Sphere
A sphere (from Greek σφαῖρα—sphaira, "globe, ball"[1]) is a
geometrical object in three-dimensional space that is the surface of a
ball (viz., analogous to the circular objects in two dimensions, where
a "circle" circumscribes its "disk").
Like a circle in a two-dimensional space, a sphere is defined
mathematically as the set of points that are all at the same distance r
from a given point, but in a three-dimensional space.[2] This distance
r is the radius of the ball, which is made up from all points with a
distance less than (or, for a closed ball, less than or equal to) r from
the given point, which is the center of the mathematical ball. These
are also referred to as the radius and center of the sphere,
respectively. The longest straight line segment through the ball, A two-dimensional perspective
connecting two points of the sphere, passes through the center and projection of a sphere
its length is thus twice the radius; it is a diameter of both the sphere
and its ball.
While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in
mathematics the above distinction is made between a sphere, which is a two-dimensional closed surface
embedded in a three-dimensional Euclidean space, and a ball, which is a three-dimensional shape that
includes the sphere and everything inside the sphere (a closed ball), or, more often, just the points inside, but
not on the sphere (an open ball). The distinction between ball and sphere has not always been maintained
and especially older mathematical references talk about a sphere as a solid. This is analogous to the situation
in the plane, where the terms "circle" and "disk" can also be confounded.
Contents
Equations in three-dimensional space
Enclosed volume
Surface area
Curves on a sphere
Circles
Clelia curves
Loxodrome
Intersection of a sphere with a more general surface
Geometric properties
Pencil of spheres
Terminology
Plane sections
Branches of geometry
Non-Euclidean distance
Differential geometry
Projective geometry
Geography
Poles, longitude and latitudes
Generalizations
Dimensionality
Metric spaces
Topology
Spherical geometry
Eleven properties of the sphere
Gallery
Regions
See also
Notes and references
Notes
References
Further reading
External links
Equations in three-dimensional space
In analytic geometry, a sphere with center (x0, y0, z0) and radius r
is the locus of all points (x, y, z) such that
Let a, b, c, d, e be real numbers with a ≠ 0 and put
Two orthogonal radii of a sphere
Then the equation
has no real points as solutions if and is called the equation of an imaginary sphere. If , the
only solution of is the point and the equation is said to be the equation of
a point sphere. Finally, in the case , is an equation of a sphere whose center is and
whose radius is .[2]
If a in the above equation is zero then f(x, y, z) = 0 is the equation of a plane. Thus, a plane may be
thought of as a sphere of infinite radius whose center is a point at infinity.[3]
The points on the sphere with radius and center can be parameterized via
[4]
The parameter can be associated with the angle counted positive from the direction of the positive z-axis
through the center to the radius-vector, and the parameter can be associated with the angle counted
positive from the direction of the positive x-axis through the center to the projection of the radius-vector on
the xy-plane.
A sphere of any radius centered at zero is an integral surface of the following differential form:
This equation reflects that position and velocity vectors of a point, (x, y, z) and (dx, dy, dz), traveling on
the sphere are always orthogonal to each other.
A sphere can also be constructed as the surface formed by rotating a circle about any of its diameters. Since
a circle is a special type of ellipse, a sphere is a special type of ellipsoid of revolution. Replacing the circle
with an ellipse rotated about its major axis, the shape becomes a prolate spheroid; rotated about the minor
axis, an oblate spheroid.[5]
Enclosed volume
In three dimensions, the volume inside a sphere (that is, the volume
of a ball, but classically referred to as the volume of a sphere) is
where r is the radius and d is the diameter of the sphere. Archimedes
first derived this formula by showing that the volume inside a sphere
is twice the volume between the sphere and the circumscribed
cylinder of that sphere (having the height and diameter equal to the
diameter of the sphere).[6] This may be proved by inscribing a cone
upside down into semi-sphere, noting that the area of a cross section
of the cone plus the area of a cross section of the sphere is the same
as the area of the cross section of the circumscribing cylinder, and Sphere and circumscribed cylinder
applying Cavalieri's principle.[7] This formula can also be derived
using integral calculus, i.e. disk integration to sum the volumes of an
infinite number of circular disks of infinitesimally small thickness stacked side by side and centered along
the x-axis from x = −r to x = r, assuming the sphere of radius r is centered at the origin.
At any given x, the incremental volume (δV) equals the product of the cross-sectional area of the disk at x
and its thickness (δx):
The total volume is the summation of all incremental volumes:
In the limit as δx approaches zero,[8] this equation becomes:
At any given x, a right-angled triangle connects x, y and r to the origin; hence, applying the Pythagorean
theorem yields:
Using this substitution gives
which can be evaluated to give the result
An alternative formula is found using spherical coordinates, with volume element
so
For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4%
π
of the volume of the cube, since V = d3, where d is the diameter of the sphere and also the length of a
6
π
side of the cube and 6 ≈ 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube
with edge length 1 m, or about 0.524 m3.
Surface area
The surface area of a sphere of radius r is:
Archimedes first derived this formula[9] from the fact that the projection to the lateral surface of a
circumscribed cylinder is area-preserving.[10] Another approach to obtaining the formula comes from the
fact that it equals the derivative of the formula for the volume with respect to r because the total volume
inside a sphere of radius r can be thought of as the summation of the surface area of an infinite number of
spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r.
At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is
infinitesimal, and the elemental volume at radius r is simply the product of the surface area at radius r and
the infinitesimal thickness.
At any given radius r,[note 1] the incremental volume (δV) equals the product of the surface area at radius r
(A(r)) and the thickness of a shell (δr):
The total volume is the summation of all shell volumes:
In the limit as δr approaches zero[8] this equation becomes:
Substitute V:
Differentiating both sides of this equation with respect to r yields A as a function of r:
This is generally abbreviated as:
where r is now considered to be the fixed radius of the sphere.
Alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In
Cartesian coordinates, the area element is
The total area can thus be obtained by integration:
The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the
largest volume among all closed surfaces with a given surface area.[11] The sphere therefore appears in
nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally
minimizes surface area.
The surface area relative to the mass of a ball is called the specific surface area and can be expressed from
the above stated equations as
where ρ is the density (the ratio of mass to volume).
Curves on a sphere
Circles
The intersection of a sphere and a plane is a circle, a
point or empty.
In case of a circle the circle can described by a parametric
equation : see plane section
of an ellipsoid. Plane section of a sphere: 1 circle
But more complicated surfaces may intersect a sphere in
circles, too:
A non empty intersection of a sphere with a surface of
revolution, whose axis contains the center of the sphere (are
coaxial) consists of circles and/or points.
The diagram shows the case, where the intersection of a cylinder and a
Coaxial intersection of a
sphere consists of two circles. Would the cylinder radius be equal to the
sphere and a cylinder: 2
sphere's radius, the intersection would be one circle, where both surfaces
circles
are tangent.
In case of an spheroid with the same center and major axis as the sphere,
the intersection would consist of two points (vertices), where the surfaces are tangent.
Clelia curves
If the sphere is described by a parametric representation
one gets Clelia curves, if the angles are connected by the equation
spherical spiral with
Special cases are: Viviani's curve ( ) and spherical spirals (
).
Loxodrome
In navigation, a rhumb line or loxodrome is an arc crossing all meridians of longitude at the same angle. A
rhumb line is not a spherical spiral. There is no simple connection between the angles and .
Intersection of a sphere with a more general surface
If a sphere is intersected by another surface, there may be more complicated
spherical curves.
Example: sphere – cylinder
The intersection of the sphere with equation and the
cylinder with equation is not just one or two
Loxodrome
circles. It is the solution of the non linear system of equations
(see implicit curve and the diagram)
Geometric properties
A sphere is uniquely determined by four points that are not coplanar. More
General intersection sphere-
generally, a sphere is uniquely determined by four conditions such as
cylinder
passing through a point, being tangent to a plane, etc.[12] This property is
analogous to the property that three non-collinear points determine a
unique circle in a plane.
Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the
plane of that circle.
By examining the common solutions of the equations of two spheres, it can be seen that two spheres
intersect in a circle and the plane containing that circle is called the radical plane of the intersecting
spheres.[13] Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real
point in common) or consist of a single point (the spheres are tangent at that point).[14]
The angle between two spheres at a real point of intersection is the dihedral angle determined by the tangent
planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of
intersection.[15] They intersect at right angles (are orthogonal) if and only if the square of the distance
between their centers is equal to the sum of the squares of their radii.[3]
Pencil of spheres
If f(x, y, z) = 0 and g(x, y, z) = 0 are the equations of two distinct spheres then
is also the equation of a sphere for arbitrary values of the parameters s and t. The set of all spheres satisfying
this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere
is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all
the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil.[3]
If the pencil of spheres does not consist of all planes, then there are three types of pencils:[14]
If the spheres intersect in a real circle C, then the pencil consists of all the spheres containing
C, including the radical plane. The centers of all the ordinary spheres in the pencil lie on a line
passing through the center of C and perpendicular to the radical plane.
If the spheres intersect in an imaginary circle, all the spheres of the pencil also pass through
this imaginary circle but as ordinary spheres they are disjoint (have no real points in common).
The line of centers is perpendicular to the radical plane, which is a real plane in the pencil
containing the imaginary circle.
If the spheres intersect in a point A, all the spheres in the pencil are tangent at A and the
radical plane is the common tangent plane of all these spheres. The line of centers is
perpendicular to the radical plane at A.
All the tangent lines from a fixed point of the radical plane to the spheres of a pencil have the same
length.[14]
The radical plane is the locus of the centers of all the spheres that are orthogonal to all the spheres in a
pencil. Moreover, a sphere orthogonal to any two spheres of a pencil of spheres is orthogonal to all of them
and its center lies in the radical plane of the pencil.[14]
Terminology
Plane sections
A great circle on the sphere has the same center and radius as the sphere—consequently dividing it into two
equal parts. The plane sections of a sphere are called spheric sections—which are either great circles for
planes through the sphere's center or small circles for all others.[16]
Any plane that includes the center of a sphere divides it into two equal hemispheres. Any two intersecting
planes that include the center of a sphere subdivide the sphere into four lunes or biangles, the vertices of
which coincide with the antipodal points lying on the line of intersection of the planes.
Branches of geometry
Non-Euclidean distance
Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e. the diameter) are
called antipodal points—on the sphere, the distance between them is exactly half the length of the
circumference.[note 2] Any other (i.e. not antipodal) pair of distinct points on a sphere
lie on a unique great circle,
segment it into one minor (i.e. shorter) and one major (i.e. longer) arc, and
have the minor arc's length be the shortest distance between them on the sphere.[note 3]
Spherical geometry[note 4] shares many analogous properties to Euclidean once equipped with this "great-
circle distance".
Differential geometry
And a much more abstract generalization of geometry also uses the same distance concept in the
Riemannian circle.
The hemisphere is conjectured to be the optimal (least area) isometric filling of the Riemannian circle.
Projective geometry
The antipodal quotient of the sphere is the surface called the real projective plane, which can also be thought
of as the northern hemisphere with antipodal points of the equator identified.
Geography
Terms borrowed directly from geography of the Earth, despite its spheroidal shape having greater or lesser
departures from a perfect sphere (see geoid), are widely well-understood. In geometry unrelated to
astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless
there is no chance of misunderstanding.
Poles, longitude and latitudes
If a particular point on a sphere is (arbitrarily) designated as its north pole, its antipodal point is called the
south pole. The great circle equidistant to each is then the equator. Great circles through the poles are called
lines of longitude (or meridians). A line not on the sphere but through its center connecting the two poles
may be called the axis of rotation. Circles on the sphere that are parallel (i.e. not great circles) to the equator
are lines of latitude.
Generalizations
Dimensionality
Spheres can be generalized to spaces of any number of dimensions. For any natural number n, an "n-
sphere," often written as Sn, is the set of points in (n + 1)-dimensional Euclidean space that are at a fixed
distance r from a central point of that space, where r is, as before, a positive real number. In particular:
S0: a 0-sphere is a pair of endpoints of an interval [−r, r] of the real line
S1: a 1-sphere is a circle of radius r
S2: a 2-sphere is an ordinary sphere
S3: a 3-sphere is a sphere in 4-dimensional Euclidean space.
Spheres for n > 2 are sometimes called hyperspheres.
The n-sphere of unit radius centered at the origin is denoted Sn and is often referred to as "the" n-sphere.
Note that the ordinary sphere is a 2-sphere, because it is a 2-dimensional surface (which is embedded in 3-
dimensional space).
The surface area of the unit (n-1)-sphere is
where Γ(z) is Euler's gamma function.
Another expression for the surface area is
r
and the volume is the surface area times n or
General recursive formulas also exist for the volume of an n-ball.
Metric spaces
More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set of points y such
that d(x,y) = r.
If the center is a distinguished point that is considered to be the origin of E, as in a normed space, it is not
mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the
case of a unit sphere.
Unlike a ball, even a large sphere may be an empty set. For example, in Zn with Euclidean metric, a sphere
of radius r is nonempty only if r2 can be written as sum of n squares of integers.
Topology
In topology, an n-sphere is defined as a space homeomorphic to the boundary of an (n + 1)-ball; thus, it is
homeomorphic to the Euclidean n-sphere, but perhaps lacking its metric.
A 0-sphere is a pair of points with the discrete topology.
A 1-sphere is a circle (up to homeomorphism); thus, for example, (the image of) any knot is a
1-sphere.
A 2-sphere is an ordinary sphere (up to homeomorphism); thus, for example, any spheroid is a
2-sphere.
The n-sphere is denoted Sn. It is an example of a compact topological manifold without boundary. A sphere
need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere (an exotic sphere).
The Heine–Borel theorem implies that a Euclidean n-sphere is compact. The sphere is the inverse image of
a one-point set under the continuous function ||x||. Therefore, the sphere is closed. Sn is also bounded;
therefore it is compact.
Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible
self-intersections but without creating any crease, in a process called sphere eversion.
Spherical geometry
The basic elements of Euclidean plane geometry are points and
lines. On the sphere, points are defined in the usual sense. The
analogue of the "line" is the geodesic, which is a great circle; the
defining characteristic of a great circle is that the plane containing
all its points also passes through the center of the sphere. Measuring
by arc length shows that the shortest path between two points lying
on the sphere is the shorter segment of the great circle that includes
the points.
Many theorems from classical geometry hold true for spherical
geometry as well, but not all do because the sphere fails to satisfy
some of classical geometry's postulates, including the parallel
postulate. In spherical trigonometry, angles are defined between Great circle on a sphere
great circles. Spherical trigonometry differs from ordinary
trigonometry in many respects. For example, the sum of the interior
angles of a spherical triangle always exceeds 180 degrees. Also, any two similar spherical triangles are
congruent.
Eleven properties of the sphere
In their book Geometry and the Imagination[17] David Hilbert and Stephan Cohn-Vossen describe eleven
properties of the sphere and discuss whether these properties uniquely determine the sphere. Several
properties hold for the plane, which can be thought of as a sphere with infinite radius. These properties are:
1. The points on the sphere are all the same distance from a fixed point. Also, the ratio of the
distance of its points from two fixed points is constant.
The first part is the usual definition of the sphere and determines it uniquely. The second
part can be easily deduced and follows a similar result of Apollonius of Perga for the
circle. This second part also holds for the plane.
2. The contours and plane sections of the sphere are circles.
This property defines the sphere uniquely.
3. The sphere has constant width and constant girth.
The width of a surface is the distance between pairs of parallel tangent planes.
Numerous other closed convex surfaces have constant width, for example the Meissner
body. The girth of a surface is the circumference of the boundary of its orthogonal
projection on to a plane. Each of these properties implies the other.
4. All points of a sphere are umbilics.
At any point on a surface a normal direction is at
right angles to the surface because the sphere these
are the lines radiating out from the center of the
sphere. The intersection of a plane that contains the
normal with the surface will form a curve that is
called a normal section, and the curvature of this
curve is the normal curvature. For most points on
most surfaces, different sections will have different
curvatures; the maximum and minimum values of
these are called the principal curvatures. Any closed
surface will have at least four points called umbilical
points. At an umbilic all the sectional curvatures are
equal; in particular the principal curvatures are equal.
Umbilical points can be thought of as the points
where the surface is closely approximated by a A normal vector to a sphere, a
sphere. normal plane and its normal section.
For the sphere the curvatures of all normal sections The curvature of the curve of
are equal, so every point is an umbilic. The sphere intersection is the sectional
and plane are the only surfaces with this property. curvature. For the sphere each
normal section through a given point
5. The sphere does not have a surface of centers. will be a circle of the same radius:
the radius of the sphere. This means
For a given normal section exists a circle of that every point on the sphere will be
curvature that equals the sectional curvature, is an umbilical point.
tangent to the surface, and the center lines of which
lie along on the normal line. For example, the two
centers corresponding to the maximum and minimum sectional curvatures are called the
focal points, and the set of all such centers forms the focal surface.
For most surfaces the focal surface forms two sheets that are each a surface and meet
at umbilical points. Several cases are special:
* For channel surfaces one sheet forms a curve and the other sheet is a surface
* For cones, cylinders, tori and cyclides both sheets form curves.
* For the sphere the center of every osculating circle is at the center of the sphere and
the focal surface forms a single point. This property is unique to the sphere.
6. All geodesics of the sphere are closed curves.
Geodesics are curves on a surface that give the shortest distance between two points.
They are a generalization of the concept of a straight line in the plane. For the sphere the
geodesics are great circles. Many other surfaces share this property.
7. Of all the solids having a given volume, the sphere is the one with the smallest surface area; of
all solids having a given surface area, the sphere is the one having the greatest volume.
It follows from isoperimetric inequality. These properties define the sphere uniquely and
can be seen in soap bubbles: a soap bubble will enclose a fixed volume, and surface
tension minimizes its surface area for that volume. A freely floating soap bubble therefore
approximates a sphere (though such external forces as gravity will slightly distort the
bubble's shape). It can also be seen in planets and stars where gravity minimizes surface
area for large celestial bodies.
8. The sphere has the smallest total mean curvature among all convex solids with a given
surface area.
The mean curvature is the average of the two principal curvatures, which is constant
because the two principal curvatures are constant at all points of the sphere.
9. The sphere has constant mean curvature.
The sphere is the only imbedded surface that lacks boundary or singularities with
constant positive mean curvature. Other such immersed surfaces as minimal surfaces
have constant mean curvature.
10. The sphere has constant positive Gaussian curvature.
Gaussian curvature is the product of the two principal curvatures. It is an intrinsic
property that can be determined by measuring length and angles and is independent of
how the surface is embedded in space. Hence, bending a surface will not alter the
Gaussian curvature, and other surfaces with constant positive Gaussian curvature can
be obtained by cutting a small slit in the sphere and bending it. All these other surfaces
would have boundaries, and the sphere is the only surface that lacks a boundary with
constant, positive Gaussian curvature. The pseudosphere is an example of a surface
with constant negative Gaussian curvature.
11. The sphere is transformed into itself by a three-parameter family of rigid motions.
Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any
rotation about a line through the origin can be expressed as a combination of rotations
around the three-coordinate axis (see Euler angles). Therefore, a three-parameter family
of rotations exists such that each rotation transforms the sphere onto itself; this family is
the rotation group SO(3). The plane is the only other surface with a three-parameter
family of transformations (translations along the x- and y-axes and rotations around the
origin). Circular cylinders are the only surfaces with two-parameter families of rigid
motions and the surfaces of revolution and helicoids are the only surfaces with a one-
parameter family.
Gallery
An image of one of the most accurate Deck of playing
human-made spheres, as it refracts the cards illustrating
image of Einstein in the background. engineering
This sphere was a fused quartz instruments,
gyroscope for the Gravity Probe B England, 1702.
experiment, and differs in shape from a King of spades:
perfect sphere by no more than 40 Spheres
atoms (less than 10 nm) of thickness. It
was announced on 1 July 2008 that
Australian scientists had created even
more nearly perfect spheres, accurate to
0.3 nm, as part of an international hunt to
find a new global standard kilogram.[18]
Regions
Spherical cap
Spherical polygon
Spherical sector
Spherical segment
Spherical wedge
Spherical zone
See also
3-sphere Hand with Reflecting Sphere, M.C. Escher
Affine sphere self-portrait drawing illustrating reflection
and the optical properties of a mirror sphere
Alexander horned sphere
Hoberman sphere
Celestial spheres
Homology sphere
Cube
Homotopy groups of spheres
Curvature
Directional statistics Homotopy sphere
Hypersphere
Dome (mathematics)
Lenart Sphere
Dyson sphere
Napkin ring problem
Orb (optics)
Pseudosphere Spherical Earth
Riemann sphere Spherical helix, tangent indicatrix of a curve
Solid angle of constant precession
Sphere packing Spherical shell
Spherical coordinates Sphericity
Zoll sphere
Notes and references
Notes
1. r is being considered as a variable in this computation.
2. It does not matter which direction is chosen, the distance is the sphere's radius × π.
3. The distance between two non-distinct points (i.e. a point and itself) on the sphere is zero.
4. Despite not being flat, a sphere is two-dimensional since it comprises only the surface of a
solid ball.
References
1. σφαῖρα (http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3
Aentry%3Dsfai%3Dra^), Henry George Liddell, Robert Scott, A Greek-English Lexicon, on
Perseus.
2. Albert 2016, p. 54.
3. Woods 1961, p. 266.
4. Kreyszig (1972, p. 342).
5. Albert 2016, p. 60.
6. Steinhaus 1969, p. 223.
7. "The volume of a sphere - Math Central" (http://mathcentral.uregina.ca/QQ/database/QQ.09.0
1/rahul1.html). mathcentral.uregina.ca. Retrieved 10 June 2019.
8. E.J. Borowski; J.M. Borwein. Collins Dictionary of Mathematics. pp. 141, 149. ISBN 978-0-00-
434347-1.
9. Weisstein, Eric W. "Sphere" (https://mathworld.wolfram.com/Sphere.html). MathWorld.
10. Steinhaus 1969, p. 221.
11. Osserman, Robert (1978). "The isoperimetric inequality" (https://www.ams.org/journals/bull/197
8-84-06/S0002-9904-1978-14553-4/). Bulletin of the American Mathematical Society. 84:
1187. Retrieved 14 December 2019.
12. Albert 2016, p. 55.
13. Albert 2016, p. 57.
14. Woods 1961, p. 267.
15. Albert 2016, p. 58.
16. Weisstein, Eric W. "Spheric section" (https://mathworld.wolfram.com/SphericSection.html).
MathWorld.
17. Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (https://archive.o
rg/details/geometryimaginat00davi_0) (2nd ed.). Chelsea. ISBN 978-0-8284-1087-8.
18. New Scientist | Technology | Roundest objects in the world created (https://www.newscientist.c
om/article/dn14229-roundest-objects-in-the-world-created.html).
Further reading
Albert, Abraham Adrian (2016) [1949], Solid Analytic Geometry, Dover, ISBN 978-0-486-
81026-3.
Dunham, William (1997). The Mathematical Universe: An Alphabetical Journey Through the
Great Proofs, Problems and Personalities. Wiley. New York. pp. 28, 226.
Bibcode:1994muaa.book.....D (https://ui.adsabs.harvard.edu/abs/1994muaa.book.....D).
ISBN 978-0-471-17661-9.
Kreyszig, Erwin (1972), Advanced Engineering Mathematics (https://archive.org/details/advanc
edengineer00krey) (3rd ed.), New York: Wiley, ISBN 978-0-471-50728-4.
Steinhaus, H. (1969), Mathematical Snapshots (Third American ed.), Oxford University Press.
Woods, Frederick S. (1961) [1922], Higher Geometry / An Introduction to Advanced Methods
in Analytic Geometry, Dover.
External links
Mathematica/Uniform Spherical Distribution
Surface area of sphere proof (http://mathschallenge.net/index.php?section=faq&ref=geometry/
surface_sphere)
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