossamers of caterpillars come close to minimal surfaces and therefore, also to
hyperbolic paraboloids.

It is, nevertheless, possible to assume that a tent membrane spanned by

a “skew quadrangle” is visually nearly identical to a HP shell (Fig. 7.71).
Fig. 7.71 on the right shows the accompanying function graph, where the
mean curvature (the mean of both principal curvatures) is indicated by the
height value. In a “minimal surface”, the graph would have a constant z value
of zero. ◂◂◂

▸▸▸ Roofs of minimal surfaces

Minimal surfaces can be characterized in two ways: physically (equilibrium
of surface tension) and geometrically (opposing principal curvatures of equal
size on the entire surface).
7.5 Minimal surfaces 329

Fig. 7.73 Minimal surfaces can be scaled without losing their useful properties.

This is why they keep their useful properties when they are scaled to huge
dimensions (Fig. 7.73 on the right shows the Olympic Stadium in Munich).
When working with other roof surfaces, one has to consider that the rapidly
increasing weight may cause other problems. ◂◂◂

▸▸▸ Sturdy sculptures with lightweight modules (Fig. 7.74)

Fig. 7.74 Minimal surface “populated” with modules morphing from triangular to hexagonal shape

Minimal surfaces can be quite useful for the creation of sturdy and lightweight
sculptures, as the following experiment shows:2 First, a minimal surface

2 www.springchallenge2012.wordpress.com (Bence Pap, Irina Bogdan, Andrei Gheorghe, Trevor

Patt, Clemens Preisinger, Moritz Heimrath).
330 Further remarkable classes of surfaces

through four attractor points was created and subdivided by means of lightweight
hollow modules. The geometry of the surface was then slightly changed ac-
cording to circulation patterns (Fig. 7.74, left) and visual axes.
The organic-looking structures pictured here boast an astonishing stiffness
(which does not compromise their light-weight quality). The geometry was
tested and improved by using the Grasshopper plugin “Karamba” (http:
//www.karamba3d.com). ◂◂◂

▸▸▸ k-noids

Fig. 7.75 The trinoid – unsurpassable beauty. In these two images, section lines parallel to the
horizontal symmetry plane are displayed.

Fig. 7.76 To the left, the trinoid is covered by non-planar parameter lines. To the right, one can
see section lines perpendicular to the vertical axis of the surface and meridian sections through
this axis.
7.5 Minimal surfaces 331

The only surface of revolution which, at the same time, is also a minimal
surface, is the catenoid. In differential geometry, a k-noid is a minimal surface
with k catenoid openings. For k = 3, the surface is called trinoid (https:
//en.wikipedia.org/wiki/K-noid). ◂◂◂

▸▸▸ “Area-minimizing surfaces”

Although speaking of a snail’s surface seems to make little sense given that it
moves and changes shape all the time, we can make the following observation:
The surface tends to minimize its area with any new position.

Fig. 7.77 Organic-looking surfaces that are computer-generated by iteratively minimizing the
surface

When we gradually change the shape of a given surface by iteration in such

a way that the area of the new surface is smaller than that of the previous
one, the surface will eventually converge towards a minimal surface. ◂◂◂

▸▸▸ An algebraic minimal surface and “something similar”

In Chapter 13, we will see that exact geometric shapes are rarely found in
nature. Sometimes, however, one finds rather good approximations of them,
as Fig. 7.78 shall show (on the right, a computer drawing; on the left, “some-
thing similar” in nature . . . ).
One of the simplest minimal surfaces is of degree nine, and it serves as an
algebraic example of such surfaces in mathematics. If we consider only the
innermost part, there are no self-intersections (the surface is theoretically in-
finitely large). If we compare the surface with a brown alga from the northern
Adriatic sea, a certain similarity cannot be denied, indicating that the algae
tend to minimize their surface area. ◂◂◂
332 Further remarkable classes of surfaces

Fig. 7.78 The comparatively simple algebraic minimal Enneper surface and some seaweed

Bending surfaces into each other

At various points in this book, we have already discussed the process of de-
velopment, that is, the flattening of a surface into a plane without distortion.
In mathematical terms, this means that the surface metrics – its angles and
also segments (measured geodesically, see p. 142) – are retained.

Fig. 7.79 Developable surfaces (e.g. bent paper strips of a rectangular or circular shape) can
be flattened into a plane, and from there, they can be bent into any other developable surface.
Klaudia Kosma applies this principle to design remarkable objects made of Möbius strips.

This applies to developable surfaces in an almost trivial way. These surfaces

can be flattened into a plane or, vice versa, be formed from a plane surface
piece. So, we only have to bend such a surface into a plane and then bend
it again to create a new developable surface. Simple examples of that can be
found in Fig. 4.87 and Fig. 5.6. Fig. 7.79 shows how an artist can be inspired
by this principle, designing objects based on the Möbius strip. The ribbons
7.5 Minimal surfaces 333

are formed by bending strips of the plane surface. In the case of the creative
clock on the left, the ribbons are not made of rectangular strips but of strips
of a circular shape, which can be joined in an effortless and yet creative way.

As you already know, doubly bent surfaces cannot be developed but, under
certain – extremely restricting – circumstances, two such surfaces can be
bent into each other. The theory underlying this claim is beyond the scope
of this book, but we still know two surfaces that fulfill all criteria: the helicoid
and the catenoid. Both are minimal surfaces and thus have the same mean
curvature. (According to one criterion for bendability, it must be possible to
match the surfaces in such a way that their corresponding points have the
same Gaussian curvature.)

Fig. 7.80 According to Bonnet’s theorem, helical minimal surfaces can be bent into each other
without destroying the surface metrics (i.e., there is no stretching or compression). If the catenoid
is bent into the helicoid, we get an infinite number of minimal surfaces with the same surface
metrics.

▸▸▸ Bending of minimal surfaces

Fig. 7.81 shows once more the famous bending of a helicoid into a catenoid
(compare Fig. 7.80). Fig. 7.82 illustrates the bending of minimal spiral sur-
faces.3 The helicoid is the only ruled surface (except the plane) that is a
minimal surface (Catalan). Thus, it does make sense to vary parameters in
the case of minimal spiral surfaces in the hope of eventually finding a ruled
minimal spiral surface. The catenoid is the only minimal surface of revolution.
This was first documented by Leonhard Euler .

3 W.Wunderlich; Beitrag zur Kenntnis der Minimalspiralflächen. Rend. Mat. 13, pp.1–15,
http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/pdf/59.pdf.
334 Further remarkable classes of surfaces

Fig. 7.81 Bending of a helicoid (orange) into a catenoid (yellow): Note the motion of the orthogonal
grid that is drawn on the surfaces – its metric is not changed!

Fig. 7.82 Bending of minimal spiral surfaces according to Wunderlich. The yellow surface is
symmetric to a horizontal plane, the orange surface is axially symmetric.

Since the catenoid is the only minimal surface of revolution, Fig. 7.83 (a
floating aluminum coin) allows the conjecture that the deformed water sur-
face is, in principle, a catenoid, because the surface tension will deform the
water surface in this way. ◂◂◂
7.5 Minimal surfaces 335

Fig. 7.83 A floating coin is very likely to deform the surface of the water along a catenoid.

▸▸▸ Bending of nudibranches and algae

Fig. 7.84 A nudibranch (Aplysia) appears to “fly” through shallow water. Parts of the surface
resemble minimal surfaces. If this is the case, the fairly firm but flexible tissue undergoes no changes
in the metric unless it is moving.

Fig. 7.85 Brown algae swaying in the occasionally strong current in shallow water

In both Fig. 7.81 and Fig. 7.82, a rectangular grid on the surfaces is drawn in
order to see that the metrics of the surface do not change during the bending
process. This is unique for minimal surfaces, and it may also be an expla-
nation for the following: Surfaces of maritime creatures like nudibranches
(Fig. 7.84) or also plants like algae (Fig. 7.85) sometimes resemble minimal
336 Further remarkable classes of surfaces

surfaces. When they glide or sway in the current, their flexible but rather
firm tissue does not need to change essentially. ◂◂◂

The second example (Fig. 7.86) seems less intuitive, and yet, it fits the context
very well: Crocheted surfaces can be bent easily without losing their metric
properties in the process.

Fig. 7.86 Pliable surfaces: Lilian Boloney crochets (!) complicated mathemati-
cal surfaces: This image shows Boy’s surface, which can serve as a model for
the projective plane. The pliable material allows for bending without signifi-
cant changes to the surface metrics. The artist was inspired by a scientific paper
(http://vis.berkeley.edu/papers/methexpview/math_exploded_view_small.pdf).
8 The endless variety
of curved surfaces

So far, we have learned about several classes of surfaces. Many other classes
of surfaces may, however, be derived from a variation of so-called parameter
representation. In the age of computer-aided creative (architectural) design,
it is becoming increasingly relevant to create completely general surfaces by
means of easy-to-control geometric instructions. Such free-form surfaces are
the topic of this chapter.
The theory behind free-form surfaces is already several decades old, and the
rapid nature of modern computers makes such surfaces interesting for any
designer working in front of a computer screen.
The problems to be solved are the following: How can we find surfaces
that pass through predefined points in space and additionally follow cer-
tain “boundary conditions”? How is it possible to work with such “control
points” and retain the ability to change their individual positions without
affecting the character of the entire surface?
The solution to these problems lies, as it often does, one dimension below:
One should define planar free-form curves that exhibit certain properties
and then transfer these properties into three-dimensional space. Thus, we
are confronted with Bézier surfaces and B-spline surfaces by way of Bézier
curves and B-spline curves.

© Springer Nature Switzerland AG 2020 337

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_8
338 The endless variety of curved surfaces

8.1 Mathematical surfaces and free-form surfaces

Function graphs

Fig. 8.1 A graph of a so-called “smooth” function

The graph of a “smooth” function (Fig. 8.1) shows a smooth curve at ev-
ery point in every direction. The tangent vectors of these smooth curves
are determined by the partial derivatives with respect to x and y, which
span a uniquely defined tangent plane that contains all tangents arising from
perpendicular sections. In this case, the given function is said to be totally
differentiable.

Fig. 8.2 Two graphs which are not totally differentiable everywhere.

Fig. 8.2 shows two functions that are not differentiable

√ everywhere. In the
figure on the left (right circular cone, z = √x + y ), the function is not
2 2

differentiable at the origin. On the right (z = ∣xy∣), the partial derivatives

exist along the x- and y-axis, but there is no tangent plane in the sense
described above.
8.1 Mathematical surfaces and free-form surfaces 339

Fig. 8.3 The “pathological” Weierstrass-℘-Function

Without going into details, Fig. 8.3 shows the famous Weierstrass-℘ -Function
and its derivative, which can also be interpreted as a function graph.

Fig. 8.4 Function graph with a “mathematical background” (simulation of the heat distribution
over the candle pictured in Fig. 1.8). The layer stripes are algebraic curves of degree-four, which
can only be represented correctly when the surface is described by an equation with exact precision.

More general surfaces described by mathematical equations

Many well-known surfaces can be described by mathematical equations. This
includes (but is not limited to) practically all surfaces that we have discussed
so far and, of course, quite a few others (Fig. 8.4 and Fig. 8.5 show three
examples).
Such “mathematical surfaces” possess one or many shape parameters, which
permits a certain variation in their visual appearance. With some mathemat-
ical dexterity, even fairly complex surfaces can be generated.
A very elegant piece of furniture by Paolo Piva is pictured in Fig. 8.6. The
idea behind it was to involve mathematics in the arts.
340 The endless variety of curved surfaces

Fig. 8.5 Surfaces generated by mathematical equations (on the right: Klein bottle in theory and
practice.

Fig. 8.6 Furniture as a mathematically defined function graph

This surface is defined by a symmetrical equation that may also be of interest

to non-mathematicians: z = c⋅(y sin x−x sin y) where c ≈ 0, 12, x, y ∈ [−π, π].
Parabolic points occur along the coordinate axes.
Artists who work with ceramics are often inspired by mathematics. In Fig. 8.7,
Martina Zwölfer presents some of her work (see also Fig. 8.5 on the right).

Fig. 8.7 Ceramics as a “playground” for mathematically defined surfaces

8.1 Mathematical surfaces and free-form surfaces 341

Different types of surfaces

Surfaces can generally be thought of as the “products” of space curves that
undergo arbitrary motions and deformations.
Surfaces of revolution, for instance, are created by the rotation of an unchang-
ing generatrix. Scaling this curve in relation to a fixed point on the axis of
rotation and proportionally to the angle of rotation yields a spiral surface.

The addition and subtraction of surfaces

Many objects that seem highly complex are the result of so-called Boolean
operations: Solids can be “added” to each other, but also “subtracted” from
each other. To the geometer, this does not necessarily indicate that a new
surface was created: Each boundary segment, no matter how small, belongs
to precisely one of the initial surfaces.

Fig. 8.8 Boolean operations in architecture and the arts: “tree house” (cube apartment in Rot-
terdam by Piet Blom), where a cube was placed on its spatial diagonal and superimposed by a
hexagonal prism. At the center: addition and subtraction. The right side pictures a tree house of
a different kind: Erwin Wurm placed a house on top of the Vienna Museum of Modern Art – a
classical Boolean subtraction is behind this effect.

Fig. 8.9 Boolean operations were used to project the initials of three great masters: Gödel,
Escher, Bach. The B is turned upside down, the E is rotated by 90○ – both are translated along
the coordinate axes, which yields the intersection of the desired residual solid (see, e. g. [19]).
342 The endless variety of curved surfaces

Fig. 8.10 Boolean operations in art objects (Eduardo Chillida)

Fig. 8.11 Extreme Boolean “subtraction”: The high-speed photograph shows an apple that has
been penetrated from the right by the bullet of an air rifle. The double-cone-shaped projectile
(encircled) is fairly easily recognizable. On the right: computer models.

Milled surfaces are also “mathematical surfaces”

During the milling process, the cutter envelops a manifold in all possible
positions – the sum of all milling positions. The surface parts that are cut
away are the differential surfaces of the original solid and this enveloped
manifold. The more complicated the shape and motion of the cutter, the
more intricate this enveloped surface. Fig. 8.15 and Fig. 8.16 show a surface
designed by Florian Gypser. When looking only at the intermediate positions
of the object in motion (the professional calls this a discrete family), one
cannot fail to notice the aesthetic quality of these “imprints” – their possible
applications in design are likewise obvious.
In practice, curved shapes are often produced by milling. Each milling stage
creates a new and occasionally complicated shape (Fig. 8.13).
How can we solve the problem?
Generally speaking, surfaces of arbitrary shape can, to some degree, be ap-
proximated by parts of standard surfaces. Due to the limited number of shape
parameters, however, the result will rarely be satisfactory.
From this point on, let us define a free-form surface such that its appearance
may be changed in an almost arbitrary manner through the variation of
parameters and generating geometric elements. Thus, it should be possible
to at least visually approximate mathematically defined surfaces very closely.
8.1 Mathematical surfaces and free-form surfaces 343

Fig. 8.12 simple cutters (cylindrical and conic)

Fig. 8.13 Surfaces as the results of a milling process . . .

Free-form surfaces play an important part in industrial design (e. g. in the

auto industry, as in ship and airplane construction) and in the modeling and
construction of interesting architecture.

Surface theory is helpful in the creation of categories

We know the following from the previous chapter: Locally speaking, there
exist only three types of surfaces – regardless of whether they are free-form or
mathematically defined: Each surface, regardless of its complexity, is curved
like an elliptic or hyperbolic paraboloid – and in some borderline cases, like
a parabolic cylinder or a plane – in the small neighborhood of a point.
It makes no difference for our prior considerations whether the surface is
mathematically defined or “free-form”. As we shall soon discover, free-form
surfaces also have to be calculated in some way – by means of relatively elabo-
rate, “recursive” algorithms, to be exact. Due to their complicated definition,
a greater quantity of shape parameters are relevant in this context.
▸▸▸ Playing with geometric forms
A torus can be generated via rotation of a circle around an axis in its car-
rier plane. When we vary the radius of the circle (e. g., harmonically by
introducing a sine function), we might get something like what is shown in
Fig. 8.18 on the left. This slightly resembles a pumpkin or a juicy tomato
344 The endless variety of curved surfaces

Fig. 8.14 . . . and an application of an altogether different kind

Fig. 8.15 Envelope or “imprint” of an object in motion

(Fig. 8.19), though such natural objects hardly ever have such extremely
symmetric shapes.
So, it might be useful to vary another parameter, namely the rotation speed
of the circle in relation to the change of the circle’s radius. In the case of
Fig. 8.18 on the right, an acceleration factor of 2.32 was taken. Why this
precise factor, which you probably did not expect? Well, it could have been
any other number, except for a simple fraction like 3/2 or 5/2, because these
fractions would then again result in too much symmetry. This goes somewhat
against classical mathematical approaches, where simple fractions usually
lead to remarkable results, but it comes closer to a simulation of natural
behavior.

Fig. 8.16 Imprint of a “discrete” family

8.1 Mathematical surfaces and free-form surfaces 345

Fig. 8.17 Partly developable surfaces (Bilbao Museum)

Fig. 8.18 From a perfect geometric shape to a pumpkin or a tomato

Fig. 8.19 Tasty tomatoes

Now, the surface resembles our pumpkin or tomato even more. . .

But what is the benefit of all this? Can we learn something from it? Answering
these questions is rarely easy. Such an approach may or may not help us
understand how nature produces such remarkable shapes. Even if we find
a good approximation of a shape by means of formulas, nature might not
have followed those rules to get there. One always hopes to discover “a good
explaination” for why nature functions the way it does. In cases where we
346 The endless variety of curved surfaces

cannot identify the proximate cause in the evolution of a specific form, we

should consider the possibility that this form may have been produced by
selective breeding of individuals with certain characteristics. ◂◂◂

▸▸▸ The shape of a virus

Viruses are extremely small. They are, in fact, the smallest and simplest
life form known. The average size of a virus could be 100 nanometres, i.e.,
a tenth of 1/1000 millimetre. In comparison, bacteria are up to a hundred
times larger. Therefore, it is possible to detect bacteria with an ordinary
microscope, whereas viruses are only visible under an electron microscope.

Fig. 8.20 World’s most infamous virus during the time this book was written in 2020 . . .

The three-dimensional structure of the coronavirus (Fig. 8.20) was discov-

ered by means of an imaging technique called cryo-electron tomography. The
spikes on the surface of the virus (to which it owes its name, which means
“crown”) are the point of contact between the virus and the host cells that
line the airways of a human’s lung. The spikes connect with the lung cells by
sliding into matching receptor sites found on those cells. Once secured, this
connection enables the virus to fuse with the host cell, injecting its genetic
material into it, and instructing the host cell to make thousands of copies of
the virus.
In computer animations, the shape of the virus can be captured by means
of purely geometrical elements: Define a sphere, look for evenly distributed
points on its surface and add a prototype of a spike there. Since such a
prototype is a surface of revolution, it is a good idea to define its meridian
by using only a few points and then joining them with a smooth curve. The
following section will explain how this can be done. ◂◂◂
8.2 Interpolating curves and surfaces 347

8.2 Interpolating curves and surfaces

There is one obvious way of approaching a definition of free-form surfaces:
It should be possible to fix single surface points as control structure and in-
terpolate the points in between. If a surface is to be given in the form of a
computer model, it may be digitized (using keyers, laser scanners, or pho-
tographs) into a wireframe model on the surface, from which the remaining
points in space may be computed.
In order to connect n + 1 coplanar points by a smooth curve, it seems logical
to position an degree-n order parabola p through the points.

Fig. 8.21 polynomial function vs. cubic spline

This approach may not be of much use to designers: Such curves tend to os-
cillate – especially at the boundaries (Fig. 8.21 on the left, curve p). Another
technique assembles the curve piece by piece from degree-three parabolas,
while the separate curves are joined at the respective endpoints with “contin-
uous curvature” (curve s in Fig. 8.21). These curves are called “cubic splines”.

Fig. 8.22 The choice of control points may be significant.

In order to achieve a smooth curve, the control points of a cubic spline should
be evenly distributed, as unexpected results may occur otherwise (Fig. 8.22).
Curves through predefined points may also be produced by so-called NURBS
(which will soon be described).
By using interpolating splines, it is possible to describe surfaces whose math-
ematical definition is no longer feasible. In Fig. 8.23 and Fig. 8.24, for exam-
ple, the cross-sections of animal horns are described by means of interpolating
splines before the horns undergo mathematical transformations (a helispiral
motion is pictured). The shell of a nautilus was created in a similar manner
(Fig. 8.25).
348 The endless variety of curved surfaces

Fig. 8.23 Approximation of animal horns using cubic splines

Fig. 8.24 Another approximation (blackbuck antelope horns)

Fig. 8.25 The shell of a nautilus is created by defining a symmetric space curve that undergoes
a spiral motion.

▸▸▸ Growth spurts

Since we uncovered the laws of growth for snail houses and antelope horns
8.2 Interpolating curves and surfaces 349

rather effortlessly in the previous part, we should also be able to find out
what rules apply for antler growth.

Fig. 8.26 Antlers regrow every year

Fig. 8.27 Photos of an antler from four different angles

Fig. 8.26 shows two different stages of growth (Wapiti), whereas Fig. 8.27
shows one and the same antler from different angles. Obviously, there is
neither exponential growth nor rotation involved in the growth, but there is
evidently one underlying rule, that guides this kind of (rather fast) growth.
350 The endless variety of curved surfaces

Fig. 8.28 illustrates how antlers grow under different parameters. One of these
parameters is the number of endings that will eventually be produced. (For
male deer this means the older the animal, the more endlings there are.)

Fig. 8.28 Simulation of antler growth (changing with the age of the deer)

The growth proceeds in several steps: So-called “sleeping positions” start

growing at a later stage, while already existing parts continue growing, pro-
ducing more “sleeping positions” for the next stage. A growth in different
stages or growth spurts, each boost with changing parameters. ◂◂◂

Direct modeling at the computer

Designers are, of course, not interested in specifying formulas and would
rather do their ingenious design work directly at the computer. The Bézier
and B-spline surfaces, which allow for the easy definition of free-form shapes,
were developed for this precise reason.
In order to comprehend the underlying principle, we shall first discuss Bézier
and B-spline curves in the plane before proceeding to the Bézier- and B-
spline-surfaces in three-dimensional space.
8.3 Bézier- and B-spline-curves 351

8.3 Bézier- and B-spline-curves

Bézier curves
Prior to B-spline curves, Bézier curves are the most important mathematical
representation of curves in computer graphics and computer-aided design
(CAD). They were introduced in 1962 by P. Bézier at Renault and are defined
by a so-called control polygon. The number of polygon sides denotes the
degree of the curve.

Fig. 8.29 the de Casteljau algorithm, inflection point, conic sections

The construction procedure for a curve point corresponding to the real pa-
rameter t was already defined by P. de Casteljau (Citroen, 1959) and goes as
follows: Subdivide all n sides of the polygon following the same ratio t ∶ (1−t)
(in Fig. 8.29, t = 0.5 was chosen on the left and t = 0.75 on the right). The
connecting segments of the new points form another polygon – with one side
fewer than the original polygon. This procedure is repeated for the new poly-
gon, which yields another polygon with n − 2 sides, and so on. After n steps,
the polygon is only composed of a single point – the curve point at the pa-
rameter t. The last segment is the curve tangent at this point. In the special
case of n = 2, this procedure always yields a parabola. Fig. 8.29 on the center
right shows a degree-four Bézier curve with an inflection point defined by a
four-sided polygon. On the far right, a degree-two Bézier curve is pictured in
black – a regular parabola.
Due to the nature of their construction, Bézier curves possess many helpful
geometric properties: They always lie within the convex hull of the control
polygon and pass through the endpoints. At the endpoints, the curves touch
the first and last segment of the control polygon. The curve does not oscillate
needlessly, which means that it does not intersect an arbitrary straight line
more often than the control polygon. Furthermore, in the cases of rotations,
reflections, translations, scaling, and parallel projections, it is sufficient to
subject the control polygon to the same transformation. The construction
procedure for curve points and tangents is invariant under affine transfor-
mations. The control points are transformed into the control points of the
image curve.
Besides the calculation or construction of points and tangents, de Casteljau’s
algorithm serves another useful purpose: the decomposition of a Bézier curve
352 The endless variety of curved surfaces

at a point t ∈ [0, 1] into two parts (Bézier curves of the same degree, Fig. 8.30).
It also provides the key for the approximated calculation of all intersection
points of two Bézier curves, or of a Bézier curve and a straight line.

Fig. 8.30 augmentation of the degree, decomposition of a Bézier curve

If a single control point of a Bézier curve is moved, then the shape of the entire
curve is affected (consider the global influence of the control points)! This is
a significant disadvantage of Bézier curves compared to B-spline curves. The
same comparison can be done with Bézier- and B-spline surfaces. It is further
problematic that certain frequent types of curves, like ellipses or hyperbolas,
cannot be precisely defined as Bézier curves (as opposed to parabolas). In
order to solve these problems, Bézier curves were generalized to so-called
rational Bézier curves.
Rational Bézier curves
Weights are assigned to each control point. If all weights are the same, then
a classical (“ordinary”) Bézier curve is given. The geometric definition of a
Bézier curve is independent of dimensions (Fig. 8.31):
Let us imagine the planar curve in a horizontal plane at height 1 and stretch
the points of the planar control polygon from the origin, with the respective
weight as the scaling factor. In the general case, we will thus get a non-
planar control polygon in three-dimensional space that will, if exposed to
the de Casteljau algorithm, yield a Bézier space curve. The central view of
this curve from the origin into the original curve’s horizontal carrier plane
produces the rational Bézier curve.
This approach allows us to define other curves apart from parabolas, such
as parts of ellipses (including parts of circles) and parts of hyperbolas (see
Fig. 8.29 on the right), which occur as central views of parabolas.
For rational Bézier curves, the properties of regular Bézier curves essentially
persist. It may, however, be added that even projective transformations will
not change the nature of the curve: It is sufficient to apply the transforma-
tion to the control polygon. The rational Bézier curve produced by the trans-
formed polygon corresponds to the projective transformation of the original
Bézier curve.
B-spline curves
By piecing together rational Bézier curves (the transition can occur at con-
stant curvature if the degree is at least 3), so-called Bézier splines may be
produced. They have the tremendous advantage that changing a single con-
8.3 Bézier- and B-spline-curves 353

Fig. 8.31 spatial interpretation Fig. 8.32 above: ordinary, below: rational Bézier curves

trol point no longer affects the entire shape on a global level. On the other
hand, the amount of data for the control points may grow to be very large, as
Bézier splines are “reluctant” to adapt to arbitrary – and especially convex
– shapes. In the general case, they thus require a large number of control
points for fine-tuning. This disadvantage was overcome by J. Ferguson (at
Boeing) in 1964 through the invention of B-spline curves, which otherwise
possess properties very similar to those of Bézier splines.
For B-spline curves, the algorithm analogous to de Casteljau’s is the so-called
de Boor algorithm. The kinship between Bézier curves and B-spline curves is
such that each B-spline curve segment can be converted into a Bézier curve.
The degree of the curve indicates how many control points around a curve
point are responsible for its position.
In order to quantify this degree of influence, a “knot vector” is given to specify
the strength of the influence that a control point’s neighbors have. This vector
must possess more components than the overall amount of control points.
If a curve is to pass through the boundary points of the control polygon,
the beginning and end components of the knot vector must be repeated
accordingly (see the following example).
Rational B-spline curves (NURBS)
If weights are provided for all points of the control polygon (as is the case
with rational Bézier curves), we get a rational B-spline-curve, which is also
called a Non-Uniform Rational B-Spline (NURBS) curve. “Non-uniform”
simply means that the knots do not have to be distributed at equal dis-
tances. The unit circle, for instance, can be specified by the control poly-
gon (1, 0), (1, 1), (−1, 1), (−1, 0), (−1, −1), (1, −1), (1, 0), the knot vector
(0, 0, 0, 14 , 21 , 12 , 14 , 1, 1, 1), and the weights (1, 12 , 21 , 1, 12 , 12 , 1). Based on this
representation of a circle, it is possible to construct surfaces of revolution.
NURBS possess properties analogous to those of ordinary B-spline curves:
local control, the convex-hull-property, and continuity of curvature (if all
weights are positive) as well as invariance under affine and projective trans-
formations. There exists a de Boor algorithm for NURBS which can be used
to calculate the points of the curve or decompose a NURBS into two parts.
354 The endless variety of curved surfaces

8.4 Bézier- and B-spline surfaces

Given a grid of quadrangles instead of a control polygon (m × n sides), Bézier
surfaces and B-spline surfaces can be constructed by analogy with Bézier
curves and B-spline curves: If Pij are the points of the grid (i = 0, . . . , m; j =
0, . . . , n), then the n + 1 points with a fixed i define the control points of a
Bézier- or B-spline curve. For each of these m + 1 curves, there exist unam-
biguously defined points for each parameter t as calculated by the algorithms
of de Casteljau or de Boor . These points may, in turn, be used as the control
points of further Bézier- or B-spline curves. We have thus defined a contin-
uous family of curves on the surface. By analogy, an additional family of
curves may be constructed by varying the index i while keeping j constant.

Fig. 8.34 Subdivision (degree augmentation + local

Fig. 8.33 Bézier surface change)

When compared to their planar cousins (Bézier curves), Bézier surfaces ex-
hibit similarly practical properties for modelers: The surface passes through
the outer edge points of the grid and is touched at this point by the planes
spanned by the respective sides of the quadrangular net. The surface lies
within the convex hull of the grid. In order to apply an affine transforma-
tion to the surface, it is sufficient to apply the transformation to the grid.
Rational Bézier surfaces (which will be discussed shortly) are even projec-
tively invariant. Once again, difficulties arise when changing a single control
point affects the entire surface. It may be helpful to increase the degree of
the surface temporarily, decompose it into parts, and continue manipulating
the separate parts only. If both surface degrees m and n are at least 3, then
it is possible to configure the surfaces so that they meet at constant curva-
ture along edges. For a designer, constant curvature is of utmost importance:
Shiny surfaces that do not have this property produce light edges.
Ruled surfaces and cylinder surfaces can easily be represented as Bézier sur-
faces by reducing one of the two degrees m or n to 1. In the case of the
cylinder, the straight lines connecting corresponding control points must be
parallel. Paraboloids occur for (2, 2) grids (but only the surfaces of transla-
tion among the (2, 2) grids are paraboloids). If the polygons of the quadrangle
grid diverge as a result of translation, then a surface of translation is created.
An exact representation of mathematical surfaces through Bézier surfaces
8.4 Bézier- and B-spline surfaces 355

is possible if the mathematical surfaces possess a parameter representation

composed of polynomial functions.
Generalized Bézier surfaces
If each grid point of a Bézier surface is specified with a weight, the result
is a rational Bézier surface. Now it is possible to represent a greater class
of surfaces in an exact manner – in fact, this applies to all surfaces whose
parameter representation can be specified as a rational function (a fraction
of polynomial functions). However, even such techniques can only encompass
the tip of the iceberg of all possible surfaces, which is why it is recommended
to work with sub-grids or sub-nets that can be edited separately.

Fig. 8.35 Rational B-spline surfaces, open and closed

Like B-spline curves, B-spline surfaces can be manipulated locally. They also
share useful properties analogous to those of B-spline curves. Fig. 8.35 shows
open and closed control grids and the corresponding surfaces.

Fig. 8.36 Rational B-spline-surfaces (NURBS), open and closed

Rational B-spline surfaces (NURBS surfaces) are the most flexible if, in ad-
dition to the control points, they can also be affected by two knot vectors for
both directions. These surfaces are, thus, included in the design modules of
all relevant CAD systems.
▸▸▸ Practical construction
After a free-form surface has been designed on the computer, problems usu-
ally tend to occur during the practical construction. Rapid prototyping per-
mits the quick and easy generation of prototypes: Three-dimensional “print-
ers” produce thin cross-section surfaces from synthetic material which can
then be glued together.
356 The endless variety of curved surfaces

Fig. 8.37 A nice application of NURBS (compare Fig. 2.32)

Fig. 8.38 The practical construction of a NURBS blueprint

Large surfaces demand adequately enormous means. A patent which specifies

the computerized control of bending machines so that an exact wireframe
model of the surface is produced from steel (Fig. 8.38) was submitted at
the University of Applied Arts Vienna (by Sigrid Brell-Cokcan and Dumene
Comploi, design by Mario Gasser, Cornelia Faißt, and Lis Ehses). ◂◂◂

▸▸▸ Nutshells made of plastic

An interesting idea by Marcus Bruckmann and Ursula Klein/schulteswien
describes the welding of congruent, circular plastic foils which can then be
inflated.
The resulting formations have Platonic solids as their base frame (Fig. 8.39),
and their shapes recall the shells of Brazilian nuts. They were modelled using
NURBS in a particularly realistic way. ◂◂◂
8.4 Bézier- and B-spline surfaces 357

Fig. 8.39 Surfaces under pressure, derived from Platonic solids

Fig. 8.40 “Inflated” cube and dodecahedron using NURBS

358 The endless variety of curved surfaces

8.5 Surface design done differently

An interplay between art and geometry

An unorthodox question posed by the ceramicist Silvia Siegl and its surprising
solution on the computer led, in turn, to a series of projects.

Fig. 8.41 The initial question . . .

At first, the problem concerned merely the rigid connection of ceramic lenses
using two orthogonal rods of equal length and the subsequent bending of the
construction into certain shapes. This was to be accomplished by using spher-
ical joints at the ends of the rods (Fig. 8.41). Initially, the geometric analysis
aimed at fitting squares with their respective corners into the joints. This
does not pose any problem in the plane, where it is simple to proceed along
the square edges following the creasing of the “lens carpet.” This creasing is
only possible if it leads to a change of position in multiple lenses.

Fig. 8.42 . . . was solved in a first step by Franz Gruber with the use of a computer.

Let us now replace the lenses by small magnetic spheres, from which four
attached magnetic rods of equal length protrude. This flexible magnetic net
can, indeed, be fitted to the desired forms. In the illustration Fig. 8.43, the
net was cast over two spheres.
The possibility of approximating free-form surfaces through “magnetic nets”
was thus discovered. Groups of four rods are thereby arranged into “skew
8.5 Surface design done differently 359

Fig. 8.43 Casting the net over a scene: a “quadrangulation” of a different kind

rhombi,” which may be decomposed into two triangles along one of the two
available diagonals. The resulting triangulation has the advantage that all
triangles are isosceles and of “similar size.”

Fig. 8.44 Load tests in theory and practice

As advantageous as the new method of triangulation may seem, the bound-

aries appear to be out of control. One may ponder what would happen if the
four corners of a rectangular magnetic net are specified in their final position.
The solution of this problem is more complicated and, indeed, there is no
exact solution in the general case. This seems to necessitate the stretching or
compression of the rods by as little as possible. Fig. 8.42 on the left illustrates
– by use of yellow and red colors – where a strong material strain may occur
in a practical construction.
The project thus reached new dimensions and culminated in a simulation pro-
gram for the manufacturer of a magnetic-geometric “toy”. Several renderings
of that program can be seen in this book (such as Fig. 3.40).
360 The endless variety of curved surfaces

A different kind of architectural draft

We have often mentioned the fact that developable surfaces play a crucial role
in architecture, as they can be constructed without the tedious technique of
triangulation and yet lose none of their curved beauty. Examples are legion:
Fig. 8.45 once again shows two lovely applications.
A further question arises: How should architectural forms be designed so that
they are guaranteed to be developable? Architects have their own methods to
accomplish this task: They work, for instance, with bendable strips of varying
materials which are then fitted into the desired arrangement (Fig. 8.46).

Fig. 8.45 Developable façades exhibit an attractive elegance. On the left: Santiago Calatrava’s
opera house in Tenerife (cone parts), on the right: Frank O. Gehry’s New Zollhof in Düsseldorf

Once a model is supposed to enter an advanced phase, it needs to be trans-

ferred into the computer program. At this point, some small compromises
have to be made: No program – so far – supports the construction by means
of bendable strips. What was previously developable now becomes a gen-
eral free-form surface. The difference may not be visually perceptible, but
when the time comes to create practical constructions, the simple construc-
tion from developable parts is largely over, and the façades will have to be
triangulated or manufactured in a complicated, doubly curved way.

Fig. 8.46 Classic sketch using strips: Lars Spuybroek, Centre de Musique Pop, Nancy (left). The
respective sketch using a custom-built computer program (right).

The University of Applied Arts Vienna attempted to develop such a soft-

ware package and published the results in [10]. It permits the use of curves
instead of surfaces during modeling (Fig. 8.46 on the right). These are then
interpreted as the midlines of “paper strips,” which are also the rectifying
8.5 Surface design done differently 361

Fig. 8.47 Design using strips and the result (left)

developables of the curves. The curves themselves can be created and ma-
nipulated through NURBS.
During practical construction, some challenges necessarily arise, but they can
luckily be solved: The midline cannot possess any arbitrary form because the
developable that is derived from it must not self-intersect. Geometrically
speaking, this can be avoided by calculating the edge of regression of the
developable. If it approaches the midline “too closely,” the designer must
not induce any further deformations in that area. This may be a restriction
during design, but it also guarantees that the shape is realistic and does not
need to be manipulated at a later stage.

Fig. 8.48 Design using strips: In conjunction with modified Boolean operations, only small num-
bers of such elements are necessary to create interesting developable shapes (Hans Hollein).
362 The endless variety of curved surfaces

Fig. 8.49 Design using strips: All surfaces are developable and therefore ease to build.

Fig. 8.50 Another example for design using strips

Fig. 8.51 The advantage: The development of the whole object is immediately at disposal.
9 Photographic image
and individual perception

This chapter is dedicated to central projection, which comes into play both
in single-eyed human vision and in photographic imaging. A simple pinhole
camera is enough to realize this kind of projection, but the way in which
the human eye resolves the image is much more complicated due to the fact
that the image plane is not flat. Ever since the painters of the Renaissance,
perspective images have been brought to a high standard. Knowledge about
vanishing points and vanishing lines (such as the horizon) is especially im-
portant in this context. Thus, we will put special emphasis on these topics.
We shall discuss several techniques for quickly achieving good results in per-
spective drawing, even though the level of difficulty remains considerable.
A complete understanding of the “intersection method” is essential for this
purpose.
Although complex perspective images are not often drawn these days, a the-
oretic understanding is required in order to rectify and interpret such images
correctly (photographs, for instance).
Finally, we shall discuss alternate central projection, where the projection
falls onto cylinders or doubly curved surfaces – which is also true for natural
vision. The position of a future spectator is the decisive element of central
projection, because the observation of a perspective image is another central
projection. One should aspire to a reasonable combination of both projections
in order to produce the desired visual effect.

© Springer Nature Switzerland AG 2020 363

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_9
364 Photographic image and individual perception

9.1 The human eye and the pinhole camera

For a long time, it was believed that the perspective images created by the
pinhole camera (and thus, essentially, by any regular camera) are virtually
identical to the visual image of the three-dimensional world that we per-
ceive with one eye closed. A famous experiment by Filippo Brunelleschi was
presented as “proof” of this claim in the year 1401(!):

Fig. 9.1 The iris and its different “apertures”

A spectator looks through a tiny hole in the wall onto a spatial object (the
object used in Brunelleschi ’s experiment was the Florence Baptistery). This
hole is then covered, and a realistic perspective painting of the spatial object
is inserted between the spectator and the object. When the hole is then
uncovered, the spectator would fail to notice a difference, thinking that they
were still looking at the original object (Fig. 9.2).
Brunelleschi even took his experiment one ingenious step further: He inserted
a mirror instead of a painting, and in this mirror, the spectator would see
a reflection of the perspective that was painted on the backside of the wall
with the hole. This construction amounted to the same effect.
In fact, if the eye is forced into a specified position with regard to the observed
object, then the visual perception is identical to that of a central projection
from that position. If this restriction is not given, then the mere change of
position would lead the one-eyed spectator to realize (even in a darkroom
9.1 The human eye and the pinhole camera 365

under otherwise perfect conditions for “illusions”) that they are looking at a
two-dimensional picture that ceases to appear realistic from other vantage
points.

Fig. 9.2 Brunelleschi’s ingenious idea interpreted by Otmar Öhlinger

Spheres are quite particular. In classical perspective (photography), the con-

tours of a sphere most often appear to be an ellipse (the remaining conic
sections rarely occur in practice). The contour ellipse is at its most eccen-
tric if the center of the sphere is farthest away from the optical axis (the
principal ray). However, if a spectator is positioned in the right place, then
the sphere contours will, without a doubt, be interpreted as circular. This is
hardly surprising: The projection cone of the ellipse through the eye is a cone
of revolution. If the observer leaves the ideal position, then the projection
cone becomes a general quadratic cone, and the observer will notice that
“something appears to be wrong with the sphere.”

Fig. 9.3 Almost circular contours of a sphere Fig. 9.4 Section through the near-spherical eye-
near the center of the image! ball

When looking at a sphere, we unconsciously move it into the center of our

field of view. The optical axis becomes the connection between the eye and
the center of the sphere. This may occur either through a rotation of the
head or through the largely equivalent rolling of the eyes. The contour of
the sphere now appears as a circle. Yet, how is this circle transferred via
central projection onto our retina? Due to the laws of optics (the vitreous
366 Photographic image and individual perception

body with the built-in optical lens being precisely being precisely an optical
device adhering to these laws), a cone of revolution whose axis corresponds to
the optical axis is to be intersected with a sphere (the near-spherical retina).
This, indeed, produces a circle on the curved retina. The optical nerves need
only transmit the image to the brain, where – based on our past visual
experience – the curved impression is imagined as a spatial object. A second
image, generated by the second eye, is simultaneously processed.

Fig. 9.5 Iris and pupil, curvature of the cornea (left); strange effects due to “image raising” (right)

This whole procedure is anything but simple. However we perceive three-

dimensional space, the following model is very fruitful for its understanding:
Actually, we do not measure lengths but visual angles. For us, a circle is not
a curve whose points are equidistant from a given center. Instead, a circle is
a curve where the lines of sight through their points enclose constant angles
with the optical axis, producing a cone of revolution. Curves that run across
the same cone of revolution are, indeed, often intuitively interpreted as circles
until a change of eye position corrects the false impression.
9.2 Different techniques of perspective 367

9.2 Different techniques of perspective

This chapter concerns the construction of perspective images which, in days
of old, provided considerable assistance during the creation of perspective
images. In the age of computer graphics, where the art of manually draw-
ing perspective images is dying out, there still exist substantial reasons for
understanding the procedures behind construction. Firstly, sketches with a
correct image may still be in demand in the future, though this may only
concern the artistically inclined. Secondly, knowledge about correct construc-
tion may help when approaching highly relevant problems today – especially
those which fall into the purview of “reconstruction” – right up to the under-
standing of 3D scanners!

Fig. 9.6 In both cases, the principal projection ray (the optical axis) is horizontal. On the left
side, it is even perpendicular to the back wall (frontal perspective).

Fig. 9.7 In both cases, the principal projection ray is directed upwards – on the left, vertically.
In both cases, the goal was to preserve the symmetry of the main directions (Cathedral of Trier).

Intersection method
A perspective image is the result of central projection. Points in space are de-
picted by projection rays that are intersected with the image plane. Technical
368 Photographic image and individual perception

objects tend to have mutually orthogonal “principal directions”. We shall first

assume the image plane to be parallel to the upright direction. This allows
us to gain a better understanding of certain procedures – a generalization at
a later point should not cause any problems.
In Fig. 9.8, the object is positioned oblique to the image plane π (see the
top view). The desired perspective image is the intersection of the viewing
pyramid emanating from the eye point E with the image plane π. Due to the
special assumption (π ∥ π2 ), the front view of the figure that is cut off from
the pyramid is identical to the desired perspective image. The front view of
E is the principal point H of the perspective image. The orthogonal distance
Eπ = EH is the distance d. The circle centered at H with radius d is called
the distance circle.

• Horizon and principal vanishing points

The base plane γ is horizontal. Each point at infinity in γ possesses an
image which is called the vanishing point. The vanishing points of all hor-
izontal directions (in γ) converge on the horizon h which passes through
H. Therefore, the horizon is a vanishing line (common to all horizontal
planes). In particular, the points at infinity of both horizontal principal
directions possess the vanishing points F1 and F2 , which can be gener-
ated by parallel translation of the directions through the eye point. This
construction makes us realize the following (see Fig. 1.40):

The vanishing points F1 and F2 of orthogonal directions lie anti-inverse with

respect to the distance circle.

⊕ Remark : The distance circle should not be missing in any perspective image.
All parts of the image that lie outside of this circle are highly distorted and would
usually appear “unnatural” to a spectator – we shall later discuss this matter
in more detail. “Classical perspective images” normally lie well-embedded in the
“half-distance circle.” This means that the viewing cone around the optical axis,
which is embedded into our scene, possesses an opening angle of more than 50○ .
If we change the image plane solely by parallel translation, the perspective image
is merely enlarged or downscaled. However, if the object is translated orthogo-
nally to the image plane, its relative position to the eye point – and thus also the
impression – will be affected. In computer programs which generate interactive
perspective images on the monitor, a parallel translation of the image plane is
equivalent to “zooming” in or out. Attempts to accomplish a usual enlargement of
the image by moving closer to the object lead to new (and increasingly extreme)
perspective images. ⊕

• Special and general points

It is particularly easy to calculate perspectives of points which lie in π.
These are simply the front views of the points. The drawing technique
9.2 Different techniques of perspective 369

Fig. 9.8 Intersection method

required for this situation is the reason why the front edge of the house
(through the point A) was assumed to lie in the image plane. Let us now
try to depict a general point P at the crown of the roof. In the top view,
its projection P c′ can immediately be found. In order to find the front
view P c′′ (and thus the perspective image P c ), we have to enlarge the
crown up to the point B in the image plane. Its point of intersection B
with the image plane lies at the undistorted height of the crown. Thus, the
image of the crown can be drawn (BF1 ). P c lies on the ordering line. It
is evident that points are ideally “reconstructed” by special straight lines,
provided that:

Straight lines are defined by the trace point and the vanishing point.

⊕ Remark : It would also be possible to reconstruct P on a so-called depth line

perpendicular to the image plane. Depth lines vanish in the principal point H.
Thus, P c also lies on P ′′ H. ⊕
370 Photographic image and individual perception

• Trace lines and vanishing lines

Since our approach worked so well with straight lines, let us attempt an
analogous one with planes. The horizontal base plane γ possesses a trace
g in π, which is called the base line. The vanishing line of γ is the horizon
h. Let us now consider the vertical plane ε defined by the left wall of the
house. It possesses a perpendicular trace e and a parallel vanishing line ecu
through F2 . All vanishing points (images of points at infinity) of ε lie on
this vanishing line. In the image, the mutually parallel oblique roof edges
thus have vanishing points on ecu (compared with Fig. 9.14).

Fig. 9.9 Sunrise and moonset

▸▸▸ Shadow constructions in perspective

In perspective, a layman may not notice a substantial difference between par-
allel lighting and central lighting. Both Sun and Moon are points (or rather,
small circular discs) on the firmament from which light rays are emitted. By
use of vertical “light planes,” it is possible to determine the shadow of a point
P given its top view.

Fig. 9.10 The “sun point” and the not-so-parallel-looking sunlight

⊕ Remark : The additional condition that knowledge about the top view of a point
must be given is usually met in architecture. The principle remains the same: Two
planar views of a point in space must be known; otherwise, its position is not
uniquely determined.
A light source, in particular, is only “fixed” once its top view is determined. If the
light source is the sun, then its top view – a point at infinity – lies on a perpendicular
9.2 Different techniques of perspective 371

Fig. 9.11 Backlight: parallel lighting on the left, central lighting (flashlight) on the right

Fig. 9.12 Sun point and sun foot in theory and practice

line through the image Lcu of the Sun and on the horizon. L′c u is called the foot of
the Sun. If the Sun is visible above the horizon, then its foot lies below Lcu – and
backlight occurs.
In the case of backlight, the Sun is not visible in the image. It can, however, be
determined geometrically (like any other point), and its (theoretical) image lies
below the horizon – this simple fact may, however, appear quite unusual to the
amateur. In both Fig. 9.13 and Fig. 9.14, the Sun is blocked by the back of the
spectator.
In the special case of sidelight, the light falls parallel to the image plane, and both
sun point and sun foot are points at infinity. All subsequent constructions are also
possible by means of continuous parallel translation. ⊕
In Fig. 9.13, the shadow of P on the base plane is determined by connecting
the point to the Sun and intersecting the resulting light ray with its top
view – the connection of the sun foot with P ′ . The horizontal straight line
through P is parallel to the base plane, and its shadow is, thus, also parallel.
Therefore it has the same vanishing point as the straight line.
The trace of the light plane λ through P in the horizontal base plane is
c
determined by P ′ and the sun foot L′c u . The vanishing line lu of the light
plane through a vertical straight line connects the sun point and the sun
foot. In Fig. 9.14, the shadow of a rod P P ′ onto an inclined roof plane
is determined. For this purpose, the light plane λ is intersected with the
building. The oblique portion of the shadow is part of this intersection figure.
In order to verify the result of the construction, it is also possible to intersect
372 Photographic image and individual perception

Fig. 9.13 light planes . . . Fig. 9.14 . . . and roof planes

the vanishing line of the light plane with the vanishing line of the inclined
roof plane.
⊕ Remark : If the Sun were replaced by another (much closer) light source, not much
would change in the construction: The only difference would be that the foot of the
light source would no longer lie on the horizon. ⊕ ◂◂◂

▸▸▸ Shadow of sidelight and general central projection

General central projections with non-vertical image planes are usually much
more difficult to construct (see Fig. 9.15). In the image, perpendicular edges
pass through a third principal vanishing point F3 , and it is no longer possi-
ble to inscribe heights by simply transmitting affine ratios. In such cases, it
is helpful to continue assuming that the image plane remains perpendicular
when we apply the intersection method. Important points of the object, how-
ever, have to be drawn in a tilted manner in the top and front view, which
requires one further side view. (More detailed instructions can be found On
the website to the book.)
However, this example should primarily concern the shadow construction
with sidelight. Let P be a point in space with the top view P ′ . This produces
c
a perpendicular line P c P ′ in the perspective image which possesses the
vanishing point F3 . We shall define sidelight as follows: Light should incline
in such a way that the shadows of vertical lines on the horizontal base plane
appear parallel in the perspective image (Fig. 9.15 on the left). Thus, the sun
c
foot L′u becomes the vanishing point of the horizontal direction. The light
rays can no longer be parallel to the image plane (as it occurred in the special
case of the vertical image plane). The sun point Lcu lies “perpendicular” above
c
the sun foot, which explains why the order line L′u Lu c passes through F3
and is, thus, horizontal. Any point on this horizontal line could serve as the
sun point. This reduces the rest of the construction to a simple recipe: The
image of the shadow Ps on the base plane can be found by intersecting the
c c
straight lines P c Lcu and P ′ L′u . ◂◂◂
9.2 Different techniques of perspective 373

Fig. 9.15 The shadows, given sidelight and a tilted image plane, are defined in the following
practical way: The shadows of perpendicular edges on the base plane should be parallel to the
image plane, and thus, appear parallel in the image. The light rays themselves, as they appear in
the image, are not parallel: They have the vanishing point Lcu .

In principle, the intersection method enables us to construct arbitrary scenes,

and by adding the aforementioned technique of shadow construction to our
toolkit, we could reasonably consider ourselves to be satisfied. Computers
calculate perspective images in precisely this way. However, in practice, it
may be too cumbersome to “derive” an object’s top and front views every
single time, especially as many cases permit a quicker procedure. More in-
structions can be found in the section concerning perspective sketching from
page 624 onwards.
Measuring points
The following method is necessary to rectify photographs. Let a photo of a
building façade or a photo of a planar landscape be given. The measurements
of the various objects are to be determined.
Let us first tackle the issue back-to-front. How would we inscribe a planar
polygon in a vertical plane ε into the perspective image (Fig. 9.16)? If, by
mere coincidence, the plane does not happen to be parallel to the image
plane, then the polygon will appear distorted in the image. Considered in
space, the image polygon can be constructed by intersecting the “viewing
pyramid” through the polygon with the image plane. Both planar sections of
the pyramid are collinear images of each other.
In the drawing, it is possible to employ a trick which also boils down to the
completion of a collineation – though in a manner different from the one
described above. If e is the trace of the carrier plane ε, then we can imagine
ε rotated about e into the image plane. This causes all points of the plane
to become circles of revolution. The rotation, however, can also be replaced
by a parallel projection into the direction of the rotation chords. The center
of the projection is a point at infinity Du . In three-dimensional space, the
polygon in ε and the rotated polygon in the image plane are related via
parallel projection (axis e, direction Du of rays) which is displayed in the
374 Photographic image and individual perception

Fig. 9.16 Parallel rotation becomes “recalibration”

perspective image as a perspective collineation. (The original polygon and

the rotated copy are congruent!)

Fig. 9.17 Image of a perpendicular line segment and the corresponding collineation

This relation can be understood by looking at a sketch of the situation in

space, where it is evident that the point Du at infinity is associated with
a vanishing point Duc = Mε on the horizon and at a distance of F2 E. The
measurement point Mε can be constructed by rotating E about F2 in a side
view. All chords of the rotation vanish at this point of the image, and a
planar perspective collineation derives from the relation between the rotated
polygon and the polygon in space. The determination of a single pair of
associated points allows us to construct images of all other points via a small
number of lines.
Fig. 9.17 illustrates the most common case, where the carrier plane of the
polygon corresponds to the horizontal base plane γ. The measurement point
9.2 Different techniques of perspective 375

Mγ for this plane is found purely planimetrically by drawing the distance

d = EH to the principal point. The axis of rotation (= axis of collineation)
is the base line g.
Both the rotated and the regular top view are related via a parallel perspec-
tivity by way of the parallel rotation chords. This establishes a perspective
collineation between the perspective image and the rotated top view (center
of collineation Mγ , collineation axis g).
⊕ Remark : The central view P c of a point P at height z which does not lie in the
c
base plane is located above the central view P ′ of its top view. Let us now consider
the point P0 which lies above the rotated top view P ′ 0 at a distance of z (Fig. 9.17).
This point can equally be constructed by intersecting the rotation chord through
P with the image plane π. Thus, P0 also lies on a collineation ray through Mγ – a
conclusion that is by no means trivial! Planimetrically, P0 can also be interpreted
as the horizontal view of P (given a distortion ratio of 1 ∶ 1). The horizontal and
perspective views are, thus, in a simple relation. ⊕
▸▸▸ Reconstruction from a photo

If the image of a rectangle is visible on the original photo, then the actual shape
of any figure in the carrier plane of the rectangle can be reconstructed.

Fig. 9.18 Planar figures with varying levels of distortion (original photos)

The goal is to find the measurement point of the carrier plane. The shape of
any planar figure can be deduced from it, even without a ruler.
When dealing with original photos (that is, photos that have not been cropped
as in Fig. 9.18), we can assume that the principal point is located at the in-
tersection of the diagonals. As soon as the image of a rectangle is provided,
the two principal vanishing points F1 and F2 can also been found. The dis-
tance circle, on which the measurement point of the plane is located, can be
reconstructed using the Thales circle above F1 F2 . ◂◂◂

▸▸▸ Spatial reconstruction with a given image of a cuboid

376 Photographic image and individual perception

Fig. 9.19 Spatial reconstruction is generally possible when we know the image of a cuboid.

This useful practical case usually concerns a photograph where the horizon h
does not pass through the principal point, i.e., where three principal vanishing
points F1 , F2 , and F3 with h = F1 F2 exist (Fig. 9.19). The principal point
(the orthogonal projection of the center C) is, then, the orthocenter of the
triangle F1 F2 F3 . C itself is the intersection point of the three Thales spheres
determined by each pair of the principal vanishing points. Again, one can find
the measurement point of the base plane on the Thales circle above F1 F2 on
the altitude through F3 .
◂◂◂

▸▸▸ Anamorphoses
In the 16th century, Hans Holbein the Younger painted his famous picture
“The Ambassadors.” What looks like a shroud on the floor turns out to be
a skull – but only if observed from an extreme angle. A figure that is only
visible from extreme positions of the observer (such as curved mirrors, as
those shown in application p. 227) is called an anamorphosis.
9.2 Different techniques of perspective 377

Fig. 9.20 A morbid detail in the famous picture by Hans Holbein the Younger

It is possible to “implant” such an anamorphosis into a perspective image

through a modified measurement point. In the painting, the skull is well-
concealed, since it only becomes visible from a low perspective and after a
45○ rotation of the image. At the National Gallery in London, the picture
has been positioned in such a way that the skull becomes visible from the
stairway (Fig. 9.20 on the right). ◂◂◂

Fig. 9.21 Anamorphoses: A speed limit sign – the street inscription on the right is not actually
a “pure” anamorphosis because the impression of a perspective distortion remains.

⊕ Remark : Fig. 9.21 (Stefan Wirnsperger) illustrates that traffic signs are often
anamorphoses. Due to the extremely low angle of observation, letters, numbers, and
other symbols used in such signs must often be printed with a certain perspective
distortion. Similar challenges arise in the design of billboards on skyscrapers, which
people usually observe from far below. ⊕

▸▸▸ Deliberate distortions in computer graphics

Perspective images are very frequently employed in computer graphics. One
of the elemental tasks of such graphics programs is the so-called clipping
procedure: Parts of the scene which lie behind the observer or too close to
the front clipping plane (parallel to the image plane through the eye) have to
be removed. This may be done through intersection with the viewing pyramid.
A fast and thus often employed method consists in applying a spatial colli-
neation to all points in the scene – which turns the eye point into the point
at infinity of the principal projection ray. This makes the viewing pyramid
378 Photographic image and individual perception

Fig. 9.22 Two different objects (top and front view) with identical side views

a coordinate cuboid, and thus, greatly enhances the efficiency of clipping

algorithms.
Fig. 9.22 on the left shows an object after collinear transformation in a top
and front view. The view from the left is now equivalent to the perspec-
tive image. Now one might maintain that it would be sufficient to inscribe
the distance from the image plane above the image points (as illustrated in
Fig. 9.22 at the center). Unlike a collineation, however, this transformation
would not be linear. The objects are “distorted” accordingly, which leads to
needlessly complex algorithms. ◂◂◂

The image of a circle

If the planar figure is a circle, then the corresponding image is a conic section,
because the projection of the circle from an eye point E generates an oblique
quadratic cone. An intersection with the image plane thus produces a conic
section (see also Fig. 3.66). It follows from the definition on p. 55 that:

Fig. 9.23 Ellipses and hyperbolas as images of circles

The perspective image of a circle is an ellipse, a hyperbola, or a parabola,

depending on whether the circle intersects, does not intersect, or merely touches
the vanishing plane.
9.2 Different techniques of perspective 379

πv ∥ π
E
g v = πv ∩ γ π
g =π∩γ

e
p

hc ec
pc eye point E, image plane π, base plane γ
vanishing plane πv ∥ π through E
yellow circle e does not intersect πv ⇒ image ellipse ec
red circle p touches πv ⇒ image parabola pc
blue circle h intersects πv ⇒ image hyperbola hc

Fig. 9.24 The position of the circle relative to the vanishing plane πv determines the type of the
circle’s perspective image. See also the special case with a perpendicular image plane (Fig. 3.66).

Fig. 9.25 Conic sections as images of circles

Let us consider an elliptical image of a circle together with a circumscribed

square of tangents, with two sides parallel to the image plane. The center
of the circle is located precisely at the intersection of the square’s diagonals.
The square’s image is a trapezoid (straight lines that lie parallel to the image
plane possess parallel images). The intersection of the trapezoid diagonals
thus yields a point equivalent to the center of the circle – or, the intersection
of the images of all diameters of the circle. However, it is not equal to the
center of the ellipse appearing as the circle’s image, as can be seen in Fig. 9.24
(yellow circle).
⊕ Remark : If the circle lies in a plane parallel to the image plane, then its image is
also a circle. The 10-euro bill contains a perspective image of a bridge which carries
circular arcs without any apparent perspective distortion. ⊕
380 Photographic image and individual perception

Circle depictions in ancient times

Fig. 9.26 Two Egyptian reliefs that are more than 3200 years old (Abu Simbel). In principle, we
are dealing with front views and “integrated side views”.

It is usually believed that realistic and correct perspective drawings with

geometrical context started with the painters of the Renaissance, while the
ancient Egyptians “only” used front views and integrated side views in the
pictures. (Note the 6-spoke wheel of the chariot in Fig. 9.26 (on the left),
which appears undistorted as a circle.)

Fig. 9.27 A 2300-year-old Greek tomb painting (Vergina, central Macedonia). A realistic perspec-
tive drawing: The wheel closer to the spectator has been drawn perfectly!

However, we have several examples illustrating that this conjecture is not

quite true. One of the most impressive counterexamples is a tomb painting in
the grave of Philipp III (the successor of Alexander the Great), drawn around
320 B.C. (Fig. 9.271 ). Not only is the painting impressive as a whole – the
wheels of the chariot are drawn almost as if their images were constructed.
On the right, a computer-drawn reconstruction of the four-spoke wheels has
been superimposed on the original image. Especially the wheel in front is
amazingly accurate – a fastidious critic may consider the other wheel to be
too small and flat, but that does not diminish the accomplishment of the
painter. The four spokes are even “conjugate diameters” of the image ellipse.
(In space, they are perpendicular to each other and the tangents in the end
points of one diameter are therefore parallel to the other.)

1 http://commons.wikimedia.org/wiki/File:Painting_vergina.jpg
9.3 Other perspective images 381

9.3 Other perspective images

Optical illusions

Fig. 9.28 Painting on walls in Pompeii: Perspectives create the illusion of additional space.

Optical illusions seem to be very old. Fig. 9.28 shows two murals in Pompeii
(the ancient Roman city was destroyed and buried under ash and pumice
following the eruption of Mount Vesuvius in 79 A.D.). One can still see
that artists painted columns and other architectural objects on the walls in
order to create the illusion of a spatial scenery behind the wall. This provides
another example proving that a certain knowledge about perspective drawing
had already existed long before the Renaissance.

Fig. 9.29 Relief perspective I

Set decoration employs the term relief perspective, which describes the illusion
of a larger space that would never fit on the actual stage. Fig. 9.29 shows a
hall of columns that appears realistic, but only if observed from the front.
382 Photographic image and individual perception

It would, indeed, be a fitting backdrop to a dramatic event such as the

assassination of Julius Caesar. A spectator in a cheaper seat will, however,
recognize that the supposed right angles are not right angles at all.

Fig. 9.30 Relief perspective II

Fig. 9.30 which features a variety of simple geometrical objects (square

prisms, pyramids, spheres, and cones) demonstrates to what extent our imag-
ination still accepts an oval as being spherical before noticing a definitive
geometric deviation.
▸▸▸ The Gaussian collineation: Virtual 3D images
Similar to the already mentioned relief perspective, we have a collineation
in optics (Fig. 9.31). When we apply a refraction at a convex lens, there are
two rules according to the laws of reflection:
Light rays h that go through the center C of the lens are not refracted
(h = h∗ ), and light rays s parallel to the optical axis are bundled in the
focal point F ∗ behind the lens. The refraction s → s∗ occurs in the principle
plane γ ∋ C perpendicular to the axis. A space point P = s ∩ h corresponds
to a (virtual) space point P ∗ = s∗ ∩ h∗ .

virtual space real space in

behind the lens front of the lens
Fig. 9.31 collineation in space

According to Gauss, this (reversible) relation P ↔ P ∗ deserves the name

collineation, i.e., straight lines g are transformed into straight lines g ∗ .
Proof : We consider g to be the intersection of a plane ε1 parallel to the axis and
the connection plane ε2 = gC with the center. The first plane is transformed into
a plane ε∗1 through F ∗ and the intersection line ε1 ∩ γ, whereas ε2 = ε∗2 remains
unchanged. Thus, g ∗ is the straight intersection of two planes. ⊙
9.3 Other perspective images 383

Planes ϕ ∥ γ are transformed into planes ϕ∗ ∥ γ. It is useful to know this

when it comes to photography. Therefore, we will come back to the Gaussian
collineation and its consequences in our geometry-based photography course
(Appendix B). ◂◂◂

▸▸▸ Two different sequences

Fig. 9.32 Real and pseudo-perspectives

Fig. 9.32 shows two arrays of Buddha statues. On the left, all figures are of
equal size, and it is possible to move freely in their vicinity. The statues on
the right, however, were stacked parallel. They can only be seen from a single
position near the cave entrance. The figures become gradually smaller and
smaller, which is barely noticeable due to the lighting effects, and this makes
the cave appear deeper than it is. ◂◂◂

▸▸▸ Geoglyphs in the desert sand

Members of the pre-Columbian Nazca culture (300 B.C. to 600 A.D.) in
the southern Peruvian coastal valleys used to decorate their pottery and
embroideries with stylized animal motifs. They are best known for scraping
gargantuan animal depictions, which often span many hundreds of meters,
into the dry desert ground. These images can only be seen in their entirety
from an airplane (Fig. 9.33 on the right) and even appear perfectly symmetric
from certain angles – a masterly accomplishment, which also led to far-fetched
speculations. Some, for instance, have claimed that the Nazca people must
have employed hot-air balloons!
It is noteworthy that the giant animal figures do not appear to be symmetric
from straight above. Instead, they seem to exhibit perspective distortions as
in a photograph shot from a high angle (or a photograph from a low angle
– see Fig. 9.33 on the left). Quite paradoxically, this leads to a semblance of
symmetry, but only from certain positions, which some commercial airplanes
luckily cross on their flight path.
Without any claim to truth, one may imagine the following story – inspired
by projective geometry – of how these figures came about:
384 Photographic image and individual perception

Fig. 9.33 In principle, these are planar figures.

Somebody – let us say a high priest – stands on a h = 15 meters tall wooden

tower (apex Z, top view Z ′ , Fig. 9.34 on the left) holding the symmetric
drawing of an animal. He then directs his subjects to lay stone blocks in a
way that would look symmetric and consistent with the image from his point
of view. Ideally, the subordinates work at some distance from the tower so
that the priest can overlook the whole construction site without needing the
ability of “panoramic view”.
This procedure would make the animal motif appear symmetric from the
skewed high-angle perspective of the tower but distorted from a vertical top
view. Now let us discuss the enlargement of the image: A rope is stretched
from Z0 = Z ′ to one of the many contour rocks P and the rope length P Z0
is applied multiple times in the course of the enlargement process (e.g., k = 3
times as in Fig. 9.34, or at even larger scales). This must involve at least two
people: One person at the center directs the other at the end of the rope,
which ensures the straightness of the ray P Z0 . At the end of the enlarged
distance, the point P ∗ is marked with another rock.

Fig. 9.34 It could have been like this.

This technique makes it possible to construct an enormous contour in a

relatively short period of time. As with the real Nazca lines, the image would
appear prospectively distorted in the top view and symmetric from a tower
9.3 Other perspective images 385

that has been scaled accordingly (height k ⋅ h meters, in our case 3 ⋅ 15 = 45

meters)!
It can further be concluded that the scaling center Z0 does not necessarily
have to be the top view Z ′ of the tower apex. In the general case, the point
from which the image appears to be symmetric does not lie directly above
the tower apex but in the elongation of the segment Z0 Z scaled with factor
k.
A further remark that might enhance the plausibility of the above specula-
tions: It would, of course, be possible to enlarge a smaller motif – perhaps
only a few meters in diameter – using the “rope method.” However, the scaling
factor would have to increase, which would, in turn, lead to greater errors
and imprecisions. Besides, why would the Nazcas create perspectively dis-
torted, non-symmetric images on a very large scale if their original images
(hummingbirds, spiders, etc.) were perfectly symmetric? ◂◂◂

▸▸▸ Baroque illusions

In the 17th century, it used to be the fashion to paint ceilings in such a way
that a spectator looking up from the center of the room would have the
illusion of a much bigger space.

Fig. 9.35 Pozzo at work

In order to produce a correct planar painting, a plane (such as the ceiling

of a room) must be painted with a perspective image that, from a chosen
point of view, creates the illusion of great spatial depth, “with all the bells
and whistles” that might be found in this imaginary space (clouds, columns,
etc.).
Andrea Pozzo was a true master of this genre. He was even capable of cor-
rectly painting barrel-shaped interiors, using a special technique (Fig. 9.35)
to accomplish this goal: He first constructed a planar “auxiliary ceiling” which
386 Photographic image and individual perception

Fig. 9.36 Andrea Pozzo’s ceiling paintings and his projection onto a horizontal ceiling plane

consisted of a square net of tight strings. He then projected this net onto the
curved interior by placing a candle in the spectator’s position. By tracing the
shadow lines cast by the candlelight on the curved surface, he got a “reference
net” into which he interpolated the perspective image for his painting. (The
illustrations are by Stefan Wirnsperger.)
The inverse of the procedure would be the following: Consider the pattern of
straight lines and the orthogonally intersecting half-circles on the cylindrical
ceiling. If this curved net is then projected into the virtual ceiling plane, it
would generate a net of parallel straight lines and ellipses. The latter are
derived as the section curves of the plane with the projection cones of the
circles, which are oblique circular cones. In order to get the same impression
of a curved painting from the projection center as in Fig. 9.36 on the left, one
might choose to observe a distorted painting (as in Fig. 9.36 on the right) on
a planar ceiling. ◂◂◂

▸▸▸ Dynamic Sphere – the illusion of a sphere

Who would not want to be under a dome with plants climbing up its sides
(Fig. 9.37) and just relax on a sofa to enjoy the atmosphere? Would this be
possible if we only had a room with a flat, low ceiling? An optical illusion
might do the trick.
Solution:
Ruth Schnell took it upon herself to create such an optical illusion. The walls
of a room with 250 m2 (Fig. 9.38 shows a computer model created by Franz
Gruber) were covered in extremely thin foil (23-karat leaf gold, ca. 100 m2
of gilded surface, grey paint) so that, at least upon entering the room, you
would get the perfect illusion: You would, in fact, see a scene like the one
represented in Fig. 9.37. Fig. 9.38 shows how the foil cut-out (Fig. 9.39) was
pasted onto the ceiling.
A very precise calculation of the extremely distorted projection was possible
with a computer, by projecting a virtual model of the room from a given
9.3 Other perspective images 387

Fig. 9.37 A semi-spherical dome with vines

Fig. 9.38 Virtual room model including the necessarily distorted leaf pattern

centre onto the flat ceiling at a very oblique angle. Fig. 9.39 on the right
shows a detail of the cut-out.
How does this effect look in reality (The Ritz Carlton in Vienna, Fig. 9.40)?
The illusion is, in fact, perfect when we enter the room. Once you start
moving inside the room, you get the impression that you are surrounded by
dynamic plant patterns. ◂◂◂

▸▸▸ We are only measuring angles!

The following idea was conceived by Franz Gruber: Let us imagine ourselves
sitting in a lecture hall, looking at images cast onto the screen by a projec-
tor. In such cases, it is advisable to avoid sitting at the sides. Under ideal
conditions, we would want to sit near the projecting apparatus (Fig. 9.41),
since this would enable us to see the pictures as they were taken.
Let us now imagine that a prankster pulls on a rope, opening a trap door
on the ceiling which causes paper snippets to rain from above – and straight
into the light cone. Spectators on the sides would no longer be able to discern
388 Photographic image and individual perception

Fig. 9.39 This is what the gold foil cut-out looks like! Right: Enlarged detail.

Fig. 9.40 This is the reality: perfect illusion upon entering the room (left, center), dynamic plant
patterns when we walk around the room.

much of the projected image. Spectators near the projector, however, would
barely notice a thing!
What actually happens is a perfect illusion: The paper snippets are illumi-
nated by the projector as soon as they enter the light cone. A small part
of the original image is projected onto each snippet at varying levels of dis-
tortion. The view is further distorted by shadows of the snippets appearing
on the projection screen. However, seen from the lens center the projected
image remains coherent, since only angles can be measured and the medium
onto which the projection takes place is unimportant. ◂◂◂

Fig. 9.41 A dia projection seen from two different viewpoints . . .

9.3 Other perspective images 389

Fig. 9.42 . . . suddenly disturbed by a shower of paper snippets

▸▸▸ Star signs (Fig. 9.43)

To find the North Star, one must first discover Ursa Major (the Big Dipper)
in the night sky. The North Star is located approximately at the five-fold
elongation of the first stars in the constellation.

Fig. 9.43 The striking constellation of Ursa Major

The spatial constellation of the stars is indiscernible to the naked eye, since
the stars may vary in luminosity and in their position on the projection rays.
Due to the enormous distances involved, we also fail to notice a difference
when observing the constellation over a half-year interval (and thus at a
distance of 300 million kilometers). ◂◂◂

▸▸▸ The spider on the sphere

How can you determine what a spider is “interested” in or in which direction
it wants to run? By placing the spider on a free moving spherical surface
and then rolling this computer-controlled ball in such a way that the spider
always ends up running “on the spot”. Let us now turn to the geometrical
part: We would like to deceive the animal by projecting images of a natural
environment onto a cylinder of revolution. These images change in synchrony
with the direction of the spider’s movements (as well as its velocity). What
do we have to consider when we set up this projection?
Solution:
The apparatus for this projection (Fig. 9.44) was designed at the University
390 Photographic image and individual perception

Fig. 9.44 Left: Different computer-modelled projections on the inside of a cylinder of revolution.
Right: The spider (or any other viewer in this position) sees the curved squares as variously sized
“entrances” for hiding.

of Vienna (Department of Neurobiology2 ). The projection is generated by

four projectors, which, for reasons of space, are positioned outside of the
cylinder. The almost point-shaped light sources emit – due to the edges of
the image – rectangular light pyramids. If we intersect the side surfaces of the
pyramids with the cylinder of revolution, we get ellipses. The “rectangular”
illuminated field on the cylinder thus has elliptical edges. The neighboring
illuminated fields overlap (Fig. 9.44 left) – an additional complication that
must be taken into account in the calculation of the images.
Several square-grid fields lying in one plane shall now be shown to the spider.
The position E of the spider’s eyes (a spider has eight lens eyes, see Fig. 9.45)
is more or less constant. If the vertical fields of the grid are projected onto
the cylinder, the images remain straight and vertical. Together with E the
horizontal fields of the grid form projection planes that intersect the cylinder
along ellipses. Squares of the grid that are farther away appear smaller in
the projection – that is, the angles of vision α and β (Fig. 9.44 right) to the
vertical, as well as horizontal, square sides change. If the spider wants to run
towards the nearest black square (= hiding place), it will automatically aim
at the square that appears to be the largest.
The more closely meshed grids that can be seen in the image emanate from
flat – and, naturally, virtual – square grids that lie farther away.
The cylindrical images must still be recalculated onto the projection plane
of the projectors. The elliptical edges are traced back on quadratic cones
through the light centre to the image that is to be projected. So, on this
image, the edges also appear as ellipses. This results in double exposures,

2 Researchproject funded by the Austrian Science Fund (FWF; PI Axel Schmid) in collaboration
with the Austrian Center for Research on Virtual Reality and Visualization VRVis (Anton
Fuhrmann).
9.3 Other perspective images 391

Fig. 9.45 Eyes of a wolf spider (tarantula). You can clearly see eight well-developed lens eyes,
which are mainly located towards the front and are remarkably efficient for spiders, which usually
rely on their tactile sense. The blue of its eyes is an interference color that is the result of repeated
reflections and refractions in the multi-layered reflective tissue (tapetum lucidum).

which can be “airbrushed” by making the corresponding image points (pixels)

darker. Additional challenges lie in solving the remaining problems (how to
control the ball) and “fine tuning”. ◂◂◂

The silhouette of a sphere

Fig. 9.46 Dew in a spiderweb: Spheres always look the same, no matter how big or small they
are. In this case, even the biggest spheres have a diameter of less than 1/2 mm.

In nature, spheres are “produced” in two completely different ways: Firstly,

tiny spheres like water droplets are formed by surface tension (Fig. 9.46).
392 Photographic image and individual perception

Secondly, the big spheres in the universe (starting with a diameter of more
than 500 km) are formed by gravitation (Fig. 9.47).
▸▸▸ Hyperbolas as silhouettes of spheres?

Fig. 9.47 When looking at big spheres, like planets, their contours may appear as parts of hyper-
bolas. To the right: Extreme wide-angle simulation.

From the photographer’s point of view, it is impossible to depict tiny spheres

like the ones shown in Fig. 9.46 in such a way that the outlines are not
ellipses (or circles, if we look at the sphere’s center), because a sphere that
has a hyperbola as its outline would have to intersect the vanishing plane
through the center of the lens, and therefore, it is simply too small (Fig. 9.48,
however, shows that – if you try hard enough – it can be possible).

Fig. 9.48 In these extreme wide-angle photographs with short distances, it was possible to capture
the silhouette of a sphere as part of a hyperbola. The photo in the middle shows a closeup of the
street lamp in the image on the right side.

From a space shuttle or similar vehicle, however, it is very easy to see that the
planet that is being circled by the shuttle intersects the vanishing plane, e.g.,
if one focuses on something like the Moon on the horizon (Fig. 9.47). In the
picture itself, it is usually very hard to determine what kind of conic we are
dealing with. We could say: If it is obvious that the planet is partly behind
the camera, the conic must have “the DNA of a hyperbola”. The limiting case,
namely the parabola, is practically not achievable: For this, the big sphere
would have to touch the tilted camera’s vanishing plane.
It seems easy to answer the question whether one can see the curvature of the
Earth from an aircraft or not in the affirmative, if we look at the photo on
9.3 Other perspective images 393

Fig. 9.49 Does one see the curvature of the Earth from an aircraft?

the left-hand side of Fig. 9.49. However, this could also be a lens error (and it
probably even is to a certain degree): Not particularly expensive wide-angle
lenses tend to create slightly bent curved lines as images of straight lines –
except that the straight line is “radial”, i.e., it intersects the optical axis. A
good test is to rotate the camera in such a way that the end points of the
alleged curves go through opposite diagonal points of the frame. If the image
of the horizon is still curved, we can be sure that our answer is correct. (In
Fig. 9.49 on the right, the horizon is considerably less curved than in the one
on the left.) ◂◂◂

▸▸▸ When silhouettes of spheres are always circles

In central projections, the silhouette of a sphere is always a conic, namely
the intersections of a cone of revolution (the tangential cone from the center
C to the sphere Σ with the image plane π (Fig. 9.50)). The only “problem”
with that is that humans expect circles as silhouettes, because if they take
a look at a sphere, they look directly towards the estimated center of the
sphere.
There is an additional reason for the circular boundary: Our eyeballs are
spherical and not flat. The retina is doubly curved. In fact, if the center
of the central projection is the center of a surrounding sphere K (which is
not exactly the case, however), then the intersection of the projection cone
would exactly be a circle. A mathematician will immediately try to project
the spherical image on the retina stereographically onto a plane π. This
special projection preserves circles and, again, we have circular boundaries
of spheres.
As shown in Fig. 9.50: In a central projection (C, π), spheres Σ usually have
elliptical contours sc . Projected onto a sphere K around C, the countour will
always be a circle cκ . In a stereographic projection onto a plane (center S,
π), the contours will again be circles cs . ◂◂◂

▸▸▸ “Smoothing” undesirably extreme perspective images

We already know that it does not matter onto which surface an image is
394 Photographic image and individual perception

Σ1 c1
K

cc1
C cκ2 cs2
S cc
2
Σ2 c2
cκ1 cs1

π
π

Fig. 9.50 Contours of spheres in central projection are always conic sections. The projection of
spheres on a sphere around the projection center always leads to circular outlines, and finally,
those circles can undergo a circle-preserving stereographic projection.

Fig. 9.51 A cropped fish-eye-lens picture. In the original photograph, the horizon lies above the
center of the image. It is, therefore, curved downwards, though not a hyperbola.

projected, if only the spectator is forced into the “right” position. Certain
circumstances may, therefore, necessitate the production of extreme perspec-
tive images.
If a photographer standing in a small room tries to capture as much of that
space as possible, an extreme wide-angle lens is usually employed. The result
might look like Fig. 9.52 on the left. If such a photograph is then projected
onto a wall, and if a spectator is forced into the same relative position as the
photographer at the time the picture was taken, the optical impression will
be realistic.
A similar problem arises during outdoor photography when a large object
is supposed to be fully contained by a photograph from a short distance
(Fig. 9.52 on the right). This, too, calls for an extreme perspective image. If
such a photo were then projected, it would appear “normally distorted” from
the perspective of a spectator sitting close to the projection screen. At larger
distances, the image’s unnatural appearance would become noticeable.
9.3 Other perspective images 395

Fig. 9.52 Extreme perspective images

Image on a sphere
The question arises whether a compromise exists that would allow the entire
object to be “contained within the frame” while still allowing the image to
appear natural from greater distances. Curiously, the answer is both yes and
no.
We would have to squeeze the “outliers” at the image edges into the picture
by curving the image surface. This comes at the cost of linearity (straight
line preservation)! In the case of the human eye, the image is projected on
a spherical image surface. It, thus, makes sense to project the image onto a
sphere. For the sake of simplicity, we shall project from the midpoint of
the spherically curved surface, which may not correspond exactly to the
projection in our eye, but it is equivalent to the “stitching together” of many
small images during the “scanning” of a room. The eyeball constantly rotates
about its center, and the many partial images are combined into a whole
image inside our brain.
After the projection of the room from the center of the sphere to a spherical
surface, we are faced with the next question: How do we save this image in
order to show it to the spectators? Either of the following solutions may be
technically feasible: The photo sensors on the electronic chip may certainly
be arranged on a sphere, just as an approximately spherical projection screen
may be produced. In both cases, however, the practical realization may be
quite costly.
⊕ Remark : As we have discovered numerous times before, the surface of a sphere
cannot be developed into the plane without distortions. There are, however, two
methods – among many others – which accomplish this task particularly well for
our purposes.
One of these methods is the stereographic projection of the sphere surface (with cen-
ter M ) from a point Z on the sphere to a plane orthogonal to the radius M Z. Such
a projection has the great advantage that sphere circumferences remain circular in
the projection (Fig. 9.54 on the left). The projection cone around the sphere is a
cone of revolution with the apex M , touching the sphere at a circle. Since stere-
ographic projection preserves circles, contours of the sphere remain circular after
the projection. Straight edges in space are first represented by circular arcs on the
396 Photographic image and individual perception

Fig. 9.53 Stereographic projections

sphere and then as circular arcs in the image plane. Fig. 9.53 illustrates how this
technique can be used to “smooth” extreme perspective images.

Fig. 9.54 Extreme views of spheres

The other method was propagated by Guido Hauck around 1900: It uses the already
mentioned cylindrical projection of the sphere, where the spherical coordinates (lati-
tude and longitude) are taken as planar Cartesian coordinates. Fig. 9.54, right shows
how the silhouettes of a sphere are distorted into irregular ovals by this method.

Fig. 9.55 Hauck’s perspective

Given a horizontal optical axis (Fig. 9.55 on the left), Hauck ’s method is well-suited
for the transformation of extreme perspectives: Vertical edges remain vertical and
straight in the projected image. All other straight edges appear to be curved, though
9.3 Other perspective images 397

not in the form of circular arcs. In general views of objects, the edges may even be
projected as curves with inflection points (Fig. 9.55 on the right). ⊕ ◂◂◂

▸▸▸ The geometry of the fish eye

Fig. 9.56 What does a fish see?

The aforementioned methods for the smoothing of extreme perspectives are

mostly applied in computer graphics. There exists a further practical solution
to this problem: the use of a fisheye lens.
This name makes sense when considering Fig. 9.58: When we look out of
water with a calm, flat surface, we can see everything above the water surface.
The image of the outside world is centered in a circle. Objects which are close
to the water surface appear downscaled and flattened towards the edge of
the circle.

Fig. 9.57 Another view from the water

This kind of geometry on the water surface will be discussed in the next
section. Fig. 9.58 shows an exception in the animal kingdom: the four-eyed
fish. Its eyes are split in two, which enables it to see undistorted images above
and below water.
398 Photographic image and individual perception

Fig. 9.58 A very strange fish eye

Fisheye lenses are often employed in door viewers. When one looks through
them, the objects outside are pressed inwards – an effect which is achieved
in photography by using fisheye lenses. Standard photographic lenses are
flat by comparison and symmetric with respect to a plane. The fisheye lens,
on the other hand, is spherically curved to a large degree and is, therefore,
capable of capturing “eccentric points” – the points whose angle between
their connection to the lens center and the optical axis may be up to 90○ .
Compared to a flat lens, light rays have a smaller angle of incidence with
respect to the surface normal, which leads to lower degrees of refraction.
Furthermore, due to the inclination of the optical axis, the image points of
eccentric points in space are pressed inwards into the image.

Fig. 9.59 On the left: circular ornament, on the right: concentric rainbow

Circles are usually pictured as non-elliptical ovals. Circles whose axes coincide
with the optical axes of the lens remain circular, because both the circles and
the lens are rotationally symmetric with respect to the optical axis.
Fig. 9.59 serves to illustrate this principle. In the left image, a circular orna-
ment was photographed “at ankle level.” In the right image, two (!) rainbows
are visible upon closer inspection. When the Sun is very low in the sky, the
axis of any rainbow (or, in this case, of both rainbows) is almost horizontal.
The primary and secondary rainbows [9], thus, appear to be nearly circular.
9.3 Other perspective images 399

Straight lines are usually pictured very distortedly while the angles that they
enclose are spared the extreme distortion of traditional perspective images
(Fig. 9.52). ◂◂◂

Straight lines g which intersect the optical axis have their images in the
projecting plane that is defined by g and the optical axis. With respect to
the image, it can, therefore, be said that the closer a straight line comes to the
lens center, the less distorted it appears. In practice, this becomes relevant
when taking pictures of scenes with distinct horizons (such as Fig. 9.60 on
the left). Due to the rotational symmetry of the lens, this principle also works
for non-horizontal straight lines (Fig. 9.61).
The center image of Fig. 9.60 shows that this knowledge can be employed to
alter fisheye photos in deliberate ways: A relatively small translation of the
principal point causes the boards of the patio to become distorted – in one
case, in the frontal area; in the other case, in a ray-like formation.
What makes fisheye photos especially fascinating is the near-infinite depth
of field: Both the sunflower and the mallow blossom almost touch the lens in
these shots. This property is also explicable in geometric terms: The smaller
the focal length, the more distantly the objects are positioned (measured in
focal lengths). We know from far-away objects that their image points are
very close to the light sensitive layer of the camera sensor.

Fig. 9.60 Diagonals remain straight (I)

Any regular photograph (points P c ) can be converted into a fisheye image

(points P f ) – and vice versa: Any fisheye photo can be transformed into
a regular perspective image. The associated points P c and P f lie on rays
through the principal point H. The distances HP c and HP f are connected
by a mathematical relation which does not depend on the polar angle of the
ray. The law of refraction allows us to calculate several “support points” of
the relation and derive further intermediate points by interpolation.
Finally, let us weigh the advantages and disadvantages of the fisheye lens.
The one great disadvantage is the “distortion” of general straight lines. After
half a millennium of “classical perspective,” this visual feature is very unusual
to our eyes. Yet, to be fair: The fish eye would have to be compared to an
400 Photographic image and individual perception

Fig. 9.61 Diagonals remain straight (II)

ultra-wide-angle lens, and the images produced with such a lens are equally
unusual.
This disadvantage can be mitigated to some degree by using the lens in a
specific way: Straight lines which pass through the principal point remain
straight, just as circles parallel to the image plane remain circular.
The advantage of a fisheye lens lies in its capacity for photographing scenes
from extreme perspectives which would otherwise be “impossible to capture.”
Fig. 9.62 on the right illustrates this very point: The inside of a cockpit can
only be wholly captured by a fisheye lens. In this particular case, the image
does not even seem to be very unnatural, because the original scene is not
dominated by long straight lines. The human head is also largely undistorted
in the photograph. The affine stretching of the circular image into an ellipse
even enhances the “landscape effect” of the image.

Fig. 9.62 Anything is possible!

Fig. 9.62 on the left demonstrates a further advantage: The toes in the image
appear so large because they were only two centimeters away from the lens –
the remainder of the scene (at the edges in particular) is downscaled, which
neatly harmonizes with the extreme magnification. What is particularly no-
ticeable, however, is the enormous depth of field offered by the lens. This, in
turn, is an effect of the very small focal length [9].
9.4 Geometry at the water surface 401

9.4 Geometry at the water surface

In this section, we deal with the physical and optical peculiarities at the
boundary surface between air and water (the latter being optically denser).
In particular, we shall consider the reflections at the top and bottom sides of
the division plane, as well as the refraction towards or from the normal of the
boundary surface, depending on the medium that one inhabits. The refraction
towards the normal happens in the optically denser medium. Knowing a point
of the light ray and another point of the refracted light ray, we calculate the
refraction point by solving a degree-four algebraic equation. In this particular
context, its solution is always unambiguous. When looking through the water
surface, a plane can never be seen in an edge view. Furthermore, a general
object inside the water cannot be precisely reconstructed, no matter how
many pictures of it are taken from outside the water.
Adapted eyes

Fig. 9.63 Adapted eyes

⊕ Remark : Water is our elixir of life – it is almost universally beloved among our
own species and among many of our mammalian relatives (Fig. 9.63 on the left
and right). Animals which frequently alternate between land and aquatic life often
possess specially adapted dichotomous eyes, such as the 5 mm long whirligig beetle
in Fig. 9.63 (at the center) or the four-eyed fish in Fig. 9.58.
The seal (Fig. 9.63 on the right) is capable of changing the focal length of its eye
lens by muscle contraction, though it tends to be rather short-sighted on land (see
also application p. 409). For the same purpose, some reptiles have a transparent or
translucent eyelid that is drawn over the eye when they descend below the water
surface. Crustaceans possess telescopic compound eyes, which apparently enable
them to see rather well above water.
There are some small animal species that can run across the water surface without
sinking, such as the predacious water strider in Fig. 9.64 on the right or the wolf
spider (not a water spider!), pictured with its egg cocoon in Fig. 9.65 at the center.
Other animals, such as the Nepomorpha in Fig. 9.64 on the left, can even move on
the bottom side of the water surface!
Like all small objects, small creatures do not immediately sink into the water be-
cause their body surface, or contact surface, is relatively large compared to their
402 Photographic image and individual perception

Fig. 9.64 Danger from above and below

Fig. 9.65 Surface tension can carry a number of objects or animals, as long as they are small
enough.

weight. If an object is k times smaller than a similar one of regular size, its surface
is k times larger compared to its weight. (This relation has very important conse-
quences in nature and can easily be proven with the example of a cube.) This is
why a sufficiently small coin (Fig. 9.65 on the left) will float on the water surface
under ideal conditions. In large creatures, the pressure exerted on the water surface
is greater than the counteracting water pressure. ⊕

Reflection at the surface

Flatly incoming light rays (Fig. 9.66) are almost completely reflected by
water, whereas total internal reflections occur underwater (Fig. 9.68). As we
shall soon see, this is only the case up to a certain well-defined threshold
angle of incidence.
The total internal reflections in Fig. 9.70 show the tiles on a vertical wall
or the floor of a basin. The right image shows the reflection of both types
of tiles, as well as the abrupt ending of the total internal reflection at the
threshold angle.
The Snell-Descartes law of refraction
The speed of light is 299 792 458 meters per second but only in vacuum. In
the air, this speed is a little bit smaller, whereas in water, it is only about
75 percent of the original velocity. The ratio n of these speeds is called the
refractive index. When ordinary (visible) light hits water, this index is about
9.4 Geometry at the water surface 403

Fig. 9.66 Perfect mirror from above (flat viewing angle)

Fig. 9.67 Partial reflection (steeper viewing angle)

4 ∶ 3. Different light components (such as the colors of the rainbow) have

slightly different indices.
The wave structure of light plausibly explains the refraction of light rays (the
wavefront tilts upon hitting the separating layer). Snellius’s law of refraction
can immediately be derived from Fig. 9.69 by

BC1 v1 BC1 AC2

= =n⇒ =n⋅ ⇒ sin α1 ∶ sin α2 = n.
AC2 v2 AB AB
If we are underwater and wish to calculate the threshold angle for the total
internal reflection, we only need to determine the inverse of the refractive
index. Apparently, Snellius’s equation does not always provide real solutions

Fig. 9.68 Perfect mirror from below

404 Photographic image and individual perception

Fig. 9.69 The wavefront explains the law of refraction.

for the exit angle. A total internal reflection occurs starting at the limiting
case
sin β = 1/n = 3/4 ⇒ β ≈ 48○ .
From then on, the water surface acts like a perfect mirror, and because the
threshold angle is slightly different for different wavelengths, a rainbow effect
can be seen at the transition area (Fig. 9.57).

Fig. 9.70 Image raising from above and below

Who is looking towards the left in Fig. 9.56? The answer is not trivial at all.
Fish A sees fish C both directly and as a reflection and fish B only directly,
including its tail fin as a reflection outside of the cone Γ’s base circle with
the critical opening angle of 2 ⋅ β. The cone Γ may be called the limiting cone
if seen from the perspective of A.

Image raising
The so-called effect of image raising causes confusion when we look into
the water. The flatter the viewing angle, the more pronounced these effects
become. Fig. 9.70 shows an extreme example. The upper crocodile (which
does not actually exist) is flatter than the lower one – which, in turn, is not
exactly where it appears to be.
9.4 Geometry at the water surface 405

Fig. 9.71 Image raising at the swimming pool

Fig. 9.71 shows a pool of constant depth, though one constantly has the
impression of standing next to the deepest spot. The distortion effects appear
at their most extreme if one moves sufficiently close to the water surface.
Projecting planes

Fig. 9.72 Can a plane be turned into a projecting plane through refraction?

In Fig. 9.72 on the left as well as in Fig. 9.72 on the right, the following
question seems to arise: Is it possible to see planes through the water sur-
face in an edge view? Fig. 9.72 at the center illustrates that the points of a
surface are always distributed on a polygon (which, in the general case, is
not convex). Only the planes which contain the normal to the water surface
through the eye point appear as straight lines.

Fig. 9.73 “Folding” of a plane into a ruled surface

Fig. 9.73 on the left shows two ruled surfaces appearing as straight lines from
the position of the eye. They are generated by straight lines which correspond
to the pencil of rays through the straight line in the division plane after the
refraction. From an algebraic point of view, the symmetric parts of the surface
406 Photographic image and individual perception

also belong to it. The viewing rays emanating from the eye point E belong,
after refraction, to the normal congruence of an ellipsoid of revolution or a
hyperboloid of revolution (Fig. 9.73 on the right). The focal surface of the
congruence is created by a rotation of the respective focal lines (diacaustics).
▸▸▸ Reconstruction with two eyes underwater
Fig. 9.74 in the top left shows how a point S in space can be perceived
at another, well-defined position – at least in the case of planes which are
normal to the surface and which also contain both eye points.
By moving both eye points closer to each other at the transition (Fig. 9.74
bottom left), the intersection point of the refracted rays moves towards a
point on the focal line that does not lie on a normal through S to the water
surface. If S moves at a constant water depth, then a locus curve whose
deepest point is between the eyes (or, in the limiting case, below the eye) is
traced. This explains the illusion of always standing above the deepest point
of the pool (Fig. 9.71).

Fig. 9.74 Reconstruction of an object underwater

Of course, the point S in the water does not generally lie in the main nor-
mal plane through both eye points. According to the law of refraction, the
refracted light rays lie in planes normal to the water surface, and thus, in-
tersect on the normal m through S. On the other hand, the refracted rays
coincide with m at various points (unless the eye points are equidistant from
m). This means that the elongations of viewing rays cannot intersect in the
points of refraction, and therefore, it is not possible to reconstruct a point
S in space precisely, no matter how many photographs are given. The error
increases with the distance of S to the main normal plane. ◂◂◂

Submerged
As a camera is submerged into the water (like in Fig. 9.75), the photographer
is faced with particularly strong refractions, which abruptly disappear as soon
as the camera is wholly submerged and we thus return to regular perspectives
– though the objects now appear enlarged by the factor 4/3. This leads us to
9.4 Geometry at the water surface 407

Fig. 9.75 The transitional layer and the world below

greatly overestimate the mass of underwater objects, which increases linearly

with the cube of an object’s length (see Fig. 9.76).

Fig. 9.76 Simultaneous vision above and below water

Refraction at curved (spherical) transitional layers

We shall now discuss the creation of a rainbow as an example of refraction
in a curved water surface (Fig. 9.77). The primary rainbow is produced by
a refraction towards the normal of the transition surface, a one-time total
internal reflection, and a final refraction away from the normal of the transi-
tion surface (Fig. 9.77 center) – all due to the presence of water droplets in
the atmosphere after a downpour. The total internal reflection at the back
side of the droplets only occurs for light rays which strike the droplets at a
flat angle. An exit angle of about 42○ produces a maximum scattering into
spectral colors. Blue wavelengths are refracted the most strongly, while red
wavelengths experience the least refraction. This causes red colors to appear
at the outer edges.

Fig. 9.77 Rise of a rainbow

408 Photographic image and individual perception

A rainbow is not actually a circle but a cone of revolution which appears

as projecting to the observer and which possesses half an opening angle of
about 42○ . Driving into the foot of the rainbow is, thus, actually possible,
but only at sunset, on a wet road producing a fine drizzle and at a 42○ angle
to the shadow direction.

Fig. 9.78 Daily illusions

⊕ Remark : Part of the light which bounces about inside the droplets will only exit
after a double reflection, producing a secondary rainbow with its red strip on the
inside that is very faintly visible at 51○ (Fig. 9.77 on the left).
The spectral colors’ varying indices of refraction cause a scattering of sunlight at flat
angles of incidence. The more energy-rich components (blue, green) are so strongly
scattered by the atmosphere that they do not even reach the observer. Due to the
curvature of the light rays, the red Sun remains visible on the horizon up to several
minutes after it should have already disappeared (Fig. 9.78). ⊕

Fig. 9.79 Mirages

Fig. 9.80 Water in the desert?

9.4 Geometry at the water surface 409

⊕ Remark : Mirages also result from continuous light refraction (as in Fig. 9.79 and
Fig. 9.80). The hot layers of air close to the desert surface or paved road are optically
less dense. Light rays which would otherwise reach the ground are refracted away
from the perpendicular and reach the spectator via a detour. The spectator thus
notices an apparent reflection in the elongation of the curve tangent. ⊕
▸▸▸ The optical system of our eyes
The human eye is a highly interesting optical device. Light rays are first re-
fracted at the cornea, proceed into the “area” of the intraocular fluids, and
then only have to move through the inner lens before finally lighting the
retina. First of all: Calibrating sharpness is only possible within a very small
deviation from the optical axis, though a greater viewing cone is automati-
cally captured by “scanning” (eye-rolling).

Fig. 9.81 Focusing and adaptation using the lens

In 1911, the Swedish ophthalmologist Allvar Gullstrand received the Nobel

Prize for Medicine for his precise model of the optical rules that determine
image construction in the eye – a model that is still used today and specifies
curvature radii (including the indices of refraction) for all optical media that
are involved. As mentioned in the application p. 278, the light rays undergo
their greatest refraction at the cornea.
Fig. 9.81 shows a computer simulation based on standard values. Light rays
which are parallel to the axis are precisely focused on the retina in several
steps. A point on the axis (center image) near the cornea first appears very
blurry. Adaptation quickly changes the curvature radii of the lens via muscle
contraction, though as close an object as the one pictured in the right sim-
ulation image can only be sharply seen by children and very short-sighted
adults. ◂◂◂

▸▸▸ How does insect vision work?

According to the rules of geometry, three-dimensional vision – or rather the
ability to estimate depth values – generally requires two separate images of
the spatial situation.
410 Photographic image and individual perception

Fig. 9.82 Fixed insect eyes produce curved perspectives.

It is sometimes stated that the simultaneous vision with two eyes and the sub-
sequent reconstruction of a three-dimensional model from the images require
exceptional brain power, of which an insect would simply not be capable.
The logical conclusion would be that insects do not possess spatial vision.
Human vision can only partially be compared to the vision of a compound
eye. The eyeball of “higher animals” moves almost spherically – it rotates
with two degrees of freedom. If we, as humans, fixate upon a not-too-distant
point, we direct the optical axes of both eyes towards it and then “calculate”
the distance of the point from the angle of the intersecting axes.

Fig. 9.83 An eye excellently adapted for spatial vision

An insect eye is “fixed.” A point in a shared field of vision is observed by ex-

actly one facet (ommatidium) of the left and right eye respectively (Fig. 9.84).
If, however, a point is visible by the facet x of the left eye and the facet y
of the right eye, then its spatial position can be determined – so to speak
– “in hardware”. No further measurement of angles or reverse calculation of
distances is required!
9.4 Geometry at the water surface 411

Fig. 9.84 The optical axes of the ommatidia intersect at fixed points.

If, for instance, an insect has 1000 facets (ommatidia) in each compound eye,
and if 200 of them share the same field of vision in each respective eye, then
there are thousands of such fixed positions that the insect may be able to
interpret without even “thinking about it” – just as the predefined joints and
the rigid chitin exoskeleton allow for several thousand leg positions which
the insect probably does not “think about” either.

Fig. 9.85 Insects are not the only ones good at judging distances.

⊕ Remark : Jumping spiders (Fig. 9.83 on the right) possess multiple pairs of eyes
which enable them to see excellently – and even in color. Naturally, only a distance
area of a few centimeters can be important to such a small an animal. Fig. 9.85
on the left shows the way in which a reptile targets its prey. The insect may not
even be aware of the immobile predator. The right image shows the superb eyes of
a flying fox (not an insectivore, by the way), which are well-suited for manoeuvring
through dense treetops. ⊕ ◂◂◂

Nature has invented and reinvented the eye many times over; so, there must
be an enormous evolutionary benefit to having good eyes. The mere fact
that insects have been using compound eyes for many hundreds of millions
412 Photographic image and individual perception

of years proves what an important step they must have represented in the
evolutionary process. From the perspective of physics, it may be stated that
insects are rather “wasteful” in their usage of light due to the absence of lenses:
Unlike in lens-based eyes, light rays are barely focused in the compound
eyes of insects. However, the following example shows that an even simpler
possibility exists.
▸▸▸ A pinhole camera is actually enough!
Seeing, in its most basic form, is less complicated than one may imagine. The
first attempts at photography show that a simple box impermeable to light
is sufficient. A small hole is made at the center of one of its sides, opposite
a light-sensitive layer which captures the outside world (Fig. 9.86, left box).
In these primitive conditions, a considerable amount of luminosity (like a
flashlight) is required in order to illuminate the photo layer adequately.

Fig. 9.86 The simple geometrical background of a camera obscura: The light rays enter a dark
box, illuminating the back wall. Right box: When we insert a mirror under 45○ , the image is
diverted to the top wall.

When we insert a mirror into our dark box (Fig. 9.86, right box), we can
divert the created image, and thus, make it visible on a waxed paper. (We
just have to keep our head and the upper part of the box shrouded by a piece
of black cloth.)
An experiment, such as the one by Marianne Kampel, Christoph Mandl, and
Markus Jagersberger, may now be conducted: The “box” may be a cuboid
tent – completely impermeable to light – with a very small hole. If an observer
is put into the tent, they would be astonished by a projection of the outside
world onto the walls and the floor (once their eyes have adapted to the dark-
9.4 Geometry at the water surface 413

Fig. 9.87 Left: The principle of the “camera obscura”. Right: A tent impermeable to light.

Fig. 9.88 Incredible: This is what the inside of the tent looks like once the eyes have had enough
time to adapt. Almost razor-sharp images of the outside world are wrapped around the inside of
the cuboid, and even around its edges!

ness). This effect can also be “proven” fairly easily by taking a photograph
from inside the tent with a very long exposure time (Fig. 9.88). ◂◂◂

▸▸▸ The small step towards a real camera

Pinhole cameras have the disadvantage that only very little light can enter
the system. When we enlarge the pinhole, the image immediately becomes
blurred. We have to insert a convex lens that collects the light on the back
plane (Fig. 9.89 on the left). When the focal length of the lens is chosen
such that a point of interest is “bundled” via a light cone on the back plane
(according to Fig. 9.31), we get a sharp (and bright) image. Remarkably, this
cone can again be “deviated” by a 45○ -mirror like the one in Fig. 9.86, which
projects a clear and sharp image onto a sheet of waxed paper (Fig. 9.89 on
the right).
This is how nature developed different eyes: A pinhole had to be covered by
a refracting lens (Fig. 9.90). Some “fine-tuning” was achieved by inserting an
additional lens, which enabled a slight varying of focal length (Fig. 9.81). The
combination of multiple lenses, as found in sophisticated optical objectives,
is probably a human invention (Fig. 9.91). ◂◂◂
414 Photographic image and individual perception

Fig. 9.89 When we enlarge the pinhole, we allow more light to enter, but we need a convex lens.

Fig. 9.90 Opening and closing the pinhole . . .

Fig. 9.91 Placing several lenses in front of another makes the situation more complicated.
10 Kinematics: Geometry in motion

Kinematics – the geometry of motion – may very well be called the “paradise
of geometers.” It is where the insights and theorems of geometry are ap-
plied to solve difficult problems. Some of the basic laws of kinematics appear
remarkably simple, the basic point being the formulation of geometrical or
physical propositions about moving mechanisms. The paths of points or the
envelope curves (or surfaces) are of particular interest, as are the consider-
ations about instantaneous velocities and accelerations.
In the plane, any system of arbitrary complexity rotates at any given moment
about a precisely determined (instantaneous) pole, which may also be a point
at infinity. From this insight, conclusions may be drawn about path tangents
and instantaneous velocities. Envelopes of moving curves can also be captured
in this way, as well as curvatures and instantaneous accelerations of path
curves, provided that certain additional information is given.
The most important types of transmissions will be categorized and studied
in this chapter, with a particular focus on the central role of the mechanical
linkages (such as the hinge parallelogram or the antiparallelogram). Many
technical challenges can be mastered by their use.
The ellipse motion, whose generalization is the trochoid motion, will fre-
quently reoccur in our considerations. Trochoid motion plays an important
role in gear theory.

© Springer Nature Switzerland AG 2020 415

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_10
416 Kinematics: Geometry in motion

10.1 The pole around which everything revolves

Constrained motion of two points

If an object can only be moved in a predetermined way, it is undergoing a
so-called constrained motion. Let us imagine a triangle ABC in the plane
(such as our set square). If we move A along a predefined curve a and simul-
taneously move B along another predefined curve b, then C (and any other
point of the “triangle system”) is left with no choice: It is constrained to move
along a path curve c, which is already predetermined.
▸▸▸ Two-point guidance with a bicycle
The bicycle in Fig. 10.1 is driving on a wavy road. It is intuitively clear that
points on the bicycle frame, such as the saddle’s point of support, “have no
choice.”

Fig. 10.1 Two-point guidance

The bicycle would move in exactly the same if its wheels did not spin but
were simply pushed along the road. The two contact points between the
wheels and the road are precisely defined, and the path curve (including the
motion) is thus determined.
⊕ Remark : As the picture shows, the cyclist is not rigidly connected to the bike.
For him, the aforementioned rules only apply to a limited degree – small deviations
from the theoretical path result, among other factors, from the pedaling motion and
from the absorption of impacts. ⊕ ◂◂◂

Two path normals are sufficient

In the first chapter (Fig. 1.8), we showed the following: Two pairs of (di-
rectly congruent) positions A1 B1 C1 and A2 B2 C2 of a triangle ABC can be
transferred into each other by a uniquely defined rotation. The center of the
10.1 The pole around which everything revolves 417

rotation P12 is located at the intersection of the perpendicular bisectors of

A1 A2 and B1 B2 . The third bisector C1 C2 automatically passes through P12 .
Let us now consider a kind of “stroboscopic pair of photographs” within a
very short period of time. Both triangle positions will be nearly identical,
but the construction of the bisector intersection point will be exactly the
same. The bisectors of two points A1 , A2 on a curve a and B1 , B2 on a
curve b are to be determined. A1 A2 and B1 B2 are, thus, the chords of the
predefined curves a and b. We can use this insight to make the following
limiting-case extrapolation: If the time interval for the stroboscopic photos
is “infinitely small,” the chords of a and b will converge towards the tangents
of the curves. The perpendicular bisectors, thus, become the curve normals
at A and B (Fig. 10.2). Their intersection yields the precise instantaneous
center of rotation – the instantaneous pole P . All other limiting positions
of the perpendicular bisectors (i.e., the path normals) automatically pass
through P .
No matter how complicated the motion of our set square becomes, if we
mark the path curves a and b of two points A und B, we can reconstruct the
motion by moving A on a and B on b. This leads us to the following always
pertaining, fundamental

Theorem of path normals: At any instant of a planar motion, all path normals
pass through a single point – the instantaneous pole. During translations, the
instantaneous pole is a point at infinity.

Fig. 10.2 Constrained motion: two-point guid- Fig. 10.3 In which direction does the saddle
ance (A ∈ a, B ∈ b) move?

▸▸▸ Two-point guidance with the bicycle (continuation)

Let us construct the path tangent of a saddle point C in Fig. 10.3.
Solution:
The path normals of the wheel contact points A and B are given by the
418 Kinematics: Geometry in motion

shape of the road. All path normals pass through their intersection point,
the instantaneous pole P . The path tangent tC of C is, thus, orthogonal to
P C.
⊕ Remark : If the bicycle moves on a planar road, it undergoes a pure translation,
where all path tangents are parallel. The instantaneous pole is the point at infinity
of the normal direction. ⊕ ◂◂◂

Instantaneous velocity
During rotations, the distribution of velocities is very simple: The center of
rotation does not move at all, and thus, it has zero velocity. The further all
other points are from the center, the faster they move. The absolute value of
the velocity increases linearly with the distance from the center of rotation.
Envelopes
If we know two path normals of an arbitrary constrained motion, we can de-
termine the path tangent for any point. However, this is not enough, because
it is also true that:

If a curve c is moved by a constrained motion, then the pedal points of the

normals drawn from the instantaneous pole on c are points of the envelope of c.

Proof : We can calculate points of the envelope by intersecting two immediately

succeeding positions of c. We shall now show that these intersection points are pre-
cisely the pedal points Ti of the normal: Like any other points, they have path
tangents which are orthogonal to the line connecting them with the instantaneous
pole P . However, following the definition of the pedal point of the normal, this path
tangent touches the curve c. This means that the points Ti are the ones that cur-
rently move along c. A “neighboring position” of c thus intersects c at the points Ti .
⊙

Fig. 10.4 Astroids and parallel curves as the envelopes of a straight line

In the most simple case, the curve c is a straight line. It thus contains ex-
actly one pedal point of the normal through the pole. Fig. 10.4 (on the left)
10.1 The pole around which everything revolves 419

illustrates that a curve is produced if the straight line c is moved such that
a fixed point X is directed along the x-axis and another fixed point Y is
directed along the y-axis. The resulting curve is also called an astroid, which
we have already discussed in a prior chapter.
In Fig. 10.4 on the right side, a second straight line c, which is parallel, and
therefore, at a constant distance d to c, is carried along. It envelops a parallel
curve to the aforementioned astroid.
Envelopes of straight lines often possess cusps. Cusps occur in astroids, con-
sidered as envelopes of a straight line, if the instantaneous pole lies on the
straight line – a situation which may arise in the course of the motion. The
cusps of the parallel curves occur at the same time.

Fig. 10.5 Paths of points and envelopes of straight lines of a cycloid motion

A special situation occurs if a moving point happens to coincide with the

instantaneous pole. The rolling of a wheel along a straight line (Fig. 10.5)
is an example of such an occurrence. This type of motion is called cycloid
motion. In the course of the motion, each point on the circumference of the
wheel will coincide with the straight line, which produces a cusp on its path
curve. Interestingly enough, a wheel spoke (half of a diameter of the circle)
envelops a similar cycloid of half the size.
The instantaneous pole of a rolling wheel is the contact point of the wheel
with the fixed straight line. For this point the instantaneous velocity equals
zero, and it lies on the path normal of the center of the circle, whose path is
a straight line parallel to the horizontal road.

Fig. 10.6 Envelope of a generic straight line

The envelope of a generic straight line during a cycloid motion is a curve

which is an offset curve of a cycloid (Fig. 10.6): In this case, one can simply
translate the straight line by a constant vector into the center of the circle.
420 Kinematics: Geometry in motion

Fig. 10.7 Envelope of a special circle

If we take a circle (instead of a straight line) in the moving system of the

wheel, we get an envelope with two branches, which stem from two possible
circle normals (respectively) through the instantaneous pole (Fig. 10.7). Let
us first consider the special case where the center of the circle M lies at the
boundary of the rolling circle. We are now faced with positions where M
becomes the instantaneous pole. In this case, every point of the circle which
was moved along is a pedal point of the normal through P . Therefore, these
circle positions are parts of the envelope.
In Fig. 10.8, the center of the circle is no longer a point at the edge of
the rolling wheel (on the left, it is inside the wheel, and on the right, it is
outside). The last two figures permit an interesting spatial interpretation: If
we interpret the separate positions of the moved circle as normal views of
spheres (Fig. 10.8 in the middle), then we can recognize the normal view of
a helical pipe surface whose pitch is (at first) precisely defined. If we then
apply an orthogonal affinity in the direction of the straight line that is the
midpoint’s path, we can vary the pitch arbitrarily.

Fig. 10.8 Envelope of general circles and a spatial interpretation

The outline of a helical pipe surface in a normal projection is composed of

parallel curves of an affinely distorted cycloid.
The cycloid is a frequently recurring curve. As Johann Bernoulli discovered
in 1696, it is the curve connecting two points A and B located at different
heights, along which a particle under influence of gravity can get from A to
B in minimal time (ignoring friction). In this particular context, it is called
the brachistochrone curve.
10.1 The pole around which everything revolves 421

Fig. 10.9 Brachistochrone curve Fig. 10.10 Cycloidal pendulum

A pendulum which swings on a cycloid in the way pictured in Fig. 10.10

can be used as an exact harmonic pendulum. The frequency of oscillation
is independent of the amplitude. In this context, the cycloid is called the
tautochrone curve. The pendulum clock, which relies on this property of the
cycloid pendulum, dates back to Huygens, who utilized the fact that the
evolute of a cycloid is another congruent cycloid. However, the error caused
by friction was greater than the increase in accuracy due to the isochronicity
of the oscillation.

Fig. 10.11 Envelopes of straight lines

Envelopes often occur in reflections and tend to produce beautiful patterns.

Fig. 10.11 shows the reflection of a CD in the LCD screen of a laptop. The
ellipses around the center of the reflection come from the CD’s circular edges.
The pointed focal line at the center, however, is only indirectly caused by
the reflection. As with the Newton rings in application p. 215, it is actually
a result of interferences. The curve remains visible – if slightly less so – after
the disk has been partially covered.
But what exactly do reflections have to do with “motion”? Well, that is a mat-
ter of interpretation. One might say: “ ‘If a light ray moves along a curve . . . ”.
The scale of the time during which this “motion” occurs is irrelevant. It may
even happen at speed of light.
Relative motion
As a spectator, it is often practical to adapt to an existing motion. For
example, in order to understand the path of the Sun on the firmament, it
is useful to switch from the “terrestrial system” into the “solar system.” The
solar system, however, is itself in motion: Not only does it move away from
the position of the Big Bang at very high speed, it also rotates about the
center of the Milky Way.
422 Kinematics: Geometry in motion

Fig. 10.12 Relative motion: The excavator is being moved by its own scoop.

Fig. 10.13 Relative motion: It is not the tool that moves but also the work piece!

If one follows this logic to its end, one must conclude that all motion is
relative. We can hold onto a wooden piece and sculpt its outer layer (as
sculptors do), or we can clamp it into a lathe and let it spin so as to change
it by means of a lathe tool (as in Fig. 10.13).
We will soon deal with more examples of relative motion (such as application
p. 444).
10.2 Different mechanisms 423

10.2 Different mechanisms

Let us shake hands!

Fig. 10.14 Simulation of the opening and closing of a palm

The act of shaking hands is a multi-layered process, and most of it happens

inside the brain, including the steering of the hand and the impressions of the
“receiver.” Being greeted by the wooden hand in Fig. 10.14, one is surprised
by the realistic nature of the feeling (it is, however, a left hand, which relates
to the android we used in the following example). The mechanism itself is
as simple as it can be: A tug on a bundle of strings pulls the slightly curved
fingers at their lower joints.
The angles of rotation of the separate fingers are slightly different, which can
be regulated by the distance of the strings’ starting points. The thumb rotates
about its “own” axis, which causes the subjectively important embracing
sensation. Such a wooden hand, as simple as it is, can already be used to
grab and reposition certain small objects (like chess pieces).
One may insert an additional moving hinge joint to each finger, which pro-
duces a so-called two-bar linkage.
Two-bar linkages
A rod AB with length r rotates about its fixed endpoint A while a second
rod BC with length s rotates about its end point B. This mechanism can
be used to reach any point within a circle with the radius r + s about A as
long as s is greater than r (otherwise, it is an annulus with an inner radius
of r − s): The position of B is derived from the intersection of the two circles
– one about A with a radius of r and another about C with a radius of s.
Two solutions exist if A and C are assumed to be fixed.
424 Kinematics: Geometry in motion

r
A
s

Fig. 10.15 Two-bar linkage: Any point C within a circle or an annulus about A can be reached.
On the right: The arm of a pepper grinder can be freely rotated in one direction.

▸▸▸ The chess-playing android

The late 17th and the early 18th century were the age of “androids” – human-
like automata whose intricate mechanical interior rendered them capable
of remarkable feats. Wolfgang von Kempelen famously developed a “chess-
playing Turk” (Fig. 10.16).

Fig. 10.16 The Chess-playing Turk and historical attempts at explaining the phenomenon

Many contemporaries have racked their brains trying to explain the supposed
artificial intelligence in mechanical terms, while the actual solution was some-
thing of a cheat: A little person who was hidden inside the box managed
to receive information about what was happening on the chess board, and
steered the left hand of the marionette so as to move the chess figures to the
intended spot.
At the time of the android’s creation, the real secret was never lifted. An
explanatory theory by Joseph Friedrich von Racknitz was verified at the
University of Applied Arts Vienna in a computer simulation, and a subse-
quent mechanical realization (Fig. 10.17) was deemed to be successful. The
hand in Fig. 10.14 was made of wood, like much of the rest of the automaton.
10.2 Different mechanisms 425

Fig. 10.17 Possible kinematic model of the chess-playing android (according to Joseph Friedrich
von Racknitz )

Without further elaboration, it is sufficient to say that the core piece was a
two-bar linkage formed by the upper and forearm. ◂◂◂

▸▸▸ The joints of articulate animals

In articulate animals (insects, spiders, and crabs being among them), mecha-
nisms such as the two-bar linkage (or their generalizations) are implemented
to perfection. The complex motions of the legs, claws, and pincers are not
even directed by the brain but by “quasi-microprocessors” located directly in
the joints!

Fig. 10.18 A multitude of two-bar linkages and their generalizations

Not quite as trivial as the limbs – and only visible in a close-up photograph:
The bees in Fig. 10.19 extend a sort of “auxiliary jaw” where their tongue
is located. Through this auxiliary jaw, the bee’s tongue can penetrate the
farthest corners of the mallow blossom pictured in the photograph. ◂◂◂
426 Kinematics: Geometry in motion

Fig. 10.19 Something close to a two-bar linkage: One can observe the mandibles (including the
tongue) of the black carpenter bee.

▸▸▸ Leonardo’s cranes

Cranes have been “commonplace” for much of the last century, and it is,

Fig. 10.20 Three sketches by Leonardo da Vinci

therefore, easy to forget just how innovative the sketches made by the genius
from Tuscany actually were. You can exercise your spatial imagination by
rotating the various wheels in your mind and considering the directions into
which the loads may be heaved. ◂◂◂

The hinge parallelogram

If a parallelogram ABCD is held at one side (for instance, AB), and if the
parallel side (CD) is left to rotate, one deals with a hinge parallelogram
(Fig. 10.21). Each point that is connected to CD describes a path circle
congruent to the path circles of C and D. Hinge parallelograms are often
found in daily life, for example, in desk lamps or in letter scales (Fig. 10.22).
Parallel translations in all directions can be accomplished by a combination
of multiple hinge parallelograms (old drawing machines in drawing offices
employed this trick). A particularly clever variant is visible in Fig. 10.23,
which shows a tripod that can be used to keep heavy cameras steadily parallel
to the ground – even while walking.
10.2 Different mechanisms 427

Fig. 10.21 A hinge parallelogram: Each path is a circle.

Fig. 10.22 The hinge parallelogram in a letter scale and in a designer lamp

Fig. 10.23 Hinge parallelograms in a portable tripod (Robert Eder)

▸▸▸ Pantograph
The pantograph, sometimes called “lazy tongs” (Fig. 10.25), consists of par-
allelograms strung together and can often be found, for instance, in the
adjustable mounting of lamps, sunblinds, or even whole platforms, as in
Fig. 10.26. ◂◂◂
428 Kinematics: Geometry in motion

Fig. 10.24 As ingenious as it is simple: How to turn a cardboard box into a lectern (by Christian
Bezdeka www.bezdeka.com): This construction is based on so-called lazy tongs. In its final position,
the lectern can even be tilted!

Fig. 10.25 Panthographs . . . Fig. 10.26 . . . in theory and practice

Antiparallelograms
If a hinge parallelogram is stretched into its longest form, its motion may
continue in two different ways: either, as outlined before, as a parallelogram
or as an antiparallelogram, with only one pair of parallel opposite sides. In
Fig. 10.28, such a hinge LABM can be seen. A moves on a circle about L
and B on a circle about M . Thus, the two longer sides are path normals, and
their intersection is the instantaneous pole P . The sum of the distances LP
and M P = AP is constant (= LA), which causes P to move along an ellipse
with the focal points L and M . For reasons of symmetry, an ellipse which is
symmetric with respect to the ellipse’s tangent in P and which has the focal
points A and B exists in all positions and belongs to the moving system.
It does not always have to be a uniform rotation . . .
The hinge antiparallelogram has an interesting application: The pole curves
are congruent ellipses which roll onto each other. If these ellipses are furnished
with gear flanks, they can be made to roll, as in Fig. 10.29.
If, instead of the principal axis of the ellipse being held at rest, the single
focal points of both ellipses are fixed, the uniform rotation about one focal
point can be transformed into a periodically non-uniform rotation about the
other focal point. Conversely, it is sometimes possible to diminish the effects
10.2 Different mechanisms 429

Fig. 10.27 Practice and . . . Fig. 10.28 . . . theory of the antiparallelogram

Fig. 10.29 Elliptic gears

of certain non-uniform rotations and even to equalize these rotations in ideal

situations.

Fig. 10.30 Elliptic gears move downwards.

Fig. 10.30 shows a very original method of propelling elliptical gears: Both
gears are placed within a cylindrical container – bounded by cylindrical discs
– and are moved downwards by gravity.
Observing elliptic gears in slow motion may, to some readers, be more calming
than an hourglass, though a certain similarity can be observed: Once the gears
have reached the bottom, the entire cylinder can be turned upside down so
that the motion proceeds anew. Notice the small pocket of air which has
crept beneath the right gear.
The crank shaft
If a point A is moved on a circle a about a fixed point L, and a point B which
is rigidly connected to A is also moved along a straight line b (Fig. 10.33), we
430 Kinematics: Geometry in motion

Fig. 10.31 Ancient gears? Columns which have fallen apart!

Fig. 10.32 Crank-shaft propul-

sion Fig. 10.33 Crank shaft of an Otto engine

are dealing with a crank shaft. Fig. 10.32 shows the crank shaft propulsion
of a workbench and Fig. 10.33 the crank shaft of an Otto engine (see also
Fig. 10.35 in the top right corner).

B1 P1 P4 B4
ω1
ω1P2 B2 B3 P3
ω2
ω2 v⃗A4
A1 A4

L1 v⃗A1 L2 v⃗A3 L3 L4
A2 A3
v⃗A2
Fig. 10.34 Construction of the velocity vector of B in four different positions

▸▸▸ How fast is the piston?

When the piston oscillates inside the cylinder of an Otto engine, its speed
changes dramatically. In its deadlock positions (highest and lowest) the ve-
locity is zero. In between those positions the velocity varies considerably. In
10.2 Different mechanisms 431

order to determine the speed in an arbitrary in-between position, we first

look for the instantaneous pole P (Fig. 10.34).
The pole P is found by intersecting two path normals: The point L of the
crank is fixed. The endpoint A of the crank rotates uniformly around L and
its orbit is a circle. Therefore, the path normal in A is the line LA. The
point B on the connecting rod is forced to run on the vertical axis of the
cylinder. Thus, its path normal is the horizontal line through B. The position
of P is now determined, and it is the only point of the system connected with
AB with a velocity of zero. In that very moment, he whole system rotates
around P . The velocity of any point of the system AB is proportional to
its distance from P . Since we know the velocity vector v⃗A of the point A
(a vector of constant length perpendicular to LA), we can determine the
velocity vector v⃗B in B via similar triangles: The angle ω from the normal
to the end points of the velocity vectors is equal. ◂◂◂

Fig. 10.35 Sawmill according to Leonardo da Vinci , with remarkably detailed solutions

▸▸▸ Sawmill by Leonardo da Vinci

Leonardo da Vinci has been known for his ability to plan mechanical con-
structions down to their smallest detail, without necessarily realizing these
ideas in practice. His sawmill (Fig. 10.35 on the left) is still in use today and
has undergone only slight changes. The crank shaft propulsion (Fig. 10.36),
propelled by water, lies at its core.

Fig. 10.36 In the sawmill, propulsion is generated in a manner comparable to that of the piston
engine.
432 Kinematics: Geometry in motion

The platform that pulls the pieces of wood towards the running saw blade is
driven by a subtle mechanism: A rod s, which is rigidly connected to an axis
a, is forced by a loop to resonate with the driving rod. Another rod t, to which
the hook h is attached, is also rigidly connected to the axis a. When swinging
back, this hook attaches to a cog, and when swinging forward, it glides over
the tips of the triangular teeth – thus, it can move the gear forward step by
step, which in itself pulls the platform over a shaft. ◂◂◂

Fig. 10.37 Path curves of four-bar linkages

General four-bar linkages (hinge quadrangles)

A four-bar linkage (hinge quadrangle) is a mechanism made of four rods. De-
pending on the length of these rods, there exists a huge number of variations
– and a corresponding multitude of applications. Once again, let LM AB be
the hinges of the quadrangle (Fig. 10.37). If the points L and M are fixed,
then the motion of the system connected to the rod AB is called a linkage
motion and the path curves are called linkage curves. The paths of the points
A and B are circles about L and M . The corresponding path normals are the
linkages (or swings) LA and M B. The instantaneous pole P can be found
at the intersection of the path normals. Each point C of the linkage system
always moves orthogonally to the associated pole ray P C.
In addition to the special four-bar linkages, the general forms are also widely
in use (Fig. 10.41). The advantage always remains the same: The devices
are easy to manufacture and to utilize, and beyond that, they are also very
robust.
10.2 Different mechanisms 433

The multiformity of the path curves is virtually unbounded, and the only
challenge lies in finding the right rod lengths for a particular problem.
Once an appropriate hinge quadrangle has been found, it might be necessary
to compensate for the differing path velocities of a system point. Many critical
points often move at very high speeds, whereas other, less important areas are
nearly fixed in dead-point positions. Here, one should attempt to configure
the propulsion of the hinge in a flexible way.

Fig. 10.38 Four-bar linkage with approximate straight line guidance (oil pump)

Fig. 10.39 Four-bar linkage with approximate straight line guidance (crane)

Fig. 10.38 and Fig. 10.39 show two special forms of four-bar linkages that are
very useful. You can see an additional auxiliary four-bar linkage (colored red
in the sketch), which can be driven by rotation. Note that in Fig. 10.39, the
point C ∗ is not directly connected to the linkage. Its path curve is congruent
434 Kinematics: Geometry in motion

to the path curve of C, and the tangent t∗ ∋ C ∗ is always parallel to the tan-
gent t ∋ C. The approximate straight line guidance is of utmost importance
in these cases.
▸▸▸ Film gripper
What is the mechanism underlying the machine in Fig. 10.40? The point A
rotates about the fixed point L. The point B is guided along a circular arc
with the center M . Therefore, M B is constant, as is AB. Point C is rigidly
connected to AB. The mechanism is obviously a four-bar linkage.

Fig. 10.40 Film gripper versions by Leo Mühlfeld (above) and Alan Schiegl (below)

The position of C is chosen in a way that the lower part of its path is almost
straight. The hook connected to C can, therefore, grip something like a film
or balls (like in the bottom image of Fig. 10.40). ◂◂◂

▸▸▸ A useful fixing mechanism (“Mole grips” or “Vise-Grips”)

Pliers are a hand tool used to hold objects firmly. Locking pliers can be locked
into position by means of an over-center toggle action. A lever is used to aid
release from the locked position (Fig. 10.41). Due to the toggle mechanism,
very high forces can be applied with this type of pliers. Grip pliers can be
released by pushing back a lever that overcomes the “dead center” and in
turn releases the workpiece.
Applications include, for example, the loosening of very tight screw connec-
tions, the fixation of workpieces, and the use of pliers as a “third hand”.
The series of images in Fig. 10.41 (left) show how the fixed position can be
adapted to the diameter of the object that has to be fixed. For this purpose,
10.2 Different mechanisms 435

Fig. 10.41 Pliers with a four-bar linkage and a fixing mechanism

a screw is used. When in the fixed position, there are maximal forces at play.
If the screw is correctly adjusted, the fixation process will require hardly any
exertion of force, and the force necessary to hold the workpiece in place is
thus even more impressive. ◂◂◂

▸▸▸ The front tires have to be rotated asynchronously!

What happens if we turn the steering wheel of a car? First, the rotation is
transferred onto a horizontal axis by a Cardan shaft, where it is converted
into a translation by a gear rack (Fig. 10.42). This propels a hinge trapezoid,
which rotates the front wheels – and quite ingeniously at that!

Fig. 10.42 Different rotations of the front wheels. Each tire has a different instantaneous speed
proportional to its distance from the instantaneous pole.

Let us now literally “track back” the process: In order to ensure optimal steer-
ing, the path normals of the wheels’ four contact points must pass through a
436 Kinematics: Geometry in motion

single point – the instantaneous pole. Pairs of wheels are mounted on an axis,
and thus, all wheels must be perpendicular to the straight lines that connect
their centers with P . This can be accomplished with a four-bar linkage. ◂◂◂

▸▸▸ Cognate linkages

Cognate linkages are linkages that allow to generate a coupler curve in
different ways. In case of four-bar linkage coupler cognates, the Roberts-
Chebyschev Theorem states that each coupler curve can be generated by
three different four-bar linkages (Fig. 10.43). These four-bar linkages can be
constructed using similar triangles and parallelograms (left image).

N∗ N

F∗ H∗ F H

G
E∗ G∗ E
C∗ C

L A M
L∗ A∗ B∗ M∗ B
Fig. 10.43 One and the same coupler curve can be generated in three different ways (construction
and different positions). The triangles LM N and ABC are similar.

The possibility of generating a coupler curve in multiple different ways has

one big advantage: there are three distinct ways to find a desired mechanism
that are most suitable to fulfill the requirements. In technology, it is often
necessary to use straight line guidances. The linkages, however, must run
smoothly or must not require space at certain positions.
F
A
B

L L
M M
G
E N N

H
Fig. 10.44 The search for an alternate mechanism for a “stepper” fitness machine. The construction
also works when the triangles LM N and ABC are degenerate.
10.2 Different mechanisms 437

Fig. 10.44 illustrates how one can replace the black fourbar linkage by either
a green or a blue one. Consider a “stepper” in a fitness center where the
trainee needs to move their legs in the given way. The black linkage does not
quite as smoothly as desired (Fig. 10.44 to the right) since there might not
even be enough space left for the movement. Consequently, one of the two
other linkages might be better suited to the task. In this special case, the
triangle ABC degenerates: C lies on the prolongation of AB. Then N has to
lie on the prolongation of LM in the same ratio. ◂◂◂

▸▸▸ A sugar bowl and a four-bar linkage

The sugar bowl pictured in Fig. 10.45 implements a classical, though barely
recognizable, planar four-bar linkage.

Fig. 10.45 The second image from the left shows which points remain fixed (L and M ) and which
have circular paths: A and B, and thus, also D.

Let us consider the three points M , A, and B, which determine the vertical
symmetry plane of the hinge. Now, let us further consider the fourth point L
of this plane (it lies at the center of the imagined, horizontal axis of the mech-
anism, which appears almost projecting in the photo). A hinge quadrangle
LM AB is thus defined.
In order to fulfill the condition that the lid should be closed in the stretched
position on the left, an imaginary, rigid triangle M BD is also moved. The
segment BD materializes through the circular lid, and the segment AB is
likewise curved for aesthetic reasons. The designers of the Biedermeier period
apparently had a penchant for hinge quadrangles. ◂◂◂

▸▸▸ How to translate a continuous rotation into an intermittent rotation

In technology, one task that needs to be performed very frequently is a ver-
sion of “stop and go”, or rather “stop and rotate”. This mechanism is used,
for example, in money counting machines, film projectors, tool changers in
CNC machines, and many more. The so-called Maltesian gear (also known
as Geneva drive) accomplishes this quite efficiently.
Fig. 10.46 illustrates how a rotating dark blue disk drives a light blue cross in
such a way that 3/4 of the time nothing happens: The rotating pin E is not
438 Kinematics: Geometry in motion

E1

E2 E5
E3 E4 C5
C4

C3
C1 C2

Fig. 10.46 The driven cross rotates 90○ and then stands still for a while.

in contact with the driven system (left). In the center image, E hooks into
the slot, drives the cross (positions E2 to E5 ), and leaves the driven system.
A point C of the driven system rotates 90○ during a full opposite rotation of
the blue disk and then then rests three times as long. ◂◂◂

If the arms could change their length

We have thus far discussed mechanisms with fixed arm lengths, which have
obvious advantages with regard to stability under heavy load.
▸▸▸ A hinge quadrangle with a single variable

Fig. 10.47 Hammer mill after Leonardo da Vinci

After the example of the robot arms, the mechanism behind Leonardo da
Vinci’s hammer mill appears to be especially interesting. The simulation in
Fig. 10.47 shows the propulsion of a hammer by the rotation of a spiral
attached to a shaft.
In each position, we are dealing with a quadrangle (A∗ , A, B, B ∗ ) with three
constant rod lengths and a variable one (BB ∗ ). The angle ∢AA∗ B may be
rigid. ◂◂◂

▸▸▸ The Stewart platform

Contemporary robotics allows us to control any arbitrary motion of a plane
10.2 Different mechanisms 439

Fig. 10.48 A triangular platform ABC which is controlled by three robotic arms of variable length
with the fixed points L, M , and N . In the top left corner: “zero position”. On the right: Certain
triangle positions are “dangerous” due to the indeterminacy of their robotic motion.

by the use of three arms of varying length (Fig. 10.48). Despite the state-of-
the-art technologies involved, there still exist “unsafe positions”, such as the
ones pictured on the right. ◂◂◂
440 Kinematics: Geometry in motion

10.3 Ellipse motion

If a rod of fixed length is directed in such a way that its endpoints U and V
follow two straight lines u and v, it undergoes an ellipse motion.

Fig. 10.49 Three closely related ellipse constructions

Such a motion is pictured in Fig. 10.49. Both guiding straight lines u and v are
orthogonal in this case. We are concerned with the path curve of an arbitrary
intermediate point C with V C = a, U C = b. Let F = u ∩ v be the origin of
a Cartesian coordinate system and ϕ the rod’s angle of inclination. Thus, C
is located at x = b cos ϕ, y = a sin ϕ. This is the parameter representation of
an ellipse with axis lengths 2a and 2b. Remarkably, the midpoint R of the
segment U V (a = b) describes a circle.
▸▸▸ Construction of paper strips
Ellipse motion, by its classical definition, can be accomplished by two or-
thogonally intersecting guidance rails even though the mechanism is only
partially practicable: The motion is jerky, and the stretched positions are
undefined, which is why the use of this “ellipse compass” has never really
become widespread.
The actual significance of this mechanism lies in its easy illustration of the so-
called reverse paper strip construction: If an ellipse’s principal vertex (major
axis 2a) and a general point P are given, then half of the principal axis a of
P needs to be inscribed onto the auxiliary axis (the perpendicular bisector
of the principal vertices) in order to get V – though this is possible in two
ways. The straight line connecting P and V yields U and thus b. ◂◂◂

We can find the instantaneous pole P of the motion by intersecting the path
normals at U and V that are parallel to the axis. Compared to R, P is always
at twice the distance from the origin, and thus, lies on a circle about F with
10.3 Ellipse motion 441

Fig. 10.50 Ellipse motion and reverse paper strip construction

a radius U V = a + b and, at the same time, on a circle about R with half

the radius (Fig. 10.49 in the middle). The motion of the rod U V could also
be forced by letting the Thales circle roll over U V within a circle of double
radius. Both circles are named Cardan circles after their discoverer Geronimo
Cardano (1501–1576).
Cardan circles roll on each other without gliding. Each arbitrary point on
the smaller circle becomes, at some instant, the instantaneous pole – the
point V , for example, does so in the point V0 on the y-axis (Fig. 10.49 at
the center) once R has rotated by an angle of ψ. Following an adaptation of
the angle-at-circumference-theorem (p. 10), V has moved on a circle of half
the size by an angle of 2 ⋅ ψ. Therefore, we always correlate circular arcs of
equal length – P̂ V on the smaller circle and P̂V0 on the larger circle – which
requires a rolling motion.
▸▸▸ Ellipses as hypotrochoids The rolling motion of the smaller Cardan

Fig. 10.51 Practical realization of the “inner circle rolling”

circle within a second circle that is twice as large can be well simulated by a
hollow wheel.(Fig. 10.51) ◂◂◂

▸▸▸ Gliding or rolling along straight lines

In Fig. 10.52 on the left, two circles with constant distance glide along straight
lines. Thus, their centers U and V move on straight lines u and v, and we
can see an ellipse motion: Every point that is rigidly connected to the moving
system has an ellipse as path curve.
442 Kinematics: Geometry in motion

Fig. 10.52 Ellipse motion, not immediately apparent: Two circles (or wheels) are moved along
two straight lines. The lengths of the (green) velocity-vectors are proportional to the distance from
the instantaneous pole P .

It does not make a difference whether the circles glide or roll: When a cyclist
moves from one plane to another like in Fig. 10.52 on the right, every point on
the bicycle’s frame is undergoing an ellipse motion. The guiding lines u and v
are the parallels to the given planes through the centers of the rolling wheels
(or gliding circles), the center O = u ∩ v of the ellipse is their intersection.
◂◂◂

▸▸▸ Revolving a Reuleaux triangle inside a square

Consider a regular triangle ABC (side length r). It can be “rounded of” by
means of three circular arcs (radius r = side length), and then has a “constant
thickness” r. Thus, it is possible to revolve it inside a square with side length
r. There are always two vertices that glide along two perpendicular sides of
the square.

Fig. 10.53 A Reuleaux triangle ABC is revolved inside a square: The edges are rounded off “almost
circularly” by elliptic arcs.

Fig. 10.53 shows three subsequent positions of ABC when revolving clock-
wise. Subsequent ellipse motions (axes u, v, center F , Cardan circles p and q)
10.3 Ellipse motion 443

“shake hands” periodically. The center R of the triangle has an almost circular
orbit, consisting of four elliptic arcs. Compare also application p. 449. ◂◂◂

▸▸▸ Drilling quadratic holes in practice

Since a Reuleaux triangle can be revolved in a square, one can extend this
idea further when thinking about how to drill quadratic holes.

Fig. 10.54 An ordinary drilling machine can be extended by two Cardan shafts . A template of
the quadratic hole guides the drill bit.

Fig. 10.54 illustrates in a convincing way how this can be done by using a
template. The drill bit, however, has to be moved along an almost circular
curve consisting of quarter ellipses (Fig. 10.53). This can be achieved by a
combination of two Cardan shafts (explained in Chapter 11, Fig. 11.6).1 ◂◂◂

Fig. 10.55 Rolling of the outer circle instead of the inner circle, realization of the pole ray . . .

▸▸▸ Milling machine for ellipses (Fig. 10.55, Fig. 10.56)

The “inner circle rolling” (hypotrochoid motion) with the given radial ratio

1 Thecorresponding animation was done by Franz Gruber, inspired by the remarkable website
https://www.etudes.ru/ru/etudes/drilling-square-hole.
444 Kinematics: Geometry in motion

(the smaller wheel about R with a radius of 1 rolls within the larger wheel
with a radius of 2) can be replaced by a much simpler “outer circle rolling”
(epitrochoid motion) by inserting an arbitrary auxiliary gear (midpoint H)
which reverses the outer wheel’s direction of rotation. The most important
aspect remains the radius of the large wheel, which is twice as large as that
of the smaller wheel about R. For the pole P , the relationship F P = 2 RP
still applies.

Fig. 10.56 . . . and a practical application

If we fix the centers of the wheels, then a point C on the diagonal of the wheel
about R describes, of course, a circle about R – but that is only the case when
we refer to the fixed system. However, relative to the rotating larger circle, C
moves on an ellipse. The respective tangent to the ellipse is always orthogonal
to the pole ray CP . It follows that an arbitrarily-sized circle through C about
a point D on the pole ray always touches the ellipse. Wilhelm Fuhs has
designed a mechanism (Fig. 10.55, 10.56) capable of realizing the pole ray on
which the cylindrical mill rotates about a vertical axis through D. If a sheet
were to be mounted onto the large wheel, the milling process would turn it
into an exact ellipse. ◂◂◂

▸▸▸ Oval mill by Leonardo

Leonardo da Vinci solved the problem of milling an ellipse in his own way
and thus invented the oval mill, which is also a beautiful example of relative
motion (reverse ellipse motion):

Fig. 10.57 Oval mill for the milling of ellipses

10.3 Ellipse motion 445

⊕ Remark : The lathe tool remains fixed. The “rotary table” consists of two plates
(indicated in green and red). The red plate rotates about its center P . The green
plate rotates along with the red one, but is meanwhile translated so that two syn-
chronously moved rails (blue in the picture) always touch a circle through P (also
blue in the picture). This causes the center of the green plate to move along a Thales
circle with half the radius of the blue circle. We shall now interpret the Thales circle
and the doubly-sized circle as the pair of Cardan circles. The green plate, which is
connected to the small Cardan circle, then performs an ellipse motion relative to
the fixed blue circle: Each point of the green table, thus, has an ellipse as a path
curve from the perspective of the fixed system. All that remains is to position a
lathe tool “anywhere”, which would cut an ellipse out of the raw material attached
to the green plate. ⊕ ◂◂◂

▸▸▸ A compact coupling

If we have parallel axes and want to transform the rotation of one axis to the
other, we can use John Oldham’s idea to use three disks as in Fig. 10.58.

Fig. 10.58 Oldham coupling: Three discs, one coupled to the input, one coupled to the output,
and a middle disc joined in between by tongue and groove.

The tongue and groove on one side is perpendicular to the tongue and groove
on the other. The middle disc rotates around its center at the same speed
as the other axes. Fig. 10.58 shows that in a projection parallel to the axes,
the orbit of the center is a Thales circle with the projection of both axes as
diameter points.

Fig. 10.59 Oldham coupling as seen from the middle disk

446 Kinematics: Geometry in motion

In Fig. 10.59, the middle disk stays fixed in the “main position”, whereas
the axes sweep harmonically in vertical and horizontal direction. Since the
distance between the axes is constant, we deal with an ellipse motion. This
is graphically indicated by showing an orbit of an arbitrary in-between point
on the connecting line. Therefore, the motion of the middle disk is also a
reverse ellipse motion. ◂◂◂

▸▸▸ Ellipses produced by a crank shaft gear

Fig. 10.49 on the right shows a third possibility of producing an ellipse motion
– by making use of a special crank shaft: If the shaft F R rotates about F
and propels the crank RU along so that U is moved along the straight line
u, then a point C on the crank describes an ellipse.

Fig. 10.60 Practical realization of the crank shaft gear

Fig. 10.60 shows a practical realization of this idea. The shaft is actuated
over a table surface. The lengths of the shaft and the crank, as well as the
distance from the gib, are adjustable. ◂◂◂

▸▸▸ Ellipse compass according to Hoecken

In the pictured mechanism (Fig. 10.61), two (yellow) rods are directed so
that their end points move along circles with the adjustable radii a and b.
After rotation about the angle ϕ, the intersection point has the coordinates
(a ⋅ cos ϕ, b ⋅ sin ϕ), and thus, lies on an ellipse.

Fig. 10.61 Ellipse compass by Hoecken

In this case, the mathematical explanation is easier than the geometrical one,
because the intersection point of the rods is, from a kinematic point of view,
not so easy to capture. ◂◂◂
10.3 Ellipse motion 447

▸▸▸ Another ellipse compass

Another beautiful ellipse compass (the last one to be discussed here) can be
seen in Fig. 10.62. It realizes the classical two-point guidance, though this is
only apparent upon closer inspection.

Fig. 10.62 Ellipse compass with “hidden” straight line guidance

A big (red) circular disk is moved parallel within a rectangular boundary.

Thus, its midpoint moves on the perpendicular bisector of the shorter side
of the boundary. A smaller (blue) circular disk is moved analogously, so that
its midpoint remains on the perpendicular bisector of the longer side of the
boundary. The distance of the centers of the circle is adjustable by slot guides,
just as the distance of the drawing pen may be varied. ◂◂◂

General ellipse motion

Up to this point, the guiding straight lines of the ellipse motion were always
mutually orthogonal. In the general case, however, it can be said that:

If a rod of fixed length is directed such that its end points U and V are moved
along the arbitrary intersecting straight lines u and v, then each point which is
rigidly connected to the rod describes an ellipse.

Proof : The intersection F = u ∩ v is the midpoint of the path curve. The circle
q through U , V , and F (midpoint R) can be interpreted as the peripheral circle
over the segment U V due to the fact that the angle ∢U F V is constant (see also
Fig. 1.7). The radius of q is, thus, likewise constant. The instantaneous pole P is
given as the intersection of the path normals in U and V , according to Thales’s
theorem it also lies on q, namely opposite F . Thus, F P is constant, and P moves
on a circle p about F . The radius proportion of the circles p and q is 2 ∶ 1, which
means that the circles p and q are the Cardan circles of an ellipse motion.
Let us now consider the elliptical path of a point C which is rigidly connected to
U V . The diameter CR of q intersects q at the points X, Y . These points describe
“infinitely thin” ellipses through F and thus segments. According to Thales’s the-
448 Kinematics: Geometry in motion

Fig. 10.63 General ellipse motion

orem, the straight lines F X and F Y are orthogonal. The given motion is, thus,
also forced if the rod XY of fixed length is moved along two straight lines which
intersect orthogonally. These are, therefore, the axes of the path ellipse of C. ⊙
⊕ Remark : The point C might, for instance, be given through its distances to U and
V . The triangle U V C can then be positioned onto U V in two ways, which, in turn,
yields two different results (Fig. 10.63 left and right). ⊕

Fig. 10.64 Revolution within a triangle

▸▸▸ Revolution within a triangle or a pentagram

Fig. 10.64 shows the revolution of a square (or “digon”) bounded by circular
arcs within an equilateral triangle. The circular arcs have their centers on
the opposite arc, producing a “curve of constant width.”
During revolution, an alternation between three different ellipse motions oc-
curs. The digon, in particular, produces traces of metal due to the frequent
revolutions. They are the elliptical path curves of the vertices. Fig. 10.65
shows the revolution of a three-leaved curve of constant width within a pen-
tagram. ◂◂◂

Fig. 10.66 illustrates the revolution of a curve of constant width within a

parallel strip. The path curves of the corners consist of pieces of straight lines
and cycloids. The path of the midpoint resembles a sine curve. Compare also
application p. 442.
10.3 Ellipse motion 449

Fig. 10.65 Revolution within a pentagram

Fig. 10.66 Revolution of a curve of constant width within a parallel strip

▸▸▸ The world’s first seven-sided coin

The British 50p coin (Fig. 10.67) was first issued in 1969. Since then, several
heptagonal coins have been introduced to the market worldwide.

Fig. 10.67 Reulaux heptagon

A geometer will immediately think of a way to construct such a curved regular

polygon when the circumcircle is given. This is, in fact, quite easy and there
are several video tutorials on the Internet that give helpful instructions. The
task of finding the correct radius for the rounding circles will be left to the
reader. The result should be a curve of constant thickness . . . ◂◂◂
450 Kinematics: Geometry in motion

10.4 Trochoid motion

The name comes from the Greek word for wheel, “trochos”. We have seen
how an ellipse motion can be forced by the revolution of a circle as long as
the radii of both Cardan circles define the ratio 2 ∶ 1. However, what happens
if this ratio is changed?

Fig. 10.68 Gardeners often exhibit an inclination towards geometry: The pictured rose patch in
Schönbrunn is a good example of that (see also Fig. 10.69 on the right).

If a circle with an arbitrary radius rolls within a fixed circle, then each point
connected to the rolling circle describes a hypotrochoid.

Fig. 10.69 Hypotrochoid motion as a generalization of ellipse motion

Fig. 10.69 shows this generalization with the ratios 2 ∶ 1, 3 ∶ 2, and 5 ∶ 1 for
the radii. The instantaneous pole is located at the point where the rolling
circles touch and has an instantaneous speed zero. The corresponding path
normal of a point’s path curve passes through it.
If the radius of the rolling circle exceeds the radius of the fixed circle, then the
rolling circle encompasses the fixed circle, and the point of contact is located
on the outside. (This situation is sometimes referred to as a epitrochoid.)
10.4 Trochoid motion 451

Fig. 10.70 Hypotrochoid or epitrochoid motion?

Fig. 10.70 illustrates this case for three different ratios of radii. The path
curves of points attached to the moving circle are stretched, have cusps, or
are engulfed, depending on whether the point lies inside, on, or outside the
rolling circle.

Fig. 10.71 Outside rolling for different radius proportions

A further generalization is given if the moving circle is definitively rolled on

the fixed circle. Fig. 10.71 shows the corresponding path curves for the same
radius proportions as before.

If a circle with an arbitrary radius rolls on a fixed circle, then any point connected
to the rolling circle describes an epitrochoid.

Trochoids are often found in geometry. If the ratio of the radii is a rational
number, then the curves are algebraic and closed. The curves are even ratio-
nal, i. e., they admit a rational parametrization. Examples include the ellipse
(of degree two) and Pascal ’s limaçon (of degree four), which often occurs as
the locus (Fig. 1.23) or normal view of a space curve (Fig. 11.3 on the right).
Fig. 10.72 and Fig. 10.73 show some more algebraic examples of trochoids.
When the moving Cardan circle is rolling on the fixed Cardan circle, the ratio
of the radii determines the algebraic degree of the path curves. Among the
path curves of moving points we always have curves with loops, cusps and
452 Kinematics: Geometry in motion

stretched curves (depending on whether the point is outside, on, or inside

the rolling circle).

Fig. 10.72 Epitrochoids: The green Cardan circle is rolling on the outside of the fixed red Cardan
circle. From left to right the ratio between the radii (red:green) is −1 ∶ 1, −2 ∶ 1, and −5 ∶ 1.

Fig. 10.73 Hypotrochoids: The green Cardan circle is rolling on the inside of the fixed red Cardan
circle. From left to right the ratio between the radii (red:green) is 4 ∶ 3, 3 ∶ 2, and 5 ∶ 2.

The black curves in Fig. 10.72 and Fig. 10.73 are multicuspidal curves. In
Fig. 10.72 on the left one can see once more the cardioid (a limaçon of
Pascal, degree 4), in the middle we have the two-cuspidal nephroid (degree 6).
In Fig. 10.73 one can see the astroid (degree 6) on the left and Steiners
tricuspidal hypotrochoid (degree 4) in the middle.
▸▸▸ Wankel’s engine
A classical technological application of trochoids is the Wankel engine. The
section of the piston chamber (Fig. 10.74) is an epitrochoid, which is thrice
traced by the vertices of the synchronously moved equilateral triangle. Note
that this epitrochoid also occurred in Fig. 10.70 in the middle. It can be
proven that epicycloids can always be generated in two different ways ([28]).
The envelopes of the triangle sides fit easily into the piston chamber. In order
to accomplish a greater compression, the central piece is thickened such that
the envelopes are as large as possible while still fitting in the chamber. ◂◂◂
10.4 Trochoid motion 453

Fig. 10.74 Wankel engine

Fig. 10.75 Relative motion of an inner planet on the celestial sphere

▸▸▸ Planetary motion

The trochoid motion was originally called planetary motion. An astronomer
observing the path curves of our neighboring planets eventually discovered
curves which are nearly trochoidal: Due to our own revolution around the
Sun, the rotation of both planets is superimposed.

Fig. 10.76 Relative motion of the inner planets

Fig. 10.76 illustrates the rotation of the planets about the Sun – in slightly
deviating planes but with highly varying velocities. If, at a specific time of
day, we mark the position in the sky at which such a planet is seen, we will
notice that it essentially moves on a great circle: the equinoctial line. If the
planet’s path plane is additionally inclined relative to the path plane of the
454 Kinematics: Geometry in motion

Earth (in the case of Mercury, this inclination equals 7○ ), then the motion
of the planet diverges from the equinoctial line.
In Fig. 10.75, Mercury’s orbit was simulated, but it was done with a threefold
inclination of its path in order to emphasize the effect. We can see that these
cases may, indeed, have something to do with trochoid motions, but certainly
not precisely so.

Fig. 10.77 The four biggest moons of Jupiter. The photo to the left shows how one can see them
with a good eye, the photo to the right was taken with a 3,000 mm lens and 1 second aperture
time. During that second, the whole system rotates relatively to the observer on Earth in such a
way that it is hard to get a sharp image without tracking the system.

Fig. 10.78 A simulation of different relative positions of the four moons

Fig. 10.77 shows a famous situation in the sky: Jupiter with his biggest
moons (Io, Europa, Ganymed, and Callisto) – almost in a row. After the
telescope was invented, these moons were discovered at the same time (1610)
by Galileo Galilei in Padua and Simon Marius (Mayr) in Ansbach. This
discovery expedited the establishment of Copernicus’ heliocentric system,
10.4 Trochoid motion 455

since it is as direct evidence for the fact that celestial bodies not only move
around the Earth but also around other celestial bodies.
The four moons align in a row. This is not particularly surprising, since
they more or less move in the so-called ecliptic, which locally appears as
a line. The order of their appearance on the line, however, is not trivial
at all: Their path curves very closely resemble circles, and since the path
circles with radius r1 , r2 , r3 , r4 appear as a straight line, each moon can be
“everywhere” in between the extreme positions ±ri . Thus, a case may arise
in which Callisto seems to be closest to Jupiter (or even behind the planet)
(compare Fig. 10.78). ◂◂◂

▸▸▸ Planetary gears: plenty of angular momentum

A screwdriver is powered by electricity from an unassuming 3.6-Volt battery
(Fig. 10.79). A small electric motor with about 3500 revolutions per minute
turns the screwdriver’s blade indirectly. The power transmission via gears
shall be analyzed.

Fig. 10.79 Left: A screwdriver is powered by a single battery. Middle: The first stage (there is a
second stage behind it). Right: The whole gear system as a “tangible” model.

Solution:
The shaft of the electric motor propels a small gear with only six teeth
(Fig. 10.79 middle). This so-called “sun gear” transfers its rotational momen-
tum to three equilaterally positioned “planetary gears”, whose 19 teeth are
not significant but simply “a byproduct”. The planetary gears are dimen-
sioned such that they are in mesh with a fixed outer gear with 48 teeth.
This causes the centers of the planet gears to rotate with one eighth (6 : 48)
of the shaft’s angular velocity. The centers of the planet gear form a fixed
equilateral triangle. Right behind it, in a second stage, there is another 6-
teethed sun gear. This gear as well propels three planetary gears (also with
a translation ratio of 6 : 48). Finally, the connecting triangle between the
planetary gears in the second stage is connected to the axis of the screw-
456 Kinematics: Geometry in motion

driver. Rotations of the electric motor thus cause a single rotation of the
screw (about 3500/64 = 55 revolutions per minute). This explains the large
angular momentum M , because for a constant power P of the motor, the
following applies: P = M ⋅ ω (ω is the angular velocity).

Fig. 10.80 Some more planetary gears. Here it is even better to see how small gears mate with
larger ones several times.

Planetary gears have many applications in technology, for example in gear

manufacturing, cable winches, and bicycle hub gears (Fig. 10.80). In these
applications, we can find a variation that is worth highlighting: If the axes of
the planetary gears are fixed and the outer gear is allowed to rotate (the photo
of the model in Fig. 10.79 was taken at the German Museum in Munich),
then the outer gear rotates significantly slower than the shaft. ◂◂◂

▸▸▸ Involute gears

The following paragraph contains a short overview on the theory of gearing.
First of all, the angular velocity ratio between two gears of a gear set must
remain constant (fundamental law of gearing), making the transmission of
power smooth and the speed variation negligible. Furthermore, it is of great
advantage when the profile of a gear does not depend on the gear it mates
with (Fig. 10.81 and Fig. 10.82). This is achieved by utilizing involute gearing.
Here the profile curves are involutes of a circle: For each tooth flank the points
of contact (“line of action”) lie on a tangent shared by both base circles. The
contact between a pair of gear teeth occurs at a single instantaneous point.
Involute gears also work well when a circular pinion gear is mating with a
linear rack gear (Fig. 10.82). Fig. 10.83 shows some other gears, probably
so-called cycloid gears.
◂◂◂
10.4 Trochoid motion 457

Fig. 10.81 Involute gears

Fig. 10.82 Rack and pinion gear

Fig. 10.83 Cycloid gears

Cycloids (trochoids) at a higher level

The trochoid motion can be generalized by letting a third rod rotate at a
constant angular velocity about the endpoint of the second rotating rod. This
approach produces a third-level cycloid with a much greater number of de-
458 Kinematics: Geometry in motion

grees of freedom, permitting the construction of many new curves. Fig. 10.84
shows two third-level cycloids from this infinite multitude.

Fig. 10.84 Third-level cycloids with the rod lengths 2, 2, 1 and angular velocities with ratio
1 ∶ 3 ∶ 4 (left curve) and 3 ∶ 2 ∶ 4 (right curve).

If k rods rotate, we speak of k th -level cycloids. Fig. 10.84 shows two third-
level cycloids. In general, it can be said that any curve can be approximated
to any desired level by higher level cycloids.

Fig. 10.85 Relation to Fourier series expansion

Without going into further detail, Fig. 10.85 illustrates how Fourier series
expansion may be simulated by the mere consideration of the y-values of a
higher-level. With such decompositions, any periodic function graph can be
approximated to any desired level.
11 Spatial motions

Planar kinematics is the basis for kinematics on the sphere. In a way, spherical
kinematics can be conducted similarly to planar kinematics. In the spheri-
cal variant, everything rotates about straight lines through the center of the
sphere. Purely spatial motions, however, make these matters much more diffi-
cult. In this chapter, both possibilities will be explored on the basis of certain
examples.
Two general positions of a spatial Cartesian coordinate system can only be
transferred into each other by a helical motion (including the special cases
of a rotation or a translation).
Kinematics often deals with the transmission of motions from one axis to
another. If these axes are parallel, then a two-dimensional problem is given.
If these axes intersect, then we are dealing with geometry on the sphere.
If these axes are skew (as in the image above), then the problem is “truly
three-dimensional”.
Besides a number of technical applications, we will try to shed light on the
motion of the Earth around the Sun, where two rotations with skew axes
are performed. To make matters even more complicated, these rotations are
only approximately proportional. Important questions arise relating to the
theory of the position of the Sun, such as at what angles of elevation the
Sun appears during sunrise and at what position on the firmament it can be
found at a given date and time. We will learn about the equation of time and
become capable of analyzing sundials with a precision of mere minutes.

© Springer Nature Switzerland AG 2020 459

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_11
460 Spatial motions

11.1 Motions on the sphere

Each motion in space with one fixed point M is a rotation about an axis
f through M (see Fig. 11.1). The path curve c of each point P within the
moving system lies on a sphere.

Fig. 11.1 The composition of two rotations about intersecting axes

In order to improve our understanding of this relation, let us consider the

pedal point N of the normal from P to r. Both P N and M N remain constant
in the course of the motion.

Fig. 11.2 Blender (including its enveloping surface): composition of rotations about intersecting
axes. On the right: Something comparable; This analogy, no doubt, contributes to a feeling of
dizziness that is felt when trying to wrap one’s head around spherical kinematics.

⊕ Remark : The path curves of the blender pictured in Fig. 11.2 are far less interesting
than the enveloped surface that is generated by the curved rod. ⊕

Let us focus our attention on spherical motions, where two rotations about
intersecting axes r and f (intersecting at M ) are composed. In that situation,
we do not even have to prove the following assertion:

Two composed rotations can always be replaced by a single rotation.

11.1 Motions on the sphere 461

Spacial motions differ greatly from planar motions. Instead of an instanta-

neous pole, we find an instantaneous axis a, provided that the transformation
is not a pure translation at this moment. In general, each motion behaves
instantaneously like a helical motion with a as its axis. Things become sim-
pler if the motion behaves like a rotation at each instant as shall be the case
in the following. The new instantaneous axis p lies in the plane spanned by
r and f and passes through M (Fig. 11.3 at the center).
If the composed rotations are proportional, then the instantaneous axis forms
a constant angle with the fixed axis, thus lying on a cone of revolution ∆f
with its apex at M . In the moving system, too, the angle enclosed by p and
r remains fixed, and thus, p generates a cone of revolution ∆r which always
touches ∆f . It can be shown that both cones perform a mutual glide-free
rolling. The path curves of single points are called spherical cycloids.

Fig. 11.3 Cone rolling, including path curves (principal views)

Fig. 11.3 shows a very special cone-rolling situation: The axes r and f inter-
sect orthogonally, and both cones have the same aperture.
462 Spatial motions

The following theorems (without proof) are important for many considera-
tions relating to spherical kinematics:

First theorem of spherical kinematics: Two positions of a shape that is moved

on a sphere can be transferred into each other by a rotation about an axis
through the center of the sphere.

At each moment, a fixed system undergoing arbitrary motion on the sphere

acts as if it were subject to a well-defined rotation about an instantaneous axis
through the center of the sphere. The spherical path normals thus pass through
a momentary pole on the sphere.

▸▸▸ The star handle in motion (a rotation about a general axis)

Fig. 11.4 Star handle in motion (Rotation about an arbitrary axis)

The star handle (Fig. 11.4) is an example of a very simple spherical motion.
The end points of the three equally long bars move on a sphere with a fixed
radius – the bars’ length. The motion itself, however, is a fixed rotation
about a likewise fixed (oblique) axis. The endpoints of the rods share their
path circle! ◂◂◂

Spherical motion can also occur as relative motion – as the following example
demonstrates.
▸▸▸ The Cardan shaft

Fig. 11.5 Cardan shaft (non-uniform transmission)

11.1 Motions on the sphere 463

By using a Cardan shaft, it is possible to transfer rotations about an axis f

onto a second, intersecting axis r. Fig. 11.5 demonstrates this technique in
a star handle, which is connected to one axis via one fork (in the shape of a
half-circle) at f and to another axis at r.

Fig. 11.6 A combination of two Cardan shafts allows a uniform transfer of the rotation.

The transfer of the rotation, however, is not uniform. In a uniform powertrain,

this produces a “wobbling” of the driveshaft, which increases in magnitude
relative to an increasing axis angle γ. A combination of two Cardan shafts
(Fig. 11.6 and Fig. 11.5 on the right) resolves this problem.

Fig. 11.7 A more sophisticated combination of two times two Cardan shafts enables a uniform
transfer of the rotation in merely any direction.

Fig. 11.7 shows how two pairs of Cardan shafts can be connected, which then
enables uniform rotations from one axis to another skew axis. ◂◂◂
464 Spatial motions

Fig. 11.8 Bevel wheels

The uniform transmission of a rotation to a rotation about an orthogonal

axis is a recurring problem in technology. In the simplest case, it can be
solved by letting two rubber-coated cones of revolution with aperture angles
of 90○ roll on each other, as in Fig. 11.3. If greater forces are involved, then
the use of cogwheels will be required. Fig. 11.8 shows a possible solution to
this problem. In practice, worm gears are often employed, as in Fig. 7.34.
▸▸▸ A sophisticated ball joint
If the axes f and r are positioned at a variable angle and the use of a
Cardan shaft is not desired (presumably because the transmission of rotation
has to be proportional), then the ball joint pictured in Fig. 11.9 may be
employed. This particular ball joint has been the subject of detailed research
by Hellmuth Stachel.

Fig. 11.9 An efficient ball joint, which could replace many bevel wheels.

⊕ Remark : Further details can be found at http://www.geometrie.tuwien.ac.

at/stachel/dres.pdf ⊕ ◂◂◂

▸▸▸ “Mad House”

A (red) circle rotates about a vertical axis. A second (blue) circle hangs
horizontally within the first circle and rotates proportionally about this hor-
izontal axis. Within the second circle, a third (black) circle hangs vertically
to the horizontal axis of rotation. It also rotates proportionally about this
11.1 Motions on the sphere 465

third axis (Fig. 11.10, see also Fig. 5.94). Which path curves are described
by points on these circles?

Fig. 11.10 Compound rotations about intersecting axes are always spherical. The shape on the
right, however, is slightly eccentric and possesses no constant gear transmission ratios.

Naturally, each rotation produces circles as path curves of points off the

axis. Superimposing the first rotation with a second one about an axis that
intersects the first results in spherical path curves, since the distance of each
point in the moving system to the common point of the axes is constant.
Each point moves on a particular sphere about the intersection of the axes.
If the angles of rotation are proportional, then (as in the case of the cone
rolling in Fig. 11.3) we get spherical cycloids with top views that are planar
cycloids of second or third level (p. 457). Fig. 11.10 on the left illustrates
this motion for three equal angular velocities. In Fig. 11.10 at the center, the
respective top view is visible.
A variant by Maria Walcher is visible on the right. It is, however, slightly
eccentric and thus does not resemble an exact spherical motion despite the
spheres involved. Due to the absence of propulsion (the outer sphere hangs
like a puppet and can be moved in much the same way), the angular velocities
are not proportional. Walcher was inspired by the motion of the eyeball. ◂◂◂

Fig. 11.11 On the left: spherically positioned “compass” on a speedboat. On the right: another
spherical motion – with appropriate lateral accelerations.
466 Spatial motions

Construction on the sphere

We live on a sphere, even though we may not recognize this at first sight:
Therefore, we also construct on a sphere whenever we construct on the surface
of the Earth. Thus, it might be said that our planar kinematics are actually
“slightly spherical”. What happens if we actually let certain simple things
happen on the sphere, such as the string construction of an ellipse (Fig. 1.29)
or its construction according to de la Hire (Fig. 3.65)?

Fig. 11.12 Spherical conic section: Some planar constructions can be carried over onto the sphere.
On the left: string construction. On the right: construction according to de la Hire.

The result is a “spherical ellipse,” or rather a “spherical conic section.” It turns

out that it makes no sense to distinguish between an ellipse and a hyperbola
on a sphere (as seen in Fig. 11.13 on the left). The spherical parabola, defined
as the locus of all points which are equidistant to a given “straight line” (=
great circle) and a fixed point, is not an exception (and looks rather like
Fig. 11.13 at the center left, where, however, a small circle is touched): All
spherical conic sections are intersection curves of the sphere with certain
quadratic cones that are concentric with the sphere. (More information can
be found in [16].)

Fig. 11.13 Spherical conic sections and intersection curves of quadratic cones and spheres around
the apex: It is impossible to make any further type differentiations.
11.1 Motions on the sphere 467

Fig. 11.14 Spherical conic sections and intersection curves of quadratic cones and spheres around
the apex: It is impossible to make any further type differentiations.

Fig. 11.15 A combination of two Cardan shafts allows a uniform transfer of the rotation.
468 Spatial motions

11.2 General spatial motion

If a system moves in space in any shape or form, then it naturally becomes
difficult to imagine. The main theorem of spatial kinematics is applicable here
(a proof may be found in [9]). It states that any spatial motion – no matter
how complicated – can be approximated at each point by an infinitesimal
helical motion about a well-defined instantaneous axis, as mentioned earlier.

Fig. 11.16 A sphere with a variable radius

⊕ Remark : Fig. 11.16 depicts a very interesting and sophisticated “gimmick” – a

“sphere with a variable radius.” Due to an ingenious system of joints, the object
is flexible in such a way that all points on the rays pass through the center of the
circumsphere. ⊕
Two rotations about skew axes
If rotations about skew axes are composed, then certain assertions about
path curves and enveloped surfaces may be attempted.

Fig. 11.17 Two composed rotations about skew axes

If the rotations are proportional, then the problem is further simplified. We

have already pointed towards the one-sheeted hyperboloid of revolution: By
using two hypoid gear s, a rotation about an axis can be converted into a
proportional rotation about another axis (Fig. 6.35).
11.2 General spatial motion 469

▸▸▸ An earnest effort on a tiny scale

Fig. 11.18 The inside of a very efficient pencil sharpener: A cylindrical mill rotates about an
oblique axis.

Fig. 11.18 shows the complicated inner life of a very efficient pencil sharpener.
The axis on which the pencil is centered stays fixed: In contrast to simpler
pencil sharpeners, the pencil itself does not undergo any rotation! Instead,
a cylindrical mill with a skew axis is rotated and moves around the pencil.
First, a crank turns the cylindrical mill by moving along two points of the
axis. On the top of the mill, there are gear flanks which force its relatively
quick self-rotation by means of a fixed, hollow gear. ◂◂◂

▸▸▸ The Stewart platform in space

In three-dimensional space, robotics allows us to control any motion of a
plane by six arms of variable length (Fig. 11.19). The points of these arms
lie on a regular three-sided prism.

Fig. 11.19 Triangular platform ABC controlled by six robot arms of variable length with the
fixed points L, M , N , and L∗ , M ∗ , N ∗ . On the left: “zero position”. At the center: The triangle
was translated horizontally. A rotation about the inscribed axis (about the center of gravity) can
easily be done. On the right: This time, the triangle was tilted, and the instantaneous axis of
rotation is inscribed (images: Georg Nawratil).

Such platforms are used, among other areas, in flight simulators. Fig. 11.20
shows something similar in an amusement park, where the “platform” consists
of a row of seats in which the hapless victims have a chance to get sick for
small change. ◂◂◂
470 Spatial motions

Fig. 11.20 An amusement park “3D shaking machine” with arms of fixed length and robot arms
of variable length for precise and safe control

Fig. 11.21 Flywheel according to Leonardo da Vinci , and a large-scale modern application.
During uniform rotation, the path curves of all points naturally describe circles. Acceleration or
deceleration brings about spatial spirals which approximately lie on a torus.

▸▸▸ And yet it moves!

Take a cube and cut off the corners, as shown in Fig. 11.22 left (note the
orientation). It is not difficult to show that each triangle 123 thus obtained
is tangent at a point N to the sphere inscribed in the cube. Now drop the
perpendiculars from N to the sides of the triangle and consider the eight blue
triangles that result.

Fig. 11.22 A surprisingly flexible (“overclosed”) mechanism by Otto Röschel

11.2 General spatial motion 471

We now proceed as follows: The eight red triangles also define two squares in
the bottom and top surfaces of the cube as well as four congruent parallelo-
grams. We now erect prisms of arbitrary but equal height above the defined
polygons. We join the prisms with “spherical double hinges”, as shown, for
example, in the picture above in the form of circular arcs. Normally, such a
multilinkage mechanism is rigid, which means it has no degree of freedom.
Because of its particular dimensions, this special mechanism is surprisingly
flexible (“overclosed”). It can move with no difficulty until the edges of the
prisms collide. The mechanism was discovered by Otto Röschel in 2002 ([15]).
◂◂◂

▸▸▸ Non-rigid connections

The construction in Fig. 11.21 on the right serves a similar purpose as the “3D
shaking machine”. During uniform rotations, an equilibrium between centri-
fugal and gravitational forces is quickly established. Phases of acceleration
and deceleration are especially interesting from the perspective of physics
due to the pronounced effects of inertia that occur.
Once again, it was the proverbial Renaissance Man who already investigated
issues of a similar kind. (In the computer simulation pictured in Fig. 11.21 on
the left, the corresponding sketches are overlaid transparently.) The blueprint
for a flying machine (Fig. 11.23 on the left) provides one further example of
Leonardo’s tremendous imagination. From a purely geometric point of view,
it is entirely correct and was probably derived from the observation of bat
flight.

Fig. 11.23 Flying in the manner of Leonardo da Vinci : In theory, this model works perfectly fine.
It can be animated and verified by methods of computer geometry. There is no doubt that its
weakest link is the human passenger, who would have no chance of generating the necessary uplift
by pulley-induced wingflaps (not to speak of maintaining it). Bats and birds are capable of flight
due to their small size, which automatically increases their relative muscle power (see [9]).

◂◂◂

▸▸▸ “As if it were turned by gears”

What happens in the above example on a tiny scale occurs in nature on a
tremendous scale: The Earth rotates about an axis of fixed direction while
472 Spatial motions

it also rotates about the Sun. The axis of the latter rotation is skew to the
Earth’s axis. For an observer in outer space, the motion of the Earth is akin
to the motion of the mill about the pencil. Spatial kinematics at its finest!
(For those interested: The Sun itself has a self-rotation, and its midpoint
describes an elliptical path about the gravitational center of the entire solar
system.)
Of course, the motion of the Earth is not caused by gears, but essentially
by a constant reaction towards gravitational and centrifugal forces. Gravity
is of particular significance in outer space – and this significance is perhaps
best exemplified by the observation that due to self-gravity, virtually every
solid object with a diameter greater than 500 km is spherical. ◂◂◂
11.3 What is the position of the Sun? 473

11.3 What is the position of the Sun?

In this section, we shall employ geometrical methods to illuminate the relative
position of the Sun in the course of the calendar year, as well as during a
whole solar day. Certain simplifications allow us to get an easy grip on the
issue and answer not only every day questions but also several important
ones from the field of architecture:
• When and where does the Sun rise and set?

• At a given day and time, from where does the sunlight come?

• How can a wristwatch be used for an exact determination of the cardinal

directions on a sunny day?

• When is the Sun exactly in the west?

As it turns out, the first and the last questions are completely equivalent.
For the mathematically inclined readers, the results will be described by
approximate equations. The quality of these approximations can be checked
by comparing them with astronomical tables.
Does the Sun not rise in the east and set in the west?
The Sun famously sets precisely in the west on only two days of the year.
During the winter half-year (and in northern latitudes), it disappears some-
where between the south west and the west – during the summer half-year,
between the west and the northwest.
Let a living room be precisely directed towards the north. At which time of
the year, and for how many minutes or hours, is the room illuminated by the
morning or evening Sun? This problem leads to precisely the same question.
The naive assumption that the Sun stands at exactly 18h “of true apparent
time” in the west – in Vienna at about 19h of Central European Summer
Time – is revealed to be false. Let us, therefore, get to the bottom of the
matter using the methods of descriptive geometry.
First of all, the facts
The Earth takes a year (365.24 days) to revolve around the Sun and con-
tinuously rotates about an axis of fixed direction, which currently points
approximately to the North star. The time for a full revolution is a little
less than 24 hours (about 23h 56m ). Four further minutes are required to
compensate for the Earth’s rotation about the Sun (“revolution”).
The path curve of the Earth is a near-circular ellipse. The relation e ≈ a/60
holds for the great half-axis a and the ellipse’s linear eccentricity e. Along
with the axis of the Earth, the normal of the ecliptic (carrier plane of the
474 Spatial motions

ellipse) defines a constant angle ε = 23.44○ . Due to the precession motion

(gyroscopic motion) of the Earth, this value changes slightly over the course
of the decades. Thus, in older books, this value is sometimes given as 23.50○ .
The idealized Earth’s surface is well approximated by a sphere. The flattening
at the poles measures only about 1/300 of its radius. A point on the surface
can be precisely given by its geographical longitude λ and latitude ϕ (Vienna,
for instance, is located at λ = 16.3○ , ϕ = 48.2○ ).
Let us now simplify several things
In order to decrease the difficulty of our problem, let us assume the following
– wholly acceptable – simplifications:

1. Time designations generally refer to the apparent time. It is defined such

that the Sun reaches its culmination point (highest point on its apparent
path on the celestial sphere) at exactly 12h noon. The time difference
between the culmination points always varies and does not amount to
exactly 24 hours. In order to see the Sun culminate at 12h on each day, we
would need to adjust our clocks all the time. The daily errors accumulate,
which leads to deviations of about ±15 minutes over the course of weeks.
This is caused by Kepler’s second law, the law of planetary motion: The
ray from the Sun to the Earth generates equal areas at equal times.

Fig. 11.24 Shadows become continuously more blurry from left to right – the Sun is not
point-shaped.

2. Earth’s orbit can be approximated by a circle (the maximum deviation of

the path ellipse measures below 1.5% of the circle diameter).
3. The Sun’s rays are parallel due to the Sun’s great distance. In the ap-
proximate model, it is, therefore, enough to calculate by using the Sun
midpoint. However, it should be kept in mind that despite its distance
of ≈ 150 million kilometers, it still appears on our firmament – due to
11.3 What is the position of the Sun? 475

its enormous diameter of ≈ 1.4 million kilometers – at a viewing angle of

≈ 0.5○ .

4. If each month is assumed to be 30 days long (and thus, each year to

have 360 days), then the Earth rotates roughly 1○ about the Sun each
day. This value, too, is subject to Kepler’s second law, and thus, varies
slightly. During its north winter, the Earth has a higher rotational velocity
than during summer. On the northern hemisphere, this causes the winter
half-year to be about six days shorter than the summer half-year! For
observations that only last for a single day, however, this relatively tiny
change of position is disregarded.

When compared to the actual values (e.g., without the various simplifica-
tions), it can be shown that, for latitudes that are not too extreme (ϕ < 52○ ),
all values will be correct to a maximum deviation of 4 minutes (though
usually not more than 2 minutes). Given an average day length of 12h = 720m ,
this amounts to a negligible maximal error of about 0.5% – and this even
though we have applied several simplifications!

a
Z N

h V
c

N . . . north celestial pole

V . . . vernal equinox
c . . . celestial equator sun
Z . . . zenith
h . . . horizon
Fig. 11.25 The axis through the celestial poles is parallel to the Earth’s axis a and inclined under
the elevation angle ϕ.
476 Spatial motions

Two important auxiliary theorems

For an observer on the Earth’s surface, the following first auxiliary theorem
can be applied:

The north celestial pole (≈ North Star) can be found in northern direction at
the elevation angle that equals the latitude ϕ (Fig. 11.25).

Proof : The normal projection of the Earth’s axis a on the horizontal base plane γ –
the sphere’s tangential plane at the vantage point B – points towards the geographic
North Pole. The inclination of the axis corresponds to the geographic latitude ϕ of
the observer (Fig. 11.26). ⊙

Fig. 11.26 The North Star, around which everything revolves (on the right: Kepler )

From now on, when we – somewhat imprecisely – refer to the “North Star”,
we mean the north celestial pole – the point on the firmament that defines
the direction of the Earth’s axis. There may not be a prominently visible
star at this precise position, but the actual North Star is only offset by
about 0.9○ . This star, the α Ursae Minoris (main star of Ursa Minor), is the
current North Star. Due to the precession motion of the Earth, the direction
of the Earth’s axis continuously changes, and so does the celestial pole. In
a few hundred years, another star will assume the function of today’s North
Star. The ancient Egyptians even spoke of multiple circumpolar stars, which
were visible at the time of the great pyramids (4500 years ago) in northern
direction at an elevation angle of ϕ.

The auxiliary theorem may even be of some use in daily life. From our per-
spective, the whole of outer space rotates about the axis through the North
Star. All stars, including the Sun, describe circles about this axis.

From a heliocentric perspective, due to the daily relative motion of the Earth
about the Sun (which is negligible due to its small impact), the direction of
the Sun’s rays s stays constant during the course of a day. Geocentrically
speaking, we can thus derive a second auxiliary theorem:
11.3 What is the position of the Sun? 477

a
N . . . north celestial pole
n ⊥ π1
V . . . vernal equinox
c . . . celestial equator
E
E . . . pole of π1

π1

V S

c
sun

Fig. 11.27 When investigating Stone Age celestial geometry, we have to take the precession
motion of the Earth into account.

The direction of the Sun’s rays s rotates almost uniformly about the Earth’s
axis a during the day and in due course generates a cone of revolution Σ with
the axis a. The half angle of aperture σ of Σ is independent of the geographic
position and varies according to the dates in the interval by 90○ −ε ≤ σ ≤ 90○ +ε
(ε = 23.44○ ).

⊕ Remark : In astronomy, we mostly speak of the complementary angle of σ, which

is called the “sun declination.” The value of σ changes slightly during the course of
24 hours, but never more than by 0.4○ (close to the equinox). ⊕
Fig. 11.28 illustrates the change of the cone of revolution during the four
seasons (summer is on the left, equinox at the center, and winter on the
right). The aperture of the cone is, as already indicated, independent of the
geographical latitude ϕ. However, the inclination of the cone’s axis (geo-
graphical latitude ϕ) significantly affects the inclination angle of the Sun’s
rays.
478 Spatial motions

Fig. 11.28 The “sun-cone-of-revolution” at northern latitudes

⊕ Remark : Within the tropics (ϕ = ±23.44○ ), the Sun alternates between north and
south at noon. In this area, if σ = 90○ − ϕ, it is precisely vertical at noon (at the
zenith).
At the moment of the equinox (σ = 90○ ), the cone of revolution “degenerates” to a
plane orthogonal to the axis a. On this day, the shadow of a point on the horizontal
base plane moves on a straight line. On all other days, the shadow point moves on a
cone of revolution. With the exception of the vicinity of the poles, this conic section
is always a hyperbola.
This situation is particularly transparent at the North Pole and the South Pole,
where, during the day, the Sun simply moves about a perpendicular cone axis. For
24 hours, it is located at the same angle of elevation (a maximum of ε). During
winter (north winter or south winter), the Sun does not appear at all. ⊕
▸▸▸ The locus of all possible sun positions
The Sun may occupy many positions in the sky during the course of the
year, but the totality of all these positions covers only a part of the visible
sky. If one imagines all cones of revolution with the half angle of aperture
σ intersected with a huge sphere centered at the viewer, one gets a set of
circles on the sphere which form a sphere layer. This layer is symmetric with
respect to the normal plane of the direction of the North Star.

Fig. 11.29 Locus of all sun positions on the equator and on the northern polar circle

All parts of the layer above the base plane (the sphere’s tangent plane at
the viewer’s position) belong to positions above the horizon. It is, therefore,
11.3 What is the position of the Sun? 479

possible to derive a lot of information about the length of days from Fig. 11.29
(see also Fig. 11.56).
⊕ Remark : Fig. 11.29 shows the sphere layer on the equator and on the northern
polar circle. It is immediately apparent that, on the equator, all days have a duration
of 12 hours. Even on the equator, the Sun is only at the zenith twice per year.
On the northern polar circle, the sphere layer clearly touches the base plane twice
– on the 21st of June (the only 24-hour day) and on the 21st of December (the only
polar night). Fig. 11.33 shows a common day-length-diagram. ⊕ ◂◂◂

▸▸▸ Determination of the angle between the Earth’s axis and the sunrays
As we have seen, the size of the date-dependent angle σ is important. We
now want to determine it by a simple of construction, to an accuracy of half
a degree.

Fig. 11.30 Construction of σ Fig. 11.31 The main orientations of the Earth

Let us imagine the path circle k of the Earth E in the base plane π1 and the
Earth’s axis a parallel to the front view plane π2 . The “zero position” of the
light ray direction may be reached at the beginning of spring (around the
21st of March) and is assumed to appear as a point in the front view.
At any given date – on the 21st of May, for instance – there is a well-defined
angle of aperture for the cone of revolution Σ. Let this date be α days after
the onset of spring (in this particular example, it is 60 days). The Earth E
is then located about α○ after the zero position.
⊕ Remark : For the sake of simplicity, let us now assume all months to be 30 days
long, and thereby, kill two birds with one stone: Firstly, we get 360 days in a year so
that each day corresponds to one degree. Secondly, the days are “weighted” in accor-
dance with Kepler’s law of equal areas. Our supposedly arbitrary calendar, where
February only has 28 days and both July and August have 31, roughly compensates
for the differences between the summer and winter half-years. ⊕
According to the terms of descriptive geometry, the half of Σ’s angle aperture
equals σ and results from a parallel rotation of the light ray direction s = ES
about the Earth’s axis a until an orientation parallel to the front view has
480 Spatial motions

been reached: The axis of rotation is a frontal line, and from this it follows
that the circle of rotation of S appears as one of its diameters. The length r of
the segment ES appears undistorted in the top view (⇒ E○ ′′ S○ ′′ = E○ ′ S○ ′ =
r).
⊕ Remark : Compared to the exact value of σ on the 20th of May, the error measures
only about one fourth of a degree, which we shall gladly accept in this case. During
the course of the year, this error never exceeds half a degree, and it is usually
substantially lower than that. ⊕ ◂◂◂

▸▸▸ Sunrise and sunset

Let us now determine the lenght of a day, or equally, the moment of sunrise
and sunset for our given date at a location with a geographical latitude of ϕ
(for instance ϕ = 48.2○ ). Fig. 11.32 shows how this calculation may proceed
in the simplest of terms: The axis a of the cone of revolution Σ is assumed
to be perpendicular. The parallel circles of Σ then appear undistorted in top
view. The horizontal base plane γ, if set against a, comprises an angle ϕ. It
cuts out two points S1 and S2 from an arbitrary parallel circle (such as the
one with the unit length for its radius), and these points represent sunrise
and sunset. The corresponding circular arcs can be interpreted as the lengths
of day and night. By inscribing a 24-hour “clock-face” on the “unit circle”,
the points in time corresponding to S1 and S2 can be read off directly.

Fig. 11.32 Sunrise Fig. 11.33 Lenghts of day on Earth

⊕ Remark : In reference tables, day lengths are usually described as being about 15
minutes longer than their theoretical values. This is because sunrise and sunset are
defined such that they correspond to the subjective impression of an observer. Thus,
the Sun is already said to rise as soon as the first light rays – and not the center of
the Sun – reach the horizon plane. The same is also true for the setting of the Sun.
During the transition from the vacuum of outer space, the sunrays are always re-
fracted into the gradually densifying atmosphere towards the perpendicular so that
the Sun (which appears to increase in redness as it approaches the horizon because
the non-red color components are scattered more strongly) remains visible on the
horizon, even though it should, in theory, have already set. ⊕ ◂◂◂
11.3 What is the position of the Sun? 481

▸▸▸ At which angle of elevation does the Sun culminate?

Fig. 11.34 How high does the Sun make it on the 21st of December? By the way, the angle
measurement on the left is a relatively simple way to verify some of our conclusions quickly.

From Fig. 11.32, we can immediately conclude the following:

The Sun reaches its maximum angle of elevation of 180○ − ϕ − σ at noon.

This angle reaches its minimum on the 21st of December. On this day, the
Sun on the 50th degree of latitude reaches a mere angle of elevation of 180○ −
50○ − (90○ + 23.44○ ) ≈ 16.5○ (Fig. 11.34). On the same day, it merely touches
the horizon on the northern polar circle (ϕ = 90○ − 23, 44○ ≈ 66.5○ ).
⊕ Remark : On the northern polar circle, the Sun rises during its summer solstice to
an angle of elevation of 2ε ≈ 47○ . If one considers that the Sun never sets on that
day, one can imagine the gargantuan quantities of solar energy which are induced
into the Earth. Under ideal conditions (on a clear sky day), this amount is greater
than the induced solar energy on the same day in equatorial areas! It stands to
reason that an incredible amount of pack ice must thaw on that day – especially on
the southern hemisphere. ⊕ ◂◂◂

The incidence of light at a given moment

Once we are equipped with the aforementioned knowledge, it should be a
small step to determine the direction of light for any given time of any given
date. This time, we draw the cone Σ(a; σ), not in a special orientation (a ⊥ π1 )
but adjusted to the direction of the North Star: let the north direction be the
y-direction of our coordinate system. Fig. 11.35 shows how the situation in
Fig. 11.32 can be restored by employing a side view. In this view, the given
time (auxilary point H ′′′ on the light ray s′′′ ) can be inscribed on the clock-
face and also transferred into front and top view by the rules of descriptive
geometry (→ H ′′ ∈ s′′ and H ′ ∈ s′ ).
We are already well-equipped to solve the questions raised in Fig. 11.36: From
the photo, it is apparent that it must have been sunrise. The geographical
latitude of the Northern Tropic is ϕ = 23.5○ . On the 21st of December on the
northern hemisphere, σ = 90○ + 23.5○ . A construction analogous to Fig. 11.32
482 Spatial motions

Fig. 11.35 Let the geographical

latitude, the date, and the time be Fig. 11.36 This picture was taken on the morning of De-
given: From which direction does cember 21st at the Northern Tropic. How late was it, and
the sunlight incide? what was the position of the Sun at midnight?

yields about 7h in the morning for the sunrise. The construction in Fig. 11.35
for midnight shows that the Sun was located precisely at the nadir at the
moment the picture was taken – the opposite of the zenith. On the same
day in the Southern Tropic, the Sun stands at the zenith (σ = 90○ − 23.5○ ) at
noon.
It is also apparent from this construction that the Sun stands exactly in the
west (or in the east) whenever the y-value of the helping point H disappears.
(In the drawing, the western direction is labeled as W .)
Let us now imagine the cone Σ rotated by −90○ about the x-axis so that it
corresponds to the geographical latitude of ϕ − 90○ . For the new cone, the
synchronously rotated point W assumes the role of the point S2 , which marks
the moment of sundown. Thus:

The Sun stands on a circle of latitude with the geographical latitude of ϕ in

the precise east and west whenever it rises or sets on the “complementary circle
of latitude” ϕ − 90○ on the other hemisphere (provided that it does the latter
at all).

▸▸▸ Where is south?

Fig. 11.35 can be read in an interesting way. If we interpret the 24-hour
clock-face as a clock, then we notice that its hour hand (corresponding to
the central angle ω) is the normal projection of the light direction onto the
clock-face, provided that the 12h -marking (central angle 0○ ) points towards
the normal projection of the south direction.
If we imagine the usual 12-hour clock-face on the clock, then the central
angle ω of the current time towards the 12-hour marking is twice as large as
on the 24-hour clock-face in Fig. 11.35 (ω = 2ω). The southern direction is,
thus, marked by the moment which lies precisely between the current time
and 12 o’clock.
11.3 What is the position of the Sun? 483

Thus, by using any analogue watch, the southern direction can be derived
from the sunlight in the following way:
First, one proceeds according to the commonly known “boy-scout rule”: One
imagines the moment S that lies between the current time and the culmina-
tion point of the Sun (around 12 o’clock, or 13 o’clock during summer time).
We also have to consider the geographical longitude: For each degree of lon-
gitude, the culmination point moves by 4 minutes so that on the Greenwich
meridian in France or Spain, 60 minutes would have to be added! Neither
must we forget the deviation of the culmination point from 12 o’clock caused
by Kepler’s law of equal areas (equation of time), which affects the calculation
by up to ±15 minutes.
Now, we take off our watch and turn it around so that the hour hand points
to the Sun. The marked position then points pretty accurately towards the
south.
For a more exact determination of the south direction, we would now have
to tilt the watch: We should be oriented towards the currently determined
south and position the hour hand at the time S so that it points in the same
direction. Now, we must merely tilt the clock-face towards the body by 90○ −ϕ
(in our latitudes about 42○ ) and repeat the procedure. The tilting action
yields a slightly modified south direction. The error, without subsequent
correction by tilting, is the greater, the earlier in the morning or the later in
the evening we make use of the boy-scout rule, or the farther south we are.
By this rule alone, the Sun would always be located precisely in the west at
6 o’clock p.m.! ◂◂◂

Fig. 11.37 A very small and versatile gem. It can be used in a similar way to the boy scout rule,
either to determine the south direction (if you know the time) or to find out what time it is (if
you know the south direction).
484 Spatial motions

▸▸▸ A tiny and versatile gem on your necklace

The gem mainly consists of a refracting semi-sphere placed on a 24-hour time
scale. Let us assume you know what time it is. Now try to get the focused
sunrays onto the adjacent point on the time scale (you usually have to tilt
the “tool”). The letter “S” then already points to the south.
If you do not know what time it is, but you know the south direction: Turn
southwards and try to get the focus onto the time scale (tilt the tool again).
Then you read the (true) time on the time scale. In Fig. 11.37, the focus
shows that it is 8 a.m. true solar time. With daylight saving time, you have
to add an hour. ◂◂◂

A constant and a varying angular velocity

All our previous deliberations referred to the apparent solar time, which is
defined as the time when the Sun reaches its culmination point precisely at
noon. We now have to consider that the path of the Earth about the Sun
is an ellipse, and the Sun’s angular velocity on its path about the Sun is,
therefore, not constant. On the other hand, the angular velocity during its
rotation about its own axis is not affected: The time for a whole revolution
about the Earth’s axis is about 23 hours and 56 minutes.
The equation of time
In the remaining four minutes (which would complete the full 24 hours), the
Earth rotates one degree more, which causes the Sun’s culmination point to
occur at a slightly different time than on the previous day – during these
24 hours, the Earth has moved by another degree about the Sun, but only
approximately due to its constantly changing angular velocity. Each day,
several seconds of error are added or subtracted, culminating in a deviation
of about ±15 minutes.
⊕ Remark : A precise calculation of this phenomenon is very elaborate. There exists
no precise equation for it – rather, one always has to fall back on Kepler ’s non-
algebraic equation of time and try to solve it approximately, see [9]. ⊕
▸▸▸ Detective games
If we are faced with a picture of a well-known object (such as a building)
on which the Sun’s shadows are visible, then we can determine the date
and the time at which the picture was taken. The date, however, is dually
ambiguous, as we shall soon discover. In order to succeed, we must know the
north direction and the geographical latitude of our scene.
Actually, we only have to do “reverse engineer” the light ray construction at
a given date and time. (Fig. 11.38):
Let s be the light ray that is given in a top view and front view. Let us now
consider the plane which appears as a line in the top view and contains the
Earth’s axis a (points towards the north). In a side view, the geographical
latitude ϕ appears undistorted (auxiliary point H ∈ s → H ′′′ ∈ s′′′ → a′′′ ∋
O′′′ ).
11.3 What is the position of the Sun? 485

In a second step, let us rotate the light ray s about a such that it becomes
undistorted (H ′′′ → H0 ∈ s0 ). Let us then rotate the projecting path circle
of H so that its axis a appears projecting in a fourth view. We have now
achieved two things: In the third view, we can see the date-characteristic
angle σ, from which we can derive the date by reversing the construction of
σ.
The angle ω of rotation appears undistorted in the fourth view, which gives
us the time (15○ correspond to an hour). From the indication, we can see
whether or not the Sun is still largely in the east. The time difference from
12 o’clock is to be subtracted or added accordingly.

Fig. 11.38 The date (σ) and the time (ω) for a given light direction s

The 53th degree of latitude was chosen in this example (Berlin, for instance,
lies on it). The construction produces σ ≈ 77○ , ω ≈ 53○ .
The angle α is to be determined via Fig. 11.30: In the front view, we draw a
cone of revolution with the axis direction towards the North Star, half of the
aperture σ, and the base circle of radius r. It is to be fitted into the scene
so that its apex E lies on the path circle k and the base circle of the cone
includes the Sun S. The top view of E leads to α (α ≈ 34○ ).
The situation then unfolds either 34 days after the onset of spring (25th of
April) or 34 days before the onset of autumn (19th of August). The angle
ω corresponds to about 3.5 hours. This makes the actual time 15 ∶ 30. Both
possible dates occur in central Europe during summer time, which means
that we need to add another hour. We must not forget the deviation from
the 15th degree of longitude, to which central European time refers. Berlin
lies about 1.5○ westwards from the reference circle, which produces a further
delay of 6 minutes. A public clock which may have accidentally been included
in the photograph would indicate about 16 ∶ 36. The clothing of the people
in the picture would provide us with further clues about the time of year: If
many people are pictured in summer clothes, it would probably have to be
the 19th of August. ◂◂◂
486 Spatial motions

▸▸▸ Two comparable and yet slightly different photographs

It takes a while to understand why in the photos in Fig. 11.39, the Sun ap-

Fig. 11.39 Same position, same time, same camera data – but different target points. The camera
was only rotated horizontally such that the Sun is depicted on the left. Now it seems to be elliptical
– and higher!

pears so differently (both in the sky and in the reflection). The pictures were
taken at the same time from the same position without changing anything
in the camera settings. Still, the Sun seems to be at a higher elevation in the
right image. The “Sun disk” also appears much more distorted in the right
image. The answer is not trivial. In order to fully understand what is hap-

s
Σ
π

C
Fig. 11.40 With respect to the rotating image plane π, the Sun is rotating about the vertical
axis with the lens center C. Thus, the adjacent image of the Sun is moving along the section
hyperbola s of a cone of revolution Σ and π.

pening, we need, like we do so often, a good geometrical sketch (Fig. 11.40).

If we take several photos at the same time, the Sun will always be in the
same location on the sky. Now we rotate the camera horizontally about an
11.3 What is the position of the Sun? 487

angle ω, i. e., about the vertical axis through the center C of the lens. With
respect to the sensor plane (or an image plane π which is parallel to it), the
Sun then rotates about the vertical axis through the angle −ω.
The orbit of the Sun during this relative motion is a circle on an upright cone
of revolution Σ. It does not actually matter whether the Sun is “infinitely far
away” or not, the image of the Sun is always on the intersection hyperbola s
of Σ with π. The contour of the Sun becomes more and more elliptical when
the Sun is further away from the center of the image. Also, the height of the
Sun will be the larger, the further to the left or to the right it appears. ◂◂◂

▸▸▸ When are we allowed to measure an angle in a perspective image?

Fig. 11.41 Two photos of vertical objects with shadows, taken at noon time. Only one will allow
you to measure the altitude of the solar noon.

As we know already (p. 481), the altitude of solar noon varies between 90○ −
ϕ−δ and 90○ −ϕ+δ, where ϕ is the latitude of the location and δ ≈ 23, 4○ is the
inclination of the Earth’s axis. Fig. 11.41 shows two photos taken at the same
location at noon. Even though it might look better, the picture on the left
is not good for measuring angles, as opposed to the wide-angle photograph
on the right: One can see that both the images and the shadows of the two
ski sticks appear parallel (side light with vertical image plane). One can also
check that the light rays (drawn in yellow) through the tips of the ski sticks
are parallel. Thus, the whole light plane through a stick is parallel to the
image plane (sensor), and all angles in this plane appear undistorted. Let us
say that the measured angle of the altitude of solar noon is approximately
20○ . Sherlock Holmes would now be able to determine either the date if the
latitude of the location was given, or vice versa, determine the latitude if the
date was given.
⊕ Remark : The fact that the camera was held in such a way that the sensor plane
was kept vertical, made the task easier. Otherwise, we would have to invest a lot of
geometrical knowledge and would need more geometrical objects in the environment
to arrive at the same result. Fig. 11.42 (a panorama photo that mainly measures
angles) would then be helpful. It would probably even be possible to reconstruct
the mountain hut and the exact relative position of the photographer. ⊕ ◂◂◂
488 Spatial motions

Fig. 11.42 A panorama photo, taken a few minutes later

The path of the Moon on the firmament

▸▸▸ The path of the Moon in the solar system
◂◂◂
M0
E0 full moon

waxing (1/2)
E3
M3
E2 new moon
M2
M1
E1 waning (1/2)
29.5 days = 4 × 7 days 9 hours

Sun M0 full moon

E0

Fig. 11.43 The Earth moves 29.5 days around the Sun and – in the average – sweeps ≈ 29○ . The
moon needs 27.32 days to rotate 360○ about the Earth, but 2.18 more days to compensate the
rotation of the Earth.

The interplay of Sun and Moon has been an object of constant fascination for
all cultures. Like essentially all planets, the Moon moves on the same path
plane as the Earth, give or take periodic fluctuations of ±5.2○ and a period
duration of 18.2 years.
It may be tempting to conclude that the path of the Moon – just as the
paths of the planets – essentially overlaps with that of the Sun. This may
approximately be true during the equinox or new moon, but the full moon’s
much higher angle of elevation during winter is still noticeable.
Whenever one talks to an astronomer about the Moon’s path, one is in-
evitably met with a smirk and a suggestion that the whole issue is not quite
as simple and that it, therefore, has to be calculated with a much greater
degree of accuracy. We shall, nevertheless, attempt to solve the problem qual-
itatively and geometrically.
11.3 What is the position of the Sun? 489

Fig. 11.44 The full moon rises at sunset and disappears at sunrise. Close to the horizon, the two
spheres can be “compared” with the size of trees and buildings which is of course misleading.

Fig. 11.45 Whenever the days or nights are at their longest, the path of the full moon deviates
considerably from the path of the Sun. The full moon stands very high at the onset of winter.

Let us have a look at Fig. 11.45 on the left: It is north winter – the southern
hemisphere is much more strongly illuminated – and we have new moon. Only
the dark side of the Moon is visible from the Earth. In the following 24 hours,
the Earth rotates about its axis while the Moon always points its same side
towards Earth. It takes 4 weeks to rotate about the Earth. When considered
from the perspective of the Earth, the sunrays and their reflections from the
Moon overcoat the same half cone of revolution. It seems that the new moon,
indeed, follows the path of the Sun (see also Fig. 11.26 on the left).
The situation is completely different during full moon (Fig. 11.45 on the
right): The Moon is now opposite the Sun, and the half cone generated by
the reflecting rays is now practically the elongation of the Sun cone. During
the self-rotation of the Earth, the full moon rises to the same height that was
essentially accomplished by the Sun at the start of summer. This explains
why, during long and clear winter nights, the full moon is a considerable
source of light from sunrise to sunset.
According to the theorem about the Sun’s culminating point on p. 481, the
Sun on the 50th circle of latitude is visible at a maximum angle of elevation
of 90○ − 50○ + 23.45○ ≈ 63.5○ . If we now apply the maximal deviation of
the Moon’s path, as it was in the winter of 2005/06, then we have almost
490 Spatial motions

reached 69○ . Such an angle of elevation is subjectively almost interpreted as

“perpendicular”. During the day, however, the Sun moves at a maximum of
90○ − 50○ − 23.45○ ≈ 16.5○ .
On the poles, the Sun is not visible for the winter half of the year. Every
few weeks, however, the Moon shows up as the half moon and reaches its
culmination point as the full moon at up to 23.45○ . This happens whenever
the full moon occurs precisely on the 21st of June or the 21st of December.
According to our previous considerations, it should be clear that the paths of
Mercury and Venus on the firmament should always be approximate to the
path of the Sun to a certain degree (see also [9]): Both planets are significantly
closer to the Sun than the Earth and are, thus, to be treated geometrically
like the new moon. Mars and Jupiter, however, can “change sides” on the
days surrounding the full moon, and their path on the firmament can, thus,
significantly deviate from the Sun’s path.
▸▸▸ What is different on the southern hemisphere?
Inhabitants of the northern hemisphere unconsciously tend to privilege the
northern hemisphere geometrical statements and examples. On the northern
hemisphere, the stars are turning counter-clockwise when we look towards the
North Star, and clockwise when we look toward the south. What is different
on the southern hemisphere?
One of the first things a traveler should do when arriving on the southern
hemisphere is to sit down in a park and verify whether the shadow of a
tree does indeed go “the other way round.” The Sun still rises in the eastern
direction and sets in the western direction, but in general, it reaches its
culmination point in the north. Due to the fact that our traveler from the
northern hemisphere is “standing on their head”, compared to their homeland,
the direction of the Sun’s rotation appears inverted. (In space, there is, in
fact, no orientation in rotations, because it all depends on the direction from
which one is looking towards the rotation axis). On the southern hemisphere,
the direction of the Sun’s rotation is inverted, and the Sun moves counter-
clockwise across the horizon when we intuitively turn towards the north in
order to observe it. If we, however, look towards the south by night, the stars
rotate clockwise around a fictive point, the south celestial pole (Fig. 11.46).
Fig. 11.47 shows how the night sky at the Cape of Good Hope (South Africa)
moves clockwise a little more than 90○ within more than 6 hours (≈ 15○ per
hour). There is no star around the south celestial pole that could be used for
orientation the way the North Star is employed on the northern hemisphere.
Therefore, one tries to find the Southern Cross or Crux. It is so popular that
it is featured on several flags representing states on the southern hemisphere,
including Brazil, Australia, New Zealand, or Papua New Guinea. Once we
have made out the constellation, the south celestial pole can be found by
elongating the axis of the deltoid by a factor of 4.2 (see Fig. 11.47). ◂◂◂
11.3 What is the position of the Sun? 491

I am looking northwards
towards the North Star:
The stars rotate ccw. I am looking southwards
towards the Sun:
The Sun rotates clockwise.

I am looking southwards
towards the southern cross:
The stars rotate clockwise. I am looking northwards
towards the Sun:
The Sun rotates ccw.

Fig. 11.46 Clockwise or counter-clockwise?

Fig. 11.47 The southern cross rotates clockwise about the south celestial pole (where, unfortu-
nately, no star can be made out with the naked eye). The yellow marking shows how the south
celestial pole may be found. The two pictures were taken more or less from the same viewpoint.

▸▸▸ Multiple circuits around Antarctica

Although it does not quite tie in with the theory of sun positions, the fol-
lowing example still fits here (and, to some extent, additionally qualifies as
an example of a spherical motion):
492 Spatial motions

Fig. 11.48 Antarctic Circumpolar Current (ACC) – on the left: depicted in a comprehensible
way; on the right: only readable for people with extremely well-developed spatial imagination.

The Antarctic continent appears completely distorted on maps like the one
in Fig. 11.48 on the right. One either needs to apply a stereographic projec-
tion from the North Pole or display a “bottom view” (or “top view,” if you
are on the southern hemisphere) of the globe. As a matter of fact, this global
current flows entirely on a sphere, which means that we need an interac-
tively twistable model of a sphere in order to be able to imagine everything
accurately! ◂◂◂
11.4 Minute-precise sundials for identifying the mean time 493

11.4 Minute-precise sundials for identifying the

mean time
In this section, we shall introduce sundials for the determination of the mean
time. Two types function with shadows of curved surfaces on the indicators.
Another sundial does its job by means of an intricate mechanism – the sundial
introduced by Oliver and Bernhardt, a variation by Riegler , an idea for a
sundial by Hofmann and another by Pilkington and Gibbs. All sundials make
use of the equation of time and are exact to the minute, given a careful enough
construction and a sufficient size. The calculations of the arrows and of the
described mechanisms will now be discussed.

True and mean time

Sundials used to be instruments for the measurement of time, and today
they still exert a great degree of fascination. Their history dates back to
antiquity, when the duration between sunrise and sunset was divided into
twelve approximately equal parts with the help of sundials. The position of
the Sun was projected onto a surface where a grid of date and hour lines
was drawn by means of a shadow cast by a rod or by a circular aperture
(Fig. 11.49). An hour used to be a variable unit of measurement that was
dependent on the time of year – a temporal hour.
In the modern division of hours, the duration between the Sun’s two consec-
utive meridian passes is divided into 24 equal parts. The meridian of a point
of observation is (with the exception of the poles) the great circle on the ce-
lestial sphere that passes through the zenith, the north point, and the south
point of the horizon. The midpoint of the celestial sphere can be assumed at
the point of observation due to its “unimaginably great” radius. As will be
shown, these lengths of days vary during the course of the year – this is the
true local time. Today, a mean time is employed, constituting a uniform unit
of time. It is measured most precisely by atomic clocks. In harmony with the
revolution of the Earth about the Sun, a second of mean time is defined by
the duration period of 133 Cs light.
Next to the true time of a location, the mean location time was first observed.
The railroad industry required the introduction of time zones and of mean
times for greater areas. Central European Time, for instance, is the mean
location time for locations at 15○ of eastern longitude. Our summer time
corresponds to Eastern European time (30○ eastern longitude).
Sundials – truly analogue clocks – measure true local time. We speak of true
noon whenever the Sun is precisely in the south. Mean time and true time
correspond to each other on four days a year (mid-April, beginning of June,
beginning of September, and the end of December). The mean noon deviates
494 Spatial motions

Fig. 11.49 Ancient clock Fig. 11.50 Tower clock

from true noon during the course of the year by the equation of time, which
we shall soon discuss – this can amount to up to 16 minutes.
Simple sundials
From the perspective of a location on the Earth’s surface, the sky seems to
rotate about a celestial pole. On the northern celestial sphere, the North Star
is located at that point at a distance of under one degree. The axis of the
firmament’s apparent rotation points towards the celestial pole. It lies in the
vertical north-south-plane and is tilted against the horizontal at the angle of
the respective geographical latitude ϕ. The axes of many sundials point in
that direction (Fig. 11.50).

Fig. 11.51 Date lines . . . Fig. 11.52 . . . and hour lines

The planar and horizontal trace of the cone (with half aperture σ) overcoated
by the light rays through a fixed point is a conic section, a hyperbola in non-
polar regions, and a straight line during equinox (Fig. 11.51). Path curves of
this kind are called date lines.
If we wish to construct a simple sundial, we must – for the full-hour moments
– intersect a pencil of planes with planes of 15○ to 15○ through a straight
line parallel to the celestial axis with the carrier surface of our clock-face
(Fig. 11.52). The intersection lines are the hour lines: The Earth rotates by
360○ during 24 hours.
Equatorial clocks are special sundials with planar, equator-parallel clock-
faces or with clock-faces shaped like cylinders of revolution whose generators
are parallel to the celestial axis. In both cases, hour lines are distributed
evenly.
11.4 Minute-precise sundials for identifying the mean time 495

The equation of time

The motion of the Earth about the Sun is complicated. In order to accomplish
a sundial that is exact to the minute, we have to work with tables, or better
yet, with a computer simulation. The Earth’s precession motion (the rotation
of its axis during the course of 26000 years, Fig. 11.27) and the nutation (small
disturbances in the precession motion) are not significant for sundials.
Fig. 11.53 shows a “caricatured depiction” of the Earth’s positions during the
equinoxes and the solstices (see also Fig. 11.27). The precession motion is vi-
sualized on the great sphere (the Earth’s axis generates a cone of revolution).

Fig. 11.53 Precession motion Fig. 11.54 Kepler equation

In order to achieve a computer simulation, we must explore the issue ge-

ometrically with a certain foreknowledge in physics. A mere application of
Kepler’s laws is possible, but not easy to implement: The path ellipse of the
Earth must be parametrized by time, which requires the solution of the tran-
scendental Kepler equation. By using the nomenclature from Fig. 11.54, this
equation is given as
√
f (α) = α − ε ⋅ sin α − t = 0 with ε = 1 − (b/a)2 ,

where this time ε is the numerical eccentricity of the ellipse. One would do
better to employ the theorem of the conservation of angular momentum, from
which Kepler’s laws necessarily derive.
From the perspective of differential geometry, the path curve with its physical
properties may be deduced from the knowledge of the initial velocity and the
corresponding direction of motion. The angular momentum is calculated with
respect to the Earth’s axis through the Sun which is orthogonal to the Earth’s
path plane.
The Earth moves on an elliptical orbit about the Sun, according to the the-
orem about the conservation of angular momentum (Kepler’s first law). The
numerical eccentricity of the path ellipse is rather small (ε=0.017). By the
same principle, the Earth must constantly be changing its orbital or angular
velocity about the Sun. A diminution of the distance from the Sun leads to a
square increase of the angular velocity with respect to the diminution factor
(Kepler’s second law: equal times – equal areas).
1. The “physical” component
While the Earth moves non-uniformly about the Sun, it also rotates with
496 Spatial motions

a uniform angular velocity about its own axis. This leads to a small daily
deviation, until the Sun once again returns to the south. This deviation
contributes to the equation of time as the daily difference “real time minus
the mean location time.”
2. The “geometrical” component:
The equation of time includes another substantial component produced by
the tilt of the Earth’s axis. The rotation about the Sun and the self-rotation
have mutually oblique axes of rotation. Thus, it is impossible to imagine both
rotations in one plane! A circular clock-face whose plane is parallel to the
equator plane with a uniform division of hours appears slightly distorted if
projected orthogonally on the path plane of the Earth. Thus, in this normal
projection, the clock-hand’s shadow on the clock-face does not rotate at
uniform angular velocity!
Let us assume, for the sake of simplicity, that the Earth rotates about the
Sun at a constant daily angular velocity of 1○ . Thus, every single day, an
angle of 1○ would have to be compensated on the elliptically distorted hour
index in order to reach the Sun’s relative position. The revolution of the
shadow on a circular clock would not be uniform!
3. “Equinox line” and path ellipse
At both equinoxes, the Earth’s self-shadow boundary passes through both
poles. Sunlight incides the Earth’s axis orthogonally. The connecting straight
line of both positions on the Earth’s path – the equinox line – encloses an
angle with the principal axis of the path ellipse. This angle changes slightly
due to the precession motion and the perihelion precession (the rotation of
the ellipse apices) – though this change has been negligible during recorded
history. It also affects the values of the equation of time, though not to a
great degree.

Fig. 11.55 The eight-shaped loop of the equation of time

In Fig. 11.55, the equation of time was solved for the Earth’s data. Each
position of the Sun at midday corresponds to a point on the eight-shaped
loop. If the angle between the equinox line and the principal axis of the path
11.4 Minute-precise sundials for identifying the mean time 497

ellipse is 90○ , then the eight-shaped loop is axially symmetric. Fig. 11.55 also
shows the self-shadow boundaries of the Earth.

Fig. 11.56 Eight-shaped loops on the celestial sphere

Fig. 11.56 shows an image of the celestial sphere on which the stars were
projected. The horizon plane for 48○ of northern latitude (Vienna), the zenith
Z, the celestial axis, the plane of the local meridian, the celestial equator
(double line), and both tropics are all inscribed. The eight-shaped spherical
curves for the full hours of mean local time are also visible. They are produced
by a rotation of the hour line for the mean midday by a multiple of 15○ .
By the way: Computer simulation allows us to solve the equation of time
for any location of any planetary system, any numerical eccentricities, and
any axial tilts. This produces many varying loop shapes, which may also be
symmetric and do not necessarily have to possess a double point.
The Bernhardt sundial

Fig. 11.57 The creation of the curved clock-hand

The 365 sunrays through a fixed point on the Earth throughout all precise
middays of the year lie on a cone – a director of this cone is the eight-shaped
loop on the celestial sphere derived from the equation of time (“eight-shaped
cone”).
498 Spatial motions

Let us now consider a circle in an equator-parallel plane (Fig. 11.57). Its axis
passes through the celestial pole. Let us now mark 24 points on the circle
separated by 15○ each and label them 0, 1, . . . , 23, 24=0.
Let us now inscribe rays towards the eight-shaped loop on the celestial sphere
through point 12 for the mean midday. In order to prevent an intersection
of this cone with the axis of the circle, point 12 is rotated from its deepest
position slightly towards the east (the directions towards the eight-shaped
loop are preserved).
During the day, a light ray which belongs to the eight-shaped cone through
12 rotates uniformly about the axis of the circle and in due course generally
envelops a hyperboloid of revolution1 . The surface of rotation enveloped by
the rotating eight-shaped cone (or by all hyperboloids) can be employed as a
curved clock-hand of a sundial for mean time. The surface of rotation is the
focal surface of the congruence of rays formed by all light rays meeting the
circle. The circle itself is a focal curve of this congruence.
According to the laws of construction, all light rays which touch the clock-
hand at mean midday and meet the circle pass through point 12. The equa-
tion of time is thus considered. In order to read a zone time on the sundial
(instead of local time), the circular hour index is rotated by the difference
angle between the respective location’s geographical longitude and that of
the zone meridian.
The described idea dates back to Oliver in 1892 and was improved by Bern-
hardt in 1964. The problem that the occurring envelope surface of revolution
is composed of two sheets due to its construction from an eight-shaped cone
is solved in this sundial by means of two different surfaces of revolution,
which are simply swapped at the solstices. Each surface is employed as a
clock-hand for a half-year. The boundary areas of the surfaces of revolution
are especially critical at the solstices, because the enveloping hyperboloids
overlap. At those moments of the year, the time can no longer be read to
minute accuracy.

Fig. 11.58 Riegler sundial

Riegler adapted the idea by Oliver (Bernhardt) by rigidly connecting both

envelopes of the surface of revolution to the plate of both clock-faces (Fig. 11.58

1 RolfWieland: Equatorial sundials with automatic time compensation. Newsletter 26 and 27 of

the workgroup concerning sundials of the Austrian Astronomical Society 2003/04
11.4 Minute-precise sundials for identifying the mean time 499

on the left). The outer sheet is pictured as a grid, the inner as a closed surface.
Depending on the half-year in which one reads the time (each running from
solstice to solstice), the shadow on the inner sheet or outer sheet is employed
for the determination of time (Fig. 11.58 on the right). The shadow of the
clock-face pane on the grid of the outer sheet further indicates the respective
date.

Cycloid sundial for mean time

We received a question online from somebody in the U.S.A. about the exis-
tence of a sundial whose shadows on a clock-face parallel to the celestial axis
display the mean time by straight hour lines. If date lines and eight-shaped
hour lines are drawn on such a plane, the mean time might be read by means
of a circular aperture – though this detail was not asked by the questioner.
A beautiful sundial that displays real time is described by Fred Sawyer .2 If a
circle rolls in a plane on a straight line, then a diameter of the circle envelops
a cycloid (Fig. 11.59).
A cylinder with a generatrix that is parallel to the celestial axis and touches
a hump of a common cycloid in an equator-parallel plane can, thus, serve as
the clock-hand for real time if the hour index lies on the apex tangent of the
cycloid hump.

Fig. 11.59 Diameter . . . Fig. 11.60 . . . of cycloids and the consequences

A sundial can be positioned so that the apex tangent is directed from the
east towards the west. The true local time between 6 o’clock in the morning
to 12 o’clock noon can then be read on one half-tangent. The true local time
between 12 o’clock noon to 6 o’clock in the evening can be read from the other
half-tangent (Fig. 11.60). The hour points are placed at equal distances. This
allows the sundial to be adapted to the real local time of a zone meridian by
simple translation of the hour index on its carrier straight line.
Now, one of both sunray cones (as they have been considered for the Bern-
hardt-sundial) are directed as follows: Their tip is moved along the hour
scale of the cycloid sundial, its generatrices for the equation of time zero
touch the envelope of the cycloid cylinder, and the ray of the equinox retains
its declination of 0○ .

2 The Cycloid Polar Sundial. Compendium, North American Sundial Society, Volume 5, No. 4,
pp. 21–24, 1998)
500 Spatial motions

Fig. 11.61 Construction and appearance. . . Fig. 11.62 . . . according to Hofmann

The result is an envelope (Fig. 11.61, Fig. 11.62) which may be employed as
the clock-hand for mean time. Self-intersections occur in the areas near the
solstices, as well as near the cusps of the cycloid. A reading of the sundial
is especially precise between 7 and 17 o’clock mean time. The intermediate
area of the clock-hand does not have to be especially adapted for this reading
– in fact, the clock-hand and the clock-face may be connected.
Translations of the hour index allow a reading of zone times, especially dur-
ing summer time. As with the Bernhardt sundial, there are two clock-hand
surfaces which are interchanged at the moments of the solstices.
The calculation of this sundial was accomplished by using the software Open
Geometry. The characteristics of the surface were calculated at uniform in-
tervals by intersecting two adjacent eight-shaped cones – for instance, at
one-minute intervals. The surface was, thus, determined by a family of char-
acteristics, and the connecting surface of two neighboring characteristics was
interpolated.
While the sundial may be visually impressive, its plan for construction may
only be provided as the answer to a given initial question. The construction
would in any case be very expensive, and the mean times would be better
displayed on other sundials.
The Pilkington-Gibbs sundials
Another approach to a heliochronometer that is exact to the minute was
given in 1906 by Pilkington and Gibbs – Gibbs was the inventor and Pilk-
ington the manufacturer. They equipped a pivotable circular disc so that, in
areas outside of the horse latitudes bounded by the tropics, it can be rotated
parallel to the equator immediately after it is brought into line (Fig. 11.63).
Near the equator, the carrier half-sphere must be positioned along a slope.
The circular disk C (gray) that functions as the clock-face of a 24-hour clock
can be rotated into itself (Fig. 11.64). An attachment A with a (red) reading
mark is fixed to it. A second attachment B (yellow) is also fixed, but in
a movable way: It can be restricted towards a circular arc (realized by a
slot guide within the disk), whose center is the normal projection of the
11.4 Minute-precise sundials for identifying the mean time 501

reading mark A on the circular disk (see also the exploded-view drawing in
Fig. 11.65). Two small drilled holes in the attachment B allow light rays to
reach the reading mark. The attachments are dimensioned so that light may
enter onto the reading mark through at least one of the two drilled holes
regardless of the time of year.

Fig. 11.63 Adjustment of the geographical latitude . . .

Fig. 11.64 . . . and the time

The sundial is adjusted on location to its geographical latitude ϕ. A very

simple instruction manual may now be given as follows: One may first adjust
the current date on the small (green) wheel D (Fig. 11.64 center, for instance
November 4th ) and rotate the circular disk until one of the light rays hits the
reading mark (Fig. 11.64 on the right). Now, the exact time may be read on
the hour index (in the particular example, it is 09:44 – without the equation
of time, it would merely indicate 10:00).
If both attachments were diametrical, then the equation of time would be
disregarded, but if the circular disk is rotated far enough so that the incident
light hits the reading mark, then the displayed time is pretty accurate: The
502 Spatial motions

deviation of each hour from the true midday corresponds to a rotation of the
disk by 15○ .
In order to take into account the equation of time, the attachment B (with
both drilling holes) must be slightly rotated – depending on the date. This
occurs, as pictured in the above example, by means of an eccentric dial inside
the half-sphere. Both attachments are connected by a rod (straight line). The
rod is pressed towards the eccentric disk by a spring. A rotation of this disk
causes the rod to be pressed in one or the other direction (Fig. 11.65 to the
right).

Fig. 11.65 Exploded-view drawing and the eccentric date dial

The dimensioning of the disk must be calculated so that the rotation angle
corresponds to the required position with the corresponding date. This used
to be done with the assistance of tables, and can nowadays be calculated by
using computers. Given an infinitely thin rod, the profile is the envelope of a
family of straight lines – and the actual profile is a curve which is an offset
(Fig. 11.65 to the right).
11.5 Kinematics in the animal kingdom 503

11.5 Kinematics in the animal kingdom

▸▸▸ Inverse kinematics

Fig. 11.66 Young cat in action, that uses its tail for counterbalance.

Over thousands of years of evolution, nature has perfected the movement of

animals. Be it a small cat that already moves its body perfectly to catch
bait (Fig. 11.66) or a dear that, in full speed, jumps through rough terrain
(Fig. 11.67).

Fig. 11.67 Leaping dear. Most terrestrial mammals can jump or leap, but some use this form of
locomotion more often than others.

There is no doubt that nature has already more or less optimized motion.
Otherwise, evolution will find a better solution via mutation, and individuals
that exhibit an even better way of moving, will reproduce more frequently
and, thus, spread their mutated genes. The question for a computer program-
mer is: How can such a perfect motion be simulated?
One possible solution for the problem is the following: First, gather reference
images and/or videos, preferably of exact side and front views (therefore,
Fig. 11.68 is not 100% ideal, even though the sequence is quite spectacular).
Next, load the model of a skeleton into professional programs that provide
modules for inverse kinematics (e. g., the free modeling and animation soft-
ware Blender ). Ideally, the model should be in a symmetrical resting pose.
504 Spatial motions

Fig. 11.68 Sprinting lion. Cats mostly use their back legs for jumping. The front legs leave the
ground before the back legs do.

Next up is the rigging process, where a bone-like structure is created that

represents the moveable joints and that will serve as a guideline for the model
to follow. After the overall structure is complete and the individual parts of
the model are assigned to the joints, inverse kinematic is applied to the limbs
and the head. In this way, just one of the joints is positioned by hand and all
others further up the joint hierarchy will be calculated by the program au-
tomatically. After adding keyframes to the animation, which are snapshots
of the joint positions and rotations, they will be interpolated in the final
animation to create continuous motion.

Fig. 11.69 Walking cat

The results can be quite spectacular: Fig. 11.69 shows the typical movement
of a larger cat. The skeleton was also used to simulate the leopard’s jump
in Fig. 3.11 (skeleton source by Florian Croquet https://floriancroquet.
com/). ◂◂◂

▸▸▸ Locomotion in mammals and large birds

Even though this is not a book about biology, the geometry explained in
this book can facilitate a better understanding of biology. A mammal moves
its legs backwards and forwards beneath its body (in contrast to a reptile
which has legs that are stuck out to the side). The most common four-
11.5 Kinematics in the animal kingdom 505

legged locomotion is the diagonal walk, being used by most hoofed animals
(Fig. 11.70) and their predators (Fig. 11.68)3 .
Giraffes and a few other animals, such as brown bears and camels, move both
legs on one side together and then both legs on the other side. When they
accelerate, there is a moment in each stride when all four legs do not touch
the ground anymore (Fig. 11.71).

Fig. 11.70 Running oryx (diagonal)

Fig. 11.71 Running giraffe: Both legs on one side together and then both legs on the other side.

Fig. 11.72 Sprinting ostrich: Outrun predators . . .

3 https://www.earthlife.net/mammals/locomotion-2.html
506 Spatial motions

Running on two legs, like humans or the ostrich (Fig. 11.72), is obviously less
complicated (ostriches are among the fastest runners and can easily outrun
their predators; it therefore does not pay off to develop “expensive” wings for
flying – the wings are only used for steering).

Fig. 11.73 Giraffe running

Fig. 11.74 Sprinting ostrich in a simulation.

The two animation sequences (Fig. 11.73 and Fig. 11.74)4 shall demonstrate
how inverse kinematics leads to practically the same results as nature has
come up with by itself. It is not surprising, since, in between the given posi-
tions, only those poses are calculated that lead to the smoothest and fastest
motion. Mathematics – originally a “self-contained microcosm” – is a science
that has always been oriented towards nature, and, thus, has “evolved” in a
direction that now enables this science to pre-calculate and simulate natural
processes.
As a final example, Fig. B.44 shows how the locomotion of frog legs (Fig. 11.75)
can be simulated by means of a simple kinematic mechanism. The inspira-

4 Giraffe: https:www.cgtrader.com/3d-models/animals/mammal/realistic-giraffe-skeleton
Ostrich: Idaho Virtualization Laboratory (IVL), https://sketchfab.com/ivlpaleontology
11.5 Kinematics in the animal kingdom 507

Fig. 11.75 Frog legs in practice

Fig. 11.76 Frog legs are being moved by a mechanism.

tion for this computer simulation came from a model exhibited in the zoo of
Vienna (Schönbrunn). ◂◂◂

▸▸▸ Gripping, biting, and shredding

Fig. 11.77 and Fig. 11.78 show three types of devices for gripping, biting,
and shredding, that can be found among arthropods.

Fig. 11.77 A tiny pseudoscorpion (≪ 5 mm). The pedipalps generally consist of an immobile
“hand” and “finger”, with a separate movable finger controlled by an adductor muscle.
508 Spatial motions

Fig. 11.78 Mighty mouths: On the left an ant, on the right a tiger beetle

The mandibles of ants are thick at the base and covered in thorns. They
can be opened to a degree of slightly over 90○ . The mandibles of tiger beetles
(genus Cicindela) differ considerably in form and function. They terminate in
dagger-like tips to cage their prey. These mandibles cannot be used as trans-
port devices, but they are a powerful apparatus when it comes to catching,
handling, and shredding prey. ◂◂◂

▸▸▸ Kinematic four-link chain in the monitor lizard head

Kinematic chains are frequently found in the animal world. The benefit of
this mechanism is, once again, that it enables the animal to move the parts
of its jaws against each other like a pair of scissors.

Fig. 11.79 Monitor lizards have a built in four-bar-link.

Such a kinematic chain can be found in the heads of monitor lizards (Fig. 11.79).
Note that both jaws open at the same time, since the red bone stays fixed.
◂◂◂

▸▸▸ How to extend the mouth to a long tube?

The sling-jaw wrasse (Epibulus insidiator) has developed a cunning hunting
trick: While catching prey, it rapidly extends its mouth into a long tube with
which it sucks up small fish.
11.5 Kinematics in the animal kingdom 509

Fig. 11.80 The sling-jaw wrasses ambush their prey with an astonishing kinematic apparatus
which contains multiple joints.

Fish estimate the level of danger by a predator’s size, speed, and distance
to them. The slingjaw wrasse is a seemingly inconspicuous, and therefore,
prey approaches without fear. The underlying kinematic mechanism, which
contains multiple joints, is very sensitive due to the length of the movable
bone structures. ◂◂◂

▸▸▸ Unfolding the wings

In beetles, the membranous hindwings are folded under heavily chitinized,
stiff forewings, the elytra. On the wing blade, beetles have several joints that
allow them to fold their wings in a characteristic way.

Fig. 11.81 Wing cases of chock3

Fig. 11.81 shows the different stages of a cockchafer (Melolontha melolontha)

unfolding its wings as it launches itself into flight. The elytra are lifted first.
Then then membranous wings, which are folded several times, are gradually
unfolded. The whole process lasts about half a second.
Once the stabilizing joints are fully extended, the wings are absolutely rigid
and impervious against aerodynamic forces that occur during flight (Fig. 11.82).
510 Spatial motions

Fig. 11.82 A cockchafer in flight

Fig. 11.83 Wing cases of flower chafers

A distinctive feature of flower chafers (Cetoniidae, Fig. 11.83) or also scarab

beetles (Scarabaeidae, Fig. 2.14) is that they do not open their wing cases
during flight. Their wings slide out from slots on the beetle’s sides and flap
in the open air, while the wing cases remain closed. By not having to lift
and position its wing cases, the flower chafer saves time and can launch into
flight very rapidly. ◂◂◂
12 A variety of filling patterns

A planar tiling is a gap- and overlap-less coverage of a plane with specially

formed panels or tiles. The simplest forms are triangles and quadrangles,
which may be regular or irregular. Regular pentagons already cause problems,
because a plane cannot be gaplessly covered by such shapes.
This is due to the violation of a basic condition: In each apex point, k n-
gons coincide, which means that the sum of interior angles in the n-gon
must be n ⋅ 360○ /k. However, the sum of interior angles of an n-gon equals
(n − 2) ⋅ 180○ (which can be proven inductively by decomposing the n-gon
into triangles). An equalization yields the condition 1/n + 1/k = 1/2.
This is impossible for regular pentagons, as there exists no natural number k
for which this condition is met. Nevertheless, there are many possibilities of
covering the plane with irregular pentagons.
The regular hexagon does not cause such problems – the cells of honeycombs
are a classical example of a hexagon tiling. However, this already exhausts all
regular n-gons where tilings are possible. There are tilings with only one basic
form (monohedral) or with many different basic forms (n-hedral). However,
the most important distinction is whether or not the tiling is periodical or
non-periodical. Coverage is periodical if there exists a translation in the plane
that would move the pattern into itself.

© Springer Nature Switzerland AG 2020 511

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_12
512 A variety of filling patterns

12.1 Ornaments and symmetry groups

Frieze ornaments
Two congruent figures with the same orientation can be brought into align-
ment through rotation and translation, while two congruent figures with op-
posite orientations require a reflection or a glide reflection (i.e., a reflection
followed by a translation along the axis of reflection).

Fig. 12.1 Seven types of frieze ornaments (singly periodic patterns)

A frieze ornament is a singly periodic pattern whose basic design arises from
elementary copies of single primary cell. To create such ornaments, we con-
sider a basic figure (inscribed in a square “fundamental cell”) and subject it
to one of these transformations (Fig. 12.1, Fig. 12.2).
From the primary cell, an ornament can be created through
● translation along the main direction or a diagonal direction,
● rotations about the midpoint through multiples of 90○ , and
● reflections in both central axes and the two diagonals (Fig. 12.3).
12.1 Ornaments and symmetry groups 513

Let us ignore pure translation along the main direction, which yields a simple,
trivial ornament, which is of no particular interest. We can now arrange two
or four congruent primary cells, either with the same or opposite orientation,
so that a frieze ornament results from repeating this enlarged pattern.

Fig. 12.2 Flowchart for the seven different ornaments created from a primary cell

Fig. 12.3 Two more examples for the seven different ornament that are cerated by a primary
“fundamental” cell.
514 A variety of filling patterns

Plane symmetry groups

One can attempt to bring some systematization to the not so obvious multi-
plicity of ways in which the plane can be tiled.

Fig. 12.4

The flowchart shown in Fig. 12.4 allows for the assignment of any tiling to
one of the seventeen possibilities shown.
In the Alhambra, in Granada, all seventeen patterns appear among the wall
tiles, and these were created long before this geometrical classification.
12.2 Periodical tilings 515

12.2 Periodical tilings

Regular n-gons as basic building blocks

Everybody knows the pattern of honeycombs (Fig. 12.5), which is based on
regular hexagons. Since every such hexagon can be subdivided into equilateral
triangles, this type of tiling also works with equilateral triangles.

Fig. 12.5 Classic tiling with hexagons (wasps and bees)

It is also possible to tile a floor more trivially, using regular quadrangles

(or squares). Other tilings with regular n-gons are impossible. In this case,
nature proves to be generous and pragmatic, as it often is (Fig. 12.6). In
the left image, the elongated plates on the bottom side of the rattle snake’s
neck appear to be considerably shortened. They barely appear to disrupt the
planar tiling that is pictured.

Fig. 12.6 On the left: hexagonal tiling (at least on the top side). On the right: Certain modifica-
tions can be discerned on the bottom side.

▸▸▸ From shark skin to flow-optimized architecture

The arrangement of the scales on the skin of fish and reptiles is of prac-
tical importance to these animals. Investigations of shark skin have shown
that water undergoes tiny turbulences as it moves across it – an impressive
achievement in streamlining.
516 A variety of filling patterns

Fig. 12.7 Analysis and architectural construction of shark skin tiling. The turbulences are pro-
duced deliberately for the purpose of ventilation.

At Studio Hadid, Mario Gasser, Peter Schamberger, and Sebastian Gallnbrun-

ner used these insights in architecture – an idea that won them the Austrian
construction prize in 2005. ◂◂◂

Inspired by such pictures, one may immediately think of interpreting tilings

as projections of spatial manifolds. A software developed for this purpose
permits the interactive construction of shapes based on magnetic repulsion
which, for instance, may very well be composed of regular pentagons, but
which may, nevertheless, diverge into a rhombic structure. Otherwise, the
result would always be a pentagondodecahedron (Fig. 12.8).

Fig. 12.8 Left and right: planar “half-regular” shapes which are essentially composed of regular
pentagons but which, nevertheless, partially diverge. At the center: Thus, bending allows tilings
of surfaces. The left – non-symmetrical – tiling by Franz Gruber seems to be unknown, and the
right one was invented by no less a figure than Albrecht Dürer.

▸▸▸ Tiling with paintings

An interesting “two-and-a-half-dimensional” tiling based on squares goes back
to Robert Lettner: He interprets square paintings as function graphs, and
thus, creates artificial “artistic landscapes” (Fig. 12.9).
The color does give the tiling an extra dimension. A color value (red-green-
blue) is assigned to each point in the plane, which is then somehow translated
into a height value. A clever coloring of the corresponding spatial points
produces interesting and often unintentional effects. ◂◂◂
12.2 Periodical tilings 517

Fig. 12.9 Tiling with squares

▸▸▸ Alteration of the sides of a basic building block

Tiling with squares or rectangles seems to be trivial. By changing border
lines, however, it can be varied such that the result is not trivial any more.
Fig. 12.11 shows how the sides of the squares of Florentine patterns are
replaced by line strips which are centrically similar with respect to the square
side.

Fig. 12.10 Simple tilings Fig. 12.11 Tiling using squares ◂◂◂

Fig. 12.12 Rhombus patterns with 3D effects (see also Fig. 2.16)

Irregular polygons as basic building blocks

Every triangle can be used to tile a plane: For any arbitrary triangle, a
reflection in a side’s midpoint produces a parallelogram from which entire
strips can be built. Such strips, laid next to each other, can cover the entire
plane.
518 A variety of filling patterns

Fig. 12.13 Non-trivial tilings using rhombuses, rectangles, and parallelograms

Tiling based on an arbitrary n-gon

Fig. 12.14 No longer trivial, but still somewhat Fig. 12.15 A square and a dodecagon as the
simple basic forms

Let an arbitrary quadrangle be reflected in a side’s midpoint. Now arrange the

quadrangle and the reflected quadrangle alternatingly side by side (Fig. 12.10)
to produce a strip. Multiple such strips can be seamlessly arranged (this is
best visualized by a color labelling of the quadrangle’s interior angles). This
method also works for non-convex quadrangles (Fig. 12.10 bottom left).
A special case is a rhombus made up of two equilateral triangles. With such a
rhombus, we can produce beautiful patterns, like one that looks like stacked
cubes (for instance, in the Basilica di Santa Prassede in Rome).

Tiling with an arbitrary pentagon

Fourteen types of convex pentagons that tile the plane are known today,
and they were all discovered from 1918 to 1985. However, there may still be
12.2 Periodical tilings 519

further undiscovered variants of convex tiling pentagons – it has not been

proven that the search is complete.

Fig. 12.16 Tiling with convex . . . Fig. 12.17 . . . and non-convex pentagons

Another way of classifying irregular tiling pentagons is one in which side

lengths are kept constant (convex and concave pentagons). Fig. 12.17 shows
instructions by Walter Jank on how to best pack skiing boots into the trunk
of a car, provided that the trunk is large enough.

Tiling with an arbitrary hexagon

Fig. 12.18

Three types of irregular convex tiling hexagons are known today. Fig. 12.19
shows a tessellation instruction which leads to irregular convex hexagons.
520 A variety of filling patterns

Fig. 12.19 Irregular hexagons

It is impossible to find tilings with n-gons, where n ≥ 7 based on a single basic

form (a proof can be found in Reimann: Tilings, geometrically considered, in
“Mathematical Mosaic”, Cologne 1977).

Periodical tilings with multiple basic forms

Fig. 12.20 Archimedean tiling with regular hexagons, equilateral triangles and squares

If all utilized basic forms are regular, one speaks of a semi-regular or Archi-
medean tiling.

Fig. 12.21 Archimedean tilings: regular hexagon and equilateral triangle (to the left), regular
dodecagon and equilateral triangle (to the right)
12.2 Periodical tilings 521

There are several different types: Square and equilateral triangle (two vari-
ants), square and regular octagon (to be seen in Fig. 12.22), regular hexagon
and equilateral triangle (two variants, one to be seen in Fig. 12.21 to the left),
a triplet of regular hexagon, equilateral triangle and square (Fig. 12.20), reg-
ular dodecagon and square, and finally, regular hexagon, regular dodecagon
and equilateral triangle (Fig. 12.21 to the right).

Fig. 12.22 Two congruent Archimedean tilings are “connected” via symmetric polyhedra.

Fig. 12.22 and Fig. 12.23 show a spatial interpretation of the “connection”
of two Archimedian tilings by means of congruent truncated frustums (these
shapes belong to the so-called scutoids, see https://en.wikipedia.org/
wiki/Scutoid). In Fig. 12.23, the three-dimensional scene is viewed from
above, again showing an interesting tiling which this time consists out of
squares and non-regular symmetric hexagons.

Fig. 12.23 Continuation of the above image; to the right a top view is displayed, revealing another
tiling with squares and hexagons.
522 A variety of filling patterns

12.3 Non-periodical tilings

We can distinguish between two types of non-periodical tilings: Aperiodi-
cal tilings, where the basic forms are suited for periodical as well as non-
periodical tilings, and quasi-periodical tilings, whose basic form is only suited
for non-periodical tilings (and then only if it adheres to certain rules).
Self-similar aperiodical tilings
If many instances of a basic form can be fitted together so that an enlarged
version of the basic form is produced, we speak of inflation. The inversion of
this procedure – the decomposition of a form into scaled-down but otherwise
equal versions – is called deflation. This pattern may then repeat itself at
multiple scales, which is called self-similarity or scale invariance.

Fig. 12.24 Chair tiling

Some examples include the chair tiling (Fig. 12.24) or the Sphinx pattern
(Fig. 12.25). From a single initial figure, the pattern is constructed by re-
peated deflation. The aperiodicity of the Sphinx pattern is caused by the
greater form being composed of many mirrored and unmirrored smaller forms.
Central aperiodical tiling
Certain basic figures can be arranged radially, or as a spiral, about a center.
The resulting tiling is non-periodical. A simple example is an equilateral
triangle with an angle of 30○ at the tip. It is then positioned 12 times about
12.3 Non-periodical tilings 523

Fig. 12.25 Sphinx pattern

a center where the tips are located, and it may, thus, be continued radially. If
this pattern is cut into two halves and translated by one or many steps, then
a spirally arranged aperiodical tiling from equilateral triangles is created.
The legs of the triangle can be replaced by congruent polygonal strips, which
produces an infinite amount of central tilings.
Quasiperiodical tilings
Each periodical tiling possesses a basic form comprising a finite number of
elements from which the whole may be patterned: but do quasiperiodical
tilings also have a basic form?
In 1966, Robert Berger proved that there exists a finite number of basic
elements which can tile the Euclidean plane only non-periodically. Berger,
thus, embarked on a search and first found a set of 20426 generating elements
for a quasiperiodical tiling. This number was greatly diminished in the follow-
ing years until finally, in 1973, Roger Penrose came upon a quasiperiodical
tiling that is based on only two elements.
One of these pairs of generators is renowned for its shapeliness and is known
as kite and dart. Another pair of generators is given by two rhombuses with
equal side lengths. The thin rhombus has an interior angle of 36○ and 144○
– the thick rhombus is likewise defined by 72○ and 108○ . Tilings of this type
are called Penrose patterns.

Fig. 12.26 Penrose patterns

The secret behind Penrose’s pair of generators lies in the golden ratio. The
kite and the dart are created from a rhombus with the angles 72○ and 108○
524 A variety of filling patterns
√
by subdividing the main diagonal at a ratio of Φ ∶ 1, Φ ∶= (1 + 5)/2 and
connecting the subdivision point to the blunt vertices. Then, all sides of the
kite have either a length of Φ or 1. If the respective edges of both shapes
are connected by two circular arcs with the radii 1/Φ (orange) and 1 (yel-
low) for the kite and the radii 1/Φ (yellow) and 1/Φ2 (orange) for the dart
(Fig. 12.27), then the following rule has to be obeyed during the construction
of a quasiperiodical tiling: Adjacent edges have to connect circular arcs of
the same color.

Fig. 12.27 Inflationary kite

Penrose patterns may also be created by deflation or inflation (see Fig. 12.26,
Fig. 12.27, Fig. 12.28): A kite deflates into two small kites and a small
dart: 1K → 2K + 1D. A dart deflates into a small kite and a small dart:
1D → 1K + 1D. (The inversion of this procedure is inflation.) In matrix no-
tation, we have
K D 2, 1
( )→M ∗( ), where M = ( )
D P 1, 1
f2n+1 f2n
Thus, the iteration of this process yields M n = ( ), where fn
f2n , f2n−1
is the n-th Fibonacci number.
1 1
If one starts with a kite shape and deflates it n times, or ( ) → M n ⋅( ) =
0 0
(f2n+1 , f2n ), then the ratio of the number of kites to the number of darts
converges against Φ. If an entire plane is completely covered, this ratio is ir-
rational. This is impossible for periodical patterns – and the quasi-periodicity
of the Penrose patterns directly follows from that observation.

It can be shown that there exists an uncountable number of Penrose patterns

made of kites and darts! The same holds true for all other Penrose patterns,
which, nevertheless, have the following bizarre characteristic: All patterns
are locally isomorphic: Each neighborhood of a pattern is also present at an
infinite number of other places!
How far away are the copies of a Penrose pattern’s finite neighborhood? The
answer to this question is also surprising: If a circular neighborhood with a
diameter of d is considered, then the distance between the boundary of the
neighborhood and its next exact copy is never farther away than d ⋅ Φ3 /2 ∼
2.12 ⋅ d.
12.3 Non-periodical tilings 525

Fig. 12.28 Inflationary rhombuses

A connection to physics: Up until 1984, only two forms of solid matter were
known – amorphous solid bodies and periodically ordered crystals. How-
ever, in the same year Dan Shechtman and his colleagues discovered a quasi-
periodical crystal formation in an aluminum-manganese alloy! Several hun-
dreds of such alloys are known today, and this third form of solid matter is
now known as a quasicrystal. The above-described Penrose pattern can be
seen as a model for “two-dimensional” quasicrystals if we imagine a crystal
through periodically layered Penrose planes.
The great surprise of Shechtman’s discovery was that until then, a crystal
was known as a solid body constructed from a periodically repeated initial
cell, but this conception turned out to be too narrow. Now, a crystal does
not merely have to exhibit 2-, 3-, 4-, or 6-fold rotational symmetry, but also
has to possess a discrete diffraction diagram (X-ray spectrum) with sharp re-
flections. It may also, for instance, exhibit 5-fold rotational symmetry, which
is the case with Penrose patterns.
There exist many other tilings that permit the construction of quasisperiod-
ical structures (e.g., the Ammann grid, octagon patterns, etc.). With these
patterns, real solid bodies may be modeled, and questions concerning the
stable arrangement of the atoms into these crystal symmetries may be re-
searched.
▸▸▸ Semi-regular pentagons

Fig. 12.29 Semi-regular pentagons and tiling with pentagons and semi-regular pentagons
526 A variety of filling patterns

In his paper Semi-regular figures between beauty and regularity ([26]) Hans
Walser defines a “semi-regular pentagon” as follows: When a rhombus is cut
off a regular pentagon (Fig. 12.29), the leftover will be a semi-regular pen-
tagon. Using this new element as tile, Walser builds various ornaments, tilings
(Fig. 12.30), crystallographic patterns, and spirals (Fig. 12.31).

Fig. 12.30 More tilings with semi-regular pentagons

Fig. 12.31 Concentric rings and spirals

Inspired by this work, Franz Gruber subsequently created Fig. 12.32 and
Fig. 12.33.

Fig. 12.32 Fivefold rotational symmetry

The combination of regular and semi-regular pentagons proved to be a “fruit-

ful” source for new types of patterns. ◂◂◂
12.3 Non-periodical tilings 527

Fig. 12.33 Iterations with an increasing number of regular and semi-regular pentagons

▸▸▸ Semi-regular dodecahedrons

Walser took the next step and generalized the idea into 3-space. He defined
a semi-regular non-convex dodecahedron.

Fig. 12.34 Iterations with more and more regular and semi-regular pentagons

The regular dodecahedron may be seen as a compound of a cube with a hip

roof on every face (Fig. 12.34, upper left). When the cube has a side length 1
a singular hip roof has a roof length a = 1/ϕ and a roof height a/2 (where
ϕ = 1.61828 . . . is the golden ratio). The semi-regular dodecahedron remains
when we cut off such a hip roof at every side of the cube. Walser describes
these related construction principles as follows: The regular dodecahedron is
positive, the semi-regular dodecahedron negative. The hip roof excess of the
regular dodecahedron corresponds to the hip roof deficit of the semi-regular
dodecahedron. Because of the relative situation between the two dodecahedra
528 A variety of filling patterns

the regular dodecahedron can sit on the semi-regular dodecahedron like the
egg on an egg cup (Fig. 12.34, second from the left).

Fig. 12.35 Regular and semi-regular dodecahedrons

Fig. 12.36 The space is filled by putting regular and semi-regular dodecahedrons together.

Since the cube is a “space filler”, the regular and the semi-regular dodecahe-
dron together can fill the space without gaps or overlapping.

Fig. 12.37 What you can create from a cube with positive and negative hips: rhombic dodecahe-
dron, regular pentagondodecahedron, cube, semi-regular dodecahedron and great stellated dodec-
ahedron.

Fig. 12.37 shows all the possible outcomes, if the hip roof’s planes are at
varying inclination. Besides the cube the rhombic dodecahedron is also a
“space filler”! ◂◂◂
12.3 Non-periodical tilings 529

▸▸▸ Filling the gaps with scutoids

Fig. 12.38 and Fig. 12.39 show the “connection” between two aperiodic tilings
by means of scutoids (p. 521).

Fig. 12.38 Two aperiodic tilings are connected by means of scutoids. Note that the single scutoids
need not have planar faces: If two connected sides of the generating polygons are not parallel, the
face is a hyperbolic paraboloid.

Fig. 12.39 Same arrangement as above, different only because of the changed distance between
the carrier planes.

Per definition, a scutoid has two parallel boundary polygons, and the ver-
tices of those two polygons are joined by either a curve or a Y-shaped con-
nection. In contrary to prisms, frustums and prismatoids, a scutoid has at
least one vertex between the two parallel boundary planes. The name “scu-
toid” was coined by the authors of https://www.nature.com/articles/
s41467-018-05376-1.epdf, who claim that “scutoids are a geometrical so-
lution to threedimensional packing of epithelia”. ◂◂◂
530 A variety of filling patterns

12.4 Non-Euclidean tilings

Several models of the hyperbolic plane exist, such as the Poincaré disk model
and the Poincaré half-space model. Let us choose the disk model here, which
has the advantage that measuring hyperbolic angles is equivalent to measur-
ing Euclidean angles. However, the metric is altogether different: Distances
steadily increase towards the boundary of the circle. The circular edge itself
is the set of all infinitely far-away points (horizon). The disk is produced
by stereographic projection of the sphere model into the hyperbolic plane
(Fig. 1.56).
Axiomatically, the hyperbolic plane differs from the Euclidean plane in that
there exists an infinite number of parallels to a straight line through a point
which does not lie on it. Another characterization may be given based on the
property that the sum of interior angles of any triangle is less than 180○ .
In the disk model, straight lines (the shortest distance between two points)
are circles which intersect the horizon orthogonally. The circles may look
like circles – however, the circle midpoint is generally not at the (Euclidean)
center. Equilateral polygons (bounded by orthogonal arcs) do not appear to
be equilateral, but are, instead, distorted due to a motion in the Poincaré
disk.

Fig. 12.40 Hyperbolical tilings by Jos Leys (www.josleys.com)

Which tilings with regular n-gons are possible?

Contrary to the situation in the Euclidean plane, the sum of angles of an
n-gon is not equal to but smaller than (n − 2) ⋅ 180○ . If k is the number of
n-gons adjacent to a vertex, then the following must be true for a regular
tiling with n-gons: n ⋅ 360/k < (n − 2) ⋅ 180 or 1/n + 1/k < 1/2. This is given for
all n ≥ 3 with an appropriate k, which means that there is an infinite number
of regular tilings in the hyperbolic plane.
For each regular n-gon, there is also a minimal value of k:
(n, kmin ) = (3, 7), (4, 5), (5, 4), (6, 4), (7, 3), and (n, 3) for each n > 7.

Fig. 12.40 and Fig. 12.41 show beautiful tilings of the hyperbolic plane. The
figure in Fig. 12.41 is defined by Bézier curves.
12.4 Non-Euclidean tilings 531

Fig. 12.41 Hyperbolic tilings with a swordfish by Johannes Wallner

What is the situation on the sphere?

Let us remember that tiling a sphere is, indeed, possible in certain special
cases. It primarily applies to the Platonic solids, see application p. 140.
▸▸▸ From a dodecahedron to an original and easily manufacturable lamp
Fig. 12.42 shows a remarkable patent (probably from the Asian region –
though the patent owner is unknown) inspired by such a tiling.

Fig. 12.42 Tiling of the sphere produces an aesthetically pleasing lamp.

In order to understand the issue, we must first abstract it. Let us consider
a regular dodecahedron and subdivide the faces, as in Fig. 12.43, into five
equilateral triangles. Pairs of triangles connected by a dodecahedron edge
form an “skew rhombus,” which is easy to manufacture. The 5 × 12 = 60
equilateral triangles are thus converted into 60/2 = 30 congruent rhombuses.
All that is now required is a bit of imagination in order to make the rhombuses
◂◂◂
attach to each other (Fig. 12.42 at the center) – and we are done!
▸▸▸ Puzzles in space
The toy industry has also undertaken geometric deliberations and invented
puzzle games which produce spheres. The fact that the building blocks are
similar but not quite congruent lies at the heart of any good puzzle game.
Take the triangulation of a sphere, as in application p. 140, combine two
532 A variety of filling patterns

Fig. 12.43 From a dodecahedron to tiling with triangles and “skew rhombuses.” It was also the
basis of patents describing a novel football and an original lamp.

triangles into skew quadrangles, and then construct an entire puzzle using
this approach.

Fig. 12.44 Tiling a sphere with nearly-congruent puzzle pieces is quite a challenge.

⊕ Remark : To simplify matters for the frustrated player, the manufacturer provided
a cylinder model of the result, which is composed of a rectangular strip and two
circular caps in its developed form (see Fig. 12.44 on the right). ⊕ ◂◂◂
13 The nature of geometry and
the geometry of nature

The title’s wordplay is perhaps an indication of this chapter’s intentions –

namely, to bridge the gap between nature and geometry. It is well-known that
many of the surfaces or solids described by geometry can be found in nature,
in both the macrocosm and the microcosm. The following question arises: Is
geometry a mere “plagiarism” of nature, or has it developed independently,
with coincidental parallels to the results of evolution?
Geometry helps us to understand, for example, why Platonic solids are “ob-
vious” solutions to many problems in nature and that if certain conditions
are given (such as regularity or symmetry), no other solutions actually exist.
Nature usually produces regularity or symmetry if an equilibrium of forces
or an optimal packing surface is sought. Evolution finds optimal forms on its
own.
More than two thousand years ago, the ancient Greeks Plato and Archimedes
could not have known that “their” solids are so frequently found on a molec-
ular scale. Pythagoras, on the other hand, defined music by means of pro-
portions. Therefore, it is no wonder that musical rules can be beautifully
visualized geometrically. Another valid question may be: Is music, as defined
by Pythagoras, rooted in “natural phenomena”?
Stanislaw Ulam [21] calls mathematics a self-contained microcosm which nev-
ertheless possesses a strong capability of capturing and modeling not only
any kind of thought process but perhaps the entire scientific process. To
some degree, this probably also applies to geometry as it is a subdiscipline
of mathematics.
© Springer Nature Switzerland AG 2020 533
G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_13
534 The nature of geometry and the geometry of nature

13.1 Basic geometric forms in nature

We have already seen it throughout this book: Nature is full of certain basic
forms. They are sometimes immediately apparent, as in snail shells, and at
other times rather “imprecise”. Although we may not have an immediate
answer for every case, we shall nevertheless attempt to get to the bottom of
these issues.
Symmetry

Fig. 13.1 Antennae of a moth, a mosquito, and a beetle (all males)

In nature, symmetry is an essential design element. Biologists refer to bi-

lateral symmetry when describing the regular arrangement of the parts of
an organism – but only if it is symmetric with respect to a central plane
(Fig. 13.1).

Fig. 13.2 Left: radial symmetry; right: only supposedly radial

If certain parts are arranged about an axis, then biologists speak of radial
symmetry (Fig. 13.2). This way, the animal kingdom can be subdivided into
Radiata (jellyfish, starfish, radiolaria, etc.) and Bilateria (bilaterally sym-
metric classes of animals).
In the kingdom of plants, blossoms exhibit radial symmetry (Fig. 13.17) much
more often than bilateral symmetry (Fig. 13.3).
13.1 Basic geometric forms in nature 535

Fig. 13.3 Simple symmetry in the kingdom of plants

Fig. 13.4 shows the imitation of a bilaterally symmetrical plant by a tropical

orb-weaver spider – a tactic that serves to attract insects (on the left: view
from above, on the right: view from below).

Fig. 13.4 Simple symmetry in the animal kingdom

Circles as distance curves and path curves

We have already explained that the most elementary and simplest curve
in nature is not the straight line (which, indeed, almost never occurs in life
forms) but the circle. These shapes are especially noticeable in the way waves
spread.
If we attempt to venture into the desert without being armed with either
a compass or ancient Bedouin tricks, we will inevitably end up running in
circles. One reason for this is that the length of our left step is never exactly
equal to the right one; another one is our inability to take into account minute
changes to the Sun’s position without making mistakes (ants and bees have
this capability, which has been proven by many interesting experiments).
When a wasp cuts a circular hole into a leaf (Fig. 13.5), it applies its jaws
at constant angles (analogous to a piece of ham being cut on the breakfast
table). Described more precisely, the jaws are always orthogonal to a pencil
of rays through the center of the circle. The circle is naturally the orthogonal
trajectory of this pencil of rays.
536 The nature of geometry and the geometry of nature

Fig. 13.5 The crime and the “culprit”. . .

The wasp does not seem to follow this pencil of rays, which – after all – only
exists in theory. It repeatedly cuts short segments of constant length into the
leaf. These segments form an angle of slightly less than 90○ to the moving
direction, and the procedure is repeated until the wasp reaches its starting
point. Thus, it does not actually cut out a circle, but a regular polygon with
so many vertices that it can hardly be distinguished from a circle.
Both the journey in the desert and the cutting of the hole produce circles by
a combination of motion and “work angle”. Circles are also distance curves:
All points are equidistant from a center.

Fig. 13.6 The “building of a fortress” by a tiny crab in tidal times

This must be how the wonderful ring-like constructions of the tiny carcinus
maenas (Fig. 13.6) are produced. The animal rolls the excavated sand (which
contains its food) into balls and moves it some steps away from the center.
Once a single ring is finished, another concentric ring is made. This tedious
work is all undone when the next tide sweeps it away.
When plants such as the circular lichen grow on the surface of a stone, it
makes sense that they should try to “augment mass” at their outer boundary.
If this happens regularly, then circular contours are automatically produced
(Fig. 13.7, Fig. 13.8).
The speed at which this lichen grows is not important for the geometrical
principle. For a layperson, it would be difficult to detect whether the lichen
13.1 Basic geometric forms in nature 537

Fig. 13.7 Linear growth (I)

Fig. 13.8 Linear growth (II)

stops expanding every few years. This is far easier to detect in trees: The tree
produces new layers around the central trunk (Fig. 13.7). The coloring or the
hardness of the cross-section suggests that, while this growth is periodical,
it does not proceed at the same rate throughout the year. In tropical woods,
these tree rings are not as pronounced, because the growth is more regular
than in colder latitudes.
Although the tree’s increase in thickness is more or less linear, it has no
influence on the form of the cross-section.

Spheres and other convex basic forms

The circles in the plane correspond to spheres in three-dimensional space.
If the carrier medium permits it, then the waves or impulses always expand
spherically. Natural objects (whether living or dead) also tend to expand
spherically at first, but a change in strategy usually occurs after an initial
growth phase. The reasons for this are manifold, though one particularly
important reason may be the following:
The surface of a sphere increases at a square of its radius, which is why
for sonic strength or brightness, for instance, decrease by the square of the
distance. However, the volume that is included inside the spherical surface
increases cubically with the radius. If a sphere has reached twice its initial
radius, then its volume, as well as its mass, has increased eightfold, whereas
its surface has only increased fourfold.
538 The nature of geometry and the geometry of nature

Minimal and maximal packets

Fig. 13.9 Half-spherical water droplets

The water droplets in Fig. 13.9 are forced into a spherical shape by the
surface tension of the water (cohesion), in opposition to the rules of gravity,
it would seem. In order for such a droplet to remain on the leaf, the contact
surface with the leaf should be as large as possible. This makes it possible
for adhesion to balance out the gravitational pull.
This aforementioned subtle equilibrium only works up to a certain scale. At
some point, either cohesion or adhesion are too small to “counteract” the
increasing mass.

A maximum of food for the offspring. . .

Fig. 13.10 Oval insect eggs (egg-shaped ellipsoids of revolution)

The shape of insect or bird eggs is interesting from a geometrical perspective.

Without going too much into detail, it may be said that bird eggs are neither
spherical nor ellipsoidal – we know this not least because of our daily break-
fast experience. The insect eggs in Fig. 13.10 are most likely not entirely per-
fect geometrical surfaces either, though they do seem to approximate spheres
and ellipsoids of revolution much better than their chicken counterparts.
13.1 Basic geometric forms in nature 539

Fig. 13.11 Ants love their plant lice.

⊕ Remark : The “cute” ladybird in Fig. 13.10 on the left is, interestingly enough, al-
most shaped like half an ellipsoid. It is thereby optimally protected against agitated
ants, which tend to protect “their” herds of aphides – from which they extract a
valuable sugary syrup – with great ferocity (Fig. 13.11). Plant lice are a favorite
dish of the ladybird and its larvae. ⊕
Don’t roll away!
The reason why eggs have a peculiar shape is to make sure that they contain
a maximum of food for the offspring in the smallest amount of space. Bird
eggs have the additional difficulty that they tend to roll out of their nests,
whereas insect eggs mostly stick to one spot.

Fig. 13.12 “Laboratory experiment”: An egg is rolled on a wooden floor (“short distance”).

In order to minimize this effect, nature simply deviates from the spherical
form, and thus, eggs simply wobble around – if they move at all.
Fig. 13.12 shows an experiment in which the rolling behavior of a chicken egg
with an angular momentum is tested. It is clearly visible that the egg moves
in a very specific swerving motion, which may prevent it from rolling too far
away. (However, this also has to do with the fluid contents of the egg).

Unrolling and development

As we have already seen, developable surfaces play a big role in technology,
being so easy to manufacture. Whenever nature “goes to work” on leaves or
blossoms, they are usually unrolled which is the geometrical equivalent to a
development.
540 The nature of geometry and the geometry of nature

Fig. 13.13 The many phases in the growth of a blossom (I)

Fig. 13.14 The many phases in the growth of a blossom (II)

The process of unrolling often happens over a longer period of time. In the
meantime, the leaf may slightly alter its shape before partially hardening.
Thus, the result is not necessarily a developable surface anymore. The many
phases in the growth of a blossom in Fig. 13.13 and 13.14 show this quite
well.

Fig. 13.15 On the left: The cross-section of a dried willow branch, on the right: coral colony
13.2 Evolution and geometry 541

13.2 Evolution and geometry

We have previously touched on the geometry of bird eggs. However, how do
birds “know” that non-spherical eggs are better for their offspring (assuming
that spherical eggs roll away more often, and thus, break more frequently)?
If a bird were to lay spherical eggs in inclined or otherwise unstable cir-
cumstances, then it might discover that an elliptical form would be more
practical if only the animal had the mental capacity for this analysis. If it
had that capacity and could somehow affect the shapes of its own eggs, it
would certainly do so in the next attempt.
While the individual may be incapable of such a feat, the species as a whole
is able to surmount this challenge given evolutionary timescales. Evolution is
as ingenious as it is simple. We must now deviate slightly from the main topic
of this book – for a good reason, as a solid understanding of evolutionary
principles is essential for the section that follows. Biologists are hereby asked
for their indulgence with regard to my generalizations, but since evolutionary
thinking does not yet enjoy widespread awareness, even 150 years after its
landmark introduction, and is even the target of some hostility, an attempt
to clarify some of the basic issues cannot hurt.

Mutation
We shall first and foremost consider the principle of “inexact replication” (mu-
tation): No animal in nature produces offspring that are completely identical
to itself, and no lifeform produces identical offspring all the time. This may
in part be influenced by the physical difficulty of “repeating” a complicated
genetic code without any error. (As a point of comparison, one may try to
memorize the first 250 decimal places of π.) The act of exact replication
is further complicated by the varying states of the environment where such
copying takes place. However, it stands to reason that even perfectly identi-
cal laboratory conditions would, sooner or later, be host to imprecise copying
procedures.
The ancestors of reptiles and birds may at some point have “started” the
colonization of land with eggs that were spherical, like those of fish. If that
was the case, then many of these eggs must have rolled away and were prob-
ably destroyed in the process. This waste may be compensated by a more
prolific production of eggs, which of course costs more energy. However, due
to the principle of inexact copying, some early reptiles may have produced
more spherical eggs, while those of other reptiles were less so. Thus, the non-
spherical eggs, being more stable on the ground, were more likely to produce
healthy offspring.
542 The nature of geometry and the geometry of nature

Inheritance
A second principle of nature applies now: Offspring usually inherit most
of their phenotypic properties from their parents. If, therefore, more parents
have been born from “less symmetric eggs,” then – depending on the strength
of this selective pressure – the next generation should exhibit a higher pro-
portion of individuals laying non-symmetric eggs. If the selective pressure is
strong, then it does not require many generations before the kinds of reptiles
which lay “classically shaped eggs” – and not those who lay spherical eggs –
become dominant.
The sum is what matters
It may now be argued – quite wrongly – that this principle should apply to
every trait of the organism, and that it should take only a few generations
before a perfect solution is found for every problem (which in the case of the
eggs would be purely geometrical). However, this runs contrary to a third
principle of nature: The best sum of adaptations is usually the successful
one, and this also depends on a constantly changing environment. If some
reptiles, for instance, continue to lay spherical eggs but, for reasons of another
trait that is affected by the same genes, are “better at evading predators”,
then this might turn out to be a more successful adaptive strategy than “not
laying spherical eggs”.
It may be the case that the individuals who are better at evading predators
also adapt to lay non-symmetric eggs without any negative effects on their
evasive strategies. For lack of a better term, this would kill two birds with
one stone. It may simultaneously occur that other, “better evaders” acquire
a new trick by way of mutations that somehow lessen the disadvantage of
spherical eggs. Whatever the situation might actually have been like, it seems
that the advantage of more or less non-spherical eggs was so significant that
this trait is now shared by virtually all oviparous land-based animals.
Phyllotaxis, or: How easily are we deceived?
What appears to be a simple explanation for the non-spherical shape of eggs
may require more profound scrutiny in other cases. A typical example is the
arrangement of sunflower seeds, which is also known as phyllotaxis. Much
has been written and theorized about this topic – and despite all of that, the
solution turns out to be as ingenious and simple as evolution itself.
A premature conclusion
Let us briefly discuss the underlying theories: Looking at a sunflower long
enough, one will inevitably find spirals in the pattern. In fact, two opposing
spirals can be discovered! It may further be noticed that the number of spirals
in a pencil is always a so-called Fibonacci number. (More on this topic can
be found in [9].) The golden mean is closely linked to these famous numbers,
as is exponential growth. (By repeatedly multiplying a number with itself,
we get a geometrical sequence which – if displayed in a coordinate system
13.2 Evolution and geometry 543

– yields an exponential curve.) We might thus be easily impressed: It seems

that sunflowers are “aware” of the golden mean and grow exponentially –
while most other plants do not!
A completely different explanation
Now on to the most likely explanation (claiming to know the definitive solu-
tion for such matters may be misguided):
A sunflower multiplies by means of its seeds, which are usually eaten by birds
and dispersed because the birds that pick them up tend to lose seeds from
time to time. One thing is clear: For this strategy to work, a sunflower has
to produce as many seeds as possible. This increases the probability that the
species may survive in the long run.
Let us imagine a plant species (a predecessor of the modern sunflower) which
may have adopted the following successful growth strategy during the course
of its evolution: “Grow your seeds by creating a principal seed at the ap-
proximate center, and position each further seed such that the direction is
changed and that the position is translated slightly away from the center.”
In order for this procedure to be quantifiable, the rotation angle would have
to be constant, as would the translation distance. Up to a certain point, this
strategy would, indeed, allow the flower to produce new seeds. The plant
would reach its natural limit when there is no more space for a new seed
because it is occupied by an already existing seed.

Fig. 13.16 Different translation distances

Among the infinite number of combinations of angles and distances, there is

certainly an optimal one. Evolution would eventually find it over the course
of many generations, as it did with the shape of eggs: Mutation is constantly
at work, changing angles and distances ever so slightly, and those flowers who
are lucky to have hit upon a good combination would carry more seeds for
their reproduction. “Their” advantageous combination would be passed on to
their progeny, which have a chance of “improving” it further.
The result is quite impressive: a very “close” approximation of a modern
sunflower (Fig. 13.16 on the left). The center is a little bit tight – the seeds
there would have to be smaller than those at the outer edges. The fact still
544 The nature of geometry and the geometry of nature

remains that this is a remarkably effective method of packing seeds into a

blossom. We can also detect the aforementioned spirals. Indeed, this method
creates them entirely without the need for exponential growth! The angle of
rotation is quite peculiar, dividing the whole revolution of 360○ at the golden
mean.
The images in Fig. 13.16 at the center and on the right show further possibil-
ities if the translation distance is allowed to decrease by a certain percentage
as the seeds move away from the center. However, this is mere “cosmetics”.
If this procedure were to be “extrapolated” onto a sphere, as illustrated in
Fig. 13.17, then the seed density would increase even further.

Fig. 13.17 Computer simulations. On the right: Optimization by an extrapolation onto the sphere
or ellipsoid.

⊕ Remark : From a daisy to a meter-high sunflower, a common explanation for some

of their specific proportions can often be found in the “golden mean”. A fact that
has fascinated many numerologists. This is not meant to be derogatory: Even such
deliberations may be mathematically valid. It can be shown that the golden mean
is the “most irrational of all numbers”:
1
Φ=1+ 1
≈ 1.6180339887.
1+ 1+ 1
1+ 1
1+...

The flower “needs” this irrationality, which makes it possible to apply the rotation-
translation-step to the seeds as often as possible. ⊕

The development is not always clearly visible

By changing the golden angle ever so slightly, we get patterns as in Fig. 13.18
on the right. Such figures are already familiar to us: For instance, the blossom
in Fig. 13.18 on the left already resembles the pattern very closely. Let us
13.2 Evolution and geometry 545

Fig. 13.18 Changing the “golden angle” by tenths of degrees

leave it at that, without coming up with any further hypotheses: In geome-

try, spirals are relatively easy to produce: Even when they are defined quite
differently, the may be very similar locally. The spiral pattern in Fig. 13.18
on the left might owe its appearance to a completely different principle than
the filling pattern of the blossom.

Fig. 13.19 Snapshot of a low-pressure turbulence in the making

Turbulences caused by low-pressure areas are another class of phenomena

where we should not be too hasty in inferring the precise reasons for their
appearance. Several parameters apply here. Conservation of angular momen-
tum must surely play a central role, but so must lateral acceleration, which
is connected to the north-south-expansion of the turbulence (different path
velocities depending on the geographical latitude!), as well as widespread
pressure drops and several other factors. Given this complexity of cause and
effect, it is an undeniably great accomplishment of modern meteorology that
accurate predictions of such large-scale weather patterns can be made up to
several days in advance.
546 The nature of geometry and the geometry of nature

13.3 Planetary paths and fish swarms

The forms of galaxies may very well be inevitable: On a large scale, outer
space is essentially dominated by only a few laws, primarily by gravitational
attraction and the law of inertia.

Fig. 13.20 Galactic spirals including their carrier surfaces of revolution

Let us hypothetically consider a galaxy like our own Milky Way, which forms
as a result of the compaction of interstellar mass and, according to the law
of conservation of angular momentum, is finally left with a constant angular
velocity or momentum.
Due to gravity, the center of this system will gradually increase in density
(and may eventually form a black hole which will continue to “accrue” mass
until it evaporates). The clusters of stars in the system will continue to move
towards the center and increase their angular velocity in the process.

Fig. 13.21 A cosmic catastrophe (devastating impact) . . .

No matter what parameters one applies, one can always find a galaxy that is
similar to a snapshot of the corresponding simulation shown in these pictures
– and this simulation will always contain spirals made of star clusters.
The Moon seems to have been formed 4 to 5 billion years ago by a collision
of the newly-formed volcanic Earth with another massive celestial body. Fol-
lowing the impact, glowing chunks abounded in our planet’s neighborhood.
Each of these chunks possessed the Earth’s angular momentum and drifted
along in spiral motion before being “caught” on a circular orbit around the
13.3 Planetary paths and fish swarms 547

Fig. 13.22 . . . and the subsequent consolidation (Moon)

planet. A gradual accretion of mass followed – probably due to the lack of

space and gravitational forces – and the Moon was thus gradually formed.

Fig. 13.23 The explosion cloud of a supernova passes our Sun . . .

It may “only” have been hundreds of millions of years prior to those tumul-
tuous events that the remains of a supernova passed our still very young Sun,
which managed to catch part of this debris by means of its gravitational pull.

Fig. 13.24 . . . and “produces” a planet.

The laws of physics then acted their part, and it did not take long for the
supernova remains (which mostly consist of metals) to circle our star on
elliptical paths (with the Sun being at one focal point). Collisions of the
orbiting matter caused its gradual accretion until the planets were born. Our
asteroid belt is probably nothing more than a not-yet-consolidated “souvenir”
of that long gone supernova.
One may now argue that these topics do not have much to do with geometry
and that such speculation is best left to physicists. On the other hand, one
may also consider it a beautiful example of separate academic disciplines
supporting one another: Without computer simulations, which depend on
computer geometry, the outcomes would certainly not look as spectacular
548 The nature of geometry and the geometry of nature

as they do in the resulting image, and the layman who is not able to solve
differential equations would have to take physicists at their word.

Fig. 13.25 “Spiral lawn sprinkler”

The following question is closer to home: When a lawn sprinkler rotates using
the repulsion of the exiting water streams, spirals are formed which resemble
their galactic counterparts: As the water moves away from the axis of rota-
tion, its angular velocity decreases by the square of the distance according
to the law of conservation of momentum. Other factors apply here, such as
air resistance and gravity, but this need not concern us here, as so much of
this phenomenon is simply the result of chance.
Let us examine another water-based scenario, but from a different perspec-
tive: The seemingly chaotic developments in outer space are not that far
removed from, for instance, the simulation of a swarm of fish. This time,
let us focus on the discoveries of biology. Small fish often exhibit a very
sophisticated swarming behavior, not unlike that of many bird species.

Fig. 13.26 Impending danger and escape

Interestingly enough, there is no alpha leader determining the direction of

the swarm. This whole choreography is rather the product of many singular
reactions. The goal of a single fish is only to remain close to its neighbors
(about a single body length). When foraging, individuals tend to be more
leisurely while they move in the direction of more food.
When a predatory fish takes interest in the swarm, the individuals in its
vicinity will naturally want to flee from the place of danger. Their neighbors
13.3 Planetary paths and fish swarms 549

Fig. 13.27 Back to the swarm . . . and immediately away from it!

may not even have realized the cause of disturbance, but they still reflexively
follow their neighbors on their paths of escape. This leads to a chain reaction
(shock wave), which may even involve individuals that are far away from the
predator.
Once the danger has passed, the fish which are farthest away will once again
seek the close company of their relatives. After a certain amount of time, and
possibly until the next predatory attack, “order” will have returned to the
swarm.

Fig. 13.28 An everyday occurance under water: Some cormorants are hunting small fish. They
try to encircle the fish, but this proves to be quite a challenge amidst all the chaos . . .

This supposed chaos is actually governed by only a few parameters, which

hardly restrict the variety of shapes that a swarm can have at any given
moment (see also Fig. 13.28).

▸▸▸ Geometry as nature’s artistic inspiration

It is not be mere coincidence that the shapes in this very interesting avant-
garde architectural sketch by Andreas Krainer (“the Emperor and the Queen,”
Fig. 13.29) resemble jellyfish, since the sketch consistently follows a geomet-
rical plan of construction that is often found in nature:
This entire sketch is based on a single space curve, which defines both the
position of the beams and girders as well as the form of the hull. Beams and
girders in the traditional sense no longer exist, and the transition between
them is completely smooth. By a multiplication and a deliberate positioning
of the generating curves into a specific pattern (such as a hexagon whose
550 The nature of geometry and the geometry of nature

Fig. 13.29 Art and nature

center and corners contain one copy each), a radial system defining the tower
is created. The relation between one tower and the next is the same as that
between one supporting beam element and the next. This produces a repeti-
tion of the same principle of arrangement, only on a higher scale. A change
of the generating space curve changes the whole sketch, which allows for an
unimaginable variety of possibilities. ◂◂◂

▸▸▸ Organic architecture

Fig. 13.30 Dome, rib vault, barrel vault

A geometrically interesting sketch was created by Matthäus Wasshuber. Pro-

ceeding from classical vault shapes (Fig. 13.30), which were themselves in-
spired by nature: He connected his vault-like modules following examples
found in the plant kingdom (Fig. 13.31).

Fig. 13.31 Inspired by the shapes of plants

13.3 Planetary paths and fish swarms 551

After a longer process of experimentation, he produced the sketches shown

in Fig. 13.32, which combine shapeliness with the capacity to carry large
weights.

Fig. 13.32 New architectural formations

◂◂◂
552 The nature of geometry and the geometry of nature

13.4 Scaling behavior in nature

To some degree, it is possible to deduce how big an animal is by “looking”
at a neutral photograph (without the background or other objects in the
vicinity being visible): An elephant’s legs are so massive because they need
to be able to carry the animal’s own weight. The tiger, being the largest land
predator, also needs to have characteristically large muscle diameters. (The
tiger’s paw is famous for its sturdiness.)

Fig. 13.33 The largest herbivore and the largest carnivore on land: Proportionally speaking, both
animals need higher muscle diameters.

Reducing the size by two orders of magnitude (Fig. 13.34) takes us to the
world of insects and other articulate animals. Their muscle diameters need
not be so large anymore: The weight of the animal has, in fact, diminished
cubically in relation to the scaling factor, while the muscular force has only
been reduced in square proportion to the diameter.
Interestingly enough, both tigers and wasps bear striped patterns – as do
many other animals. In the case of the wasp, this may be explained as a
warning signal, but on the tiger it is a form of camouflage. Comparable
geometrical patterns are employed in different ways here.

Fig. 13.34 Large herbivores and carnivores on the macro scale: Proportionally speaking, both
animals need far smaller muscle diameters.
13.4 Scaling behavior in nature 553

It may be arguable to what degree a giraffe actually resembles the seahorse

in Fig. 13.35, but consider the following: Geometrically speaking, there is a
limited amount of possibilities of “fitting” volumes and functionality into a
confined space. Given the immense variety of lifeforms, one should, therefore,
expect to stumble upon certain similarities eventually. Apart from the curious
shape of their heads, however, the giraffe and the seahorse have very little in
common. They each live in completely different conditions, to say nothing of
their different scales.

Fig. 13.35 There is certainly a geometrical resemblance, including the mane or the dorsal fin, but
that is about it.

Geometrical similarities often give the misleading impression that the underlying
laws that produced these shapes must have been comparable. This may not have
been so. We do not even have to mention the strange laws that govern physics
at the quantum scale to conclude that, from a mathematical and physical
perspective, small-scale events do not have to proceed the same way on a large
scale.

Fig. 13.36 Scaling behavior: It is difficult to determine the size of such a stone formation. This
is why in the left image the stone formation looks like a big lizard being fed.

This scaling behavior is less obvious among inorganic objects: As in Fig. 13.36,
it does not seem to matter for a weathered rock whether it measures several
meters or just a few centimeters.
554 The nature of geometry and the geometry of nature

Fig. 13.37 Ice crystals – only slightly enlarged

Fractal forms – the “frayed silhouette” which seems to repeat itself over and
over again – occur both on large and on small scales. It has been proven that
it is virtually impossible to ascertain the approximate size of a cloud from a
mere photograph.

Fig. 13.38 Glaciers and screws of ice crystals

When comparing Fig. 13.37 to Fig. 13.38, we may notice that frozen water
tends to produce fractal contours of some aesthetic value at every scale. It
is no wonder that such forms have always inspired artists to produce novel
creations (Fig. 13.39).

Fig. 13.39 Art and nature: Zaha Hadid, Ice Storm

While modern architecture is certainly concerned with the creation of attrac-

tive forms whose variety is often taken from nature, the ideal case combines
13.4 Scaling behavior in nature 555

mere aesthetics with the implementation of certain useful insights. Fig. 13.40
shows the implementation of the “shark skin principle” (application p. 516)
in the creation of an architectural structure – in this case, the forms of sep-
arate scales of shark skin have been taken as templates. Their advantages in
aerodynamics have been proven by realistic simulations.

Fig. 13.40 Another creation from Studio Hadid: the scales of shark skin as the underlying principle
which manifests itself both in detail (application p. 515) and in the building as a whole and which
is supposed to yield aerodynamic advantages both on great and small scales.

In the architectural sketch by Martin Zangerl for the Centre for Digital and
Media Technology in Los Angeles, a prototype of a spatial knot (a minimal
surface) is used to transform a voluminous structure into a “bony,” open
structure (Fig. 13.41, bottom left) by means of variation. This procedure
occurs both on a small scale and on a large scale (top left, right), so that
parts of buildings with either completely closed or highly open hulls may
emerge.

Fig. 13.41 From voluminous museum halls to open offices

556 The nature of geometry and the geometry of nature

13.5 Musical harmony through the eyes of geometry

Many mathematical and geometrical relations can be found in music. We
shall focus on three particular examples here. Along with parts of the text
itself, these examples are taken from [1]. An entire appendix of [9] is dedicated
to the topic of music. The images were created with the programming library
Open Geometry.
Readers who do not possess any prior knowledge of music theory are sim-
ply invited to enjoy the mathematical balance inherent in music and the
geometric visualizations that it enables.1
▸▸▸ Displaying a tonal system on a torus
A torus is cut into 12 slices (Fig. 13.42). On each slice there are 12 spaces
(to keep it simple, these are identified only once in the image), which may
either be interpreted as chromatic-enharmonic locations of the tonal system
(12 major and minor keys) or as enharmonically changed tones.

Fig. 13.42 Displaying a tonal system on a torus

In 12 convolutions, the torus is entwined by a rotoid, which intersects the

slides in every point, producing 12 enharmonically closing circles of fifths. ◂◂◂

▸▸▸ The “grid of keys”

By means of a grid of congruent circles, Fig. 13.43 visualizes a circle of fifths

1 Thereis a significant difference between English and German notation (http://en.wikipedia.

org/wiki/Key_signature_names_and_translations):
C, D, E, F , G, A have the same signature in both languages. The letter H, however, is
denoted by B in English;
Instead of Cis, Dis, F is, Gis, Ais (German), one writes C ♯, D ♯, F ♯, G ♯, A ♯;
instead of Ces, Des, Es, Ges, As, B (German), one writes C ♭, D ♭, E ♭, G ♭, A ♭, and B ♭.
13.5 Musical harmony through the eyes of geometry 557

with all notes and chords for the respective key, including the circle of the
parallel key as well as the subdominant, tonic, and dominant scale degrees.

Fig. 13.43 Grid of keys in the plane

Enharmonic closure has been attempted in Fig. 13.44. Geometrically speak-

ing, this involves the development of Fig. 13.43 on a cylinder of revolution.
A transformation onto a torus or a sphere is also possible. ◂◂◂

Fig. 13.44 Closed grid of keys Fig. 13.45 Tonal system

▸▸▸ Three-dimensional circle of fifths

Fig. 13.45 shows an attempt at connecting the closed circle of fifths in equal
temperament (red surface) with the “spiral of fifths,” which is open upwards
and downwards (it is actually a helix, to be geometrically precise). The base
tones lie on the central plane (red), as well as in the ascending spirals of the
and areas (light beige, cream) above and in the descending spirals of
the and areas (dark red).
558 The nature of geometry and the geometry of nature

The enharmonically changing tones lie on the vertical green lines. The con-
necting straight lines of corresponding tones form two skew closed helical
ruled surfaces (see Fig. 7.30). ◂◂◂

▸▸▸ Tonal centers

Fig. 13.46 A tonal system with columns. On the right: new tonal center

Fig. 13.46 shows the C-major tonality as an example of a tonal system. The
columns in the inner circle display the triads which correspond to this scale.
In the outer circle, the typical sounds of the extended C-major tonality can be
found on top of the chromatic tones (D-flat-major as the Neapolitan chord,
B-flat-major as the double subdominant, F-minor as the minor form of the
subdominant, D-major as the double dominant, C-minor as a variation of
the tonic, and B-major as the dominant of the dominant parallel E-minor).
Every sound or tone represented by a column or a sphere may become the
new tonal center (Fig. 13.46 on the right), which happens by a procedure of
modulation. ◂◂◂

▸▸▸ Inverted retrograde canons

Fig. 13.47 A canon by J.S. Bach which may be played forwards and backwards but also by
inverting the notes on the central line. After one circulation, the notes have been inverted. The
second melody nevertheless harmonizes with the first and should, in fact, be played simultaneously.

Fig. 13.47 shows a Möbius strip designed by Franz Gruber with the use of
our own in-house software, with which a “marquee” displaying a canon by
J.S. Bach has been visualized by means of a texture. An animation of this
intriguing object can be found on the website to the book. ◂◂◂
14 More applications

In this chapter, we consider additional applications that may be interesting

for the reader. Sometimes these applications are independent of the previ-
ous chapters, more often they use insights gained from previously discussed
topics. First, we will deal with multiple reflections or refractions, with illu-
sions, and with so-called “impossibles”. Then we go into further detail with
inversions. Finally, we will return to the topic of mathematical patterns like
Voronoi diagrams and fractals, which seem to be omnipresent in nature. The
above picture is a close-up of a diatomeen that measures only some 1/20 mm
in diameter. These tiny microalgae, which are found in the oceans, water-
ways and soils of the world generate a large amount of the oxygen produced
on the planet. One can detect amazing patterns under a scanning electron
microscope.

© Springer Nature Switzerland AG 2020 559

G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3_14
560 More applications

14.1 Multiple reflections, illusions, and impossibles

▸▸▸ Multiple reflections in two orthogonal mirrors
With multiple reflections in two perpendicular planes, we can create a re-
markable effect, whereby left and right – and not, as in the ordinary mirror,
the front and the back – are reversed. What exactly is happening here?
For the sake of simplicity (Fig. 14.1), two planes ξ and η are assumed to
be vertical coordinate planes with the equations x = 0 and y = 0. Each of
the planes initially generates a virtual – usually reversed – counterpart Ωx
(change of signs in the x-values) or Ωy (change of sign in the y-values). The
two objects are indirectly congruent and – regarding the issue of perception
– are of equal rank as the original Ω.

Ωyx Ωxy

Ωx

Ωy

Fig. 14.1 Several images in two mutually per- Fig. 14.2 Several images in three mutually per-
pendicular mirror planes pendicular mirror planes

Consequently, Ωx has a virtual counterpart Ωxy through reflection in η, and

Ωy has a virtual counterpart Ωyx through reflection in ξ. However, because
of the special position of the mirror planes (ξ ⊥ η), the two new objects
are identical: Ω∗ = Ωxy = Ωyx . Furthermore, we can only get a partial view
of them in the respective “mirror window”. The part labeled with Ωyx is
produced through reflection in the right mirror window of the visible virtual
object Ωy in the left mirror window.
The two mirror windows touch each other along the edge s = ξ ∪ η (according
to our choice, the z-axis); so, they merge into a single window which consists
of two mutually perpendicular rectangles. This allows us to see our object
Ω∗ . It was created by double-reflection and is, therefore, directly congruent
to Ω. The two directly congruent objects Ω and Ω∗ can be transformed into
each other by a half-turn about the intersection s of the mirror planes or by
an axial reflection in s. Thereby left and right are interchanged at first, and
the characters are still readable as usual.
14.1 Multiple reflections, illusions, and impossibles 561

In the photo in Fig. 14.1, the “auxiliary objects” Ωx and Ωy are entirely
visible. In human vision, one normally only sees Ω∗ in the image – comparable
to an ordinary mirror image, but left and right are reversed. ◂◂◂

▸▸▸ Reflection of a water wave

Even though water waves do not have the exact same properties as light
waves, we can tackle the problem of multiple reflections in a similar way:

Fig. 14.3 A water wave in a container propagates and hits a wall. From then on it is geometrically
useful to take the virtual reflected wave into account.

Imagine a wave propagating in a pool. As long as it does not hit a wall, the
simulation is easy (Fig. 14.3 to the left). Once the wave hits a wall, there will
be interferences. They can easily be controlled when we consider a “virtual
counter wave” emanating from the wall (Fig. 14.3 to the right).

Fig. 14.4 Two walls: Now we have to consider three “virtual counter waves”.

If there is a second wall perpendicular to the first one, considerations become

more tricky. Clearly, the second wall will create an additional virtual counter
wave which interferes with the wave pattern in the pool. Additionally, those
parts of the wave that reach the rectangular corner will be doubly reflected
(like a ball on a billiard table). The effect of this can be determined by
considering a third virtual wave that is a secondary reflection of the reflected
waves on both walls. ◂◂◂
562 More applications

▸▸▸ What exactly happens between two parallel mirrors?

Consider the confusion that is created by two parallel mirrors in Fig. 14.5:
By changing the distance settings, the photographer was able to focus on
any of the available originals or reflections. Visually, the observer is unable
to distinguish between original and virtual objects, so that it is not even
guaranteed that the person depicted is actually seen as a real person and not
just their mirror image – this depends on the focal length that is actually
used. That is to say, it depends essentially on the aperture of the viewing
cone.

Fig. 14.5 Three similar scenes, but in each of them, a different reflection is in focus. Can you see
where “the original” is?

Let A be the position of the observer (Fig. 14.6) standing between two mirrors
σ1 and σ2 (distance d apart). The camera, which is directed towards the axis
a, can only capture those things that are within the cone of vision ∆.

P3

2v
∆
a
P1
u σ1

u
P0
d
v
A σ2

P2
Fig. 14.6 For each virtual point, a new one arises at the “speed of light”.

Now, assume that there is another person (symbolized by a point P0 ) standing

between the mirrors, at a distance u from σ1 and at a distance v from σ2 :
14.1 Multiple reflections, illusions, and impossibles 563

u + v = d. Thus, two virtual mirror images P1 and P2 arise at the distances u

and v behind σ1 and behind σ2 . The two points are a distance 2u + 2v = 2d
apart. P2 is visually indistinguishable from a real point and is reflected in σ1
to a point P3 . This lies at distance 2d from P0 . If one continues in that way,
one obtains two congruent series of points P0 , P3 , . . . and P1 , P5 , . . . whose
points are separated by a constant distance of 2d.
In Fig. 14.5, these series are clearly visible. (In one of them, the person can be
seen from the front, in another from behind.) In both Fig. 14.5 and Fig. 14.6,
the observer and the multiple reflections are not contained in the cone of
vision. It is noteworthy that the autofocus of the camera has no problem
focusing on any reflection, which again demonstrates that a purely optical
mirror image is indistinguishable from reality. The distance to which the
autofocus adjusts equals exactly that of the “reflected” person in the room
– and not the distance to the mirror. The repeated reflection causes a loss
of brightness, so that reflections that are further away appear less clearly
(darker). ◂◂◂

▸▸▸ Analyzing a selfie (Fig. 14.7)

Let us now look in detail at what can be seen in the photo inside an almost
fully-mirrored elevator cabin.

Fig. 14.7 Can we now explain every single virtual image?

First of all, the photo does not show the real person but only virtual images
of the person. Due to the design of the camera, the person pushes the trigger
button of the camera with the right index finger. So, all the virtual images
where the person appears to use the left index finger are reflections of order
564 More applications

1, 3, 5, . . . (red circles), and the other ones of are of order 2, 4, 6, . . . (blue

circles).
The mirror images that are circled in blue are actually composed of two
partial reflections of even order: The edge of the cabin wall is the border
line in between. However, note the following: This edge is acutally always a
mirror image of the real edge behind the person.
What about the image circled in yellow? According to what was previously
said, it is a composition of three reflections, and in total it is a point reflection
at the corner of the cabin. Even though the finger pushing the trigger cannot
be seen, it will be on the right-hand side.

Fig. 14.8 Distorted mirror images in an amusement park

As “homework” you can try to analyze the residual reflections, e. g., those in
the upper mirror. After that you could examine the selfies in Fig. 14.8, which
were taken in an amusement park (Fig. 14.8). ◂◂◂

▸▸▸ Vision underneath the surface (Fig. 14.9)

In Chapter 9 (p. 403ff), we discussed Snell’s window and the total reflection
at the water surface outside this window.
Let us analyze the situation in Fig. 14.9, which is a drawing by my daughter
Sophie: First of all, Fish A (a grouper) can clearly see all the other fishes
around it. Additionally, due to the law of reflection, it will see – though
in a distorted form – everything that is outside the water, which appears
compressed into the circular Snell’s window. This circle is the base of a cone
of revolution Γ with apex A. As a reflection on the surface, we get another
cone of revolution Γ∗ with apex A∗ . Everything that is outside of Γ∗ will – for
fish A – also be visible as a total reflection. Thus, the puffer fish C appears
twice to our grouper. The third fish B swims mainly inside Γ∗ . However, its
dorsal fin is partly outside. So, A will see B’s dorsal fin as a total reflection
immediately outside the limiting circle c.
14.1 Multiple reflections, illusions, and impossibles 565

B∗

A∗ C∗
β0

Snell’s window R

β0
C
A B

Fig. 14.9 Who sees whom? Fish A looks at its environment. What does it see? This question is
by no means trivial!

Fig. 14.10 Photographic proof in the bath tub and another experiment with a fish whose dorsal
fin is notoriously famous.

In Fig. 14.10, the situation with the dorsal fin is slightly different, since the
fin is partly above the surface. The tip of the fin outside the water is, thus,
visible in Snell’s window. The photo to the left was taken with an underwater
camera. However, the fish was a plastic model and the situation was recreated
in a bath tub. In a calm sea, sharks can also see clearly what is happening
outside the water as they swim with their dorsal fin above the surface. When
there are more waves, sharks have been observed poking their heads out of
the water in order to get a better view (“spy hopping”).
As a final task, we might ask ourselves what a backswimmer (Fig. 14.11) can
see in its usual position “hanging” from the water surface. It must have an
overview of the whole environment – both underneath and above the surface.
◂◂◂
566 More applications

Fig. 14.11 Backswimmers (Notonectidae) are powerful aquatic insects that hang upside down
from the water surface.

▸▸▸ Winding to the left or winding to the right?

Fig. 14.12 This is a snail and its reflection. Viewed from above, the shell winds clockwise. In the
mirror, however, it winds counter-clockwise.

Snail shells or mussel shells that “expand in three dimensions” can be either
left- or right-handed. Almost all snail shells wind in the clockwise direction
(mathematically negative) when viewed from above. This is not the case for
the “snail king”. Its shell “winds to the right”.
14.1 Multiple reflections, illusions, and impossibles 567

Fig. 14.13 Spirula spirula (also named little post horn squid, a species of deep-water squid) has
a shell inside its body. Drawing by Ewald Rübsamen.

For the shell of the little post horn squid in Fig. 14.13 – at least at first
glance – there is no view from above or below, because the spiral of the shell
is visible from both sides. Therefore, you cannot say in which direction the
spiral is wound.

Fig. 14.14 Christmas tree worms have two “crowns” in mirror-symmetric forms.

Fig. 14.14 shows strange organisms that are part of the species Spirobranchus
giganteus, which lives on corals. Here we see spiral surfaces that wind both
in clockwise and counter-clockwise direction. ◂◂◂
568 More applications

Illusions and impossibles

▸▸▸ Hexagons and cubes

Fig. 14.15 Typical illusions with planar hexagons.

Fig. 14.15 shows a common illusion created with tilings of planar hexagons:
Each hexagon is split into three rhombs that are painted in different colors.
For the human brain, this creates the impression that we are looking at cubes
or similar polyhedra.

Fig. 14.16 Planar pattern or a cube-packing?

In Fig. 14.16 a pattern was created by computer. The question whether the
displayed object is two-dimensional or three-dimensional cannot be answered.

Fig. 14.17 Left: A photo of a reflecting polyhedron; middle: the explanation; right: look twice, or
maybe even three times!

Fig. 14.17 shows a series of macro photos of reflecting polyhedra (cubes),

which are arranged similarly to the ones in Fig. 14.16. Look closely at the
14.1 Multiple reflections, illusions, and impossibles 569

pictures from left to right. It is almost impossible to concentrate. Hint: The

labeling in the rightmost photo is correct, even if you may think it is not. The
depicted object is a reflector on the spokes of a bicycle (application p. 62).
◂◂◂

▸▸▸ Zoetropes

Fig. 14.18 A disk with a number of equally distributed objects that display in-between motions
of the object. When rotated at a certain angular speed, the brain is “fooled” and sees only one
object in motion.

Fig. 14.19 Here a pile of intersecting cubes is rotated. The pile, however, is created with a system:
Corresponding cubes are twisted slightly with each rotation. With the right angular speed of the
disk, the cubes in the pile appear to revolve.

A zoetrope is a pre-film animation device that creates the illusion of motion.

A sequence of still images depicting objects at different stages in motion is
shown at such a high frequency that the human brain is led to believe that
the object is moving. ◂◂◂

▸▸▸ The Freemish crate, also called Escher’s cube

An “impossible” is an object which implies an apparent contradiction with
regard to the visibility of its constituent parts. A classic example can be
seen in the left image of Fig. 14.20. The second image in Fig. 14.20 shows
how such an object could actually be constructed, exhibiting the illusion of
impossibility only from a certain well-defined perspective – in this case, only
in a normal projection. If we had a perspective image (as a result of a central
projection), we would need another opening in the object; otherwise, the
trick would be revealed (Fig. 14.20, image to the right). ◂◂◂
570 More applications

Fig. 14.20 Left: a classic impossible. Middle: The underlying trick is revealed. Right: problems
with a central projection.

▸▸▸ The Penrose triangle, also known as the Penrose tribar

Fig. 14.21 shows how to create another impossible. Let us first look at a

Fig. 14.21 An impossible triangle (from left to right): (1) the underlying idea, (2) its secret is
revealed, (3) removing one and a half cubes from one rod, (4) the trick is revealed again.

cube from a perspective that allows its diagonal to appear as a point. During
normal projections, this produces a very regular shape. (The projections of
the edges form an equilateral hexagon.) Let us now group additional cubes
around three of the original cube’s edges (as in the picture to the left), so
that they form three square rods, which together produce a sort of skewed
triangle. If we now omit the upper cube and one half of the next cube, the
entire structure will appear to be closed in normal projection (Fig. 14.21,
third from the left). This is confusing, as it thwarts our expectations of
the visibility of the object. The trick is revealed again when we change the
direction of the projection ray (Fig. 14.21, image on the right). ◂◂◂

▸▸▸ Does this work in perspective images, i.e., in photographs?

The question is whether an object like the tribar in Fig. 14.22 could be
photographed in such a way that the “error” would be impossible to detect.
We have to be aware of the fact that a geometrically perfect normal projection
cannot, to date, be simulated by a single photo. By employing an extremely
telescopic lens (with a focal length exceeding 1,000 mm), we can approach
14.1 Multiple reflections, illusions, and impossibles 571

Fig. 14.22 This matter is even more difficult with a perspective image (sizes change!).

a normal projection relatively well. Still, we must change the sizes of the
objects in order to avoid errors when taking pictures from a specific eye
point. In the following applications, there will be an explanation as to how
this can be achieved. ◂◂◂

▸▸▸ Perspective illusions

Fig. 14.23 shows a typical illusion. It only works perfectly from a certain
position of the eye (in this case on the principal normal of the frame). In this
case, a normal projection would be counter-productive (image in the middle).
The spatial scene consists of a set of spheres. The further away the spheres
are, the bigger they have to be for the illusion to work. The image on the
right shows how the spheres are arranged. ◂◂◂

Fig. 14.23 Typical illusion (left to right): eye point in special position, normal projection, spatial
situation. Note that the closer spheres have to be smaller when they are further away.

▸▸▸ Rolling upwards?

On the Internet, one can find fascinating examples of optical illusions, like the
one created by Kokichi Sugihara in which balls appear to roll uphill (https:
//www.youtube.com/watch?v=hAXm0dIuyug). Fig. 14.24 shows a similar idea,
which was programmed by Franz Gruber. How does it work?

Everything depends on the fact that your eye point has to be forced in
572 More applications

Fig. 14.24 The sphere rolling uphill.

Fig. 14.25 Explanation for the “Kokichi Sugihara” illusion

a certain position. If, from this position, you think that you are looking at a
“stable and balanced construction”, your brain will immediately interpret the
heights of certain points in the construction and it will tell you: The platform
in the middle has the highest position.

Fig. 14.26 Another comparable illusion and its explanation

However, the construction is not symmetric and stable. Its vertices lie some-
where on the projection rays through the vertices of the alleged construc-
tion. Among the infinitely many possibilities, you should now choose a non-
symmetric construction where the platform is in a lower position. With more
complicated objects, one should employ spatial perspective collineation to
calculate the corresponding object. ◂◂◂
14.1 Multiple reflections, illusions, and impossibles 573

▸▸▸ Inside or outside?

In the course of the renovation of the Foundation of Francisco Giner de los
Ríos in Madrid, Cristina Díaz Moreno and Efrén García Grinda created a
remarkable environment: This garden is a space that is determined by and
changes through the way someone is looking at it and participating in this
space. The proposed spatial system consists of a series of rooms, some roofed
and others open to the elements, which redefine the existing garden and
combines the old buildings and the new ones.

Fig. 14.27 Varying views from different positions

Fig. 14.28 The overall concept (on the right) and detailed planning

This garden is not green but purple, coral and silver, resembling the Castil-
ian plateau like Giner saw it. It is defined not only by the changing colors
of the plants but by the changing filigree of the cladding and the overlap-
ping succession of leafy envelopments – the vegetation, the outer wall, the
reflections in the glass, the bearing structure, people in movement, the stairs
and the permanent exhibition system. The size and position of the rooms is
determined by our moving eyes, based on a precise study of the lines of sight
of the new route inside the garden. If we regard our eyes as projectors, rather
than screens for capturing information, then we can use them to shape the
diagonals, the overlaps of the different planes of the garden. ◂◂◂
574 More applications

14.2 More about inversions in space

Inversion in a sphere
In Section 1.5, we have already discussed the inversion in a circle. When r
is the radius of the inversion circle, then a point P with distance d to the
inversion center C will be transformed into a point P ∗ on P C with distance
r2 /d. We can also say:
P P ∗ = r2 .
We had the following main properties: The transformations works in both di-
rections. Even though it is not linear, it preserves circles and angles. Straight
lines have to be considered as generalized circles, so that in general, straight
lines are transformed into circles through the center C. Conversely, circles
through the center C are transformed into straight lines.
When we generalize this transformation into 3-space, we replace the inversion
circle by an inversion sphere. Again, the formula P P ∗ = r2 defines the trans-
formation, and it is revealed that additionally to the already mentioned main
properties, the transformation turns spheres into spheres and preserves an-
gles in space. Planes can be considered as generalized spheres with infinitely
large radius. So, planes are also transformed into spheres that contain the
center C. Conversely, spheres that contain C are transformed into planes.

torus cyclide

inversion
sphere

Fig. 14.29 Inversion in a sphere: A ring torus is transformed into a ring cyclide.
14.2 More about inversions in space 575

▸▸▸ Inversion of a torus

In Section 6.5, we considered Dupin cyclides, which are algebraic surfaces
of degree four with special properties. By analogy with the ring torus, we
introduced the “ring cyclide” as an inversion of a ring torus in a sphere
(Fig. 14.29).
A ring torus is – like any surface of revolution – enveloped by spheres of
varying sizes centered on its axis. Two of those spheres that touch the torus
along a parallel circle and contain the center of the torus are thus transformed
into “infinitely large” spheres through the center C, i.e., into planes. As a non-
trivial result, a ring cyclide has two exceptional tangent planes. If we invert a
spindle torus or a thorn torus, we get the other types of cyclides (the needle
cyclide and the double-horn).
◂◂◂

▸▸▸ Inversion of cylinder and cone of revolution

Fig. 14.30 Inversion in a sphere: A cylinder of revolution is, in general, transformed into a needle
cyclide.

Astonishingly, the needle cyclide and the double-horn can also be generated
by inverting a cylinder of revolution or a cone of revolution, respectively.
Fig. 14.30 shows that the needle cyclide occurs when we apply the inversion
to a cylinder of revolution. The pinch point is the inversion of the infinitely
far “apex” of the cylinder and is thus in the center of the sphere. The rulings
of the cylinder are transformed into circles that lie in the connection plane of
the line with the inversion center. Here it is again quite clear that the cyclide
has two exceptional tangent planes (which are best seen in the upper-left
image of Fig. 14.30).
Inversion also works well with cones of revolution. In the general case, this
will produce a double-horn (Fig. 14.31). One cusp is the center C of the
sphere; the other one corresponds to the apex of the cone. ◂◂◂
576 More applications

Fig. 14.31 Inversion in a sphere: A cone of revolution is, in general, transformed into a double-horn
cyclide.

▸▸▸ An almost extraterrestial figure

By means of inversion, one can create a nice and non-trivial figure (Fig. 14.32):
We start with a given number of congruent spheres and arrange them in a
circular way. (Their centers are then vertices of a regular polygon.) Now they
fit into a torus. When we apply an inversion, the two circles of the torus in
the symmetry plane are transformed into two new circles. (One has a certain
degree of freedom by choosing the center C of the inversion sphere.) The
spheres are also transformed into non-congruent spheres which touch each
other and form a chain. However, the centers of these spheres lie on an el-
lipse (compare Section 5.2). Without inversion, creating such a figure would
be a very difficult task to accomplish.

Fig. 14.32 A non-trivially closed chain of spheres or circles

14.2 More about inversions in space 577

Fig. 14.33 Plausible generation of crop circles, courtesy of Franz Gruber

⊕ Remark : The figure was created after someone had shown the author photographs
of crop circles like the one depicted in Fig. 14.33, upper left. Even though it has
been repeatedly demonstrated how humans can produce this kind of hoax (https:
//en.wikipedia.org/wiki/Crop_circle), many still believe that the patterns are
messages from “aliens”. Fig. 14.33 shows sample screenshots from a simulation by
Franz Gruber which shows how this pattern can be produced efficiently. A give-away
clue are the – always present – tracks left by agricultural machines, because people
who produce the patterns have to move across the wheat field along such tracks. ⊕
◂◂◂

▸▸▸ The loxodromes on a sphere

C
Σ∗
s∗

Σ
Fig. 14.34 Inversion of a logarithmic spiral s →s∗(inversion center C), which is simultaneously
a stereographic projection from C: The carrier plane Σ of s is transformed into the sphere Σ∗
through C.

Another example illustrating the usefulness of the inversion in a sphere is the

following: The famous logarithmic spiral intersects all rays emanating from
578 More applications

its center at a constant angle (Fig. 14.34). We choose the center C of the
inversion sphere on the normal through the center of the spiral s. Then the
carrier plane Σ of s is transformed into a sphere Σ∗ centered on the normal
that goes through C. The points of the inverted spiral s∗ lie on rays through
C and on Σ∗ . The rays are transformed into meridian circles of Σ∗ , and
due to the angle-preserving property of the inversion, s∗ intersects all those
meridian circles under a constant angle.
We have already discussed such loxodromes at the end of Section 6.3. By
means of inversion, we arrived at the result in a very efficient manner. The
task also relates to the topic “stereographic projection” (Section 5.4): The
center C of the inversion sphere is the North Pole of Σ∗ , so-to-speak, and
this North Pole is now the projection center for the stereographic projection.
⊕ Remark : The inversion of a logarithmic spiral in a circle around the center of the
spiral is again a logarithmic spiral: It is angle-preserving, and logarithmic spirals
are loxodromes of all radial rays. ⊕ ◂◂◂

▸▸▸ Inversion of a clothoid

c∗

e
e∗

Fig. 14.35 A clothoid c (black) with evolute e (blue) is inverted in a circle around the inflection
point, which generate pseudo spirals c∗ (red) and e∗ (green). A series of osculating circles of c∗ is
drawn to show how evenly the circles are distributed.

A clothoid has the remarkable property that the curvature radius changes in
proportion to the arc length (see also Fig. 4.5). The inversion of a clothoid c
leads to a curve c∗ which is “optically similar” and may be called a “pseudo
14.2 More about inversions in space 579

spiral”. Moreover, the evolute e, which is the locus of the centers of all os-
culating circles, is another kind of pseudo spiral. When we invert e, we get
another pseudo spiral e∗ that looks almost like a clothoid (Fig. 14.35). ◂◂◂

▸▸▸ Sterographic projection of a clothoid

Fig. 14.36 A clothoid is stereographically projected onto a sphere.

One more example relating to the topic of stereograhic projection: As a con-

sequence of the definition of a clothoid, all osculating circles are arranged
such that they never overlap at least for one half of the curve. If we project
the curve onto a sphere (e.g., above the inflection point), this remarkable
pattern of circles is transformed into a non-trivial pattern of circles on the
sphere without any overlapping (Fig. 14.36). ◂◂◂

▸▸▸ Inversion in a cylinder (axial involution)

Let us once more go back to the inversion in a circle. With the radius r of
the inversion circle, a point P with distance d to the inversion center C will
be transformed into a point P ∗ on P C with distance r2 /d (P P ∗ = r2 ).
Now we generalize this transformation into 3-space in another fairly natural
way: We replace the inversion circle by an upright cylinder of revolution
(Fig. 14.37 to the left). A space point P is inverted in the cylinder as follows:
We consider the top view P ′ of P , invert it in the base circle of the cylinder,
and go back in the same height as P .
Let us determine the properties of this kind of inversion. Clearly, straight
lines and planes perpendicular to the axis are fixed as a whole. A coaxial
580 More applications

Fig. 14.37 Left: The axial inversion of a straight line is a cubic circle. Right: A helispiral with an
Archimedian spiral as top view is transformed into a spiral with a hyperbolic spiral as top view.
Its carrier surface of revolution is of degree four.

cone Ψ (apex C) with meridian with z = kx is mapped to a surface of revo-

r2
lution Ψ∗ with the regular hyperbola z = kx (Fig. 14.37 on the right).

However, what is, for instance, the image s∗ of a straight line s? The top
view s′ is transformed into a circle through C, and s∗ lies on a cylinder of
revolution that contains the axis of the inversion cylinder. The curve itself
can be identified as an upright cubic curve. Generic planes are “even worse”.
Whereas, in the case of an inversion in the sphere, the image is a sphere
through C, the plane transforms into a surface that is far more complicated,
namely a surface of degree three. We see: The axial involution is not nearly
as useful as the involution in a sphere. Still, one can produce nice images
like the one in Fig. 14.37 on the right, where a helispiral is transformed into
a spiral winding up the afore mentioned surface of revolution Ψ∗ . The top
view is inverse to an Archimedean spiral, and therefore, a hyperbolic spiral.
◂◂◂
14.3 Voronoi structures 581

14.3 Voronoi structures

Fig. 14.38 A leaf under the magnifying glass

Fig. 14.39 The wings of a dragonfly

Fig. 14.40 Cracks in dried mud

Let us assume a plane on which a set of points has been scattered. Our
task is to subdivide the plane into convex regions (whose centers are de-
582 More applications

fined by the given points) such that the closest point of any location within
a convex region is always the centre of the same region. This can be ac-
complished by calculating the perpendicular bisectors of all possible pairs
of points (Fig. 14.41 left). For each point, the perpendicular bisectors to all
adjacent points produce a convex region. If all convex regions are combined,
a so-called Voronoi diagram is produced (Fig. 14.41 middle).

Fig. 14.41 Tesselation of the plane with convex polygons, starting from an arbitrarily chosen
point cloud. Right: the surface of the fruit of a philodendron.

▸▸▸ Iterative Voronoi structures

Fig. 14.42 Optimization of the tesselation: The random point set is iteratively improved by
replacing the original centers with the barycenter of the adjacent convex polygon.

In Fig. 14.42 on the left, randomly distributed centers were first chosen,
from these centers a Voronoi diagram was derived. It is noteworthy in this
case that the sizes of the individual cells vary substantially. If one wanted
to equalize the cell sizes iteratively, one could employ the following strategy:
As the (red) centers do not precisely overlap the centroids (green) of their
respective region, one could, in a next step, replace the red centers with green
centroids (Lloyd’s algorithm) and re-calculate the Voronoi diagram. It would
14.3 Voronoi structures 583

have the same number of cells at roughly the same locations. However, the
cell sizes would be significantly more balanced (Fig. 14.42 on the right). We
can speak of “iterative Voronoi structures”.

Fig. 14.43 This photo shows the bottom side of a fungus whose structure strangely resembles an
(iterated) Voronoi diagram.

Fig. 14.43 shows the bottom side of a fungus – associations with a Voronoi
diagram are unavoidable. ◂◂◂

▸▸▸ Some classic examples in nature

Fig. 14.44 Intriguing basalt formations

The pictures in Fig. 14.44 show intriguing basalt formations at the famous
echo cliffs in Iceland. Basalt in its extrusive form is a volcanic rock formed
by the cooling of extrusive basaltic lava at or very near the surface. The
contractional fractures form during the cooling of thick lava flow. The topol-
ogy of these columns’ lateral shapes forms a random cellular network closely
resembling a Voronoi diagram. Hexagons appear to be dominant in the cross-
section. However, one also frequently encounters pentagons, heptagons, and
other polygons. The slower the cooling process, the larger these polygons are.
◂◂◂
584 More applications

▸▸▸ Voronoi diagrams on the sphere

Fig. 14.45 Voronoi diagrams on a sphere, starting with random points (left). Even with extremely
unevenly distributed cells, each iteration “improves” the mesh, and eventually, the results almost
always look astonishingly similar.

The recipe for the creation of Voronoi diagrams also works on curved sur-
faces, e.g., on a sphere. In the spherical case, the perpendicular bisectors of
all possible pairs of points are great circles on the sphere, and the convex
polygons are convex spherical polygons. Physically, one can always interpret
such a diagram as the result of tensions within the surface, with the centers
of the cells being the “attractors”.
Fig. 14.45 on the left shows a diagram determined by a set of randomly chosen
centers on the sphere. Even though it already looks natural, the geometrical
quality of the mesh can be significantly improved by letting the centers float
such that they become the spherical barycenters of the corresponding cells.
After just a few iterations, the cells are distributed much more evenly on the
sphere.

Fig. 14.46 Different results of iteration processes with different numbers of cells. For 60 random
cells, e.g., the result usually comes close to a “soccer ball” (12 pentagons, 20 hexagons).

Fig. 14.46 shows three results of such an optimization process for three dif-
ferent numbers of attractors on the sphere. Even though they are still almost
on the same location as the randomized starting position, some numbers pro-
duce meshes on the sphere that strongly resemble some well-known figures.
14.3 Voronoi structures 585

In the image to the right, the result comes very close to a “soccer ball” (a
projection of the corresponding Archimedian solid). ◂◂◂

▸▸▸ Iterated Voronoi diagrams as a tool to find good point distributions

Obviously, this method is an extremely useful tool for the numerical deter-
mination of point distributions on curved surfaces. It is also likely to be a
principle of nature, affecting, for instance, the distributing of macro struc-
tures on the skin of animals.

Fig. 14.47 A caterpillar. With some imagination we could recognize a cylindrical shape on the
left and the shape of a torus on the right.

For example, the venomous hair of some caterpillars (Fig. 14.47) serves as
a layer of protection against predators. The following principle is essential
from the perspective of the caterpillar: How should the hair on its body be
distributed to provide optimal protection?

Fig. 14.48 Randomly distributed points on a cylinder and final (non-trivial!) equilibrium

Let us approximate the caterpillar’s body using a cylinder of revolution (im-

age on the left) and that of the rolled-up caterpillar with a torus (image to
the right).
Two further examples (Figure 14.50 and Fig. 14.51) shall illustrate that the
conclusion “Voronoi is everywhere” is not far-fetched. ◂◂◂
586 More applications

Fig. 14.49 Randomly distributed points on a torus and final equilibrium

Fig. 14.50 Are meshes on an armadillo Voronoi diagrams?

Fig. 14.51 Does a sea star produce Voronoi diagrams?

14.3 Voronoi structures 587

▸▸▸ Voronoi diagrams in 3-space

Finally, the concept of Voronoi diagrams can be extended to three-dimensional
space: We choose a number of cell kernels (this may, for instance, happen by
random chance), and let the faces of each cell (which are always convex) lie
in the symmetry planes to the neighboring kernels (Fig. 14.52 on the left).
Again, we can improve the “evenness” or “balance” of the diagram by replac-
ing the originally chosen kernels with the barycenters of the generated cells.
If we repeat this procedure a few times, the diagram converges towards an
optimal result (Fig. 14.52 to the right). These diagrams, which are defined
purely mathematically, come close to “real foam” – their structure merely
depends on the first set of kernels that are selected.

Fig. 14.52 Creation of an optimized three-dimensional Voronoi diagram. Left: random start.
Right: equilibrium after several iterations.

As part of an experiment, foam was generated within a plastic cube, which

produced a structure reminiscent of a Voronoi diagram. First, the cell kernels
were distributed unhomogeneously (Fig. 14.53 to the left), and after longer
periods of time, the foam slowly reorganized itself into a larger and more
stable mesh (Fig. 14.53 on the right).

Fig. 14.53 Foam in a small cube is reorganizing itself into a stable mesh.
588 More applications

Fig. 14.54 Sculpture in the Austrian Alps

◂◂◂

An alternative algorithm
Even though these are less geometrical, it shall be mentioned that, besides
the “iterative Voronoi algorithm”, there are other algorithms that can be used
to solve distribution tasks. Fruchterman and Reingold, e.g., proposed an iter-
ative force-directed algorithm for drawing graphs that exerts attractive and
repulsive forces on the nodes in [2]: Nodes connected to each other through
an edge attract each other, while repulsive forces are applied to all pairs of
vertices. In each iteration the maximum displacement applied to a vertex is
reduced such that the adjustment becomes finer over time. Since then force-
directed algorithms have found wide application in graph drawing (see [23])
and have also been adjusted for application in geometry (see, e.g., [25]).
14.4 Hexagons on a sphere 589

14.4 Hexagons on a sphere

▸▸▸ A compound eye under the magnifying glass

Fig. 14.55 Compound eye of a moth and an even closer look at it (REM photos, courtesy of
Science Visualization, University of Applied Arts: Rudolf Erlach, Alfred Vendl).

Fig. 14.56 “Pro forma” compound eye of a millipede (left photo), a dragonfly’s highly developed
compound eyes, which has thousands of facets (right photo). Note the different sizes of the facets:
the denser they are, the better is the vision of the animal.

If we take a closer look at the compound eyes of insects (Fig. 14.55), we see a
lot of hexagons. From a purely geometrical point of view, it is an impossible
task to cover a sphere with exactly congruent hexagonal spherical tiles. As
is so often the case, nature finds a pragmatic solution, where the condition
590 More applications

“exactly” is replaced by “as closely as possible”. Moreover, the surface that is

covered is not always exactly spherical (Fig. 14.56). ◂◂◂

▸▸▸ How to tile a sphere with hexagons and some pentagons

Interestingly, if one is patient enough, it is possible to find pentagons on
compound eyes (Fig. 14.57, circled in yellow).

Fig. 14.57 Detail of Fig. 14.55: There are a few pentagonally shaped facets on the compound of
eye of an insect.

Fig. 14.58 Tiling the icosahedron recursively and then projecting the result on a sphere.

For an approximation of a compound eye, one can, therefore, work with

refined Platonic solids like the icosahedron (Fig. 14.58): We place hexagonal
grids over the twenty regular triangles of the icosahedron and project the
result onto a concentric sphere. The result are “a lot of” hexagons on the
sphere (with slightly different sizes) plus twelve pentagons that stem from
the vertices of the icosahedron. A variation of this simulation (Fig. 14.59)
additionally works with small spheres placed on the centers of the spherical
polygons, and it looks quite convincing. ◂◂◂
14.4 Hexagons on a sphere 591

Fig. 14.59 From left to right, the tiling is refined, just as in Fig. 14.58. At the centers of the
polygons, spheres are inserted, and these spheres intersect neighboring spheres.

Fig. 14.60 Hexagonal grids on a hemisphere

▸▸▸ How to tile a sphere exclusively with hexagons

If we do not want to use pentagons, the task is more complicated. It is
impossible to cover a sphere with hexagons of equal size, but what if we
use different sizes? If this is possible, would the algorithm not only work on
spheres but also on more general surfaces? Given the number of faces on
a sphere (or hemisphere respectively), our approach may be the following:
Find a set of more or less balanced distributions of the given points (this
can be done via formulas or with the above mentioned algorithm), and then
determine the spherical Voronoi diagram. We may draw our inspiration from
a photo of a wasp’s honeycomb (Fig. 14.61).
By interpreting the hexagons as pyramids with the center as apex (Fig. 14.62),
we already get something that resembles a compound eye, or, if you prefer,
a wasp’s honeycomb (Fig. 14.61). ◂◂◂

Fig. 14.64 shows a variation of Fig. 14.62, where the points are not chosen
arbitrarily, but rather by means of mathematical functions. The result still
looks quite natural and resembles Text ergänzen
592 More applications

Fig. 14.61 Wasp’s honeycombs.

Fig. 14.62 This image is reminiscent of compound eyes or wasps’ honeycombs.

Fig. 14.63 Remarkable variation of the centers of the cells

▸▸▸ How to cover a surface with squares or circles

Singly curved surfaces can always be easily with meshes that result in an
equilateral and orthogonal grid when the surface is developed. On doubly
curved surfaces, however, we can never find a mesh consisting of “squares on
the surface”. Nevertheless, modern geometry sometimes has to fulfill tasks
that would have been considered as “unsolvable” in earlier times.
14.4 Hexagons on a sphere 593

Fig. 14.64 Remarkable variation of the centers of the cells

We want to find grids that fulfill the following two geometric conditions
simultaneously: All lines
(1) intersect orthogonally and
(2) locally have an almost constant grid size in both directions.
Principal curvature lines on doubly curved surfaces (Fig. 6.52, Fig. 14.66 on
the left), e.g., fulfill the first condition, but not the second one. The force-
directed iterative algorithm that is used to find the solution shall only be
sketched briefly. It is described in detail in [18].

Fig. 14.65 Surfaces on the left: no border rules. Surfaces on the right: grid closed “in one direction”

If the grid does not need to follow boundary rules, the degree of freedom
allows the algorithm to produce meshes that almost precisely fulfill both
orthogonality and aspect-ratio-preservation (Fig. 14.65, left).
If the grid has to be closed “in one direction”, orthogonality can be fulfilled
reasonably well, but the grid size varies considerably (Fig. 14.65, right and
Fig. 14.66, center and right).
Fig. 14.67 illustrates how well the algorithm works if there are no restictions
to the border. In order to test the quality of the result, one can, for instance,
inscribe circles into the grid of squares. In Fig. 14.68, the grid is not visible
at all; only the circles are drawn.
594 More applications

Fig. 14.66 Left: the exact solution of the curvature lines on an ellipsoid; center and right: two
orthogonal grids with fewer restrictions

Fig. 14.67 A doubly curved surface is covered by a near-othogonal grid.

Fig. 14.68 A doubly curved surface (in this case a cyclid) is covered by circles.

Fig. 14.69 Orthogonal grids are aesthetically pleasing.

Even though these are not geometrically perfect solutions, architects and
designers might be happy with the result (Fig. 14.69). ◂◂◂
14.5 Fractals 595

14.5 Fractals
The term fractal was coined in 1974 by the mathematician Benoît Mandel-
brot, who uses it to label certain naturally occurring or artificial objects and
geometrical patterns ([20]). Basic properties of fractals are “self-similarity”
and “scale invariance” (that is, you can zoom in any number of times and the
fractal still looks the same at each scale).

Fig. 14.70 Koch snowflakes and clouds

▸▸▸ The Koch snowflake

The classic example of a fractal in the strictly mathematical sense is the
so-called Koch snowflake (Fig. 14.70, Fig. 14.71).

Fig. 14.71 Fractal curves have a dimension somewhere between one and two. Here, the Koch
snowflake (dimension 1.26) is becoming increasingly two-dimensional.
596 More applications

Fractal curves have a dimension somewhere between one and two (since their
shape never fills the whole surface). Fig. 14.71 shall illustrate that. Just to
give an example, which you need not understand: The Koch snowflake has a
fractal dimension of 1.26, which is quite close to 1 but can still be difficult
to grasp. ◂◂◂

▸▸▸ Plane-filling curves of dimension two

Fig. 14.72 Hilbert curves increasingly fill the plane with each iteration step.

Fig. 14.73 Hilbert curves can be created with different patterns (but always using the same
system).

In 1891, David Hilbert discovered a curve that will eventually fill 2-space
completely through folding and bending (Fig. 14.72, Fig. 14.73), and there-
fore, has a “fractal dimension” 2. ◂◂◂

▸▸▸ Soft coral (gorgonia)

A biologist will immediately think of some soft corals (“sea fans”), where
flat skeletons are formed from a flexible, horny substance called gorgonin.
14.5 Fractals 597

The skeleton provides space for polyps with tentacles, which catch plankton
and particulate matter for consumption (filter feeding). Obviously, the more
space-filling the skeleton is, the better the food supply is (Fig. 14.75).

Fig. 14.74 Soft coral (gorgonia) during the day

Fig. 14.75 Soft coral (gorgonia) at night: The polyps catch plankton with their tentacles.

It must be emphasized again that nature does not always obey perfect math-
ematical rules. Evolution develops its own rules which are very often reminis-
cent of what mathematicians come up with. The goal of the process, however,
is the same for Hilbert curves and gorgonia fans. ◂◂◂

▸▸▸ Camouflage in coral environments

What might be the purpose of the fairly similar surface-covering lines on the
tumphead wrasse (Fig. 14.76)? Well, as this fish is active by day and often
hides within corals, the most probable explanation is camouflage. ◂◂◂

▸▸▸ Fractals on spheres and similar surfaces

The surface of the humphead wrasse is not entirely flat, but the concept of
filling 2-space also works on curved surfaces, e.g., on a sphere (Fig. 14.77,
598 More applications

Fig. 14.76 In between the Gorgonia, a humphead wrasse is hiding. Maybe the fractal-like patterns
on its skin serve to conceal the fish.

Fig. 14.77 Left: Hilbert curves on a sphere. Right: brain corals often possess spherical shapes.
The similarity is undeniable.

Fig. 14.78 Increasing the depth of iteration on the sphere

left). Brain corals (some species of hard corals) have such patterns on them
(Fig. 14.77, right).
14.5 Fractals 599

To provide, once again, an idea of how the iteration works, Fig. 14.78 shows
the first four iteration steps. ◂◂◂

▸▸▸ The Mandelbrot set (Fig. 14.79)

It was only in the 1980s that fractals reached their current level of popular-
ity with the impressive pictures by Benoît Mandelbrot. The first computers
with the capacity to display graphics enabled the aesthetically fascinating
visualization of hitherto unknown fractal worlds.

Fig. 14.79 The famous Mandelbrot set. Left: A bounded sequence leads to a “good point”. Right:
Here the chosen sequence is not bounded.

Let us consider the complex function f (z) = z 2 +c. Then, we define a sequence
of complex numbers z0 , z1 , z2 , . . . as follows: z0 = 0, z1 = f (z0 ) = c, . . . , zn+1 =
f (zn ) = zn2 + c. If the sequence zn remains bounded (Fig. 14.79 left), then the
number c is considered to be “good”, and we mark it with a black or blue
point in the Gaussian plane. One can say that the Mandelbrot set is the set
of all parameters c for which the recursion zn+1 = zn2 + c remains bounded if
one chooses z0 = 0. ◂◂◂

▸▸▸ Julia sets

Julia sets (named after Gaston Julia, who investigated the behavior of those
sets decades before the computer age)can be created with the same recursive
formula as the Mandelbrot set. Instead of choosing z0 = 0, however, z0 can
be any point in the Gaussian plane. When one chooses, for instance, a point
z0 along the border of the Mandelbrot set (Fig. 14.80), the shape of the
corresponding set changes rapidly.
With some imagination, one may be reminded of the rapidly changing Julia
sets when we look at the huge fire in Fig. 14.81 and the clouds in Fig. 14.82
– even though, of course, they are no mathematical structures. It is, further-
more, remarkable that such natural shapes often change permanently and
there is no clearly recognizable contour. ◂◂◂
600 More applications

Fig. 14.80 Julia sets with starting points z0 close to the border of the Mandetbrot set. The shape
of such a set somehow resembles the “local shape” of the Mandelbrot set.

Fig. 14.81 Solstice fire or Julia sets?

Fig. 14.82 Clouds or Julia sets?

14.5 Fractals 601

▸▸▸ Tree fractals

Fig. 14.83 Tree-branching in practice

Fig. 14.84 The Pythagorean figure is adapted and used for the generation of fractal tree-like
figures.

In Fig. 14.84 on the left, the Pythagorean figure is repeated in fractal-like

manner (recursive scaling). The resulting figure shows no sharp contour line
as the depth of the recursion grows. Zooming in does not change this.
A nice generalization can be found in the model Fig. 14.84 on the right, which
was designed by F. Gruber and G. Wallner. Fig. 14.83 shows a “real tree” in
all its beauty. ◂◂◂
602 More applications

Fig. 14.85 Trees and clouds

▸▸▸ Ferns
The computer graphic in Fig. 14.86 on the left is based on an idea by
Michael Barnsley according to which the algorithm repeatedly applies one
of four (exactly defined) affine transformations to a single point. This itera-
tive procedure moves the point continuously, producing a fractal of this kind
(https://en.wikipedia.org/wiki/Barnsley_fern).

Fig. 14.86 Fern in theory and practice

A fern is a good example of a fractal-like structure in nature. Take a look

at the computer-generated shape of Barnsley’s fern: Most botanically knowl-
edgeable observers will correctly identify it as a fern and agree with its fractal
designation, as the basic structure repeats itself over and over again at differ-
ent scales. Let us now consider the photograph of an actual fern (Fig. 14.86
on the right). Here, the fractal nature is not as clear as in the computer
simulation. After the second iterative step, the self-similarity is hardly rec-
ognizable. If the bottom side of the fern leaf is magnified many times over,
it becomes clear that the fanning-out of the fractal-like leaf structure serves
14.5 Fractals 603

the purpose of covering the maximum possible area with sporangia. It is also
a good strategy for optimizing photosynthesis. ◂◂◂

▸▸▸ Fractal expansion

Fig. 14.87 Fractal expansion (“string of pearls”): algorithm filling the plane in a fractal manner.

Let us consider the following algorithm (Fig. 14.87): A circle is given at the
start. We now create a second circle within a given rectangular boundary
and keep moving the new circle towards the old one until they touch each
other. Now, a third circle is introduced at a random location. If it is closer
to the first circle, it is moved towards it by the same procedure as before.
Otherwise, it is moved likewise towards the second circle. After many circle
generations of this kind, a tattered sort of “string of pearls” is produced,
which branches out at several points. This algorithm tends to fill the plane
in a fractal manner. However, nature probably does not follow this exact
recipe.

Fig. 14.88 The eye of a crocodilefish: A fractal pattern works as camouflage.

604 More applications

Areas that are filled in a non-regular manner are often encountered in the
natural world. For instance, in the growth of lichen or in camouflage ef-
fects. Figure 14.88 shows the magnified eye of a crocodilefish (Cymbacephalus
beauforti). ◂◂◂

▸▸▸ The Menger sponge

Fractals can also be created in 3-space: A simple example of that is the
construction of a so-called Menger sponge, which can be described as follows
(Fig. 14.89 and Fig. 14.90):

Fig. 14.89 Fractal in 3-space (Menger sponge): With each iteration, each existing cube is trans-
formed into 20 smaller cubes.

Fig. 14.90 Next iteration: One can see that the space is not entirely filled. We will not provide
an explanation, but mathematicians say that the sponge’s dimension converges to ≈ 2.73.

Take a cube C0 and divide it into 3 × 3 × 3 = 27 smaller cubes C1 of equal size

(by dividing every face of C0 into 9 squares). From the cubes C1 , remove the
six cubes in the middle of each face plus the one in the very center of C0 .
Repeat this procedure for all the remaining 20 cubes C1 . You will get a new
structure with 202 = 400 cubes C2 , etc. ◂◂◂
14.5 Fractals 605

▸▸▸ Sierpinski fractals

Fig. 14.91 Sierpinski pyramid (physical model). Right image: multiple reflections in the spheres,
each of them is a fractal of its own.

The “Sierpinski pyramid” (Fig. 14.91) is 40 cm tall, weighs 25 kg, and was
built by Christoph Pöppe from metal spheres with the help of the IWR
(Heidelberg University). Four spheres that are strung together produce a
tetrahedron.

Fig. 14.92 Several iterations of rounded Hilbert curves in space.

Waclaw Sierpinski (1882-1969) also discovered a recursively defined sequence

of closed curves that fills the plane in the limit. The sequence of figures in
Fig. 14.92 shows various recursion levels of rounded Hilbert curves. ◂◂◂
606 More applications

▸▸▸ Percolation

Fig. 14.93 We divide a cube into tiny voxels and randomly “occupy” a certain percentage rate
of those cubes. What is the threshold at which we get a closed random path from one side to the
opposite one?

If we consider a cube, divide it into tiny cubes (voxels, i.e., the three-dimensional
analogue of the two-dimensional pixel), and randomly “occupy” a certain per-
centage rate of those cubes, the others stay empty (Fig. 14.93). Finally, we
just consider strings that are coherent. What percentage of the cube cells
must be occupied on average so that there is a continuous path from one side
of the cube to the other side? In other words, at what point is there a con-
nected path of cells from the top of the cube to the bottom? This percentage
is called the percolation threshold.

Fig. 14.94 Below the percolation threshold (image to the left) and above the threshold (image to
the right).

If a cell is occupied with probability p, the size of the clusters increases with
increasing p (Fig. 14.94). If p increases beyond a certain threshold, the largest
cluster will permeate the entire system from top to bottom as well as from
left to right and back to front. The sponge-like structures are reminiscent of
voxel models of complex forms. ◂◂◂
 A Geometrical freehand drawing

This appendix is intended for people who believe – as I myself do – that, de-
spite the quality of today’s computer “renderings”, correct freehand drawing
is still of principal importance for geometric communication. I would even
go a step further than that: Because perfect computer renderings are state
of the art, great freehand drawings are becoming an increasingly rare and
therefore, increasingly impressive skill!
This appendix is written so that it may be read before acquiring the knowl-
edge contained in the rest of the book. We will, from time to time, make
reference to previous contents, but even without these references, the course
is largely self-contained. This should permit a “geometric layperson” to pro-
duce qualitative correct drawings within a short period of time.
There is a simple but useful saying that goes as follows: “When a problem
takes till dawn, an auxiliary figure must be drawn!” As a first step, the
solution should be inscribed into this auxiliary figure. The more useful this
figure thus becomes (due to geometrical correctness), the easier it is to figure
out which steps must be undertaken in order to reach the existing solution.
Many drawings in this appendix were done by the participants of my course
about “artistic perspective”. A lot of them received a final artistic finish by
Christian Perrelli. An experienced eye may even detect slight errors in the
drawings: This, too, contributes to the charm of freehand drawings, and
owes much to their origin in our imperfect imagination – and besides, we
have already seen enough perfect renderings anyway.
© Springer Nature Switzerland AG 2020 607
G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3
608 A course in freehand drawing

A.1 Normal view vs. oblique view

In this appendix, we shall illuminate three types of visualization in more
detail: the oblique view, the normal view, and the perspective view (central
view) – the latter being far more difficult than the oblique and the normal
views.
Why the oblique view is not our ultimate goal
The section about the oblique view will be the shortest by far.

Fig. A.1 A house, a cube with inscribed circles, and a sphere in the oblique projection

Most people have drawn oblique views at some points in their lives. Most
probably oblique projections, also called front views: Assuming a non-distorted
frontal view, images of polyhedral forms are thus easy to construct. Straight
lines proceeding towards the back are drawn at a constant angle α (such
as α = 45○ ), and depths are always shortened by the same factor (e. g. by a
half). Thus, the image of a house is quickly constructed by means of a cuboid
(Fig. A.1 on the left). What’s more, it is easily possible to inscribe the im-
ages of circles into the images of squares, as in Fig. A.1 at the center. The
images of the foremost and the backmost circles remain circles. The ellipses
appearing as the images of the other circles are each defined by the four edge
midpoints and the edges’ images are tangents there. However, it should be
noted that the vertices of the ellipse are not known. It is further not eas-
ily possible to specify the contour of a sphere, and if we simply inscribe it
without construction (Fig. A.1 on the right), then it is surely wrong!
A second approach to producing visually transparent oblique views is based
on building objects above their undistorted, oblique top views (for instance,
with an angle α = 30○ ). The results of this method can be seen in Fig. A.2.
The heights are either shortened (by half, for instance) or undistorted. This
yields a bird’s-eye view.
In the case of cubes with circles inscribed onto their faces, the lowest and
the highest circles are undistorted, and the remaining ones are displayed as
A.1 Normal view vs. oblique view 609

Fig. A.2 A house, a cube with inscribed circles, and a sphere in bird’s-eye view

ellipses bounded by circumscribed parallelograms. As before, the axes of the

ellipses are unknown. Spheres leave us with the same problem as before.
As practical and as easily explicable as the above-mentioned approaches are,
they all share a severe disadvantage:

The usual oblique views (frontal projection and bird’s-eye view) yield undistorted
images. Thus, they are ill-suited to visualize spheres and other surfaces of
revolution.

Isometry
In theory, given an oblique view, one may freely choose the images of the
coordinate axes and the distortion ratios along them. These assumptions
relatively successfully compensate the aforementioned disadvantage.
Let us start with a special assumption which produces a normal view, the so-
called isometry: Here, any pair of the axes’ images encloses an angle of 120○ ,
and all segments in the direction of the coordinate axes remain undistorted.
(Actually they are equally distorted, since isometry literally means “equal
metric”.)

Fig. A.3 A house, a cube with inscribed circles, and a sphere in isometry
610 A course in freehand drawing

In Fig. A.3, our test objects are displayed isometrically. The house now ap-
pears to be “more realistic,” and the sphere on the right now possesses a
circular contour, which is obviously the most significant improvement. The
depiction of the cube now has the disadvantage that the images of two op-
posite vertices of the cube coincide at the center of the image. The ellipses
appearing as the images of circles are even congruent. The principal axes of
the ellipses are orthogonal to the images of the coordinate axes, which gives
the artist a great advantage during construction, because the symmetry axes
of the ellipses are known.
Due to the strict definition of the axis angles, isometry leaves no choice when
depicting an object. The horizontal base plane is always seen at the same
angle. Let us summarize:

Isometric depictions are well-suited for quick and transparent sketches of poly-
hedral as well as round bodies, as long as they are not exactly like cubes.
However, they permit little variation of the viewing angle.

Fig. A.4 Isometry of an icosahedron (on the right: Sophie’s spaghetti model)

As an exercise, let us draw the isometric view of an icosahedron. Without

prior knowledge about its geometric construction, such a body is practically
impossible to draw. However, If one is aware that an icosahedron may be
defined by three mutually orthogonal “golden rectangles” (side ratios of ≈ 8 ∶
5), then an optimal isometric view may finally be attempted with confidence.

The advantage of a normal projection

From theory, we know that images of axes may be chosen (almost) freely. In
this case, there always exist precisely defined distortion ratios for each axis
direction, by which a normal view, and not just a general parallel view, may
be accomplished. The advantage:

Each normal projection depicts circles as ellipses which are symmetrical to the
image of the circle axis and sphere contours are circles. This means that the
objects in view appear natural.
A.1 Normal view vs. oblique view 611

In general, one deals with three different distortion ratios for the three axis
directions. Apart from the increased difficulty of constructing objects in such
a way, the determination of these ratios requires a construction which a
freehand artist cannot do with an adequate enough amount of accuracy.
We are already aware of a very special type of normal projection: isometry.
There, all distortion ratios are identical (1 ∶ 1 ∶ 1) due to equal angles enclosed
by the images of the axes. This makes the application of distortion ratios
unnecessary, since the result would just be scaled.
However, there exist other choices of angles between the images of the axes
which correspond to very simple distortion ratios.
The engineer’s view – dimetrical and descriptive
If one tries to find those angles between the axes where where the distortion
ratios correspond to specific values, such as 1 ∶ 1/2 ∶ 1, then the image of the
y-axis, for instance, must deviate from the horizontal axis by slightly less
than 45○ – by about 42○ , to be precise. The image of the x-axis and that
of the z-axis are symmetric to that of the y-axis due to identical distortion
ratios along the x- and z-direction. The image of the x-axis thus deviates
from the horizontal by about 7○ (Fig. A.5).

Fig. A.5 A house, a cube with inscribed circles, and a sphere in the engineer’s view

This configuration is known as the engineer’s view and offers similar advan-
tages to isometry, but it is also well-suited for “cube”-shaped objects (Fig. A.5
at the center). Of course, there also exists a symmetric solution where the x-
axis forms the aforementioned 42○ . The distortion ratio along the x-direction
is thus about one half, while there is distortion at all in the y-direction.
The general solution
The engineer’s view is also a well-defined normal view. Should neither isome-
try nor the engineer’s view provide a “useful impression” of a specific object,
then there is also a general method that, even in freehand drawing, yields
relatively accurate results. Prior to drawing, we can give good estimations of
the results, which gives the artist an additional advantage: Our desire is not
to determine what technique works best by “trial and error”, but to proceed
towards our goal in an economic way.
One starts with the image of a unit circle in the horizontal base plane (mid-
point U , radius 1 unit). The principal axis of the ellipse that is the circle’s
612 A course in freehand drawing

Fig. A.6 Different images of the unit circle

image is orthogonal to the image of the z-axis according to the rules of nor-
mal projection (Fig. A.6). Depending on how “bulgy” the circle’s image is,
the base plane is more or less oblique.
In the next step, the image of the x-axis is chosen arbitrarily.
The image of the y-axis is given automatically by the third step: There exists
a circle tangent t where the x-axis intersects the unit circle (point X) – in
the image, it is a tangent to the ellipse. Since the horizontal tangent t is
orthogonal to the x-axis (both considered as lines in space), it is parallel to
the y-axis. The y-axis intersects the unit circle at the point Y . The segments
U X and U Y are the “unit segments” on the axes. This already makes it
possible to inscribe arbitrary figures onto the base plane.

If the images of a horizontal circle and one horizontal axis direction are given,
then the image of the missing horizontal axis direction and the distortion of the
top view are also defined.

What still remains to be determined is the degree by which the heights are
distorted. From a qualitative point of view, we can state the following: The
more bulgy the ellipse (image of the base circle) is (axis lengths 2a = 2r
and 2b), the more the z-values are shortened. This statement is instantly
quantifiable, since we already know the silhouette of the unit sphere about
U – it is the circle about U ’s image that touches the image of the unit circle
in the principal vertices.

Fig. A.7 General views of the globe

A.1 Normal view vs. oblique view 613

The unit circle is the equator of the sphere. By analogy, we can inscribe the
sphere’s zero meridian through the point X. Its axis is the y-axis, and the
principal axis of its image is orthogonal to the image of its axis – as in the
case of the equator. The position of the point X determines the “bulginess”
of the ellipse that is the meridian’s image. The image of the North Pole Z
can be found at the intersection of this image ellipse with the image of the
z-axis. U Z is the unit segment for the y-coordinates. (For those interested:
One can also find the image of the North Pole by pivoting a focal point of
the image of the unit circle by 90○ or by intersecting the sphere’s silhouette
with the parallels to the image of the z-axis at a distance of ±b (semiminor
axis of the equator’s image) – the North Pole has the same height.)

A general normal projection is best defined by the image of the unit sphere’s
equator. By inscribing the sphere’s zero meridian, one can determine its north
pole and thus, one has full control over all distortion ratios.

In order to consolidate the aforementioned in our minds and improve our skill
of drawing ellipses, we shall now sketch general views of the globe (Fig. A.7).
This type of “construction”, which considers the right distortion ratios, has
the advantage that we can specify the direction from which the drawn object
is viewed. The images of the equator and the zero meridian of the unit sphere
give a good impression of what the rest of the image looks like. Such a test
sketch can also be drawn for the image of the “unit cube.”

▸▸▸ Circles with axes parallel to the coordinate axes

Let us sketch a scene that consists of several cylinders of revolution (Fig. A.8).

Fig. A.8 Standing and lying cylinders of revolution

The images of the top circles of an upright cylinder of revolution are always
ellipses whose principal axes are orthogonal to the image of the z-axis. Lying
cylinders are to be depicted in the same way:
614 A course in freehand drawing

• The images of the boundary circles are ellipses whose principal axes are
orthogonal to the image of the axis of the cylinder.

• The diameter of the cylinder is equal to the length of the principal axis.

• The search for another further image point will show how bulgy the circles’
images actually are. If, for instance, the bottom cap or the top cap lie in a
plane parallel to the yz-plane, then such a point can be found by parallel
translation of the images of the y- and z-axes through the principal vertices
of the ellipse that appear as the circle’s image.
⊕ Remark : This construction is plausible: Let A and B be those endpoints of
a diameter of the circle k whose images are the principal vertices of the ellipse
k n . The two axes y and z are orthogonal lines in space, so that the intersection
of the parallels through the endpoints A and B lies on k according to Thales’s
theorem. We thus get a point on the ellipse that appears as the image of k. In
cases of symmetric axis positions, we have also found the auxiliary vertices of
the circle’s image. ⊕ ◂◂◂
A.2 Don’t be afraid of curved surfaces 615

A.2 Don’t be afraid of curved surfaces

Unfamiliar at first. . .
Freehand drawing is strange at first: Instead of employing a ruler at every
turn, we force ourselves to draw straight lines by hand. No doubt, we would
also feel more confident if our circles were made with the help of a compass. It
does, in fact, require a considerable amount of experience to make a freehand
drawing of a circle with a given radius.
. . . but soon highly advantageous!
Except for sphere contours, perfect circles rarely occur in freehand drawing.
One most often deals with ellipses and is thus, as a freehand artist, at a defini-
tive advantage compared to those who have to rely on rulers and compasses.
The true advantage lies in the fact that – with some spatial imagination and
geometric aptitude – we can quickly surmount many obstacles that would
needlessly occupy a “constructing” artist for a much longer time.
Furthermore, it is often in in hindsight, only after the result has been “esti-
mated” in a sketch, that an exact construction seems to be much easier.

Freehand drawing that is based on geometric facts is an important aspect of

the later derivation of exact solutions.

In light of this, every person that deals with geometry should be trained in
this craft even in the absence of any artistic aspirations.
It does not get any more difficult
The next advantage is that the transition from polyhedral building blocks to
elegant, curved surfaces can happen smoothly – no pun intended – without
an apparent increase in the degree of difficulty.

During construction with a ruler and a compass, we quickly hit a brick wall
and usually fail due to our “tools”! During freehand drawing, however, we may
concentrate on our geometrical thinking.

▸▸▸ One-sheeted hyperboloid of revolution

The surface can be generated by the rotation of a straight line about a skew
axis. There exists a second family of straight lines on the surface. The contour
and the two families of straight lines are to be depicted.
Let us start with a horizontal base circle, which we want to subdivide into
eight equal parts. This may be done by first selecting an arbitrary x-axis.
The orthogonal y-axis is parallel to the tangents of the x-axis points on the
616 A course in freehand drawing

base circle. The two axes define a square which is circumscribed to the circle.
Along with the coordinate axes, its diagonals yield the evenly distributed
points 1, . . . , 8 on the base circle.

Fig. A.9 Variations of a one-sheeted hyperboloid of revolution

Let us depict a congruent cap circle at an arbitrary height. Then let us

keep the eight points but shift the labels by, for instance, three points. This
yields eight straight lines on the surface and eight tangents of the contour
hyperbola. By lifting the eight points into the other direction, we may derive
eight further tangents. Thus, the contour seems to be adequately defined.
During the final composition of the image, one should consider that a change
of visibility occurs at the contour.
A variation may be given by using the affine mapping of the base circle image
with the undistorted circle position. A circle may easily be decomposed into
equal parts. ◂◂◂

▸▸▸ Beam connection

Two recumbent cylinders of revolution with equal radii and orthogonally
intersecting axes are to be depicted. (Fig. A.10)
Let the x- and the y-axis be the axes of two cylinders. It helps if the images of
the axes of the cylinder axes are chosen as unsymmetrical as possible relative
to the image of the z-axis, just as in the engineer’s view. There is a sphere
inscribed to both cylinders simultaneously, which is centered at U . We shall
try to depict this sphere first.
The contour generators of both cylinders touch the circular contour of the
sphere. Let us cap them off with circles by picking their midpoints on the
cylinder axes at arbitrary positions. The principal axes of the corresponding
ellipses that represent circles in the image are orthogonal to the images of
the cylinder axes. Further points may be derived by a parallel translation of
the axes.
A.2 Don’t be afraid of curved surfaces 617

Fig. A.10 Beam connection

The intersection of the cylinders consists of two congruent ellipses. One must
consider that an arbitrary (non-circular) ellipse in space may also be depicted
as an ellipse, but that, in general, no axis of the ellipse will be orthogonal to
the image of any coordinate axis. Thus, the objective will consist of deter-
mining as many points and tangents of the ellipse as possible.
Let us consider the highest and the lowest generators of both cylinders. They
intersect each other in two points, C and D. The generators which lie in the
xy-plane intersect in the points A1 , A2 and B1 , B2 , which are symmetric
with respect to the origin. At those locations, the ellipses which form the
intersection of the cylinders possess z-parallel tangents. C and D belong to
both ellipses. The corresponding tangents are parallel to A1 B1 and A2 B2 . As
an additional information, it should be considered that the ellipses’ images
touch the contour generators of the cylinder. The respective contact points
are of great importance, because they mark the transition from the visible
to the non-visible part of the curve. ◂◂◂

▸▸▸ Pencil tip (Fig. A.11)

A regular hexagonal prism is to be conically “tapered” (which happens when
the pencil is sharpened).
The vertical faces of the prism will intersect the cone of revolution along the
hyperbolas. We shall, at the very least, locate the highest points of these
hyperbolas.
Let us choose the horizontal base circle of the cone of revolution, into which
we shall inscribe a regular hexagon. The tip may be chosen arbitrarily on the
z-axis. The tangents from the image of the tip to the image of the base circle
are the contour generators of the cone. The choice of the x-axis (point X on
the circle) entails the direction of the y-axis (parallel to the tangent in X).
This time, the images of the axes may very well be symmetric. We can derive
618 A course in freehand drawing

Fig. A.11 Pencil tip

the neighboring points of the hexagon, relying on a well known property of

the regular hexagon, by intersecting the y-parallel with the circle through
the midpoint M of U X. The remaining points are given by mirror-symmetry
with regard to U .
In the yz-plane, the highest point H of the section hyperbola can be con-
structed directly: It lies precisely above the side midpoint and on the gen-
erator of the cone. All other vertices are located equally high above the
remaining midpoints of the sides. The corresponding tangents are parallel to
the hexagon sides. During sketching, one should take into account that two
of the six arcs touch the contour of the cone and change from the visible to
the non-visible part of the surface.
The pencil lead shares a circle of intersection with the cone, which is located
at the appropriate height and is coaxial with the base circle.
⊕ Remark : During freehand drawing, as well as during precise construction, one may
always stumble upon inconvenient perspectives (such as projecting intersections).
For instance, the drawing attempt in Fig. A.11 on the left may not be optimal. One
may be consoled by the fact that freehand drawing tends to be quick. There is never
a lot of time lost, and one can, therefore, easily start anew. ⊕ ◂◂◂

▸▸▸ Helical stairs (Fig. A.12)

In a front view, a helix with an upright axis appears as a sine curve. It is
easy to draw such a curve: It is created by means of a harmonic oscillation
and a simultaneous uniform translation in an orthogonal direction.
Let us draw such a curve. In order to simplify our task, let us also draw the
contour generators of the carrier cylinder. The axis of the cylinder appears
as the mid-parallel of the two contour generators (Fig. A.12 on the left). The
curve’s period length equals the pitch of the helical motion. As an exercise,
let us also depict a second helix, which is created by reducing the radius of
the original helix to about a quarter. It appears as a sine curve with the same
A.2 Don’t be afraid of curved surfaces 619

Fig. A.12 Normal view and front view of varying helices (helical stairs)

frequency but a smaller amplitude. In Fig. A.12 at the center, a variation

of the original is visible: a helical staircase appears on the right. In order
to accomplish roughly 16 equally distributed points on the helix, we choose
the planes that correspond to a height difference of 1/16 of the pitch. The
16-part subdivision can be found during construction by repeatedly halfing
the pitch. ◂◂◂

▸▸▸ Surfaces of revolution

Surfaces of revolution may be produced by a rotation of the meridian during
which the points on the meridian move on the circles of latitude. Tangents
of the meridian circle generally generate cones of revolution with their apices
on the axis of rotation. (In some cases, the cone degenerates to a cylinder of
revolution which is coaxial with the surface of revolution. The cone may also
collapse to a plane which is orthogonal to the axis of rotation.)
In order to sketch the given surface of revolution in a visually accurate way,
it is paramount to first inscribe the meridian along with the axis of rotation
(Fig. A.13 on the left). Or, for the purpose of control, the symmetric meridian
may be included. We denote special points on it: These include, above all,
the points with axis-parallel tangents but also any inflection points which
may arise. The tangents in the boundary points are also important.
The normal views of the respective circles of latitude are now drawn in the
normal projection (Fig. A.13 on the right). The respective heights are short-
ened in relation to the obliqueness of the axis. If the obliqueness is only slight,
one may even use the same heights during freehand drawing. We use the im-
ages of the apices of the cones to draw the tangents at the ellipses that are
the images of the corresponding circles of latitude. The contact points corre-
spond to the contour of the surface of rotation – the tangents of the ellipses
are also tangents of the outline of the surface. At the points with minimal
620 A course in freehand drawing

Fig. A.13 Surfaces of revolution: a torus can be seen on the right.

or maximal distance from the axis of rotation (where touching cylinders of

revolution occur), the principal vertices of the image ellipse are the contour
points of the surface. ◂◂◂

▸▸▸ Blossom of a lily

Fig. A.14 Blossom of a lily

Advanced or the artistically ambitious may try their hand at the following
example: A bell-shaped surface of revolution (Fig. A.14) should be “serrated”
so that something similar to a bellflower or lily emerges – and since nature
itself rarely produces absolutely perfect surfaces, we are allowed to be a little
bit imprecise in our freehand drawing. Nevertheless, it is advantageous to first
sketch the surface of revolution without the indentations and thus determine
the silhouette.
In another step, we subdivide the boundary curve into five approximately
equal sections (nature seems to have a penchant for five-fold symmetry) and
A.2 Don’t be afraid of curved surfaces 621

estimate the corresponding meridian A through the points 1, . . . , 5 of the

subdivision.
In a last step, we inscribe the circle of latitude, which shall limit our inden-
tations. The midpoints of the points defined by the intersections with the
special meridian are the endpoints B of the section lines. These points are
now carefully connected to the starting points A. The images of the section
curves may touch the silhouette in some cases. If parts of the original silhou-
ette of the surface of revolution remain, they may form the silhouette of the
flower. ◂◂◂
622 A course in freehand drawing

A.3 Shadows
In order to give our freehand sketches a realistic appearance, we may now
add shadows, but we must proceed carefully: Shadow constructions are com-
plicated and, therefore, “dangerous.” A realistically drawn false shadow can
do more damage than good for a better understanding of the image. We shall
thus limit ourselves to shadows of simple scenes. Furthermore, it is better to
start one’s shadow exercises with sunlight (parallel shadows), rather than
using light sources directly within the scene.
Convex bodies possess neither ridges nor indentations. Thus, they do not
cast shadows on themselves. Cuboids, cylinders of revolution, and cones of
revolution are simple examples. For these three prototypes, we shall now
attempt to sketch parallel shadows in simple scenes.
▸▸▸ Parallel shadows of a cuboid

Fig. A.15 Cuboid with shadow

Let us choose the light position, ideally, by choosing the shadow Ps of a point
P of the top surface. Then, all light rays are parallel to the direction P Ps .
Furthermore, all top views of the light rays are parallel to the connection of
the top view P ′ of P with the shadow Ps in the base plane. Thus, we can
construct the shadow of a further point Q of the cap surface by drawing the
parallel to the light ray through Q and the parallel to the light ray’s top
view through Q′ . The triangles P P ′ Ps and QQ′ Qs are similar in any case
and even congruent in the special case of equal heights. P Q and Ps Qs should
be parallel and of equal length, because P is parallel to the base plane. (The
“light plane” through the edge is intersected by two parallel planes.)
The shadow provides important information about the cuboid’s penumbra.
We will try to rely on our spatial imagination during its construction.
A.3 Shadows 623

⊕ Remark : A little hint for the shading of a cuboid: Surfaces in penumbra are
“slightly darker”, but they, nevertheless, exhibit surface texture (such as that of a
building wall). Shadows, on the other hand, are “no longer aware” of their origin,
exhibiting the texture of the surface on which they are cast (such as a lawn or
asphalt). ⊕ ◂◂◂

▸▸▸ Parallel shadows of cylinders and cones

Shadows of cylinders of revolution and cones of revolution are almost as easy
to draw as those of cuboids: Their contour with regard to any projection
(parallel or central) consists of straight parts. The direction of the light can

Fig. A.16 Cylinder and cone with shadow

be given by defining the shadow of the center M of the cap circle or the cone’s
apex S. In this way, we once again obtain the characteristic triangle, which
can be employed for the construction of further shadow points. In the case
of the cylinder of revolution, the shadow of the cap circle should be centered
at the shadow Ms of M : It is congruent to the image of the cap circle. The
common tangents of the elliptic image of the base circle and of the shadow of
the top surface provide the straight-line component of the shadow boundary.
The penumbral shadows that are parallel to the axis start in the contact
points with the base circle. In the case of the cone, one only needs to draw
the tangents to the image of the base circle from the shadow Ss of the apex S
in order to obtain the straight shadow boundaries. From the contact points,
the straight penumbral boundaries then run towards the cone’s apex. ◂◂◂
624 A course in freehand drawing

A.4 Perspective sketching

There is an entrance exam for the University of Applied Arts Vienna where,
among other parameters of artistry, the drawing skills of the applicant is
to be determined. Years ago, it used to include the following exercise: The
applicants had to sketch a staircase inside the University. Fig. A.17 shows a
fisheye photograph of this staircase (from above). Is it actually possible to
sketch the staircase correctly?

Fig. A.17 Overview with the fisheye lens, general and special perspective image

Reducing the level of difficulty

A wide-angle photograph can be seen in the middle. The task seems to be
quite complex, and we may not even notice slight errors in the sketch. It would
surely do no harm to simplify the view slightly (as in the picture on the right).
No examiner would require that every point be geometrically transparent
on a candidate’s first attempt. However, systematic considerations may well
improve the result.
The method of Renaissance artists
Perspective images are the result of a central projection from an eye point
to the image plane. This serves to simulate the impression of natural sight.
In a simple wood engraving, Albrecht Dürer showed his contemporaries, as
well as the generations that would follow, how lines of sight may be inter-
sected with the image plane:
The viewing ray appears with the help of a taut thread through a loop E – the
eye point. One person places the thread at the point that is to be displayed.
Another person marks the intersection point with the image frame by means
of a crosshair. By closing the lid for a short period of time, the image point
may be fixed.
The most important terms
In order to proceed with our analysis, we need to employ some special “ter-
minology”: The eye point E is located at a distance d in front of the principal
A.4 Perspective sketching 625

Fig. A.18 Wood engraving for the intersection method

point H of the perspective image. The straight line EH is the viewing axis
or principal ray. A straight line parallel to the image plane π is called the
frontal line, and a straight line orthogonal to π (and thus, parallel to the
viewing axis) is called the depth line.
The following theorem is important for drawing and applicable here:

Perspective images of parallel straight lines intersect, with the exception of

frontal lines. Depth lines vanish in the principal point.

Frontal lines are very practical in perspective images:

Affine ratios are preserved on frontal lines. The angle between two frontal lines
remains undistorted in the perspective image.

The horizontal plane on which we stand is called base plane. Its intersection
with the image plane is called base line. The location of our feet is called
standpoint, and the height of the projection center is called eye height. The
horizon is produced when all points at infinity of the base plane are projected
to the image plane. Thus:

The perspective images of horizontal parallel straight lines intersect at the

horizon. In cases of a horizontal perspective image, the horizon contains the
principal point.

Frontal perspective image

Let us now connect the scene to a Cartesian coordinate system with three
axis directions x, y, and z. The xy-plane shall serve as the base plane. The
z-axis is then upright.
If the principal ray (the viewing direction) has the direction of a coordinate
axis, the image plane is parallel to the respective perpendicular coordinate
plane. This is called a frontal perspective. Two axes are frontal lines and span
a frontal plane, in which figures are undistorted (Fig. A.19). The third axis is
626 A course in freehand drawing

Fig. A.19 Special positions of the coordinate system. On the left a frontal perspective painted by
M. K. Steiner (see also p. 607). In reality, the painting is more than 1.5 m2 tall. If you position
yourself one or two meters in front of the principal point (the “intersection” of the railway tracks),
your brain is tricked into believing that you are actually there – slightly above the ground.

the depth line and vanishes in the principal point. Aside from the “contents”
of the frame, Fig. A.18 is a classic example of frontal perspective.
▸▸▸ The “square-grid method”
A method that was popular in the Renaissance approximated a complex top
view by means of a square grid.
Let us tile the base plane with square tiles, starting at the base line. This
makes the sides of the squares either parallel or orthogonal to the image
plane. In a perspective image, the image of this grid of squares is to be
drawn as follows: Firstly, the points on the base line equal their perspective
images, and secondly, the depth lines all vanish in H. We may still choose
an arbitrary vanishing point D1 in a diagonal direction. The other vanishing
point D2 is symmetric with respect to the principal point H.

Fig. A.20 Square-grid method: The classical method from the Renaissance (left). A wide-angle
photograph which clearly depicts the vanishing points of the square diagonals. The photo exhibits
a slight “barrel distortion”, which often occurs in lenses of a very short focal length (right).

The distance of the points D1 and D2 from H is equal to the distance EH of

the image plane’s eye point. (We have drawn parallels to all 45○ straight lines
through the eye point E.) For this reason, these points are called distance
points.
A.4 Perspective sketching 627

We may now “interpolate” the top views of arbitrary points. The appropriate
height may be drawn as follows: The top view is to be translated orthogonally
to the base line (which – in the perspective image – equals a projection from
H to g). Above the top view that was projected to g in this way, we may
now measure the undistorted height and finally project it back. ◂◂◂

Perspective images with a horizontal viewing axis

We no longer wish to look in the direction of a coordinate axis, but neverthe-
less, let the viewing axis remain in a horizontal position. The upright straight
lines still remain as frontal lines, and their images are mutually parallel. The
x-axis and the y-axis vanish at two points F1 and F2 on the horizon.

Fig. A.21 Subdivision of segments (special multiples) Fig. A.22 Cuboid objects

Cuboid objects, which are positioned with many edges parallel to one co-
ordinate axis (Fig. A.22), are very easy to sketch. Vertical segments should
be drawn, as previously described, with the use of orthogonal projection to
the image plane. They can also be easily subdivided. Yet, the following wish
is not easily attainable in freehand drawing: the desire for exact drawing of
segments along the coordinate directions. However, since there are simple
methods for drawing affine ratios in a perspective view, this need not be a
showstopper.
Fig. A.21 shows how fast midpoints or the doublings of horizontal segments
may be derived by drawing diagonals into overlying vertical rectangles.

Fig. A.23 General subdivision of segments

The general method is illustrated in Fig. A.23: An arbitrary auxiliary van-

ishing point B is to be selected on the horizon, and the segment P Q is to
be translated on a parallel to the base line. The result is a segment P0 Q0
628 A course in freehand drawing

which can generally not be seen in an undistorted state, but which may
be employed to inscribe the desired affine ratio. One may afterwards move
back again. (This method works because it can be interpreted as a parallel
projection in space.)
⊕ Remark : Consider the following hint: P0 Q0 is more likely to appear undistorted
the nearer B is to the so-called measurement point of the segment (which is the
vanishing point of the rotational chords during parallel rotations with respect to the
image plane). During sketching, one may approach the measurement point of the
x-direction (with its vanishing point at F1 ) by bisecting the segment approximately
between the principal point H and the vanishing point of the y-direction (F2 ). ⊕

Fig. A.24 Roman aqueduct and a variation by Zorica Nikolic Aigner

Fig. A.25 Catenaries everywhere

We are now at a point where we can draw images such as the one in Fig. A.24
on the right, though we should not be afraid of curved lines. The elliptic
images of the circular arcs are well determined by their vertical starting tan-
gents and the horizontal tangent at the highest point, which passes through
a principal vanishing point in the image.
A.4 Perspective sketching 629

When sketching curves, one should take care to draw the tangents at endpoints
as well as the tangents of special points (such as the highest, lowest, left-most
and right-most point) first.

In case that shadows have to be added to the perspective image, it is always

an advantage to follow the directions that lead to the principal vanishing
points. This strengthens the “vanishing-point appearance”, which is so crucial
for our spatial understanding.
The “suspension bridge” in Fig. A.25 was produced in the same way. For
illustrative purposes, the tangents at the points P and Q of the catenary and
its intersections T are inscribed exactly below the lowest point.
Scaffolding
Capturing of curved lines by means of points and tangents (line elements) is
an important and legitimate part of sketching. Whenever we wish to depict
complicated objects, it is always advisable to imagine a sort of scaffolding
that captures the locations of important points of the object.

Fig. A.26 On the left: An almost undistorted bicycle (frontal perspective). Center: the most
common view from the front. On the right: strongly distorted (being photographed by a lens with
very short focal length).

Fig. A.27 Freehand drawing of bicycles

Let us consider a bicycle (Fig. A.26), which (to be honest) is not even that
complicated, since it essentially “lies in a single plane.” If the bicycle pole
630 A course in freehand drawing

is imagined to be horizontal, then its end points, along with the midpoints
of the wheels, form an almost equilateral trapezoid. The pedals are located
roughly halfway between the hubs. This is already a good starting point, and
once it is sketched, the “rest” appears almost by itself. Fig. A.27 shows two
bicycle variants by Stefan Wirnsperger and Otmar Öhlinger. The following
must be considered when drawing the wheels:

Fig. A.28 Interior views of a tunnel

Rules for the depiction of circles in a perspective image:

• The image of a circle is an ellipse (in general, and sometimes also part of
a hyperbola), and thus, it possesses two symmetry axes. The center of the
circle is not seen at the center of the image curve (ellipse or hyperbola).

• One should, if possible, sketch the images of the foremost and backmost
points of the circle, including their perpendicular tangents, as well as the
highest and lowest points. The images of the corresponding horizontal tan-
gents pass through the vanishing point of the horizontal diameter.

Fig. A.28 is meant to illustrate that images of circles may very well be parts
of hyperbolas. When viewing a torus-shaped tunnel (see also Fig. 6.77), one
is, in fact, located within the right circular road edge, viewing the scene such
that nearly all visible circles become parts of hyperbolas. If the tunnel ex-
hibits a high curvature, then the inner edge of the road may be an ellipse. This
happens whenever the plane which is perpendicular to the viewing direction
through the eye point (vanishing plane) does not intersect the circle.

▸▸▸ A helical staircase in a perspective view

Hieronymus Rodler published a book about perspective drawing in 1531 (five
years after Albrecht Dürer). His explanations sometimes lacked a geometrical
background. The drawing in Fig. A.29 is based on Rodler’s detailed instruc-
tions for drawing helical staircases.
A.4 Perspective sketching 631

There are crucial differences between the correct perspective on the left and
the drawing on the right. As an exercise, the reader is invited to find the
main differences between the two drawings. One might also compare them
to the fisheye photograph below.

Fig. A.29 Studies of a helical staircase. Top right: a perspective drawing by Hi-
eronymus Rodler , on the left: a “modern alternative” by Christian Perrelli. Bottom
right: This fisheye photograph seems to bear some resemblance to Rodler’s draw-
ing. However, there are crucial differences. Most importantly, there are hardly any
straight lines in the fisheye photograph (except where the straight lines intersects
the optical axis).

◂◂◂
632 A course in freehand drawing

Shadows in perspective images

Shadow constructions of central views are no more complicated than those
of normal views. However, let us remind ourselves that the novice should not
attempt to draw shadows of complex scenes. Again, these words of caution are
due to the noticeable and distracting nature of incorrectly drawn shadows –
especially as they go against the primary goal of perspective sketching: visual
descriptiveness.
The following theorem is of fundamental importance:

The perspective image of a light ray through any given point passes through
the sun point, and its top view passes through the sun foot point.

Fig. A.30 Front lighting on the left, back lighting on the right

Let us differentiate between front lighting, back lighting, and side lighting
(the limiting case).
The situation of front lighting is the easiest to understand (Fig. A.30 on
the left): The Sun S is depicted as a point above the horizon. In cases of a
horizontal viewing axis, the image of its foot point S ′ lies exactly below S
on the horizon h. We may derive the shadow Ps of a point P by intersecting
P S with P ′ S ′ . In the image, it also follows that Psc = P c S c ∩ P ′c S ′c . In the
drawing, we may omit the symbol for the central projection (the c ) as long
as we are able to distinguish – at least in the mind – between the spatial
situation and the perspective image.
Now let us take a look at the back lighting situation (Fig. A.30 on the right):
Nothing seems to change from the perspective of construction, if we accept
that the sun point, as a theoretical point, may also be below the horizon,
without the scene being set at night. We can, in fact, get the image of the
Sun by connecting the Sun to the eye point (this would be a ray of sunlight
A.4 Perspective sketching 633

Fig. A.31 Simple scene with shadows

that would hit the back of our head) and intersecting the connecting straight
line with the image plane (which is always vertical and in front of us).

Fig. A.32 Side lighting: easier to construct and visually appealing

⊕ Remark : In the practice of sketching, the Sun’s mid- and foot points are usually
far outside the range of the drawing sheet. This is no problem for freehand artists:
As long as we know the approximate position of these points, we may still inscribe
the important respective sections of the connecting straight lines. ⊕

Reflections
Only very small sleights of hand are needed in order to enrich our perspective
images with simple reflections. Let us restrict our attention to horizontal and
vertical reflection planes (water and the classic example of the mirror on the
wall). We would do well to remember the following rule:

Each mirror produces a virtual counterworld, which can be observed through

the mirror window.
634 A course in freehand drawing

Fig. A.33 Simple scene with side lighting

There exists an “opposite point” P ∗ for each point P , which, by the rules
of reflection, is located on the normal of the mirror plane. The distances of
both points from the mirror plane are equal.
The reflection in water is especially easy to draw (Fig. A.34): We simply
have to find the top view P ′ in the water plane of a point P . The segment
P ′ P may now be inscribed downwards. If the viewing axis is horizontal, then
we may immediately recognize this “doubling” in the image.

Fig. A.34 The reflection in a wa- Fig. A.35 Perpendicular mirrors: the circumscribed
ter surface is easily drawn. cuboid is the secret to our success.

The reflection in a perpendicular wall (Fig. A.35) is no more difficult than

in a horizontal surface. The path s of the reflection plane and the direction
of the reflection normal (vanishing point S) must be known. Let P be, once
again, a point with a top view of P ′ . Let us project the segment P P ′ in
the direction of the reflection normal. This gives the points 1 and 2 in the
mirror plane (1 at the intersection with the path s and 2 above it). Above
A.4 Perspective sketching 635

the midpoint 3 of the segment 12 (whose half is also visible in the image),
we can find the top view of the reflected point. The reflection point P ∗ is
located directly above it. Let us keep 1 → 2 → 3 in mind.
Three principal vanishing points
So far, we have only considered perspective images with a horizontal viewing
axis. Let us now deal with the general case.

Fig. A.36 Three principal vanishing points

The three coordinate directions vanish in the image through the principal
vanishing points F1 , F2 , and F3 . The horizon h = F1 F2 , if viewed from above
or below, tilts in the direction of the upper or lower edge of the page. Perpen-
dicular straight lines no longer appear to be parallel. A cuboid can quickly
be sketched (Fig. A.36), but what about its shadow on the base plane? Now
we have to adapt the previously defined rules, and never forget that the Sun
and its top view are also connected by a vertical, so that:

In a general perspective, the sun point S and the sun foot S ′ lie on a straight
line through the vanishing point F3 of all vertical lines.

We may again distinguish between back lighting (Fig. A.36) and front lighting
(Fig. A.30) depending on whether or not the sun point lies above or below
the horizon. The special case of side lighting does not seem to bring as many
636 A course in freehand drawing

advantages as with a horizontal viewing axis, because now the light rays only
appear parallel in the top view (according to our rule, the sun point lies on
a horizontal through F3 ).

Fig. A.37 Front lighting: Different styles, with the same theory behind them.

Fig. A.37 shows a sketch and the “finished” general perspective image (Stefan
Wirnsperger), both of which show correct cast shadows. The perspective
image on the right shows no construction lines. The sketch clearly shows
that it is necessary to think things through very carefully.

Fig. A.38 Always pack sophisticated objects into cuboids, and give a suggestion of shadows!

Fig. A.38 shows other perspective images with three principal points. In such
sophisticated drawings, trying to draw an accurate rendering of cast shadows
would be too risky: So, only add a subtle suggestion of shading. Nevertheless,
it is always useful to pack more complicated objects into minimal bounding
boxes,since dealing with cuboids is what we know best.
B Geometry and photography

As we have already seen, geometry is not just a theoretical mental game

that can be played by a select few. Its applications are manifold, and one of
the disciplines most obviously connected to geometry is photography. Pho-
tographs represent the “classical” perspective. Our knowledge about perspec-
tive helps us to create certain effects deliberately. For those readers who have
only skimmed the theoretical content of this book, some of the geometrical
statements outlined in the previous chapters will be repeated.
Conversely, many readers may already be familiar with some of the topics
and techniques that will be discussed in this chapter. It might still be useful
to illuminate these matters from a geometrical standpoint, if only to clarify
the following questions:
How can we employ simple tricks to produce images which could otherwise
only be taken with the use of expensive shift lenses? Why is the choice of the
focal length not only a question of “view”, but also matters with regard to
where the photograph is going to be used? When are ultra-perspectives not
only permitted but actually necessary? Why does the depth of field depend
on the focal ratio? What changes in underwater photography?
Photographs illustrating these questions can be found throughout this book
and are not merely limited to this appendix. This reinforces the intercon-
nection between subjects that only appear to separate at first. Anyone who
wishes to approach geometry with the help of photography (and vice versa)
is most welcome to do so!
© Springer Nature Switzerland AG 2020 637
G. Glaeser, Geometry and its Applications in Arts, Nature and Technology,
https://doi.org/10.1007/978-3-030-61398-3
638 B Geometry and photography

B.1 Focal lengths and viewing angles

Fig. B.1 The famous Piazza San Marco in Venice: 17-mm focal length, full frame sensor. The
image seems to be unnaturally distorted. However, we will see later that this may, under some
circumstances, be necessary.

Fig. B.2 Left: the same subject, but shot with 400mm focal length. The focal length requires you
to move further away. The effect is completely different than in the first image. Right: the same
subject, shot with a 15-mm fisheye and a full-frame sensor.

Today’s lenses (zoom-lenses in most cases) are extremely complicated. Before

the advent of digital photography, by far the most popular types of nega-
tives were 24mm × 36mm in size. The lenses of non-digital reflex cameras
usually had focal lengths in the approximate range of 20–400 mm (with some
expensive exceptions).
B.1 Focal lengths and viewing angles 639

Modern digital cameras very often have a sensor that is much smaller than
24mm × 36 mm, and the range of the focal length is accordingly smaller.
(The only exception are some professional “full frame cameras.”)

Focal lengths and the viewing angle are connected

The physical term focal length used to be (and still is, given certain condi-
tions) a measure for the geometrical term angle of view – the angle between
the lines of sight from the lens center through the endpoints of the image
diagonal. The smaller the focal length, the greater this angle is (Fig. B.3).
If the angle is greater than 50○ (which in full-frame cameras corresponds to
a focal length of about 35 mm), then one is usually faced with a wide-angle
lens. Lenses with angles above 85○ (focal length of about 24 mm) are al-
ready counted among the super-wide-angle or ultra-wide-angle lenses, and
those with angles above 180○ (yes, this is possible!) are usually called fisheye
lenses (below 14–16 mm). If, on the other hand, the image angle is smaller
than 20○ (about 85 mm), then one is dealing with a so-called telescopic lens.

Fig. B.3 Different viewing angles ϕ (measured in degrees) correspond to different focal lengths f
(measured in millimeters). The formulas in this chapter interpolate data given by various camera
producers. The general rule is the following: the smaller the focus, the larger the viewing angle.

⊕ Remark : The focal length is now widely considered to be a confusing measure

for the image angle. Non-full-frame digital cameras based on the reflex camera
principle often have a “crop factor” between 1.5 and 1.6 (sometimes even 2). If a
zoom lens is labeled with 18-200mm, then on a full-frame camera, this corresponds
to 18×1.5-200×1.5mm = 27-300mm. This is due to the smaller size of the CCD (the
light-sensitive layer or chip). However, exceptions confirm the rule: Some expensive
reflex cameras do not crop at all (full-format sensor), while some have a crop factor
of 2.
In digital cameras which are not reflex cameras (and are, thus, incompatible with
the “good old” lenses), one may usually find focal lengths that correspond to about
one fourth of the values of full-frame cameras. One might, nevertheless, produce
640 B Geometry and photography

comparable image angles. Here, the size of the CCD appears to be significantly
smaller (at the cost of image quality). ⊕
One might be tempted to conclude that lenses with different focal lengths
or different image angles produce different images. From the perspective of
geometry, however, the following important theorem holds:

The chosen perspective depends only on the position of the lens center and
on the direction of the optical axis (inclination of the camera), and not on the
chosen focal length. A telescopic-lens photo is merely a “enlargement” of the
image center.

▸▸▸ Entire lens systems work like one single converging lens
An objective lens consists of several lenses and lens groups (Fig. B.4). If we
attempt to analyze the direction of light in such a lens, we have to admit that
the matter does not seem to be as trivial as it first appears. First, we can
determine that rays which enter into the lens system in parallel are eventually
focused on the light-sensitive layer (assuming that the distance is set to ∞.)
If we translate the direction to the point P far in the distance through the
image point P c , then it has to pass through a fixed point E on the optical
axis – if this were not the case, then we would not be dealing with a central
projection. So, let us have confidence in the lens manufacturer’s ability to
synchronize their 10 to 15 lenses in precisely such a way.

Fig. B.4 The complicated light path through a modern lens. Still, the whole lens system works
like one single lens with the optical center E (corresponding to the eye point of the perspective).

⊕ Remark : The question of why lens manufacturers go to such elaborate lengths is

easily answered: It serves to correct the many divergences from the perfect image.
It starts with the simple physical fact that different spectral colors are refracted in
a different way. A simple lens, thus, produces “rainbow edges”. ⊕ ◂◂◂

When, for a certain distance setting, P does not lie very far away, the light
rays that are emitted by P and enter the lens system will not converge at
exactly the same point on the photo layer. The point will thus appear as a
blurred spot on the image. If P is an important point, the distance between
B.1 Focal lengths and viewing angles 641

the frontmost and the backmost lens block should be altered so that a single
image point P c on the photo layer is created.
⊕ Remark : This modification of the distance setting will cause the eye point of the
perspective – given a fixed camera body – to shift. This is an important aspect to
consider when you wish, for instance, to take a series of photographs of an object
with a fixed camera at varying distance settings in order to produce an “infinitely
sharp image” with the appropriate software. ⊕
642 B Geometry and photography

B.2 3D images in photography?

Are photographs central projections?
Photography is a source of fascination for many people – especially for math-
ematicians and geometry enthusiasts. In geometry classes, we sometimes say,
in simplified terms: Photographs correspond to central projections (perspec-
tives) of space. This means a reduction of three-dimensional space into the
two-dimensional plane. This can be “proved” by the fact that it is possible to
reconstruct the photographed three-dimensional scene quite accurately from
a number of photographs. It is quintessential to use high-quality lens systems
that reproduce straight edges as exactly straight.
Even though this idea works quite well for scenes with larger technical ob-
jects, such as polyhedra (e.g. a furnished room or a building), it poses huge
problems in the context of macro photography, that is, photographs of objects
that measure just a few centimeters or even less than that.

An impossible photograph

Fig. B.5 Left: impossibly sharp? Right: “normal macro sharpness”.

The flies in Fig. B.5, which measure about 1 cm, are fundamentally different
in terms of the photographic techniques used to depict them. The picture
of the fly on the left (with some disgorged digestive juice) is in sharp fo-
cus overall. An insect photographer would be puzzled: Even with the use
of highly expensive equipment with the best macro objectives in existence,
macro flashes and the highest aperture number possible (as in the picture
on the right, where miniscule droplets of water can be seen on the complex
eyes), it seems impossible to take such a photograph.
B.2 3D images in photography? 643

This poses two questions: Why can an object as small as a fly not be rendered
in sharp focus overall, and how does it seem to be possible after all? This
subchapter will explain the situation from a mathematical/geometrical point
of view.
The lens formula

Fig. B.6 Lens equation

In physics, the method of operation of a lens (or a well-aligned lens system)

is explained as follows: Let P be a point in the real world. It emits (reflects)
light rays in all directions. Two of them will have easily predictable properties.
The principal ray through the lens center C will not be refracted, while the
ray parallel to the optical axis will go through the focal point F ∗ after the
refraction. Behind the lens (the lens system), the two rays – and all others
– meet in the pixel P ∗ . From Fig. B.6, using similar triangles, we can derive
the lens formula:
1 1 1
= + . (1)
f d d∗
Here, f denotes the focal point, and d and d∗ denote the oriented distances
of the point in space P and the pixel P ∗ from the symmetry plane through
the lens center C, respectively.
Using this method, we can determine the corresponding pixel for each point
in space.

The Gaussian collineation

Let us define d = kf . Using the lens formula, we then get
k d
d∗ = ⋅f = . (2)
k−1 k−1
644 B Geometry and photography

According to the intercept theorem, we get the following relation between

the distances of a point in space P and its pixel P ∗ from the center C:
1
CP ∗ = ⋅ CP . (3)
k−1
Even though the mapping P ↦ P ∗ works in both directions, a camera will
only be able to depict the half space whose points lie more distant from the
lens than the focal distance f (k > 1): Points in the plane through the point
F perpendicular to the optical axis are projected onto far points, because
the denominator equals k − 1 in that case.
It is easy to show that the mapping P ↦ P ∗ preserves straight lines: Let g be
an arbitrary straight line in space. It can always be defined as the intersection
of two special planes ε and ϕ, with ε being the connecting plane of g with
the center C and ϕ being the plane through g that is parallel to the optical
axis. ε is transformed into itself (ε∗ = ε) because we can think of the plane as
a pencil of principal rays. The other plane, ϕ∗ , can be conceived of as rays
parallel to the optical axis that transition into a pencil through the focal
point F ∗ and the intersection line of ϕ with the symmetry plane. The image
g ∗ of g is the intersection of ε∗ and ϕ∗ and therefore a straight line. Our
mapping P ↦ P ∗ is thus a collineation – which is the technical term for
images that preserve straight lines.1 This insight goes back to C.F. Gauss.

How does this relate to photography?

Using the simple formula (3), we can very easily transform spatial objects
made of a number of points into equally spatial objects. How does this relate
to photography, which, after all, produces a two-dimensional result?
Let us consider a point P at a distance d from the symmetry plane (collineation
plane). If the plane π of the sensor of our camera happens to be located at
a distance d∗ from the collineation plane, the pixel P ∗ will lie in π. Thus,
the intersection of the object that we aim to reproduce with the “plane of
sharpness” through P at a distance d parallel to the collineation plane is
rendered sharply.2 All other points are rendered more or less sharply.
The extent of blurring is dependent on a number of different parameters,
as we shall see. One crucial parameter is the distance of the photographed
object proportionally to the focal length.

1A note for insiders: This mapping is a very special form of perspective collineation, because the
center lies in the collineation plane (the symmetry plane). Such a collineation is called elation.
2 Using our common sense, we may have assumed that in a photographic image, we can render

sharply all those points that have a certain constant distance from the lens center (thus lying
on a sphere around C), which depends on the distance d∗ from the sensor plane. However,
according to the lens formula, this is not the case, as all of those points lie in a plane, the plane
of sharpness at a distance d.
B.2 3D images in photography? 645

Photographing elephants vs. flies

Let us first consider the image of a “large” object from a great distance.
(What is meant here is the relation of the object size to the focal length f
(Fig. B.7).)

Fig. B.7 Elephant photography (depicting large objects from a great distance): The corresponding
virtual 3D image is “almost flat”, i.e., all image points are close to the sensor plane. The image in
the sensor plane is more or less “sharp”.

For points at a great distance d = k ⋅ f (k >> 1), the image distance d/(k − 1)
(formula 2) does not vary a great deal.
The collinear virtual object behind the lens will thus be strongly oblate,
which means that there will be minor blurring of those points that do not lie
precisely in the plane of sharpness. Taking a sharp full-size photograph of an
elephant thus poses no problem.
The smaller the object that we want to photograph is, and the closer this ob-
ject is to the “forbidden” vanishing plane (Fig. B.8), the more the expression
f /(f − d) = 1/(1 − k) in the transformation formula (3) will vary. This means
that the points will become blurrier, which becomes a genuine problem.
A brief remark: Short focal lengths f apparently have a positive effect on the
focus depth, as a fly or a snail becomes larger in proportion to f . Cameras
with small sensors have a correspondingly shorter focal length.3

3 Intechnical specifications, the focal length (e.g., 100mm) of a lens is often provided in terms of a
35mm equivalent, which means that, for a “full frame sensor size” of 24mm×36mm, the lens has
a focal length of 100mm. However, if the sensor only has a size of, for example, 6mm×9mm, the
same visual impression can be achieved with a focal length of just 25mm. In this special case,
we have a crop factor of 4. Nowadays, it is possible to take amazingly sharp macro photographs
with good smart phones, owing to their extremely short focal lengths and correspondingly tiny
sensors (with much larger crop factors). However, this leads to the problem of having a large
number of pixels in the tightest of spaces, which inevitably leads to a loss of quality.
646 B Geometry and photography

Fig. B.8 Macro photography: When a small object close to the lens is photographed, the 3D
distortion of the corresponding virtual image is considerable. Only points in the focal plane π will,
therefore, be depicted in sharp focus.

Geometry vs. physics

In geometry, the matter seems to be trivial: Let us intersect the light ray
through the lens center with the sensor plane. Seen from the perspective of
physics, this, of course, does not work: A single light ray is not sufficient to
expose the sensor. We will thus have to install a circle-shaped opening in the
collineation plane – the aperture. All light rays emanating from a point P in
space will then lie inside a skewed circular cone through the opening, which
is itself refracted in a skewed circular cone (Fig. B.8).
Real halfspace

Virtual halfspace
P

Plane of
CoC collineation sharpness
plane

P∗ sensor plane

Fig. B.9 The Circle of Confusion (CoC) in the sensor plane p can be interpreted as the intersection
of an oblique cone passing through the circular aperture opening c.

The entirety of light rays in this refracted skewed circular cone exposes the
sensor plane in a dot-shaped way only if P lies on the plane of sharpness. In
all other cases, there is a so-called circle of confusion on the sensor (CoC for
short).
We could now assume that we only need sufficient light (flash) in order to
keep the aperture as small as possible. (We speak of a high aperture number
in such cases.) However, this is only possible up to a certain limit. (The
aperture should be bigger than 1 mm at any rate.) If we reduce the size
further, the wave properties of light further complicate the matter: It leads
B.2 3D images in photography? 647

to diffraction on the edges of the aperture, which results in inconvenient

diffraction blurring. Optimal results can be obtained by using the optimum
aperture indicated by the manufacturer of the lens. Photographers know that
exceeding the optimum aperture will reduce the image quality.

Focus stacking
In the picture of the elephant, we hardly ran into any problems – if a pho-
tographer targets a point that lies approximately at the end of the first third
of the desired distance range, the picture will be sufficiently sharp.4
In macro photography, however, a lack of sharpness is a serious problem –
especially if we are not dealing with artistic but scientific images.

Fig. B.10 Two mating dragon flies, photographed several times with varying lens centers (and
focal points). The first and the last photo of the series can be seen on the left. The focus-stacked
image on the right can be considered sharp in the zone that is defined by the two animals. The
background still remains blurred, which is, aesthetically speaking, a big advantage. Thus, focus
stacking can also be considered as an artistic tool.

Nowadays, a technique called focus stacking has been established. In essence,

it works in the following way: The camera takes a number of images of a
scene in the shortest interval possible, varying the distance of the plane of
sharpness. By using this method, we get an image series where different
layers of the object are in focus one after the other. The theory of image
processing is quite advanced by now, and software used for this purpose is
able to distinguish sharp from blurry pixels.
In a second step, one sharp image is created from this entire image series.
How well this works even without a tripod is shown in Fig. B.10: On the left,

4 Artistic
photographers often encounter the opposite problem: They deliberately want to work
with blurred areas. In such cases, using larger focal lengths and a wide-open aperture is rec-
ommendable.
648 B Geometry and photography

we see the first and the last picture of such a series by way of example; on
the right, we see the final product.5

The plane of sharpness scans the object

For any position of the sensor plane (distance d∗ ), there is, thus, exactly
one plane of sharpness (distance d) in the Gaussian collineation. Its position
follows from the lens formula (1):

d = f d∗ /(d∗ − f ). (4)

In commercially available cameras, the position of the sensor plane is fixed,

and the position of the lens center C moves forward and backward on the
optical axis. If we focus on points that are very far away, C lies in front of
the sensor plane at a distance f (with d∗ = f , or equivalently d = ∞). If
we photograph a two-dimensional shape – such as a drawing – that lies in a
plane parallel to the sensor plane at a distance of s = d + d∗ and if we focus
it, the center C will have a position at a distance d∗ from the sensor plane
that we want to calculate.
With the formula (4), the following applies:
2 2
s = d∗ /(d∗ − f ) or d∗ − sd∗ + sf = 0 (5)

The ambiguous solution of this quadratic equation is

√
∗ s s2
d = ± − sf .
2 4
In order for the expression below the root not to be negative, s ≥ 4f must
apply. This is, indeed, always the case because of the requirement d > f . (If
d = d∗ = 2f , the solutions coincide.) Furthermore, both solutions are always
valid, even though one would – for practical reasons – tend to stick to one
algebraic sign when computing a series of camera positions.
If the lens system lies within the computed distance d∗ in front of the fixed
sensor plane π, our two-dimensional shape will firstly be rendered in overall
sharp focus on the sensor, and secondly, it will appear similar, and thus
perspectively undistorted, even though it will not retain its original size.
Let us now turn back to three-dimensional objects. If we make a series of
photographs while purposefully varying the distance of the lens center, we
will get photographs in which one section line of the object is rendered sharply
and largely undistorted (it is merely scaled). Fig. B.11 illustrates this matter:
In a simulation of this process, a cube was mapped by means of layers, which

5 An additional advantage of this method is the fact that we usually do not get an “infinitely”
sharp image, but that there is a certain layer rendered in sharp focus. Blurry backgrounds
facilitate the isolation of objects from the background and prevent the viewer’s gaze from
getting distracted by unnecessary details.
B.2 3D images in photography? 649

Fig. B.11 When the center C of the lens system (and therefore also the focal point F ) moves
along the optical axis (with fixed sensor plane π), the plane of sharpness which corresponds to the
sensor plane moves in parallel. Its intersection lines with the spatial object are depicted in sharp
focus, but in an absolute size that depends on the distance Cp.

can be triangles, quadrangles, pentagons, or hexagons. However, if we now

simply put these pictures on top of one another, we will end up with strangely
distorted perspectives (see also Fig. B.12 on the left).
Straight lines – such as the edges of the cube – are depicted as curved (it is
possible to show that they form parabolas.) The reason for this is, of course,
that, owing to the different distance of the lens center from the sensor plane,
the section lines are scaled proportionately to this distance.

Converting the scanning process into normal/central projections

Strictly speaking, we scanned our object in the way outlined above, though
we must bear in mind that only those section lines were recorded that were
visible from the respective center. Let us now conduct some skilful scaling
in order to compute the relations in the visible three-dimensional real space
during the scanning process.
Let t be a line segment in the plane of sharpness, and thus in space, and
t∗ is the line segment parallel to it in the sensor plane π. According to the
intercept theorem, the following applies: t ∶ d = t∗ ∶ d∗ . With the scaling factor

λ = d/d∗ , (6)

we can thus calculate the genuine length t from the image length. If we
scale all pictures of the series, with the factor changing from one picture to
another, we get a normal projection of the object (see Fig. B.12, center).
This is remarkable insofar as it is impossible to achieve a normal projection
650 B Geometry and photography

with a single photograph, except if we used an astronomical telescope with

an almost infinite focal length.6

Fig. B.12 A spatial object (in this case a red cube) is “scanned” by a “sweeping focal plane”. Only
the intersections with the plane are depicted in the sensor plane. If the corresponding images of the
section are not scaled, the result is neither a central nor a normal projection (image on the left).
Scaled with the factor λ in Equation 6, the image is a normal projection (middle image); scaled
with the factor µ in Equation 7 as in the image to the right, the result is a perfect central projection
(= “perspective”). In the latter case, the center of the perspective can vary almost arbitrarily.

In order to achieve an exact central projection (perspective), which exclu-

sively relies on unambiguously recorded points, we will choose a position as
the center where the lens center used to be when the last still visible points of
the object to be depicted appeared sharp (d = dm maximal ⇒ d∗ = d∗m mini-
mal). The ith picture will then not only be scaled according to the formula
(6), with the factor λi = di /d∗i , but additionally with the factor d∗m /dm :

di d∗m
µ= ⋅ (7)
d∗i dm

If we now feed the image series into a software designed to recognize sharp
pixels, we will get a picture of graphical precision that is sharp and perspec-
tively correct, as can be seen in Fig. B.14.
The square stamp in Fig. B.13 passes the relatively strict geometrical tests
that can be used in order to test the correctness of a perspective, and the
same is true of the “geometric still life” in Fig. B.14. This is important because
we do not usually photograph miniature geometrical figures in practice, but
rather living beings and natural objects like in Fig. B.15. Creating a full-size
sharp photograph of the stamp would be a classic task of macro photogra-
phy, while the still life is already a medium-sized scene and thus easier to
photograph in overall sharp focus.

6 However, there is, one limitation: Even from various positions on the optical axis, we cannot
always see as much of a surface as we would in a genuine normal projection. Just think, for
example, of a spaceship directly approaching the moon. From its position, we will never be
able to see 50% of the moon’s surface, the way we do from Earth.
B.2 3D images in photography? 651

Fig. B.13 A square stamp of typical macro size Fig. B.14 A typical geometric scene (cube,
is focus-stacked. With the application of the cor- square, cylinder of revolution). From the geo-
responding scalings of the images, the result is metric point of view, the stacked image fulfills
a geometrically perfect perspective. all requirements of a single central projection.

Fig. B.15 Stacked photograph of a male mayfly (genus Cloeon) with its so-called “turban eyes”.
A turban eye is an additional part of the compound eye, whose large ommatidia are all directed
upwards. With them, the males can detect females flying above them in the air during the mating
dance.
652 B Geometry and photography

Outlook
Speaking from a mathematical/geometrical point of view, it is important to
note that much more can be done with a series of pictures of an object, owing
to the fact that these pictures provide information of the object’s location
in space – one example being 3D models. In any case, focus stacking on
the macro level should make it possible to achieve results that would rival
those of laser scanners if the pictures were done under laboratory conditions
– though the technology behind it is simpler, quicker, and cheaper.

Fig. B.16 To the left, only three photos were stacked, which allowed for major parts of the praying
mantis to appear sharp. The blurred background of the stacked photo is deliberate. To the right
eight pictures were stacked (10 mm large fruit of a burdock).

If the objects that we want to photograph happen to be small animals, we are

faced with the additional problem that these seldom tend to freeze in place,
and they usually move at least their feelers or individual limbs (Fig. B.16,
left.) In such cases, the image series should be processed in no more than a
tenth of a second, which will probably be possible in a few years time, if we
bear in mind the rapid technological advancement in recent years. Currently,
commercially available cameras still need a full half second for a complete
series of 8 to 10 pictures – the bottleneck does not occur when the pictures
are saved but is a result of the continuous re-adjustment of the focus.
From an aesthetic point of view, we often only need two or three pictures of
an insect for impressive photographs. In Fig. B.16, it was important to focus
on the mandibles and eyes of the praying mantis – it is secondary that the
rest of its body appears blurred.
B.3 Stereoscopy 653

B.3 Stereoscopy

Optical illusions with one eye closed

We already know the following from Chapter 2: When we look at what seems
to be a cuboid (colored yellow in Fig. 1.16; points P , . . . ) with one eye
closed, the cuboid may actually turn out to be some other form of poly-
hedron (colored red in the image), which is somehow related to the cuboid
(Fig. B.17): For this optical illusion to work, the polygons formed by the ver-
tices P ∗ , . . . of the “placeholder polyhedron” (that is, the polyhedron that we
would see with both eyes open) should also be flat, because, even with one
eye closed, non-flat surfaces are easily recognized by the different shading
effects. Among the infinitely many forms that the placeholder polyhedron
might take, polyhedra with perspective collinear distortion are particularly
suitable. The following statements apply to these polyhedra: Corresponding
points (e.g., vertices) lie on viewing rays (projection rays) through the optical
center C, and planes (e.g., the carrier planes of side surfaces) correspond to
planes ("plane preserving").

P∗

π
P

Fig. B.17 Polyhedra with identical images

Binocular vision
If we look at the same cuboid with both eyes L and R (Fig. B.17), we are
less likely to be deceived by a collinearly distorted polyhedron. Yet, we will
see that, even with both eyes open, optical illusions are still possible due to
the brain’s capacity for visual imagination, which often make us “see what
we we want to see”. Since the depth (measured as the distance from the
vanishing plane through the two eye points perpendicular to the main viewing
direction) varies, the viewing angles for the individual points differ by a lesser
or greater degree. When we take two photographs with parallel optical axes,
654 B Geometry and photography

PL
PR

R
L

Fig. B.18 Parallaxes during binocular vision

we refer to these viewing angles as parallaxes: The difference between the

right image PR and the left image PL increases as the depths of the points
decrease. PL and PR lie on a line that is parallel to the axis connecting the
two eyes (Fig. B.18).
When we alternately close our left and right eye, the parallax distance of
the thumb on the outstretched hand creates the effect that is used in the
so-called “jumping thumb method”, where the thumb appears to jump in
relation to the seemingly fixed, distant background. Since the length of the
arm is to some extent proportional to the pupillary distance, the jumping
thumb method can be used as a measuring unit to estimate the distance of
an object or the size of objects when their distance is known. (As a rule of
thumb: The distance of an object is about ten times as large as the zone
width that is measured for this object with the jumping thumb method.)
There are no parallaxes in the starry sky, which is why we cannot directly
determine the distance of stars with optical means. However, by using the
diameter of the Earth’s orbit (300 million km), we can obtain images with
measurable parallaxes at least for relatively close stars in a half-year interval.

Stereoscopic vision
We speak of “stereoscopic vision” when we simultaneously look at two slightly
different flat perspectives (such as two photographs) that are arranged in such
a way that the intersection of the viewing rays (projection rays) creates the
illusion of a three-dimensional scene. For instance, if the two images of the
cuboid in Fig. B.18 are projected onto a screen in such a way that distance
B.3 Stereoscopy 655

between the intersection points of the two camera axes and the screen equals
the average human pupillary distance of 60-65 mm, and if the position of
our two eyes coincides with the lens centers that are virtually generated by
the magnification of the stereoscopic image’s center, then we get the same
spatial situation as in photography and our brain is capable of combining the
two images to an illusion of the original spatial object. The illusion can be
improved with tricks ensuring that the left eye only sees the image produced
by the left camera and the right eye only sees the right image. (In the past,
this was achieved by anaglyph glasses with red and green lenses; nowadays,
polarization glasses or shutter glasses are used.)
The brain readjusts images with collinear distortions
The correct creation and positioning of the two images is crucial: Only if both
images are calibrated and adjusted to the respective viewer, we can expect
our brain to recreate an exact illusion of the spatial situation. In most cases,
the cuboid will not appear as an exact cuboid, although most viewers may
still identify it as such.

Fig. B.19 Through “cross-viewing” (also known as the cross-eyed method) and by
additionally varying the distance (zooming) if necessary, most people are able to
see a three-dimensional object.

Fig. B.19 is meant to be a test image for our readers. Depending on the size
of the stereoscopic pair of images, we can use the so-called “cross-view” to see
the spatial object in the middle “grow out” of the image plane. The flower
was photographed twice from a close distance. The optical axes of the two
images are parallel and less than 1 cm apart. (This distance is enlarged when
we print the images or project them onto a screen!) If a test subject manages
to create a better spatial illusion by enlarging or reducing the stereoscopic
images – for instance, on a computer screen or by means of a projector –
then this is due to the fact that the axis distance has been adjusted to the
pupillary distance. By turning the head slightly (and moving the pupillary
axis parallel to the screen or printed images), the proportions of the three-
dimensional image can change slightly without disturbing the spatial illusion.
656 B Geometry and photography

Now we would like to show that what we see are generally spatial collineations
of the original object. Let us start with the simplest case: As in Fig. B.18, we
project in such a way that the distance between the intersection points of the
camera axes (in a stereoscopic image like the flower in Fig. B.19, these would
be the centers, or rather the diagonal’s intersections points, of the original
photographs) equals the average pupillary distance of human eyes (±65 mm
for men, ±62 mm for women). With this scaling, the position of the two eyes
is precisely determined to correspond to the lens centers of the two cameras.
But what happens if we move our eyes along the camera axes?

yellow: original object PL

right image: cross-section
PRPL blue: collinearly new eye points L∗ , R∗
distorted object PR
closer to the projection screen

P∗ P P∗ P

R∗
R R∗ R
L∗ L

L∗ L
pupillary distance = distance
between the two image centers

Fig. B.20 When – with the correct adjustments for pupillary distance – viewing
subjects change their distance to the stereoscopic image, the object is only stretched
or compressed.

Fig. B.20 shows (theorem of intersecting lines/intercept theorem!) that the

changes to the stereoscopic image’s spatial result are relatively minor: The
original object is only stretched or compressed (orthogonal affine distortion).
This special collineation preserves parallels and division ratios. The latter do
not even change when the viewing subject moves around the room (while
maintaining a parallel pupillary axis). The distortion is mitigated by the fact
that, for instance, when the viewing subject sees a compressed image, they
stand closer to the supposed spatial object, and this extreme perspective
creates an even stronger illusion of depth.

A robust illusion
If the pupillary distance does not coincide with the distance between the cen-
tres (the diagonal’s intersection points) of the two original images, then the
reconstructed spatial object will still only be linearly distorted (perspectively
collinear to the original). We can find that this is also the case when the left
and the right image are moved closer towards each other. Fig. B.21 on the
right demonstrates that, even in this general case, the viewing subject will
B.3 Stereoscopy 657

barely notice these distortions, because they are looking at the new object
from a different perspective.
yellow: original object
blue: collinearly
PL distorted object
right image: PR PL
PR perspective with the center
P P
close to the new
eyepoints L∗ , R∗ P∗
P ∗

R L∗ R∗

R∗ L
L∗
Neither the position nor the
pupillary distance are properly
adjusted here, but the blue
object appears (almost) correct.
Fig. B.21 The general case

We want to prove for the general case (as depicted in Fig. B.21) that repre-
sentations of the original←→virtual object are always linear and thus preserve
straight lines: Let g be a straight line in real space. Corresponding to g, there
are two image straight lines gL or gR at the respective intersection of the im-
age plane with the visual planes through L and R. Now we project from the
two new points L∗ and R∗ , and from the resulting new visual planes, we get
the intersecting straight line g ∗ . However, the axis L∗ R∗ must be parallel
to the axis LR (which automatically implies that the two realigned images
are at the same height). Otherwise, the new viewing rays would no longer
intersect each other and the spatial illusion would collapse.
The reconstruction from two perspectives proves to be rather easy. This also
applies to non-classical perspectives, like, e.g., the fisheye perspective which
is used in popular action cameras: Those cameras produce spatial fisheye
perspectives/the spatial perspective of fish! This is much less noticeable in
non-technical object that do not feature pronounced right angles (Fig. B.19).
To put everything into perspective. . .
Not everyone is equally impressed by stereoscopic photographs and videos,
regardless of the results that are presented above. Some people can see
stereograms instantly and from any position, whereas others may require
customized conditions and also, on occasion, complain about dizziness and
headaches after some time.
658 B Geometry and photography

B.4 When should we use which focal length?

A significant difference

Fig. B.22 Change of distance (constant principal ray)

After we have learned that the focal length or the image angle have no effect
on perspective, it no longer seems to matter as much whether we take our
pictures with a wide-angle or a telescopic lens. This approach is correct: A
telescopic lens only helps us to exclude objects from the image which are not
important to us and instead fill the frame with our primary photographic
objective. If, on the other hand, the goal is to get “as much into the picture
as possible”, it may be advisable to use a wide-angle lens.
If we approach an object closely (small distance), then we need a big viewing
angle (a wide-angle lens) in order to picture it completely.
However, the object will then appear to be highly distorted, which may in
some cases be the intention of the photographer. If, for instance, the viewing
angle is 50○ , then this entails an “appropriate” distance (about twice the
diameter of the object) and the use of a “normal lens” (with a focal length
of about 50 to 60mm).
The farther away we step from the object, the less noticeable the distortion
becomes: The perspective moves ever closer to the principal point. We would
now have to enlarge the image considerably (by using a telescopic lens) in
order to picture the object at the same size.
⊕ Remark : In ultra-telescopic photographs (which are very common in nature pho-
tography – the gnu in the African savanna being just one example), the image comes
close to a normal view. This has both advantages and disadvantages: The dramatic
effect of a wide-angle photograph may be gone, but the distances are also shortened.
One may get the false impression that the gnus and the gazelles – which, indeed,
tend to graze together – are practically on the same spot (Fig. B.23 on the right).
⊕
B.4 When should we use which focal length? 659

Fig. B.23 An ultra-wide-angle photo (focal length of 17mm) can be seen on the left, where
the trunk is already extremely close to the lens. A telescopic photo is pictured on the right for
comparison (see also Fig. B.22), and a flattening of distances becomes noticeable.

Fig. B.24 Depth amplification with a wide-angle lens (28 mm) and depth shortening with a
telescopic lens (670 mm). The boat is about 400 m away (this estimation is based on the diameter
of the sun, which spans an image angle of about 0, 5○ [9]), but is certainly many kilometers away
from the islands that are visible in the background.

Subjective changes in proportion

The advantage is that the proportions of the animals in the picture are
much closer to their “true proportions”. A gnu may have a back which slants
downwards, but the true angle of inclination may best be seen in Fig. B.22 on
the far right. Sometimes, it may also be an advantage to be a bit further away
from the animals. The elephant in Fig. B.23 on the left must have mistaken
the camera for food. The photo on the right was already done from a more
respectful distance.
From one extreme to the other. . .
Normal projection is an extreme case of a telescopic photo. In such images,
the depth values – the distances from the observer – are no longer important
(Fig. B.24).
. . . or: How a broom closet may be turned into a dining hall
When taking photos of small rooms, one may no longer have the choice of
an arbitrary focal length. The dilemma persists that while a wide-angle lens
enables the photographer to encompass as large a portion of the room as
possible, it gives the room a strange appearance (Fig. B.25 on the left): It
660 B Geometry and photography

Fig. B.25 Wide-angle lens vs. regular lens for interior photography

looks as though one were standing in a long and unnaturally stretched hall.
This may be of interest for real estate agents who wish to sell a 10m2 -room
for a steep price . . .
Unintended caricatures and passport photographs

Fig. B.26 Wide-angle vs. telescopic lens: Portraits should not be done with wide-angle lenses!

When taking pictures of human faces, the “magnification effect” of propor-

tions inherent in wide-angle lenses becomes very striking (Fig. B.26). Thus,
when we are asked to take a passport photograph, we should definitely em-
ploy a telescopic lens!
Even though the heads of most people are not precisely spherical, the pictures
in Fig. B.26 may give an indication of what could go wrong when we use a
wide-angle lens . . .
Which focal length should we use underwater?
When we take pictures of large underwater creatures, wide-angle lenses have
to be employed in any case (Fig. B.27). Each meter of distance affects the
visibility. At a distance of 10 or 20 meters from the camera, just about every
color dissolves into gray and becomes drab, and not even the best flash gun
can alter this fact!
Stronger telescopic lenses are virtually never used underwater (Fig. B.27) –
and for many good reasons. First of all, the refraction between water and
B.4 When should we use which focal length? 661

Fig. B.27 An extremely small focus of 14mm was used for this picture, and the adult dolphin
almost touched the lens.

light (in the lens and the camera) already “lengthens” the focal length by
about a third so that even wide-angle lenses quickly become “normal lenses”.
Secondly, one is only capable of seeing a few meters ahead, even in crystal
clear water – and finally, color fastness decreases rapidly with distance.

Fig. B.28 Flash guns can be counterproductive when plankton is present.

In Fig. B.28, a whale shark plows through plankton, which also causes the
view to be highly limited. Flash guns are completely useless (and may, in fact,
illuminate floating particles which would block the view). Fig. B.28 on the
right: Remoras which accompany the whale shark can be captured from up
close and appear in their true colors as they are illuminated by a flashlight.
The solution for the “whale-shark problem”: Move as close as possible to the
object you want to photograph and use an ultra-wide-angle lens (Fig. B.29).
662 B Geometry and photography

Fig. B.29 If the water is full of plankton, the only way to get good photos is to use ultra-wide-angle
lenses (f = 14 mm, full frame sensor).

Nowadays, such “action cameras” (even with fisheye effects) are sold at com-
paratively affordable prices.
Depth is crucial
Colors appear as they actually are only when the pictured object is very
close to the observer, and only when we employ a flashlight or are close to
the surface: A distance of only one meter in one meter depth is enough for
the light to have to travel for two meters through water (one meter to the
pictured object and another back to the observer) – which already causes
parts of the red color components to be “swallowed”. Due to the fact that
artificial light sources are not present in nature, red color tones may act as
camouflage underwater.

Fig. B.30 The same turtle – on the surface and in 10 meters depth
B.4 When should we use which focal length? 663

Underwater and from a distance of more than two meters, flash guns are use-
less or even counterproductive (Fig. B.28). Thus, one has to rely on sunlight.
Fig. B.30 on the left illustrates that more or less true colors can be expected
close to the water surface when the distance of the photographer is not too
large (1.5 meter). Fig. B.30 on the right shows the same turtle in 10 meters
depth. Now, the sunlight has to travel 11.5 meters.
Fig. B.31 on the left illustrates that in 20 meters depth, all red and orange
colors act as camouflage and that a weak flash gun does not help either.

Fig. B.31 On the left: one meter distance to the stonefish, weak flash gun, picture taken during
the day in 20 m depth (see also Fig. B.32). On the right: a flashlight from a close distance helps
to detect creatures that would otherwise be nearly invisible.

Macro photography underwater

Underwater macro photography may be difficult for a diver because one has
to be able to remain very stable without the help of gravity. However, since
one gets very close to the object and almost always uses flash lights or flash
guns due to the lack of direct sunlight, the colors are almost true to life in
the photograph (Fig. B.31 on the right, Fig. B.32).

Fig. B.32 The closer one gets to the predator (on the left a scorpionfish, on the right the same
stonefish as in Fig. B.31), the more one begins to understand the effect of the camouflage.

Fig. B.33 illustrates that when we take pictures at night (or at greater
depths), a flash gun is the only available light source. True-to-life colors
are then easier to capture from less than half a meter distance.
664 B Geometry and photography

Fig. B.33 On the left: At night, the only light source is the flash gun. On the right: impressively
camouflaged eyes of the crocodilefish (note the side view of the eye in the back).

Finally, Fig. B.34 shall illustrate that when we take pictures of really small
objects, there is almost no difference to macro photography on land: We have
a very shallow depth of field and should try to get as many points of interest
into focus and/or use a high-focal ratio. Generally, one should always stop
down strongly in this size range. (Illumination does not pose a problem due
to the short distance – quite the contrary: Many automatic flashes tend to
create overexposure at very short distances.)

Fig. B.34 A “genuine” close-up. It is important to note that the polyps on a hard coral need to
lie on the focal plane due to their tiny size, even if one takes photographs with a closed aperture.
A rush job is unlikely to produce a satisfactory result. The picture here reveals that there were
other little animals crawling on the coral as well.
B.5 Primary and secondary projection 665

B.5 Primary and secondary projection

Fig. B.35 Primary perspective of a fairly harmless sand tiger shark taken with an ultra-wide-angle
lens. Its snout almost touches the camera. When we look at the pictures as they are printed here,
in such a small size and from a distance of 40–50 cm, they look unnatural. However, if we enlarge
the photographs and force the spectator into a closer position in front of the diagonal intersection,
we will manage to create a “natural” impression.

Ultra-wide-angle images are occasionally necessary!

At first, the extreme wide-angle picture in Fig. B.35 may appear exaggerated
– and it is, indeed, when we look at the printed image in this book from a
distance of 50 cm or more.
You are encouraged to try the following experiment: Shut one eye and po-
sition the other eye as close as possible above the center of the image (at
a distance of about 10 to 15 cm but no closer than that because your eye
could no longer focus at such close proximity). From this position, you will
probably agree that the image no longer appears so distorted.
This may be explained by the fact that it is, indeed, possible to move this
close to an object in nature. From this proximity, it would no longer be
possible to view the whole scene with a single image – you would have to
move your head or at least your eyes in order to perceive other details of your
surroundings. However, in relation to the very large object, the eyes remain
in practically the same spot.
Let us now go back to the significantly downscaled image: If you try to move
your uncovered eye in its extreme position, close above the image center,
its motion would be equivalent to the described head motion – and because
666 B Geometry and photography

the other eye is covered, you would not be able to assess distances anyway.
We have thus managed to scale down the same situation that actually hap-
pened in three-dimensional space. Let us call the spatial situation primary
perspective and the image situation the secondary perspective.
When we accept these assumptions, the following rather obscure theorem
holds true:

In order to get a realistic impression of a photograph, primary and secondary

perspectives should coincide. In other words: While taking the picture, the
photographer would already have to ponder whether the person looking at the
photograph would be able to assume the camera’s original position.

Fig. B.38 on the right may – in some circumstances – be a necessary method

of depiction: Imagine a representation of trees exhibited in museum through
which visitors are allowed to walk. There is one massive tree on a high wall
that is supposed to be pictured such that a visitor passing by in close vicinity
to it may experience the illusion that they can look up at the tree’s canopy.
For this purpose, the trunk of the tree in the lower part of the image must
be thick enough to produce a realistic spatial effect!
Cinema screens and portable TVs
The issue of corresponding perspectives is, in my opinion, important enough
to warrant another example: An underwater photographer takes pictures of
reef sharks or of another, less dangerous species, such as the sand tiger shark
in Fig. B.35 (the primary perspective in Fig. B.36 left). Although sharks are
usually smaller than a human being, they appear highly distorted when they
move closer to inspect the ultra-wide-angle lens.

Fig. B.36 On the left: Primary perspective – the camera is close to the reef shark’s snout. Center:
A dramatic, but geometrically correct secondary perspective – people sitting in the first row will
actually get a pretty realistic impression of the situation! On the right: The image angles no longer
correspond to the primary situation if the picture is seen from further back.

The images captured by the photographers are a sensational success, and

the film is screened in cinemas (Fig. B.36 center and right). Viewers in the
front rows are forced into a secondary perspective that closely resembles the
primary perspective.
After some time, the film is also broadcast on television. We first get a small
picture in a printed TV guide (Fig. B.37 on the left), which stimulates our
B.5 Primary and secondary projection 667

Fig. B.37 Unrealistic secondary perspectives: On the left, the photo of the reef shark looks
strangely distorted, and even in the center, there is distortion as a result of the small viewing
angle. Right: The size of a picture in a magazine matters. The reef shark needs a spread!

interest: It shows an unnaturally distorted shark with giant teeth. Viewers

who then see the moving images not only on a small television screen but
also from a large distance will easily miss out on the dramatic effect of the
original picture (Fig. B.35). They may even sneer at the way the shark’s
snout is moving from side to side too quickly. . .
Therefore, if you intend to print pictures on a small scale for reasons of space
(as in this book, for instance), then you should choose a greater distance (and
a telescopic lens) when taking the photos. Smaller geometrical illustrations
are best done in normal projection. However, if the pictures are to be printed
on a large scale (a photography magazine is a good example), then you may
and ought to employ a wide-angle lens (Fig. B.37 on the right).
⊕ Remark : This insight may not be so useful for paparazzi: They may want to
publish full-spread photos of their victim’s latest and greatest secrets and are thus
more likely to opt for an ultra-wide-angle lens. . . ⊕

Fig. B.38 An extreme wide-angle lens was used in both pictures. On the left, we may be able to
“rectify” the scene by using our existing knowledge. On the right, we may make an estimation of
the trunk’s thickness.

⊕ Remark : In Fig. B.38 on the left, the extreme wide-angle photo makes for an
impressive state room. Our knowledge about columns and chandeliers helps us to
estimate its true dimensions. In the right image, an ordinary tree resembles an
African baobab with an extremely thick trunk. This might be interpreted as a
“distortion of facts”. On the other hand, trees do look this way if we approach them
up close and look up. If a poster-sized print of this photograph were to hang at
668 B Geometry and photography

a height of two meters, a person walking under the picture may, indeed, get the
impression that they were walking through a forest.

Fig. B.39 Two extreme wide-angle photographs. Left: fall scene with 15–20 m high trees. Right:
about 30–40 cm high “thicket” of bushes.

Fig. B.39 shows two scenes that may seem comparable. In terms of actual size,
however, they have a similarity ratio of 50 ∶ 1. Ultra wide-angle lenses are rarely used
for very short distances, which is why we expect the plant in the right photograph
to be much larger.

Fig. B.40 We may also use extreme wide-angle photos in order to send a message or to express a
stylistic statement. Such photos elicit responses in either case, since objects which are experienced
as being up close usually generate some kind of reaction. Things which may actually be tiny or of
“regularly sized” appear gargantuan.

Fig. B.40 shows two further extreme perspectives, both of which convey a certain
“message” and may, for instance, be used for stylistic reasons. If one wishes to “drive
a point home” and demonstrate the enormous performance of one’s creation, it may
be valid to exaggerate the visual effects slightly. . .
⊕
Leonardo’s dilemma
Leonardo da Vinci faced the following problem as he designed his “Last Sup-
per”: The painting measures ten meters in width. The observers would walk
next to it at a relatively short distance and look at the various details of the
picture. If they stood still, then they would need an “ultra-wide-angle lens”.
B.5 Primary and secondary projection 669

Fig. B.41 Leonardo’s dilemma: Should he distort the heads in extreme perspectives or picture
them “normally”? Leonardo wisely chose the latter, because he knew that people were more likely
to walk along the 10m painting than stand on a specific spot to view the painting from the “ideal
position”.

Consequently, all objects would have to be highly distorted – not only the
table and the room, but also the heads of the apostles!
Leonardo knew that human beings are most perceptive of unnatural distor-
tions when it comes to heads and facial features. Thus, he settled on a com-
promise: He distorted the still-life environment to a very significant degree
but left the heads completely unaffected!

Fig. B.42 A fisheye perspective of a scene with people along a large front: Here, the objective
corrects the distortions and contractions at the edges “in its own way”. Note the deliberate usage
of light effects – in this concrete example, the scene is illuminated by bounce lighting from below,
which lends the scene a certain mystique.
670 B Geometry and photography

B.6 From below or from above?

A horizontal lens axis approximates our usual way of seeing pretty well. Our
brain evolved in the “savanna” and is used to processing stereoscopic images
of objects which lie at a comfortable distance and are approximately of “our
own size” – neither too big nor too small.

Fig. B.43 A frog’s perspective from a bird’s-eye view or a bird’s perspective from a frog’s-eye
view?

If the size of an object exceeds our field of vision, then we have to tilt our
visual apparatus – either upwards (when looking up at skyscrapers) or down-
wards (when looking at small objects in front of our feet). Over time, the
terms frog’s-eye view and bird’s-eye view have become established. The use
of these terms is fine as long as we remember that many animals must per-
ceive images very differently from the way humans do and that this is not
merely a difference in scale.
According to the rules of central projection, our scene is augmented by one
further principal vanishing point: the vanishing point of all vertical straight
lines.
To critical eyes, images with a pronounced vanishing point tend to appear
“more artistic”. The following “message” is thus conveyed by the photogra-
pher: The object itself – or my position above it – is very high.
So-called slightly tilted edges are almost “painful” to the trained eye. They
occur when the vanishing point of the vertical straight line lies far beyond
the image (Fig. B.46 on the left). In such cases, the vertical edges of the
object appear to be “nearly” parallel. For example, the left edge images turn
out to be not quite parallel to the left paper edge, and the right ones, which
are rotated in contrary motion, are not quite parallel to the right paper edge.
B.6 From below or from above? 671

Fig. B.44 Worm’s-eye view and bird’s-eye view inside the elliptic towers of the Sagrada Família
in Barcelona

Either a shift lens . . .

Professional photographers are well aware of this problem. In the era of
analogue photography, professionals or amateurs with thick wallets helped
themselves by employing so-called shift lenses and a tripod: This special lens
enables the photographer not only to take pictures in the direction of the op-
tical axis, but also to “shift” this direction slightly. From a geometrical point
of view, a regular central projection is produced. However, one can still fix
the camera axis horizontally while simultaneously “looking up” towards the
edges of the building. According to the laws of geometry, the edges remain
parallel in the image.
. . . or some geometrical know-how
The dawn of digital photography has changed various things, and arguably
for the better: A primary perspective is captured digitally and may later be
changed in an image editing software – a process which may be similar, but
is still a lot easier than the darkroom of yore, where a comparable amount of
photographic “trickery” took place. Even rectangular images can be deformed
in such a way that the edges appear as a general quadrangle, whereas the
images of vertical corners appear to be parallel. If we now crop the edges into
a rectangle, our wizardry should go unnoticed.
Unnoticed? A “geometrician” could easily prove that the new image must
have been altered. A small number of tests would be enough – for instance,
spatial angles become oblique angles if the distortion is corrected properly.
672 B Geometry and photography

Fig. B.45 Either strongly vanishing vertical edges. . .

Fig. B.46 . . . or exactly parallel vertical edges!

The change which has taken place is, in fact, two-dimensional and thus had
no regard for the depth distances of the singular points. Thus, it cannot
possibly be exactly correct. For this reason, we should only resort to this
trick “in emergencies” – for instance, if we have returned from our vacation
with a very special photo of architectural treasure which we cannot possibly
replicate.
If the following principle is considered before taking the photo, we may be
spared a lot of work later on: We should not only take a picture of the
relevant object (such as the tall building in Fig. B.48) but also of many other
important areas next to it – as long as the optical axis is horizontal. We may
even accomplish this without fancy equipment like a bubble level by simply
pointing the camera at the heads of people who are situated at a constant
height in front of the building. In post-processing, we only have to crop away
the “digital junk”. . .
B.6 From below or from above? 673

Fig. B.47 Two extreme perspectives: On the right, the optical axis is horizontal. When such
extreme wide-angle lenses (14 mm) are used, one or two degrees in terms of inclination angles
result in radically different images. If the subject is worth it, one should work with a tripod and
take one’s time.

Fig. B.48 Architecture photos: “slightly vanishing” (tilted) on the left. The bottom half, which
was also photographed on the right, is later cropped from the image.

▸▸▸ An ad showing long female legs

By now we know that objects appear to be more distorted the further away
they are from the optical axis and the closer they are to the lens center. In an
ultra-wide-angle photo, heads should, therefore, always be near the principal
point. However, do the same rules apply to legs?
Let us first consider Fig. B.50. The original picture of the wine glass is seen
on the left, and only a section of it is pictured on the right. If the picture
were to be put on a billboard, then the relevant section would certainly be
chosen – the rest would not be interesting, and besides, it might be a good
idea to catch people’s attention through the extreme distortion.
This brings us back to the topic at hand. Commercials often show women
with unusually, sometimes almost “grotesquely”, long legs. Fig. B.51 shows
the kind of tricks that may be employed to accomplish this effect. ◂◂◂
674 B Geometry and photography

Fig. B.49 Three attempts were made to keep the optical axis horizontal by pointing the camera
at a far distant point which lies on eye level.

Fig. B.50 The “evacuation” of the principal point

Fig. B.51 Let us now use an ultra-wide-angle lens. The legs should be turned towards the lens.
The apparent length of the legs now only depends on what section of the image is being displayed.
The flippers reinforce the impression of elongation.
B.7 Collinear distortion of a photo 675

B.7 Collinear distortion of a photo

Fig. B.52 An extreme distortion of a square grid. At the center: with the help of an ordinary
image editing software – not line-preserving (collinear). On the right: collinear though the exact
application of mathematical equations.

Let us mention a second method by which effects like Fig. B.51 may be ac-
complished: We can modify photos by collinear transformation, which “turns”
arbitrary quadrangles into different but equally arbitrary quadrangles (such
as rectangles). This collineation is linear, which means that straight lines are
always transformed into straight lines.
Fig. B.53 on the left shows an intriguing photo which was taken halfway
below and halfway above water. It can instantly be seen that both halves
are central projections, albeit with different parameters. The verticals in the
upper half vanish in a principal vanishing point, while the vertical edges of
the bottom half inside the swimming pool are nearly parallel. On a computer,
a collinear transformation was applied to the right image so that the upper
vertical lines of the skyscraper and the tree are “bent” to appear parallel to
each other.

Fig. B.53 A swimmer’s perspective on the so-called “Badeschiff” – a boat with a pool on top –
at the Danube Canal in Vienna
676 B Geometry and photography

However, now a section of the image which fits entirely into a trapezoid must
be chosen (image on the right with a red border). A person who only sees
the final result may not recognize that the photograph has been manipulated
to a certain degree and actually depicts an illusory world which no longer
resembles reality. However, somebody with a trained eye might notice that,
for example, the edges underwater are no longer exactly parallel.

Fig. B.54 On the left: A snail crawls across the perspective image of a cube. On the right: A
collineation was applied to the entire photo so that the cube appears “normal” once again; but
what about the snail . . .

In Fig. B.54 on the left, a snail crawls across the perspective image of a
cube. The projection of the cube undergoes a second collineation (secondary
perspective) by the mere act of photography. On the right, this collineation
is “canceled out” by applying another collineation to the entire photo. As
a result, the cube looks “normal”, but the image of the snail is completely
distorted.
Such inconsistencies are quite common these days and may hint at the soft-
ware used for post-processing. Fig. B.52 shows the admittedly extreme trans-
formation of a square grid by using a renowned software package. The result
is clearly not linear, and the grid subdivisions do not appear where they
should be if their location is based on reality. The right image could be the
photo of a grid from an extreme position.

Fig. B.55 Taking many pictures of a whole projection screen: If one does not stand immediately
next to the projector, then the projected image will be distorted in a collinear way. Nevertheless,
the rectification of a general planar quadrangle can be done precisely. This allows the detail marked
by the red border to be reconstructed with remarkable accuracy, despite the limited resolution of
the projection!
B.7 Collinear distortion of a photo 677

It should be noted that it is very well possible for a collinear distortion or

rectification of a planar figure to be precise. Fig. B.55 illustrates this claim by
a practical example (photo of a medical lecture by Zvi Ram). The following
is a geometrically interesting fact: The virus has the form of an icosahedron
with a diameter of 1/10,000 mm!

Fig. B.56 The famous Gloriette in Vienna. On the left: Original photo; on the right: Distorted
image. The “error” is negligible, since the façade is nearly two-dimensional. To get a picture like
the one on the right, an expensive shift lens or a drone to raise the camera a few meters in the air
is necessary.

Fig. B.57 On the left: An interesting snapshot of the baptistery in Florence, taken with a compact
camera (format 4:3) at night. On the right: A geometrically acceptable manipulation of the photo
(format 3:2). The slight bend of the tower on the right is also visible in the original on the left.

Let us consider Fig. B.56 and Fig. B.57 as two final examples: If you have
taken an original photo of the Gloriette in Vienna or the baptistery in Flo-
rence – with a moderately priced compact camera or a mobile phone – then
surely nobody would reproach you for your slight “alterations”. In Fig. B.57
the full moon is visible behind the dome, see also p. 365.
Contrary to Fig. B.56, an exact investigation could, rather easily, bring to
light the manipulation in Fig. B.57 on the right: This second scene is much
“more three-dimensional”.
678 B Geometry and photography

B.8 The problem with the rolling shutter

▸▸▸ Can you trust photographs?
The photograph Fig. B.58, which could almost qualify as art, shows what can
happen when a flying longhorn beetle (Stictoleptura rubra) is photographed
using a very short exposure time (1/2,500 s). The body of the beetle barely
moves, while its elytra (wing cases) move significantly faster than its body
(they flap during flight). The wings themselves, on the other hand, oscillate
at a high speed relative to the beetle’s body.

Fig. B.58 This photograph is not manipulated and shows a flying beetle with wings pointing
in opposite direction, accompanied by two “spare wings”. In extreme cases like this, the observer
immediately realizes that something must be wrong.

Fig. B.59 shows a beetle of the same species above a plane (a car roof). The
author has taken an abundance of photographs showing beetles with wings
on opposite sides. Can we take these photographs as “proof” that insects can
assume such positions during flight?
Luckily, Fig. B.59 shows that, again, there must be something wrong: The
shadow of the beetle does not correspond to the spatial situation. Take a
closer look at the shadows of the wings in the right image. The left wing, for
instance, is again isolated from the body. All this is due to the delay of a few
milliseconds while scanning the many “lines” of the sensor. ◂◂◂

▸▸▸ When the shadows do not match

To recreate the above described effect with a technical device, several pho-
tographs were taken of a miniature helicopter, roughly the size of a giant
dragonfly and with rapidly spinning rotor blades (Fig. B.60).
B.8 The problem with the rolling shutter 679

Fig. B.59 A flying beetle of the same species, again with wings pointing in opposite direction,
this time with “wrong shadows”. A certain amount of geometrical experience can help to notice
this “shadow problem”.

Fig. B.60 A model of a helicopter with fast rotating rotor blades

Although the images were shot using an extremely short exposure time of
1/32, 000 s, the rotor blades, which are actually symmetrical and not curved,
appear distorted (as opposed to the labels on the blades, which can be read
quite clearly)! ◂◂◂

▸▸▸ What every “high-speed photographer” needs to know

The so-called rolling shutter effect is an effect that is well-known in the film
business and that can occur in photographs or videos of rapidly moving
objects. Most cameras cannot expose the whole sensor surface to light at
once. Instead, imagine exposure occurring in a line from one side of the
image to the other, either horizontally or vertically – a process which takes
a short, though not inconsiderable, amount of time.
680 B Geometry and photography

The shutter time of most camera models is usually considerably longer than
indicated. Since the exposed sensor is scanned line by line and each scanning
of a line takes only a fraction of the shutter time, the exposure time for each
line is reduced accordingly. However, it takes the device the same (very short)
amount of time to move to the next line, which is once again exposed very
briefly. During that period of time, the wings may have covered a certain
distance, which then causes effects like skews, wobbles, and smears. ◂◂◂

▸▸▸ The problem lies in the proximity to reality

Image errors may appear so life-like that one is tempted to take them at
face value. Especially in high-speed photography, supposedly realistic images
must be taken with a pinch of salt! Certain tests should be carried out to
check if such image errors are actually possible. Shadows provide the best
starting point for this kind of “reality check”.

Fig. B.61 Smaller birds like sparrows have a wing frequency around 15 hertz. This can sometimes
lead to rolling shutter effects. Which one of the photos might be “wrong”?

Fig. B.61 shows a selection of the hundreds of photos the author has already
taken of flying sparrows. After careful consideration, it turns out that the
two photos on the upper right show wing positions that are impossible. The
rolling shutter effect appears predominantly when the birds are tilted in flight
to one side and the wings in oscillation are at maximum speed. Then, the
shutter reaches the wing on top first and in the time it takes for the shutter
to reach lower parts of the image, the wings have already changed position.
◂◂◂
B.8 The problem with the rolling shutter 681

▸▸▸ The man that shoots faster than his shadow

Referring back to the “wrong shadows” in application p. 678 (Fig. B.60), one
might even remember the comics about Lucky Luke, the “man who shoots
faster than his shadow”.

Fig. B.62 The “woman, whose mirror image shoots faster than herself” (in landscape format, to
demonstrate the “ideal” conditions for the rolling shutter effect to appear)

Inspired by drawings like the one at http://static.tvtropes.org/pmwiki/

pub/images/050711_lucky_luke.jpg, my daughter Sophie painted the “wo-
man, whose mirror image shoots faster than herself”. The painting looks like
a photograph that would, in principle, be possible to achieve - but only with
a very slow rolling shutter. The following application will show how such a
slow shutter might work. ◂◂◂

▸▸▸ An extremely slow rolling shutter

When a copy machine or a scanner stores an image, a rod-shaped sensor
within the machine moves continuously from left to right. In every position,
the sensor scans a very small rectangular part of the image. All scanned lines
are eventually put together to one final image.
Usually, the scanned image is static, and therefore, the scanner works like a
camera. Since the scanning process takes a while, one might get the idea to
move the original image around. In Fig. B.63, the covers of two books where
682 B Geometry and photography

Fig. B.63 Two books rotated on the surface of a scanner during the comparatively slow scanning
process

rotated quickly during the scanning process. The result is quite remarkable,
and it has only little to do with the original. In the image to the right, the
original was rotated so fast that some regions are even depicted twice (take
a closer look at the word “The”). ◂◂◂

▸▸▸ What does the image of a rotating straight line look like?
Let us now tackle the given problem from the geometrical point of view and
only look at the simplest case. Kinematically speaking, we are looking for
a curve that is created as follows (Fig. B.64): A straight line r rotates with
constant angular velocity around a point on itself, whereas another straight
line s (the shutter) moves horizontally downwards at a constant speed. The
intersection point Q belongs to the resulting curve q.
r
q q
r r
q

s Q
s Q
s Q
Fig. B.64 The image q of a rotating straight line r is the well known trisectrix of Hippias, also
called the quadratrix of Dinostratus. From left to right, the ratio between angular velocity and
translation speed in the kinematic generation increases.

This problem has already been investigated by the ancient Greeks more than
2, 000 years ago, being one of the oldest examples for a curve that is created
through motion. Hippias of Elis called it trisectrix and used it around 420
BC to trisect an angle. Approximately 70 years later, Dinostratus used it to
square the circle (and called it quadratrix). Depending on the ration between
B.8 The problem with the rolling shutter 683

angular speed and translation speed, the curve can have quite different shapes
(Fig. B.64).

Fig. B.65 Left to right: Increasing ratio between angular velocity and scanning speed

When the line r does not run through the rotation center, things do not
change essentially. Fig. B.65 shows the computer simulation of Fig. B.63
with typical matches.

Fig. B.66 Same ratio between angular velocity and shutter speed: The result depends on the
instantaneous position of the rotor blades. Note again the artefacts.

In Fig. B.60, we see a photo of the extremely fast rotating rotor blades of
a helicopter, taken with a very short exposure time. Here, again, computer
simulations (Fig. B.66) shall convince the reader of the correctness of our
theoretical considerations. ◂◂◂

▸▸▸ Photo finish

When in a sporting race two or more competitors cross the finishing line
almost at the same time, this can sometimes lead to controversies: Even if a
photographer, for instance, has taken the shot of their life at the exact right
moment, when the supposed winner was crossing the line, the photo could
still be misleading. If in Fig. B.67 the camera had been held in landscape
684 B Geometry and photography

mode instead of portrait mode (with the rolling shutter going downwards),
the red car could have been declared the winner, because, within the time
the shutter needed to reach the red car (a couple of milliseconds), the red
car would have moved further along the track.

Fig. B.67 Photo finish: The two cars finish at nearly the same time. In a photo taken with a
normal camera in “landscape orientation” and very short aperture time, the red car would probably
be the winner even under perfect conditions, due to the rolling shutter effect.

The better solution is to film in high speed mode with a camera in portrait
mode (rotated 90○ ). Now the shutter moves from right to left (or vice versa)
and the microsecond (1/1000 of a millisecond) to record just one line with
only one pixel width can be neglected. The final photo now consists of the
whole series of photos as follows: From each single frame, we only take the
pixels on the scan line in the middle (the scanning shutter is therefore consid-
ered to be fixed in the middle of the image). All those very narrow rectangles
are arranged frame by frame (the parameter is then the time). The resulting
“photo” can appear unrealistic (faster objects appear shorter) – look at the
distorted car tires in Fig. B.67 – but it guarantees the correct photo finish
(at least to that part of a microsecond we mentioned earlier). ◂◂◂

Possibilities to avoid the rolling shutter effect

The simplest, but by far most expensive way to avoid the rolling shutter effect
is to use a camera with a “global shutter” which stores all pixels simultane-
ously. At the time this book was finished, there were only very few cameras
of that design on the market, all of them at a price that is not affordable for
amateurs. For single photographs, a photo flash can be used. Then, the whole
shutter is open for a comparatively long time, like, for instance, 1/250 of a
second. During that time span, a flash is fired in a 1/30, 000 second. If the
environment is not too bright, only the artificial light of the flash illuminates
the object. High-speed filming in low resolution (max. 2 MB), using certain
affordable cameras on the market, also allows to reduce the effect to a high
degree (Fig. 11.82).
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[6] G. Glaeser: Geometrie und ihre Anwendungen in Kunst, Natur und Technik,
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[7] G. Glaeser: Geometry and its Applications in Arts, Nature and Technology.
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[17] G. Glaeser, B. Odehnal, H. Stachel: A Spatial Theorem of the Angle of Circum-
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[30] M. Berger: Geometry I, II. Translated from the French by M. Cole and S. Levy.
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[34] P.M. Gruber, J.M. Wills (eds.): Handbook of Convex Geometry. Vol. A, B..
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[36] W. Kühnel: Differential Geometry. American Mathematical Society, Provi-
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[37] L. Piegl, W. Tiller: The NURBS Book . Springer, Berlin, Heidelberg, 1995.
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[39] H. Pottmann, J. Wallner: Computional Line Geometry. Springer, Wien/New
York, 2001.
[40] J. Richter-Gebert: Perspectives on Projective Geometry: A Guided Tour
Through Real and Complex Geometry. Springer, Berlin, Heidelberg, 2011.
[41] D.J. Struik: Lectures on Classical Differential Geometry. Dover Publications,
New York, 1961.
[42] J.M. Wills (ed.): Handbook of Convex Geometry. Vol. A, B.. North Holland,
Amsterdam, 1993.
Literature 687

Picture credits
The book contains more than 2000 computer renderings, photos, and freehand draw-
ings, distributed over approximately 1,000 figures. The following figures have been
used (with courtesy of the authors), without displaying the names of the photogra-
pher / artist:
2.12 Photo on the left: Reiner Zettl
3.71 Photo on the left: Eberhard von Nellenburg, photo on the right: Cédric Thévenet
4.83 Liddy Scheffknecht, oculus, 2012, Sonnenlicht, Erdrotation, Papier, Lampe, ©
Liddy Scheffknecht and Bildrecht Wien
5.20 Gerald Zugmann
5.21 Gerald Zugmann, Paula Goldman
6.53, 10.26 Statikbüro Bollinger/Grohmann/Schneider, Metallbau Pagitz
6.96 Matthias Ecker
6.114 Photo on the right Karin Pfaffstetter-Odehnal
6.115 ISOCHROM.com, Vienna
7.67 Photo on the right: Christian Perrelli
9.4, 9.82 Harald Andreas Korvas (sodwana.uni-ak.ac.at/korvas)
9.11 Photo on the left: Othmar Glaeser
9.28 Photo on the left: Elisabeth Halmer
9.28 Photo on the right: Torre Annunziata / Villa di Poppea, credit: B. Andreae,
Neue Forschungen in Pompeji und den anderen vom Vesuvausbruch 79 n.Chr. ver-
schütteten Städten (1975) Abb. 26, Database: HeidICON - Klassische Archäologie,
Ruprecht-Karls-Universität Heidelberg
A.30 Photo on the left Othmar Glaeser
9.87, 9.88 artpool.cc (www.artpool.cc/marianne, www.artpool.cc/grisei)
13.39 Helen Binet Studio, 24a Bartholomew Villas, London NW5 2LL, U.K.
B.57 Romana König
Index

A asteroid belt, 547

ability, 68 astigmatism, 279
absolute points, 194, 276 astroid, 23, 418, 419, 452
accompanying tripod, 134 astronomy, 1
acoustic, 240 asymptotes, 197
adhesion, 279, 538 atomic clock, 493
affine mappings, 234 auxiliary apex circle, 121
affine ratio, 30, 58 auxiliary view, 67
age of rocket science, 264 axioms, 8, 28
air resistance, 200 axis of affinity, 119, 120
air rifle, 342 axis of collineation, 119, 375
Alberti, 2 axis of perspectivity, 118, 119
Alexander the Great, 380 axis of symmetry, 84
algorithm, 141
Alhambra, 514 B
aliens, 577 B-spline curve, 352
altitude theorem, 36 Bach, 558
aluminum, 209 back lighting, 632, 635
aluminum layer, 215 back view, 69
ambient space, 52 backlight, 371
amusement park, 470 ball joint, 464
analogue, 493 bamboo stool, 174, 251
analogue photography, 671 Bangalore, 222
anamorphosis, 228, 376 baobab, 667
android, 424 baptistery, 677
anemone, 280, 291 barrel vault, 166, 550
angle, 59 base line, 370, 625
angle bisector, 14, 33 base plane, 39, 370, 625
angle of circumference, 9, 301 basic building block, 517
angle of elevation, 481 Basilica di Santa Prassede, 518
angle of view, 639 bat, 471
angular momentum, 495 beam connection, 166
angular velocity, 484, 496 Bedouin, 535
animal figures, 383 beer mat method, 18
animate, 99 beetle, 534
annular sector, 178 bellflower, 282, 283, 620
annulus, 423 bending machine, 356
ant, 508 bending procedure, 173
anti-complementary triangle, 14 Berger, 523
anti-inverse, 36 Berlin, 237, 485
antiparallelogram, 428 Bernhardt, 493, 497, 499, 500
antipode, 219, 220 Bernoulli, 420
antiquated, 99 bevel wheels, 464
aperiodic tiling, 529 Bézier curve, 351, 352, 530
aperiodicity, 522 Bézier spline, 352
apex, 93, 185 Bézout’s theorem, 168, 253
apex of a star, 46 bicycle, 416, 417, 629
apices of the ellipse, 127 Biedermeier period, 437
Aplysia, 335 Big Bang, 52, 421
Apollonius of Perga, 30 Big Crunch, 52
apparent solar time, 484 bilateral symmetry, 534
apparent time, 474 Bilateria, 534
approximate polyhedron, 281 Bilbao, 300
approximate straight line guidance, 433, 434 billiard ball, 62
approximation techniques, 227 binormal, 134, 207
arc length, 128, 129, 167 biology, 1, 548
arched bridge, 191 bird’s perspective, 670
Archimedean screw, 305 bird’s-eye view, 608, 609, 670
Archimedean solid, 87, 123 Blender, 503
Archimedean spiral, 316 blending surface, 289, 291
Archimedean tiling, 520 blind hole, 308
Archimedes, 7, 22, 109, 306, 533 bluebell, 271
architectural firm, 184 Boeing, 353
architecture, 1, 56, 95, 151, 163, 182, 209, bolt, 307
245, 278, 291–293, 300, 341, 370, Bolyai, 49
473 Bonnet’s theorem, 333
articulate animal, 552 Boolean operations, 361
artificial light, 99 bottom view, 69, 166
ashtray, 142 bounce lighting, 669
assignment problems, 68 boundary circle, 269
Index 689

boundary surface, 401 chess figure, 424

Boy’s surface, 336 chess-playing Turk, 424
boy-scout rule, 483 Chillida, 342
braces, 84 chitin exoskeleton, 411
brachistochrone curve, 420 choreography, 548
brain power, 410 chromosomes, 307
brandy glass, 142 circle, 303
Brianchon, 192 circle conoid, 295
Brianchon point, 193 circle of Apollonius, 31
brightness, 537 circle of circumference, 19
brown algae, 335 circle of fifths, 556, 557
Bruegel, 316 circle of latitude, 268
Brunelleschi, 152, 236, 364, 365 circle of longitude, 50
bubble level, 672 circle reflection, 42
Buenos Aires, 222 circle’s involute, 127
bumpy road, 192 circular, 275
burdock, 652 circular conoid, 249
burning mirror, 239 circular helical surfaces, 311
butterfly wing, 216 circular sector, 177
circumcenter, 13, 43, 127
C circumcircle, 13
C-major tonality, 558 circumpolar star, 476
Calatrava, 296, 300 circumsphere, 108, 113
calendar year, 473 Citroën, 351
city map, 59
camera obscura, 413 civil engineering, 1
camouflage, 662, 663 classical spiral motion, 317
Cape of Good Hope, 490 clearance hole, 308
car body, 265 clipping, 377
carcinus maenas, 536 clock-face, 494
Cardan circle, 441, 447, 450, 452 clothoid, 129, 578
Cardan shaft, 435, 443, 462, 463 Cloud Gate Chicago, 227
Cardano, 441 cockchafer, 509
cardioid, 452 cockpit, 400
carnivore, 552 cogwheel, 464
carpenter bee, 426 cohesion, 538
carrier cylinder, 303 collinear mappings, 234
carrier line, 6, 25 collineation, 252, 382, 676
carrier plane, 33 collision, 73
carrier surface, 264 color fastness, 661
carrier surfaces of revolution, 546 columns, 197
cartography, 217, 219 Commodore Amiga, 99
Cassini oval, 31, 275 complex surfaces, 339
cast-shadow boundary, 147 composite photograph, 60
cat, 60, 66 compound computation of interest, 318
cat’s eye, 62 compound eye, 140, 401, 410
Catalan, 114, 333 computational geometry, vi
Catalan solids, 114 computer geometry, 96, 283, 471
catenary, 190, 192, 629 computer graphics, vi, 274, 397
catenoid, 281, 326, 327, 330, 333 computer simulation, 319, 409, 424, 495, 497,
caustic, 213, 231 547
CCD, 639 conchoid, 23
ceiling paintings, 386 cone of constant slope, 203
celestial axis, 494 cone of revolution, 33, 176, 234, 270, 478, 575
celestial equator, 497 cone rolling, 461
celestial pole, 476, 494 confocal conic sections, 187
celestial sphere, 493 conformal, 220
cell nucleus, 307 conformality, 225
center of collineation, 119, 375 congruence of rays, 498
center of curvature, 261 congruence relations, 45
center of projection, 54 congruence transformations, 88
center of the globe, 221 congruent, 7
central lighting, 370 conic, 37
central perspectivity, 118, 122 conic section, 23, 30, 47, 196, 199, 494
central projection, 162, 236, 367, 670 conic section construction, 196
central shadow, 120 conic wheels, 179
central similarity, 14 conjugate diameters, 380
central view, 608 connect, 38
centrically similar, 517 connecting developable, 210
centrifugal force, 180 connecting line, 34, 142
centroid, 13 connecting plane, 220
ceramic lens, 358 connecting surface, 180
ceramics, 340 conoid, 295, 296
Cetoniidae, 509 conservation of angular momentum, 545, 546
chain of spheres, 576 conservation of momentum, 548
chain reaction, 549 constrained motion, 416
chair tiling, 522 construction drawing, 86
chandelier, 100 construction of the unfolding, 68
channel surface, 288, 318 contact circles, 35
chariot, 380 contact lenses, 279
chemistry, 1 contact point, 38
690 Index

continuity of curvature, 353 cylindrical projection of the sphere, 396

continuous line, 84
contour, 137
contour cusp, 280
D
contour ellipse, 365 Dürer, 516
contour generator, 618 Díaz Moreno, 573
contour hyperbola, 616 Díaz Moreno, vi
contour line, 137 daisy, 544
contour point, 137 Dandelin, 164, 187
contour polygon, 96 darkroom, 671
control grid, 355 date lines, 494
control points, 347 de Boor, 353, 354
control polygon, 351, 352 de Casteljau, 351–354
convex, 93 de la Hire, 121, 466
cooling fan, 305 dear, 503
cooling tower, 250 deflation, 522
coordinate directions, 635 degree of symmetry, 285
coordinate grid, 219 degree-three curve, 255
coordinate origin, 68 degree-two curve, 185
coordinate plane, 67 deoxyribonucleic acid (DNA), 307, 314
coordinate system, 39 depth amplification, 659
copy machine, 681 depth line, 369, 625
coral colony, 540 depth of field, 399, 400
coral fish, 60 depth shortening, 659
corkscrews, 307 Descriptive Geometry, 53
cormorant, 549 desert, 408
cornea, 409 design, 1, 182, 238, 239, 282, 283, 292, 296,
corner of a cube, 63 342, 343, 347, 350, 354, 355
coronavirus, 346 determination of visibility, 72
corrective lens, 279 developable, 175, 295, 309
coupler curve, 436 developable surface, 38, 283, 360
course angle, 222, 276 development, 68, 104, 332
crane, 426, 433 diacaustics, 406
crank shaft, 430, 446 diagonal direction, 626
crank shaft gear, 446 differential equation, 191
crank shaft propulsion, 431 diffraction, 216
crease pattern, 116 digital camera, 639
crescent moon, 157 digital photography, 671
crocodilefish, 664 digon, 448
crop circles, 577 dimension, 5, 34
crop factor, 639 dimensional jump, 46, 50
cross-view, 655 diopter, 280
crosshair, 624 directing plane, 295
crotch, 7 direction of a straight line, 29
Crustaceans, 401 directions, 249
Crux, 490 directly congruent, 8, 88
cryo-electron tomography, 346 director, 188, 197, 295, 296
crystal formation, 525 discrete family, 342
Cuba, 160 distance, 368
cube, 102, 103, 105, 341, 518, 528, 608, 609, distance circle, 29, 368
676 distance line, 32
cube apartment, 341 distance of the jump, 61
cubic circle, 147, 255, 256 distance of translation, 302
cubic parabola, 256 distance plane, 32
cubic splines, 347 distance points, 626
cuboctahedron, 222 distance sphere, 152
cuboid, 75, 635 distance surface, 32
culmination point, 474, 483 distortion, 677
curvature, 129, 139 division of hours, 493
curvature jump, 265 division plane, 401, 405
curvature line, 262 dodecagon, 518
curvature radius, 409 dodecahedron, 103, 105, 106, 141, 357, 516,
curvature tangent, 261 531
curve normal, 126, 133 dome, 550
curve of constant width, 448, 449 dome construction, 236
curve of regression, 295 dominant parallel, 558
curve on the sphere, 285 dot product, 96
curve tangent, 133 double cone, 176
cusp, 135, 419 double cushion, 62
cyclide, 232, 287 double exposure, 390
cycloid, 304, 419, 420, 499 double helix, 307, 314
cycloid gear, 456 double point, 183
cycloid motion, 419 double reflection, 63
cycloid sundial, 499 double screw, 306
cylinder, 162 double solution, 29
cylinder axes, 616 double subdominant, 558
cylinder generator, 165 double-horn, 575
cylinder model, 532 downpour, 407
cylinder of revolution, 32, 161, 173, 234, 291, drill bit, 443
303, 575 driveshaft, 41, 463
cylinder surface, 354 driving rod, 432
Index 691

drizzle, 408 extreme position, 665

dual spaces, 35 eye height, 625
dumbbell, 179 eye lens, 401
Dupin, 261, 287 eye point, 624
Dupin cyclide, 575 eye position, 366
Dupin indicatrix, 327 eyeball, 365
duration period, 493
Dürer, 624, 630 F
facet, 410
E factor of proportion, 302
E-minor, 558 factor of similarity, 8
Earth’s axis, 484 families of straight lines, 247, 250, 251
Earth’s circumference, 264 family of circles, 275
east, 473, 482, 485, 498, 499 female legs, 673
eccentric, 465 Ferguson, 353
ecliptic, 80, 455, 473 Fermat, 19
edge centroid, 13 Fermat point, 19
edge of regression, 208, 295, 361 Feuerbach, 17
egg cocoon, 401 Fibonacci, 542
eight-shaped loop, 497 Fibonacci number, 524
Einstein, 52 fine arts, 1
elastic band, 117 fireworks, 25
elbow, 84 fish scales, 216
electric motor, 455 fisheye image, 399
elements, 192 fisheye lens, 397, 398, 639
Elements of Euclid, 28 fitness machine, 436
elephant, 552, 659 fixed circle, 450
elevation angle, 476 fixing mechanism, 435
elixir of life, 401 flashlight, 139, 662
ellipse, 45, 48, 122, 192, 195, 352, 365, 390, flat point, 135, 140
400, 421, 428, 608–610, 612, 613, flat washers, 84
615–617 flight path problem, 222
ellipse compass, 440, 446, 447 flight simulator, 469
ellipse motion, 440, 446, 448, 450 Florence, 677
ellipsoid, 198, 254 Florence Baptistery, 364
ellipsoid of revolution, 237, 262, 284, 406 Florentine pattern, 517
elliptic cylinder, 173, 234 flowchart, 513, 514
elliptic paraboloid, 242 flower, 117
elliptic plane, 50 flower chafer, 509
elliptic point, 140 flying fox, 411
elytra, 509 focal curve, 498
engineer’s view, 611 focal length, 399, 629, 639, 640
enharmonically, 556 focal line, 214, 231, 421
Enneper’s surface, 332 focal mirrors, 187
envelope, 23, 129, 272, 418, 421 focal plane, 213
envelope of straight lines, 419 focal point, 29, 186
enveloping cone, 145 focal point property, 239
Epibulus insidiator, 508 focal properties, 186
epitrochoid, 450, 451 focal ratio, 637
epitrochoid motion, 444 focal rays, 187
equation of time, 459, 483, 484, 493–499, 501, focal surface, 213, 498
502 focusing effect, 214
equator circle, 269
equilateral, 101 fool’s gold, 104
equilateral hyperbola, 130 foot of the rainbow, 408
equilibrium, 18, 141 foot of the Sun, 371
equinoctial line, 453 football, 532
equinox, 80, 477, 488, 494–496 Foster, 325
Eratosthenes, 264 four-bar linkage, 432, 435
Escher, 20, 274 four-eyed fish, 401
Escher, Maurits Cornelis, 232 Fourier, 458
Euclid, 10, 28, 29, 49 Fourier series expansion, 458
Euclid’s theorem, 36 free-form surface, 337, 342, 360
Euler, 89, 101, 333 Frenet frame, 134
Euler characteristic, 101 frieze ornament, 512
Euler line, 14, 17 frog’s-eye view, 670
Euler spiral, 311 front lighting, 632
evolute, 126, 127, 147, 421, 579 front view, 63, 67, 69, 83, 84, 113, 167, 373,
evolute developable, 146 378, 484, 608
evolution, 542 frontal line, 625
exact straight translation, 41 frontal perspective, 367, 625
excavator shovels, 306 frontal plane, 625
excenter, 15 frontal projection, 609
excircle, 15 Fruchterman/Reingold, 141
exit angle, 407 fruit bowl, 238
experience, 366 frustum, 54, 178, 219
exploded drawing, 502 Fuhs, 444
explosion sketch, 183 full moon, 156, 488, 489, 677
exponential function, 191 full section, 85
exposure time, 200 full-format sensor, 639
external dimension, 85 full-frame cameras, 639
692 Index

Fuller, 124, 222 helical minimal surface, 333

function graph, 339, 340, 458 helical motion, 135, 315, 459
fundamental cell, 512 helical pipe surface, 313
fundamental theorem of algebra, 169 helical ruled surface, 558
helical stairs, 618, 619
G helical surface, 299
Gaspard Monge, 67 helicoid, 323, 326, 327, 333
Gaudí, 191 helicopter, 101
Gaudí, 248, 296, 305 heliochronometer, 500
Gauss, 49, 382 helispiral, 316
Gaussian collineation, 382 helispiral motion, 321, 347
Gaussian curvature, 333 helix, 37, 114, 135, 144, 174, 299, 302, 303
Gaussian image, 285 herbivore, 552
gear flanks, 310 hereditary material, 307
gear rack, 435 hexahedron, 103
general theory of relativity, 52 high-speed photograph, 342
generating polygon, 93 highlights, 139
generator, 175, 295 highway entrance, 311
geodesic, 253 hinge, 297
geodesic curve, 142–144, 180 hinge joint, 423
geodesic line, 135, 174, 206 hinge parallelogram, 426
geographical latitude, 477, 484 hinge quadrangle, 432
geography, 1 hinge trapezoid, 435
geoid, 236 Hippias of Elis, 682
Geomag, 101, 106 hoax, 577
geometry in a star, 46, 50 Hoecken, 446
Gherkin tower, 325 Hofmann, 493, 500
Gibbs, 493, 500 Holbein the Younger, 376, 377
giraffe, 505, 553 hollow bodies, 98
glass façade, 209 hollow wheel, 441
glasses, 278 horizon, 368, 625
Global Positioning System (GPS), 159 horizontal projection, 59
global shutter, 684 horizontal view, 375
globe, 28, 159, 160, 175, 217, 218, 220–223, horse latitudes, 500
236, 264, 612, 613 hot-air balloon, 383
Gloriette, 677 hour index, 496, 498–501
glowing chunks, 546 hour lines, 494
golden mean, 542, 544 HP shell, 86
golden ratio, 523 HP surface, 248
GPS-satellites, 160 Huygens, 421
Granada, 514 hyperbola, 29, 86, 122, 123, 170, 175, 186,
gravitational lensing, 28 192, 195, 197, 199, 245, 252, 352,
gravity, 472, 546 478
hyperbolic paraboloid, 86, 242, 245, 263, 297,
great circle, 141, 150, 223, 253, 453 298
great circle on the celestial sphere, 493 hyperbolic plane, 50
great dodecahedron, 107 hyperbolic point, 140
great icosahedron, 107 hyperbolic spiral, 137
great stellated dodecahedron, 107, 117, 528 hyperbolically curved, 140, 270, 295
grid of keys, 557 hyperboloid, 234, 245, 247, 248, 251, 262, 281
grid point, 355 hyperboloid of revolution, 245, 247, 250, 270,
grinding, 252 406
groin vault, 166 hypersphere, 225
ground control station, 152 hypoid gear, 252, 468
ground plane, 70 hypotrochoid, 450
growth, 537 hypotrochoid motion, 443
Gullstrand, 409
gyroscopic motion, 474 I
icosahedron, 105, 106, 140, 141, 610, 677
H iff (if and only if), 77
half pipe, 171 image construction, 409
half-ellipse, 156, 157 image diagonal, 639
half-ellipsoid, 100 image editing software, 671
half-section, 84, 85 image plane, 54
half-torus, 273 image point, 54, 399
hall of columns, 381 image raising, 404, 405
handle point, 135 image situation, 666
hard coral, 664 impeller, 306
hardware, 410 implicit equation, 238, 300
harmonic oscillation, 89, 300 incidence relation, 45
harmonic pendulum, 421 incircle, 14
harmony, 556 indicatrices, 261
Hart, 41 indirectly congruent, 90
Hauck, 396 individual, 541
Hauck’s perspective, 396 industrial Design, 238
Hawking, 50 infinite, 25, 34, 139, 150
hazel leaf roller, 145 infinitely, 23, 39, 57, 261
head motion, 665 infinity, 5, 34
headlights, 242 inflation, 522
Heidelberg, 123 inflationary kite, 524
helical developable, 310 inflationary rhombuses, 524
Index 693

inflection point, 97, 135, 351 lens formula, 643

initial capital, 318 lens system, 278
initial radius, 537 Leonardo da Vinci, 2, 67, 426, 431, 438, 444,
inner circle rolling, 441, 443 470, 471, 668
insect eye, 410 letter scale, 427
insphere, 108 Leys, 530
instantaneous axis, 461, 462, 468 light center, 96
instantaneous pole, 417, 431 light edge, 354
instantaneous velocity, 418 light flare, 162
intercept theorem, 59, 120 light games, 215
interference, 215, 421 light path, 640
interference color, 391 light plane, 370, 372, 622
interpolating curves, 347 light rays, 52, 370
intersect, 38 light sensitivity, 273
intersection, 282 light source, 390
intersection curve, 183 light year, 23
intersection method, 367, 369, 625 lighting, 95
intraocular fluid, 409 lily, 271, 620
inverse kinematic, 504 limaçon, 23
inverse kinematics, 506 limaçon, 204
inverse matrix, 121 limaçon of Pascal, 130, 452
inversion, 40, 42, 578 limaçon of Pascal, 451
in a circle, 574 limestone shell, 284
in a sphere, 574 limiting cone, 404
of a cylinder, 575 limiting position, 126, 134, 417
of a torus, 575 line at infinity, 26, 27, 35, 45, 48, 56
inversion sphere, 41 line element, 194, 629
inverting linkage, 41 line of action, 456
involute, 131 line segment, 25
involute gear, 252, 310 line space, 35, 37, 39
involute gearing, 456 linear combinations, 184
involute of a circle, 130 linear mapping, 55
iris, 366 linkage motion, 432
isometry, 609 lion, 504
isosceles, 101 Lipkin, 41
load capacity, 191
J Lobatschewskij, 49
jet streams, 159 locus, 23
jump, 259 locus curve, 31
Jupiter, 454, 490 logarithmic spiral, 127, 130, 577
long distance beams, 242
K longhorn beetle, 678
longitude, 222
Kapoor, 227 longitudinal section, 84
Kepler, 107, 200, 270, 476, 484, 495 loops of dimensions, 85
Kepler equation, 495 loss of information, 42
Kiepert, 17, 19, 20, 119 low-pressure turbulence, 545
Kiepert’s theorem, 19 loxodrome, 276, 578
kinematics, 10, 92, 194, 415, 459, 462, 466
kissing number problem, 153 M
knot vector, 353
Mad House, 464
L magic cube, 88
magnetic repulsion, 516
label, 173 main normal plane, 406
lacrimal fluid, 280 main theorem of spatial kinematics, 468
ladybird, 539 Maltesian gear, 437
ladybug, 198 marionette, 424
lampshade, 199 marquee, 558
large scale, 265 Mars, 490
large-scale weather pattern, 545 masonry drill, 311
laser scanner, 347 mathematician, 264, 303
lateral acceleration, 465 mathematics, v, 6, 28, 39, 102, 270, 339, 533
lathe tool, 422, 445 matrix notation, 121
latitude, 222 mayfly, 651
law of inertia, 546 mean curvature, 327, 328, 333
law of planetary motion, 474 mean location time, 493
law of reflection, 62, 564 mean noon, 493
law of refraction, 404 measurement of angles, 50
Law of Right Angles, 45, 75, 76, 79–81 measurement point, 628
Le Corbusier, 182 measuring inaccuracy, 285
LED lamp, 304 mechanical engineering, 1, 2, 282, 292, 310
left and right view, 83 medial triangle, 14
left helical motion, 302 median, 13
left side view, 69 Mediterranean, 9
left-handed, 316 Melolontha melolontha, 509
Leitner, 240 mental arithmetic, 22
lemniscate, 130 Mercury, 326, 454, 490
length of day, 480 meridian circle, 274, 578
lens, 279 meridian circular helical surface, 312
lens eye, 391 Meusnier, 260, 261
694 Index

microcosm, 87, 533 offset curve, 131, 419

mid helix, 313 oil pump, 433
midsurface, 322 Oldham, 445
mill, 469 Oldham coupling, 445
mill flanks, 311 olive presses, 307
milling machine for ellipses, 443 Oliver, 493
milling process, 343 oloid, 208, 210, 267, 285
minimal spiral surface, 333 Olympic Stadium, 328
minimal surface, 293, 326, 555 ommatidia, 411, 651
minor key, 556 one-sheeted hyperboloid, 245, 247, 468
mirages, 409 one-sheeted hyperboloid of revolution, 246,
mirror plane, 90 249, 264, 615
mirror window, 633 ophthalmologist, 278
Möbius, 332 optical illusion, 381, 653
mole grips, 434 optician, 278
molecular biology, 1, 307 orb-weaver spider, 535
molecule, 307 orca, 73
Monge, 67, 282 ordering line, 67
Moon, 546 orders of magnitude, 552
moon crescent, 155 orientation, 7
moonset, 370 origami, 115
Morley, 17, 20–22 original photo, 375
Morley’s theorem, 20 ornament, 512
mosquito, 534 orthocenter, 13, 376
most irrational of all numbers, 544 orthogonal distance, 32
moth, 534 orthogonal trajectory, 535
multicuspidal curves, 452 orthonomic curves, 130
multiple section, 84 oryx, 505
murals, 381 oscillate, 347
muscle contraction, 401 oscillation curves, 300
muscle diameter, 552 osculating circle, 126, 127, 133, 146, 242, 259–
music, 1 262, 278, 578, 579
mutation, 541 osculating parabola, 260
mystique, 669 osculating plane, 134, 135
Möbius strip, 208 osculating sphere, 240, 241
ostrich, 506
N Otto, 430
Otto engine, 430
nadir, 482 outer circle rolling, 444
Napoleon, 42, 43 outer space, 293, 472, 476, 480, 546, 548
nautilus, 348 outline, 138
Navigation system, 158 outline generatrices, 196
Nazca culture, 383 outline of a sphere, 154
needle cyclide, 575 oval mill, 444
neighboring apex, 166 own weight, 552
neighboring generators, 248
neighboring point, 127
nephroid, 294, 452 P
Nepomorpha, 401 Packaging material, 117
Neutrino, 52 pair of angle bisectors, 15
new moon, 157, 489, 490 Palmenhaus, 166
New Year’s Eve, 25 panoramic view, 384
Newton, 153 paper snippets, 389
Newton ring, 421 paper strip construction, 440
Nile delta, 9 paper strip method, 194
nine-point-circle, 18 Pappus, 193
non-Euclidean geometry, 49 parabola, 86, 122, 123, 175, 199, 200, 241,
normal congruence, 406 298, 351, 352
normal developable, 134, 206 parabolic headlamp, 233
normal direction, 285 parabolic headlights, 242
normal projection, 162, 236 parabolic point, 140
normal section, 139, 206, 261, 262 parabolic trajectories, 199
normal view, 97, 608, 658 paraboloid, 234, 355
north direction, 277, 481, 484 paraboloid of revolution, 241, 242
North Pole, 156, 478, 578 parachute, 73
North Star, 389, 476, 490, 494 parallel circle, 268
north winter, 478, 489 parallel curve, 419
northern direction, 476 parallel curves, 131
Northern Tropic, 481, 482 parallel key, 557
nozzle, 84 parallel lighting, 370
null circle, 150 parallel pair, 32
numerologists, 544 parallel perspectivity, 118
nut, 84, 307 parallel postulate, 51
nutation, 495 parallel projection, 58, 163
parallel section, 9
O parallel shadow, 61, 120, 163
parameter interval, 299
oblique circular cylinder, 171, 234 parameter representation, 247, 255, 299, 300,
oblique projection, 59, 608 440
oblique view, 59, 608 parasitic points, 283
octahedron, 105, 106, 141 partial section, 85
offset, 302 Pascal, 192, 193, 196, 451
Index 695

Pascal line, 193, 196 point space, 35, 39

path circle, 149 point-shaped light source, 199
path ellipse, 495 polar, 34, 37
path normal, 418 polar developable, 146
path tangent, 418 polar night, 479
paving slab lifter, 127 polar plane, 35, 37, 41
peacock, 268 polar winter, 156
peacock’s feather, 8, 90 polarity, 34, 39
Peaucellier, 41 polarization, 37
pedal point, 418, 460 polarization at a circle, 130
pedestrian bridge, 300 polarization of a space curve, 37
pelvis height, 61 pole, 34
pencil, 617 polycarbonate, 215
pencil of lines, 46 polynomial function, 347, 355
pencil of rays, 535 Pompeii, 381
pencil sharpener, 469 poster-size, 667
pencils of parallels, 298
pendulum clock, 421 potato chips, 174
Penrose, 523 power, 16
pentagondodecahedron, 516, 528 Pozzo, 385, 386
pentagram, 107 precession motion, 474, 495
penumbra, 622 predatory fish, 548
penumbra shadow, 99, 103 preservation of affine ratios, 59
penumbral boundaries, 623 preservation of circles, 225
pepper grinder, 424 preservation of parallelity, 59
perception, 363 primary cell, 512
period duration, 488 primary normal image, 285
periodic pattern, 512 primary perspective, 666
peripheral circle, 447 principal apex circle, 121
perpendicular bisector, 11, 16, 127, 417 principal curvature, 279, 327, 328
perspective, 56, 236, 608, 658 principal line, 75
perspective affinity, 118 principal normal, 134
perspective collineation, 118 principal point, 368, 624, 626, 658
perspectivity, 46 principal ray, 365, 625, 658
perspectivity ray, 118 principal vanishing point, 368, 635, 670
phase-shifted, 303 principal view, 63, 67, 68, 70, 72, 107
photo finish, 684 principle of duality, 37
photograph, 1, 55, 66, 160, 376, 394, 399, priority list, 98
400, 666 prism, 93
photography, 54, 58, 365, 383, 398, 637, 638 problem of flight paths, 222
photography magazine, 667 profile view, 67
phyllotaxis, 542 projecting, 37
physical reflection, 42 projection center, 57, 625
physicist, 236, 547 projection cone, 122, 199, 365, 386
physics, 1, 2, 13, 28, 49, 52, 293, 412, 471, projection curve, 303
525
Piazza San Marco, 638 projection plane, 54
Pilkington, 493, 500 projection ray, 54, 367
pine tree trunk, 32 projection rays, 654
pipe connection, 168 projective geometry, 5, 45
pipe elbow, 164, 165 projective plane, 48
pipe surface, 288, 318 projector, 390
piston chamber, 452 Proklus, 121
piston engine, 431 prolate ellipsoid, 236
pitch, 618 propeller, 305
Piva, 245 proportional angle of rotation, 313
pixels, 391 proportionality, 302
plan view, 63, 67, 69, 83, 113, 167 prosthetic arm, 235
planar perspective affinity, 119 pumpkin, 268, 343
planar perspective collineation, 119 puppet, 465
plane at infinity, 27, 57 puzzle, 531, 532
plane of reference, 68 pyramid, 93
plane space, 5, 35, 39 pyrite, 103
plane stance, 27 Pythagoras, 16, 152, 533
planes of symmetry, 236
planetary gears, 455
planetary motion, 453 Q
planimetrically, 46 quadrangle grid, 354
plankton, 661 quadrangulation, 359
plant species, 543 quadratic cone, 47, 122, 234
platform, 469 quadratic surface, 234
Plato, 533 quadratic transformation, 40
Platonic solids, 87
Pliers, 435 quadratrix, 682
Plücker, 300 quadratrix of Dinostratus, 682
Plücker’s conoid, 299 quarter-torus, 280, 291
Poincaré, 530 quasi-periodical, 525
Poinsot, 107 quasi-periodical tilings, 522
point at infinity, 23, 93, 221, 415 quasi-periodicity, 524
point of contact, 126 quasicrystal, 525
point of culmination, 2
696 Index

R rotational surface, 42
rack and pinion gear, 457 rotoid, 556
radar corner reflector, 62 rotor, 101
radar screen, 241 row of points, 250
radial symmetry, 534 rubber membrane, 326
Radiata, 534 Rubik, 88
radio telescopes, 241 rule, 54
radius of curvature, 126, 129 Rule of Visibility, 96
rainbow, 398 ruled helical surface, 308–310
rainbow colors, 215 ruled surface, 294, 305, 355
rainbow effect, 404 ruler, 375
range of points, 46
rapid prototyping, 355 S
rational Bézier curve, 352 saddle point, 263
rattle snake, 515 saddle surface, 140
real-time, 99 Sagrada Família, 248, 670
receiver, 241 sand dune, 203
rectangular strip, 206 sand tiger shark, 665
rectification, 677 satellite dishes, 241
rectifying developable, 134, 206, 207, 360 sawmill, 431
rectifying plane, 134 Sawyer, 499
reef shark, 666 scaffolding, 629
reflection, 12, 90, 512 scale, 59, 471, 550
reflection in a circle, 40 scaling, 552
reflection in a perpendicular wall, 634 scaling factor, 17, 552
reflection in a sphere, 42 scanner, 285, 681
reflection point, 635 scarab beetle, 509
reflections in perspective, 633 Scarabaeidae, 509
reflex camera, 638, 639 scattering, 407
refraction point, 401 Scherk, 327
refraction towards the normal, 401 Scherk surface, 326
refractive index, 403 Schönbrunn, 166, 450
regular dodecahedron, 527 Schwarz minimal surface, 327
Reichstag dome, 237 scientific capital, 9
relative motion, 422, 444, 462 scorpionfish, 663
relative position, 68 screw axis, 305, 317
relief perspective, 381 screw parameter, 302
Renaissance, 53, 363, 380, 381 screws, 84
Renault, 351 scutoid, 521, 529
reptiles, 401 sea urchin, 284
repulsion, 141, 548 seahorse, 553
repulsive forces, 141 seaweed, 201
retina, 409 secant, 16, 126, 133
Reuleaux triangle, 442, 443 secondary perspective, 666
reverse ellipse motion, 444, 446 sections, 84
reverse engineering, 283, 285 seed, 543, 544
reverse paper strip construction, 440, 441 self-intersection, 182, 308, 361
revolution of a circle, 450 self-rotation, 469
rhombic dodecahedron, 112–114, 528 self-shadow boundary, 147, 162, 496
rhombic icosahedron, 114 self-similarity, 522
rhombic triacontahedron, 113 semi-regular dodecahedron, 528
rhombus, 112, 518 semi-regular pentagon, 526
rhumb line, 276, 288 semi-regular tiling, 520
rib vault, 550 semimajor axis, 77
ribs, 84 series of points, 563
Ricardo Bofill Levi, 248 set of points, 37
Riegler, 493, 498 set of tangents, 37
Riemann, 49 set square, 22, 417
right helical motion, 302 shadow, 372, 623
right helicoid, 298–300, 305, 307, 317 shadow boundaries, 99, 162
right side view, 67, 69 shadow point, 478
right-handed, 308, 315 shadow polygon, 99
ring cyclide, 287, 574, 575 shadow profile, 100
ring sector, 219 shark skin, 515, 555
ring torus, 574, 575 Shechtman, 525
road construction, 203, 311 Sherlock Holmes, 60
robber fly, 140 shift lenses, 671
robot arm, 470 ship path, 222
robotics, 438, 469 shock wave, 549
rod-shaped light source, 242 side lighting, 632, 633, 635
Rodler, 630, 631 side view, 70, 372, 378
rolling circle, 450 sidelight, 371, 372
rolling shutter effect, 679 sign of the distance, 90
roof edge, 370 sine, 300
rose patch, 450 sine curve, 136, 167, 303, 324
rotary table, 445 sine law, 31
rotation, 88, 91, 512 singular reaction, 548
rotation angle, 543 skew, 72
rotation component, 303 skew axes, 468
rotation-translation-step, 544 skew axis, 246
Index 697

skew generator quadrangle, 297 steel cable, 313

skew quadrangle, 244 steering wheel, 129, 435
skull, 376 Steiner, 19
slender moon crescent, 157 stellated dodecahedron, 107
sling-jaw wrasse, 508 step pyramid, 94
slope line, 321 stepper, 437
slot guides, 447 stereographic projection, 219, 395, 492, 577,
small circle, 150 578
small stellated dodecahedron, 107 stereoscopic image, 655
snail, 676 stereoscopic vision, 654
snail shell, 69 Stewart platform, 438, 469
Snell’s window, 564 stone pebbles, 7
Snellius, 403 stonefish, 663
snow shovel, 306 straight line guidance, 433, 436
soap bubble, 326 straight open ruled helical surfaces, 308
soccer ball, 109 streamlining, 515
street, 59
solar day, 473 string construction, 29
solar system, 472 stroboscopic photos, 417
solstice, 160, 495, 498–500 sub-net, 355
sonic strength, 537 substitute construction, 36
sound box, 162 sugary syrup, 539
soundscapes, 240, 290 summer solstice, 481
south, 478, 482, 483, 493, 496 summer time, 493
south winter, 478 sun collector, 242
Southern Cross, 490 sun foot, 372
southern hemisphere, 481, 490 sun gear, 455
space, 52 sun point, 171, 372, 635
space curve, 38, 341 Sun’s culmination point, 484
space curve of degree four, 169 sundial, 494
space of perception, 28, 29, 39 sunflower, 543, 544
spatial affine transformation, 184 sunlight, 483
spatial cross, 166 sunrise, 370
spatial diagonal, 341 super-wide-angle, 639
spatial imagination, 166, 622 supernova, 547
spatial interpretation, 195 supersphere, 239
spatial Pythagoras, 152 supplementary angle, 19, 30, 33
species, 541 surface, 537
spectral color, 407, 408 surface degree, 354
specular point, 229 surface metrics, 332
speed of growth, 302 surface normal, 137, 285
sphere, 149, 234, 537 surface of revolution, 32, 289, 341
sphere cap, 278 surface of the Earth, 140
sphere contour, 37, 365 surface tangent, 260, 261
sphere layer, 478 surface tension, 326, 538
sphere model, 50 swarm of fish, 548
sphere packing, 110 swimming pool, 173, 312, 405
sphere reflection, 42 swordfish, 531
sphere rhumb line, 277 symmetry, 195, 534, 620
spherical conic section, 32, 466 symmetry axes, 610, 630
spherical cycloid, 461, 465 symmetry plane, 135, 242, 285, 576, 643
spherical generator image, 285 synthetic proofs, 14
spherical kinematics, 180 Syracuse, 7
spherical section, 150 system of axioms, 49
spherical surface, 537 system of linear equations, 194
spherical tangent image, 285
spherical triangle, 51 T
Sphinx pattern, 522 tacnode, 174
spider’s eyes, 390 tactile sense, 391
spikes, 346 tangent, 38, 126, 133
spindle torus, 151, 152, 198, 278, 575 tangent distance, 16, 165
spine circle, 277 tangent of a curve, 126
spine curve, 289 tangent plane, 137, 150, 258, 285, 575
spiral developable, 318 tangent surface, 134, 295, 309
spiral motion, 348 tangential surface, 175, 206
spiral right helicoid, 318 tangerine, 217
spiral surface, 286, 341 tapetum lucidum, 391
Spirobranchus giganteus, 567 tarantula, 391
splitting of hairs, 198 tautochrone curve, 421
spotlight, 57 technical drawing, 84, 162
spout, 174 telescopic lens, 78, 658
spy hopping, 565 television, 666
square side, 517 Temelin, 250
square wheel, 192 temporal hour, 493
square-grid fields, 390 tennis ball curve, 285
St. Charles’ Church, 122 terminator, 81, 147, 157, 158
standard surfaces, 342 tessellation, 20, 519
standpoint, 625 Tetra Pak carton packages, 117
star, 46 tetradecahedra, 110
star handle, 462 tetrahedron, 105, 141
star polyhedra, 107 texture, 558
698 Index

Thales, 9, 10, 31, 223, 447, 614 Vienna, 122, 159, 220, 237, 341, 473
Thales circle, 18, 31, 375 viewing, 95
Thales sphere, 152, 376 viewing angle, 658
Thales’s theorem, 16 viewing axis, 625
theorem of Brianchon, 193 viewing cone, 368, 409
theorem of intersecting lines, 8, 30, 656 viewing pyramid, 368, 377
theorem of Kawasaki, 116 viewing rays, 653
theorem of right angles, 75, 77 Villarceau, 275, 276, 287
thermal radiation, 239 Villarceau circle, 277
Thompson problem, 141 virtual counterworld, 633
thorn torus, 575 virus, 346
tiger, 552 vise, 307
tiger beetle, 508 Vise-Grips, 434
tiling, 110, 522 vises, 307
Tokyo, 221 visual appearance, 239
tomato, 343 vitreous body, 366
tonal center, 558 Vogt, 64
tonal system, 556 volume centroid, 13
tonic, 558 von Kempelen, 424
top view, 67, 373, 378, 484 von Racknitz, 424, 425
topographic projection, 81 Voronoi diagram, 205
torsion, 135
torus, 167, 212, 272, 281, 470, 556 W
torus cap, 280 waning moon, 157
torus chain, 273 Wankel engine, 452, 453
total internal reflection, 403, 407 Wapiti, 349
totally differentiable, 338 wasp, 536, 552
tower clock, 494 water basin, 122
trace point, 59, 369 water droplets, 538
train, 179 water slide, 312
transcendental, 299 water spider, 401
translation, 88, 302, 512 water strider, 401
translation component, 303 water surface, 230, 397, 401, 404, 406, 407
translation surface, 242, 297 waterproof, 117
tree rings, 537 wave peak, 7
triacontahedron, 113 wave structure of light, 403
triads, 558 wavefront, 239, 241, 404
triangle, 7 waxing moon, 157
triangulation, 360 Weierstrass-℘-Function, 339
triaxial ellipsoid, 236, 257 west, 160, 473, 482, 483, 499
trihedron, 91 wheel spoke, 62
tripod, 90 whirligig beetle, 401
trisectrix, 682 whispering bowls, 240
trisectrix of Hippias, 682 wide-angle lens, 394, 658, 659
trochoid motion, 450 Wieland, 498
tropic, 478 willow branch, 540
true local time, 493 winding point, 128
trunk, 519 windmill, 208
Tunis, 221 wire, 327
turban eyes, 651 wireframe model, 356
turbulence, 545 wobbler, 267
Tuscany, 426 wolf spider, 391, 401
two-bar linkage, 423 worm gears, 464
two-point guidance, 416 Wren, 246
two-sheeted hyperboloid of revolution, 57, Wunderlich, 217, 334
246
U X
xyz-coordinates, 67
ultra-wide-angle, 639, 665
umbilical points, 257 Y
umbra shadow, 99, 103
umbrella, 327 year, 473
unit cube, 613 Yin and Yang, 11
unit points, 59
unit sphere, 612 Z
upside down catenaries, 191 zenith, 157, 478, 479, 482, 493
zero circle, 189
V zero meridian, 613
zero-dimensional, 6, 34
Vaduz, 210
vanishing plane, 55, 378 zone meridian, 498
vanishing point, 55 zone time, 498
Varadero, 159, 160 zoom-lens, 638
variety of shapes, 549
vector calculus, 152
Venus, 490
Vernes, 159
Verrazzano-Narrows Bridge, 191
vertex centroid, 13
vertex of the pencil, 39
vertical motion, 192