UNIT 2

MAP PROJECTIONS
Structure
2.1 Introduction 2.6 References and Suggested
Expected Learning Outcomes Further Reading
2.2 Map Projection and 2.7 Laboratory Exercises
Its Properties Exercise 3: Construction of
2.3 Classification and Cylindrical Map Projections
Significance of Map Exercise 4: Construction of
Projections Zenithal Map Projections
2.4 Summary Exercise 5: Construction of
Conical Map Projections
2.5 Answers

2.1 INTRODUCTION
In geography map making is becoming more important because the concept
of geography has undergone several significant changes in recent times. The
study, now, involves cartographic analysis of relevant data so that the
conclusion arrived at are scientifically correct and mathematically accurate.
Techniques have been developed to cartographically express various
statistical conclusions, so that the result becomes presentable. Many
achievements have been made; techniques have been developed; still a map
maker has to discover many novel ways to achieve the desired end.

Cartography is concerned with the designing, constructing and producing

maps. Map projection is one of the techniques through which the spherical
shape of the Earth is represented on the plane surface of the paper. You are
introduced to map projections in this Unit. Section 2.2 explains meaning and
the properties of map projections. Classification and significance of projections
are explained in Section 2.3.

Further, we have a dedicated Section on Laboratory Exercises at the end of

the unit, which facilitate to learn the skills in making map projections. You have
been introduced to some exercises with suitable examples. This will enable
you to draw map projections and their utility in representing the real world
phenomena. Broadly the map projections are categorized in to three types
such as cylindrical, zenithal and conical which are explained through hands-on
Exercises 3 to 5, in Section 2.7.

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Expected Learning Outcomes
After studying this unit, you should be able to:
 Define a map projection;
 Describe classification of map projection;
 Discuss the significance of map projections; and
 Draw different types of map projections including cylindrical, zenithal and
conical in the part of laboratory exercises.

2.2 MAP PROJECTION AND ITS PROPERTIES

We may easily construct a map for a small region/area, but creating a flat map
for the entire nation/world is a complex undertaking. So, how do you create an
accurate flat map of a very large area? Geographers particularly
cartographers all across the world have created various ways and procedures
for creating such maps throughout history. However, it is clear that no map
can accurately portray the genuine form of the Earth. The cartography is a part
of geography that deals with the study of maps and map making. A
cartographer is someone who creates or draws maps. For making/drawing an
accurate map we generally use map projections. Then, let us understand what
a map projection is? Yes, the Earth is a three dimensional surface. The
transformation of Earth’s three-dimensional surface into two-dimensional
plane on a map using mathematical formulae is called a map projection.
Therefore, we can understand that the map projection transforms a spherical
surface to a flat surface.

Cartographers have to devise complex graphical, geometrical and

mathematical methods of transforming the Earth. In the case of map
projection, we project the network of latitudes and longitudes of a globe on a
plane surface. A globe is a true representation of the Earth or it is true map of
the Earth. Since a map represents a flat surface and a globe, a spherical
surface, the shape of the network of the parallels and the meridians that we
obtain after representation is what projection is all about. Other terms which
are more suitable for this network are a graticule, or a grid or a net, or a mesh.
But the term ‘projection’ is most commonly used in practice.

We can define “a map projection is a systematic network of lines of latitudes

and longitudes on a plane surface, either for the whole Earth or part thereof”
(Singh and Singh, 1979). In other words, map projection is a method of
representing a parallels and meridians of the Earth on a plane surface at
certain scale, so that any point on the Earth’s surface may correspond to that
on the drawing. The globe is true map of the Earth, which is divided into
various sectors by the online of latitudes and longitudes. The network of these
lines is known as a gratitude. On the globe the meridians and parallels are
circles. When they are transferred on the plane surface, they become
intersecting lines, curved or straight. Thus, a map projection may be defined
as the transformation of the spherical network of latitudes and longitudes on a
plane surface, irrespective of the method of transformation.

A globe is the true representation of the Earth and it gives three dimensional
effect of the Earth. Although size, shape, distance and direction of an area
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correctly represented on a globe. Therefore, it shows very accurately the
shape and size of the continents and the oceans. But the use of globe is not
always convenient. Maps are more convenient and easier to carry than the
globe. Thus, the globe which is of three dimensions has to be transformed into
two dimensional surfaces very carefully in order to maintain the area, shape
and the direction of places on map. However, the resulted map will not,
represent the Earth truly and it will have some limitations in form of errors. As
such it is not possible to achieve all properties simultaneously that are
required to make a perfect map.

On the other hand, globes have many practical disadvantages. They are
difficult to reproduce, cumbersome to handle, awkward to store and difficult to
measure and draw on. All those drawbacks are eliminated when a map is
prepared on a flat surface. The spherical surface must be transformed to a
plane (flat) surface. The transformation of a spherical surface into plane
surface is a hard nut to crack without modifying the surface geometry. There
are many transformations in geometric qualities of the globe in the process of
it being transformed into a map. The crux of the problems as already
enunciated is the transformation of a spherical surface into a plane surface,
many possibilities have been explored to solve this complex puzzle. However,
we have a satisfactory classification by adopting the constructional approach a
projection can be obtained through projecting light. A globe made of glass with
parallels and meridians marked upon it, so that the shadows of these lines are
cast on the paper. The projection with the help of light will give a shadowed
picture of the globe which will be distorted in those parts which are farther
from the point. The obtained projection is called perspective or geometrical
projection. But if we modify the layout to a great extent with the aid of
mathematics is, therefore, known as no-perspective or non-geometrical
projection. A large part of a spherical surface cannot be represented on a flat
surface without shrinking, breaking or stretching it somewhere. The major
alternations have to do with angles, areas, distances and directions. As Misra
(1969) suggests it is possible to develop projections which have one or more
properties through not all of them. They are: conformality, equal area,
equidistant, and azimuthal.

2.2.1 Conformality or Orthomorphism

In the construction of map projection true shapes can be preserved only if the
parallels and meridians intersect each other at right angles and the scale is
the same at a point. Preservation of the right angle and the maintenance of
the same scale both along the parallel and meridians make a projection
conformal. An orthomorphic projection is also known as a conformal
projection. Both the term conformal and orthomorphic means "true shape". As
the area scale varies from point to point, large areas are rather imperfectly
represented with respect to shape. Hence, no projection can provide true
shape to areas of wide dimensions.

2.2.2 Equal Area or Equivalence

In this group of projection the graticule is prepared in such a way that every
quadrilateral on it may appear proportionately equal in area to the
corresponding spherical quadrilateral. To have the property of equivalence, a
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projection must possess two properties. The parallels and meridians should be
drawn to scale, and the spacing between the parallels and meridians should
be true. This type of projections is derived mathematically. It is also known as
Equal area or Equivalent or Homolographic projection because areas are kept
correct. Equal area property is maintained by making proper adjustment of
scales along the parallels and meridians.

2.2.3 Equidistant
The distances between specific locations are well preserved, which is an
important feature of equidistant projections. Equidistant means that one point
is equal to all other points, or that a few points are equal to others. Remember
that none of the equidistant projections are valid from all points to all other
points. Scale however is not uniform throughout the map. Scale is
appropriately maintained when the length of a line on a map is the same as
the length of the same line on the globe, regardless of whether it is a large or
tiny circle, straight or curved line. Hence, scale will be uniform along the lines
whose distances are true.

2.2.4 True Bearing or Azimuthal

The azimuthal projection attempts to present true bearings or azimuths. The
projections which show directions or bearing correctly are called azimuthal
projections. The direction of all points from the centre of the map projection
remains correct. These projections are, therefore, known as azimuthal
projections. In such projections, directions of all point on a map, as taken from
the central points, will be the same as between corresponding on the ground
(Misra, 1969).

Thus, the nature of all projections is so complex that they often possess one
or more common properties. There is no projection which can be grouped in a
single class. Moreover, it is difficult to obtain a rational classification of map
projection. There can be as many as classifications may be suggested
depending on different bases (Singh 1979). Let us study the classification of
map projections after answering the given short answer question below.

SAQ 1
a) What is the meaning of map projection?
b) What are the properties of map projection?

2.3 CLASSIFICATION AND SIGNIFICANCE OF

MAP PROJECTIONS
Projections are drawn by various methods and the resulting projections differ
from one another. It can be classified on the basis of a number of criteria.
Generally, the more commonly adopted criteria for this purpose are:
a) The nature of the developable surface;
b) The properties of the projection; and
c) The method of derivation or the source of light in the globe.
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Let us look into further sub-divisions of classification.

A. Based on the Developable Surface Used

1. Cylindrical Projection
2. Conical Projection
3. Zenithal Projection or Azimuthal
4. Mathematical or Conventional
B. Based on the Property or Qualities
1. Equal Area or Equivalent or Homolographic Projection
2. True Shape or Orthomorphic or Conformal Projection
3. True Bearing or Azimuthal Projection
4. True Scale or Equidistant Projection
C. Based on Construction or Derivation or Source of Light or Position
of View Point
1. Perspective
2. Non-Perspective
3. Mathematical or Conventional
D. Based on the Position of Tangent Surface
1. Polar
2. Equatorial
3. Oblique
E. Based on Conventional Projection
1. Mollweide's Projection
2. Interrupted Mollweide's Projection
3. Sinusoidal Projection
4. Interrupted Sinusoidal Projection.
5. Globular projection
6. Hammer's projection

Perspective projections are those which can be derived by the image of the
network of parallels and meridians of a globe on any developable surface.
Most of the zenithal projections are perspective projections. These projections
are also called natural projection.

The non-perspective projections are drawn without the help of the shadows
from an illuminated globe. Their construction depends upon mathematical
principles. They acquire special property such as equal area, correct shape,
true scale, etc. The lines forming the network are straightened or curved and
the spacing between the parallels and meridians are reduced or enlarged to
make a perspective projection equivalent, orthomorphic or azimuthal. Such
projections are called non-perspective projections.

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Fig. 2.1: Classification of Map Projections.

A developable surface is one which can be flattened and which can receive
lines projected or drawn directly from an assumed globe. The developable
projections are classified as cylindrical, conical and zenithal (Fig. 2.2).

Fig. 2.2: Developable Projection Surfaces a) Zenithal b) Cylinder and c) Cone.

Cylindrical projection can be derived through the use of a cylindrical

developable surface. When the globe is covered by a cylinder made of paper
touching the equator and the parallels and meridians are projected on it. To
derive a projection on a cone, we will have to wrap a cone around a globe. If
we place a light in the centre of the globe, it will cast shadow of a geographic
network as the inner surface of the cone. The shadow is drawn and the cone is
cut open and laid flat to get conical projections. When a plane paper is put on
the globe touching it on one point and the graticule is projected on it, we get
polar zenithal projection.

The position of light source is also of very great importance, when it can be
placed at the centre of the globe, or at any other point of the equator, or at any
point outside the globe. Accordingly, the Zenithal projections can be divided
into three classes:
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Gnomonic – When the light is at the centre of the globe.

Stereographic –When the light is placed at the periphery of the globe at a

point diametrically opposite to the point at which the plane surface touches the
globe.

Orthographic – when the light is at infinity so that the rays of the light are
parallel.

According to point position of the plane of Perspective Zenithal Projection,

each class having three sub-classes. Thus, there are nine types of perspective
zenithal projection shown in Figure 2.3.

Fig. 2.3: Types of Perspective Zenithal Projection.

In addition to these perspective zenithal projections, there are a number of

non-perspective zenithal projections. Only two of them which are most useful
and are also quite popular in atlases are: Zenithal Equidistant Projection
(Polar case) and Zenithal Equal Area Projection (Polar case). The spherical
surface of the globe cannot be projected on planer surface accurately, no map
projection can be absolutely accurate or a true representation of the Earth.
Some inaccuracies do occur in all projections. On the basis of the global
criteria the projection can be classified as equivalent, orthomorphic and
correct bearings. Such projections can be developable as well as non-
developable. They can be Cylindrical, Conical and Zenithal.

The above classifications of map projections reveal that there is no way of

classifying projections into initially exclusive classes. According to specific
requirements, a projection can be drawn in a manner so that the desired
quality may be achieved. The important qualities or properties sought in any
projection are classified as: equal-area projection, true shape or orthomorphic,
true bearing or azimuthal, true-scale or equidistant projections.

SAQ 2
Differentiate between perspective and non-perspective map projections.

2.4 SUMMARY
In this unit you have studied:

 Definition of map projection as it refers to the transformation of Earth’s

three-dimensional surface into two-dimensional plane on a map using
mathematical formulae is called a map projection.

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 Cartographers have to devise complex graphical, geometrical and
mathematical methods of transforming the Earth.
 Projection properties and types of projections and their significance.
 The construction of cylindrical, zenithal, and conical map projections.
 The construction of different map projections such as Simple Cylindrical
and Mercator’s projections in Exercise 3, Polar Zenithal Equidistant and
Polar Orthographic map projections in Exercise 4, and Polyconic projection
in Exercise 5 from Section 2.7.

2.5 ANSWERS
Self-Assessment Questions (SAQ)
1. a) A map projection is the mathematical transformation of the Earth's three-
dimensional surface into a two-dimensional plane on a map.
b) The four properties are including conformality, equal area, equidistant,
and azimuthal.
2. Perspective projections are those which can be derived by the image of the
network of parallels and meridians of a globe of any developable surface.
The non-perspective projections are drawn without the help of the shadows
from an illuminated globe.

2.6 REFERENCES AND SUGGESTED

FURTHER READING
1. Khullar, D.R. (2015): Essentials of Practical Geography. Jalandhar: New
Academic Publishing Co.
2. Misra, R.P. and Ramesh, A. (1969/1986): Fundamentals of Cartography.
New Delhi: McMillan.
3. Monkhouse, F.J. and Wilkinson, H.R. (1971): Maps and Diagrams.
London: Methuen & Co Ltd.
4. Robinson, A.H., Morrison, J.L., Muehrcke, P.C., Kimmerling, A.J. and
Gupltill, S.C. (eds.) (1995): Elements of Cartography. New York: Wiley and
Johnson.
5. Sarkar, A. (2008): Practical Geography: A Systematic Approach. Kolkata:
Orient Black Swan.
6. Sharma, J.P. (2013): Prayogik Bhugool (Hindi Medium). Meerut: Rastogi
Publications.
7. Singh, G. (2004): Map Work and Practical Geography. Delhi: Vikas
Publication House.
8. Singh, L.R. and Singh, R.P.B. (2003): Elements of Practical Geography.
Delhi: Kalyani Publishers.
9. Singh, L.R. and Singh, R. (1979): Map Work and Practical Geography.
Allahabad: Central Book Depot.

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2.7 LABORATORY EXERCISES
Let us now perform the laboratory exercises.

You have already studied the map scales and its construction in the previous
Exercises 1 and 2 of the Unit 1. In the construction of map projection, the
scale has great importance. It is generally expressed by a fraction which is
called Representative Fraction (R.F.).

R.F. can be expressed in different units for example 1:1,000,000. It means

that one unit of the map is equal to the million units of the ground. The scale
as R.F. is used in map projections so that it may be translated into any
standard of measurements.

We know that equatorial circumference of the Earth is about 25,000 miles

(40,075 km) and the mean radius 4000 miles (6371 km). As a round number, it
may be regarded 250,000,000 inches because other figures in thousands will
little affect ordinary scale.

The length of equator of the globe will be equal to the circumference 2R.

In case R = 1 inch, the length of the equator will be equal to:

22
2 1  6.3 inches approximately.
7

The laboratory exercises will enhance the skills in construction of map

projections. You are required to complete all the exercises given in the
laboratory and submit the same to the counselor/lab instructor as per the
given instructions. It is expected that after completion of these exercises, you
will gain hands-on experience and skills in understanding map projections.

The following three exercises are based on map projections dealing with the
Construction of Cylindrical Map Projections (Exercise 3), Construction of
Zenithal Map Projections (Exercise 4) and Construction of Conical Map
Projections (Exercise 5).

Requirements: To perform the exercises, you may be required the following

materials/tools. Some of them are indicated here as optional.

 Ruler or Scale
 Pencil/Pen
 Sharpener and Eraser
 Compass
 Divider
 Protractor
 Set-squares
 Scientific Calculator
 Drawing Sheets or White Sheets

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After completing the exercises, you need to submit the practical record to the
concerned academic counselor/lab instructor for its evaluation.

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 EXERCISE 3

CONSTRUCTION OF
CYLINDRICAL MAP
PROJECTIONS
Structure
3.1 Introduction
Expected Learning Outcomes
3.2 Construction of Simple Cylindrical Map Projection
3.3 Construction of Mercator Projection
3.4 Practical Exercises

3.1 INTRODUCTION
You have learnt the concept, meaning and use of map projections. You have
also studied the classification of map projections. Before performing the
exercise work you need to learn the basic concept and the calculation of scale
for constructing map projections. In fact, map projections portray the three
dimensional (3D) Earth surfaces into two dimensional (2D) features on a flat
sheet. Cylindrical projections are one of the types of map projections.

A globe is presented to be enclosed in a developable cylinder. A cylinder

wrapped around the globe to touch it along the equator. Light is placed at the
center of the globe, to cast the image of the graticule of the transparent globe
on the cylinder then the true cylindrical projection is obtained. The common
properties of cylindrical projections are explained below:

 In this projection, the parallels and meridians are drawn as straight lines
cutting each other at right angles.
 The exaggeration of the parallel scale as well as meridian scale would
continue to increase away from the equator.
 On the projection the scale would be true only along the equator.
 The poles cannot be shown because their distances from the equator
become infinite.
 The meridians are equal spaced but the intervals between parallels
increases towards the poles.
 All the parallels are equal in length.
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 Distances are correctly represented along the equator.
 This projection is rarely used as the exaggeration of area and shape is too
much in higher latitudes and only a narrow strip along the equator is
correct.

In this Exercise 3, you will learn the construction of Simple Cylindrical Map
Projection and Mercator’s Projection. The well-known types of other
cylindrical map projections are Cylindrical Equal Area, Gall’s, and Transverse
Mercator’s projection.

Here, we will learn two methods: Method 1: Mathematical Construction and

Method 2: Graphical Construction. Graphical method simply works on basic
principles of geometry, therefore, one can easily understand it but it provides
approximate values for constructing a graticule. On the other hand,
mathematical method uses trigonometrical functions for calculating radii, the
length of parallels and meridians, and distances between them. It is also
important to know that some projections can only be constructed with
mathematical methods. Therefore, it is essential to understand basic
trigonometric formulae for successful construction of map projections. We
have given trigonometric formulae and logarithmic tables at the end of
exercise for the reference.

Expected Learning Outcomes

After doing this exercise, you should be able to:
 Understand the method of construction of cylindrical map projections;
 Construct simple cylindrical map projection; and
 Construct Mercator’s projection.

3.2 CONSTRUCTION OF SIMPLE CYLINDRICAL

MAP PROJECTION
Simple cylindrical projection is also known as Plate Carree’s Projection or the
Cylindrical Equidistant Projection because in this projection both the parallels
and meridians are equidistant. They are drawn as straight lines, cutting one
another at right angles. As the distance between the parallels and meridians is
the same, the whole network represents a series of equal squares. All the
parallels are equal to the equator (2R). The properties and uses of this
projection are explained below:

Properties:

 Al the parallels and meridians are straight lines.

 Both the parallels and meridians are equally-spaced.
 Parallels and meridians intersect one another at right angles.
 The scale along the equator is correct, so that distances are shown
correctly along the equator.
 The scale along the equator is true but exaggerated at an increasing rate
away from the equator. This leads to great distortion in shape particularly
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in high latitudes. This projection is not appropriate for representing areas

in the higher latitudes.
 The scale along meridians is correct.
 The projection is neither equal-area nor orthomorphic.

Uses:

 This projection is of little uses because it can show correctly only a narrow
strip along the both sides of the equator that the area can be called
correct that too by courtesy. The distances are correct in a narrow step
only along the equator. It is highly unsatisfactory for a world map due to its
demerits.
 The projection suffers from defects of both area and shape. This
projection may be used for areas which extend along or near the equator
as Africa and South-America.

Let us start to learn the construction of graticule for Simple Cylindrical Map
Projection by doing the following example.

Example 3.1: Construct a Simple Cylindrical Map Projection for a world map
on a scale of 1: 250,000,000 and projection interval is 10°.

Solution:

METHOD-1: MATHEMATICAL CONSTRUCTION

In the preliminary stage, we have to calculate the values for constructing the
map projection. Then, the calculated values are to be used for making the
projection in the next stage.

Calculations of the Projection:

Step 1: Calculate the radius (R) of the generating globe for the given scale.

Let us note down the R.F.

We have now obtained the value of radius i.e., R = 2.56 cm.

Step 2: Record the latitudinal extent of the globe, i.e., 90N  90S and the
interval between the parallels i.e., 10. The following parallels have to be
drawn.

90N, 80N, 70N, 60N, 50N, 40N, 30N, 20N, 10N, 0, 10S, 20S, 30S,
40S, 50S, 60S,70S, 80S, 90S.

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Step 3: Note the longitudinal extent of the globe i.e. 180W-180E and the
interval between the meridians i.e. 10. So the following meridians are to be
drawn.

180W, 170W, 160W, 150W, 140W, 130W, 120W,110W, 100W, 90W,

80W, 70W, 60W, 50W, 40W, 30W, 20W, 10W, 0, 10E, 20E, 30E,
40E,50E, 60E,70E, 80E, 90E, 100E, 110E, 120E, 130E, 140E,
150E,160E, 170E, 180E.

Step 4: Calculate the length of equator using the formula:

2R = 2 x 3.14 x 2.56 = 16.07 cm.

Step 5: Calculate the length of the division (d) on the equator at 10 intervals
using the standard equation as follows:

Now, the length of division on each parallel for spacing the meridians
i.e. d = 0.44 cm.

The length of division on central meridian for spacing the parallels is also the
same length of d = 0.44 cm.

You have now all the calculated values. Let us start the construction of the
projection. Keep ready with you a drawing sheet, pencil, ruler, protractor and
divider etc.

Construction of the Projection:

Step 6: Take a drawing sheet and place it on plane surface. With the help of
pencil and scale, draw a pair of straight lines on the sheet. The lines must be
intersected with 90 angle. The horizontal line is to be treated as equator 0
(W-E) and other vertical line is central meridian.

The equator would be 16.07 cm (Step 4).

Step 7: Set the divider with a measurement of d = 0.44 cm (Step 5) and mark
the division on the equator in east and west directions. Now the equator of line
is divided by the value of d for spacing the meridians. The length of the
interval along the meridians is also the same i.e. d = 0.44 cm.

All the parallels and meridians are placed at equal distances as result of which
it is also known as equidistant projection.

Step 8: After successful completion of the construction of projection, you have

to finish it with proper labelling as given below. All the graticules and the scale
of projection must be written legibly.

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Simple Cylindrical Projection (R.F. 1: 250,000,000)

METHOD 2: GRAPHICAL CONSTRUCTION

Find out the radius of the reduced Earth according to scale. We have
calculated the radius i.e., R = 2.56 cm. Now, draw a circle with the
measurement of the radius (2.56 cm) from the centre (O).

With the help of protractor mark the angle on the circle for the given interval
(10). Then, draw line through the marking that will represent 10N (OXZ).

Then, measure the arc length (XZ) which will be the value of a division on the
parallels and the meridians both. Mark the intervals along the equator using
the obtained value. Erect perpendicular at both ends of the line so that one of
these perpendiculars is a tangent to the circle.

You have to write labels and the scale of projection as given below for
finishing projection.

Simple Cylindrical Projection (R.F. 1: 250,000,000)

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3.3 CONSTRUCTION OF MERCATOR’S
PROJECTION
Mercator’s projection is a cylindrical projection. This projection was designed
by Gerhardt Kramer of Holland in 1569. Kramer is a Dutch word which is
called Mercator in Latin and this projection is popularly known as Mercator’s
Projection. This projection is also known as Cylindrical Orthomorphic
Projection. The projection is derived by the lines of equally spaced longitudes
and horizontal lines of latitudes. Like all other cylindrical map projections, the
scale along parallels is exaggerated away from the equator. In order to
maintain correct direction and shape Mercator increased the scale along
meridians in the same ratio as the increase in parallel scale.

Properties:

 All the parallels are drawn as straight lines which are parallel and are
having the same length as the equator.

 All meridians are drawn as straight lines which are perpendicular to the
equator and parallel to one another.

 The parallels and meridians are intersected at right angles.

 The distances between the parallels are wide apart moving towards the
poles. The poles cannot be shown in this projection.

 Shapes are represented more accurately whereas areas are distorted

greatly toward the Polar regions.

Uses:

 Any straight line drawn on this projection represents an actual compass

bearing, hence it is useful to sailors. These true direction lines are known
as rhumb lines. Scale is true along the equator or along the secant
latitudes.

 The main advantage of this projection is that it can be used to map the
regions near the equator.

 More suitable for a world map and widely used in preparing atlas maps.

 Oceanic currents, temperature, rainfall, winds and their directions are

appropriately shown on this map projection.

Example 3.2: Construct a Mercator’s projection for the world map on RF

1:500,000,000 scale at 20º interval.

Solution:

METHOD-1: MATHEMATICAL CONSTRUCTION

In the preliminary stage, we have to calculate the values for constructing the
map projection. Then, the calculated values are to be used for making the
projection in the next stage.

52
Unit – 2 Map Projections Exercise 3 - Construction of Cylindrical Map Projections
…………………………………………………………………………………………………………………………
Calculations of the Projection:

Step 1: Calculate the radius (R) of the generating globe for the given scale.
Let us note down the RF.

We have now obtained the value of radius i.e., R = 1.28 cm.

Step 2: Let us note the latitudinal extent of the globe i.e., 90ºN-90ºS and the
interval between the parallels i.e., 20º. The following parallels have to be
drawn.

80ºN, 60ºN, 40ºN, 20ºN, 0º, 20ºS, 40ºS, 60ºS, 80ºS

Step 3: Also record the longitudinal extent of the globe i.e., 180ºW-180ºE and
the interval between the meridians i.e., 20º. So, the following meridians are to
be drawn.

180ºW, 160ºW, 140ºW, 120ºW, 100ºW, 80ºW, 60ºW, 40ºW, 20ºW, 0º, 20ºE,
40ºE, 60ºE, 80ºE, 100ºE, 120ºE, 140ºE, 160ºE, 180ºE

Step 4: Calculate the length of a division (d1) on the equator at 20ºinterval

using the standard equation as follows.

Now, you got to know the length of a division on equator for spacing the
meridians i.e., d1 = 0.45 cm.

Step 5: Calculate the length of a division (d2) on the central meridian (0º)
using the standard equation as follows.

To get the value, you shall prepare the following table with given interval
i.e, 20º for determining the division using the above equation. You can use the
scientific calculator to get the Logarithm (log) and Tangent (tan) values
directly or refer to the annexures provided at the end of this exercise.

53
Block - 1 Fundamentals of Cartography
……………………………………………………………………………….…………………………………………
R Constant
(cm) value (cm)
20º (N&S) 55 1.4281 0.1547 1.28 2.3026 0.456
40º (N&S) 65 2.1445 0.3313 1.28 2.3026 0.976
60º (N&S) 75 3.7320 0.5719 1.28 2.3026 1.685
80º (N&S) 85 11.430 1.058 1.28 2.3026 3.118

You have now all the calculated values. Let us start the construction of the
projection. Keep ready with you a drawing sheet, pencil, ruler, protractor and
divider, etc.

Construction of the Projection:

Step 6: Take a drawing sheet and place it on plane surface. With the help of
pencil and scale, draw a pair of straight lines on centre of the sheet. The lines
must be intersected with 90° angle. A horizontal line is to be treated as
equator and other vertical line as central meridian.

Step 7: Set the divider with a measurement of d1 = 0.45 cm and mark the
divisions on the equator in E and W directions. Now, the equator line is divided
by the value of d1 (refer to Step 4), for spacing the meridians. Take a scale
and pencil, and draw straight lines perpendicular to the equator by connecting
each of these division points. These lines will represent meridians.

W E
0.45 cm

Now, the meridians have been drawn for both W and E directions (from 0º to
180º with 20º interval).

Step 8: Let us now start working on the central meridian. Mark the divisions on
central meridian in North and South directions with the help of the divider.
Using the values of dɸ from the calculated Table (refer to Step 5), the
divisions are done for spacing the parallels.

Step 9: Straight lines are drawn perpendicular to the central meridian

connecting through these divisions. These lines will represent parallels.

54
Unit – 2 Map Projections Exercise 3 - Construction of Cylindrical Map Projections
…………………………………………………………………………………………………………………………
N
3.12 cm

1.7
0.98
0.46 cm
W E
0.45 cm

Step 10: You have now successfully completed the construction of Mercator’s
projection. But you have to finish it with proper labelling as given below. All the
graticules and the scale of projection must be written legibly.

R.F. 1:500,000,000

Now, you have understood the construction of Mercator’s Projection and

Simple Cylindrical Projections.

You should now complete the following laboratory tasks for your practical
record before submitting it. You can also utilize the dataset given at your study
center for doing exercises. After completing the exercises, you need to submit
the practical record to the concerned academic counselor/lab instructor for
evaluation.

3.4 PRACTICAL EXERCISES

Exercise 1: Construct simple cylindrical projection for a world map whose
scale is 1:400,000,000 at 15° interval.
55
Block - 1 Fundamentals of Cartography
……………………………………………………………………………….…………………………………………
Exercise 2: Construct simple cylindrical equal area projection for the globe on
RF 1:300,000,000 scale at 20º interval.

Exercise 3: Construct Mercator’s projection for the globe on RF

1:500,000,000 scale at 20º interval.

Exercise 4: Construct Mercator’s projection for the globe on RF

1:320,000,000 scale at 10º interval.

56
 EXERCISE 4

CONSTRUCTION OF ZENITHAL
MAP PROJECTIONS
Structure
4.1 Introduction
Expected Learning Outcomes
4.2 Construction of Polar Zenithal Equidistant Projection
4.3 Construction of Polar Zenithal Orthographic Projection
4.4 Practical Exercises

4.1 INTRODUCTION
In the previous exercise, you have been introduced to the method of
construction of cylindrical map projections. In this Exercise 4, you will learn how
to construct Zenithal map projections for representing the Earth’s surface areas.

Zenithal map projections are obtained by projecting the lines of latitude and
longitudes on a plane surface, which is tangent to the globe at a point. They
are known as Azimuthal or Zenithal projections. This is a unique property
possessed singularly by the projections of this group. Further, it can be used
for mapping any part of the world and for any purpose, because a plan surface
can be tangent to the globe at infinite points. In Zenithal projections, the
directions of all points from the centre of the map projection remain correct. It
has two broad divisions: 1) Perspective Zenithal Projection and 2) Non-
perspective Zenithal projection.

All the perspective types of Zenithal map projections are derived by supposing
a plane surfaces tangent to the globe. The plane can be a tangent to a globe
and can occupy several positions, e.g. at one of the poles, or at any point on
the equator, or at any other point on the globe. The position of light is of great
importance because the distances between various lines of latitudes and
longitudes will be determined by the relative positions of the point to be
projected and the position of the source of light. The light can be placed at the
centre of the globe, or at any point of the equator or at any point outside the
globe. In addition to these perspective zenithal projections, there are a number
of non- perspective zenithal projections. Only two of them which are most
useful and are also quite popular in atlases are: Zenithal Equidistant
Projection (Polar case) and Zenithal Equal-area Projection (Polar case).

K. Nageswara Rao 57
Block - 1 Fundamentals of Cartography
…………………………………………………………………………………………………………………………
One of the important properties of Zenithal map projection is that they show
the correct bearings, correct directions or azimuths of all the points from the
centre of the globe. This is a unique property possessed singularly by the
projections of this group.

Here, you will learn the construction of Polar Zenithal Equidistant Projection
and Polar Zenithal Orthographic Projection. Accordingly, you will learn two
methods for constructing the projections: Mathematical and Graphical.

Expected Learning Outcomes

After doing this exercise, you should be able to:
 Understand the method of construction of Zenithal map projections;
 Construction of Polar Zenithal Equidistant projection; and
 Construct Polar Zenithal Orthographic projection.

4.2 CONSTRUCTION OF POLAR ZENITHAL

EQUIDISTANT PROJECTION
Polar Zenithal Equidistant projection is a non-perspective Zenithal projection.
The details of this projection are as follows:

Properties:
 All parallels are drawn as concentric circles.
 Parallels are spaced at their true distances apart from one another.
 Meridians are straight lines.
 Either of the poles is the centre of the projection.
 Distances as well as directions are correct from the central point to the
other point, because the meridianal scale is correct.
 The scale along parallels is not correct, and the distances along parallels
are exaggerated, because meridians on the projection diverge more
rapidly than on the globe so that away from the centre areas are also
exaggerated.

Uses:
 This projection is commonly used for preparing general purpose map of
the polar areas or, occasionally, a hemisphere. The distances and
bearings in different directions from the centre are correct and this map is
useful for showing missile ranges, radio waves and air routes with
reference to the centre of the projection.

Now let us learn the construction of Polar Zenithal Equidistant Projection

with the help of Example 4.1.
Example 4.1: Draw a network of a Polar Zenithal Equidistant projection for the
Northern Hemisphere covering latitudes 10°-90°N on a scale of 1:500,000,000
when the graticule interval is 10°.
Solution:
Step 1: Calculate the radius of the generating globe (R) for the given scale.
58
Unit – 2 Map Projections Exercise 4 - Construction of Zenithal Map Projections
…………………………………………………………………………………………………………………………
1
𝑅𝐹 = 1: 500,000,000 =
500,000,000

1
𝑅𝑎𝑑𝑖𝑢𝑠𝑅 = (𝑐𝑚) × 𝐸𝑎𝑟𝑡ℎ𝑟𝑎𝑑𝑖𝑢𝑠
500,000,000

1 64
= (𝑐𝑚) × 640,000,000 = = 1.28 𝑐𝑚
500,000,000 50

We have now obtained the value of radius i.e., R = 1.28 cm.

Step 2: Note the latitudinal extent i.e. from 10ºN to 90ºN and the interval i.e.,
10º. So, the following parallels are to be drawn.

10ºN, 20ºN,30ºN, 40ºN, 50ºN, 60ºN,70ºN, 80ºN

Step 3: Note the extent of meridians i.e., from 0º to 180º E/W and the interval
is 10º. So the following meridians are to be drawn.

0, 10E, 20E, 30E, 40E,50E, 60E,70E, 80E, 90E, 100E, 110E,
120E, 130E, 140E, 150E,160E, 170E, 180E/180W, 170W, 160W,
150W, 140W, 130W, 120W,110W, 100W, 90W, 80W, 70W, 60W,
50W, 40W, 30W, 20W, 10W

Step 4: Calculate the length of a division (d) on the central meridian using the
standard equation as follows.
2𝜋𝑅 2 × 3.14 × 1.28 8.04
𝑑= × 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 (°) = × 10 = = 0.22 𝑐𝑚
360° 360 36
22
𝑤ℎ𝑒𝑟𝑒 𝜋 = = 3.14 (𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡); 𝑅 = 1.28 𝑐𝑚 (𝐹𝑟𝑜𝑚 𝑆𝑡𝑒𝑝 1)
7

Now, you got to know the length of division on central meridian for spacing
the parallels at 10º interval i.e., d = 0.22 cm.

You have now all the calculated values for constructing the map projection.
Let us start the construction of the projection. Keep ready with you a drawing
sheet, pencil, ruler, compass, protractor and divider, etc.

Construction of the Projection:

Step 5: Take a drawing sheet and place it on a plane surface. With the help of
pencil and scale, draw a pair of straight lines on centre of the sheet. The lines
must be intersected with 90 angle. The horizontal line represents 90E-90W
and the other vertical line indicates 0-180 meridians. The point of
intersection represents 90N pole.

Step 6: Using the divider, mark the divisions with the measuremnt of 0.22 cm
(d) (refer to Step 4) on these lines. Now, with the help of compass, from
intersection point the concentric circles are drawn connecting through these
markings. These circles will represent 80ºN, 70ºN, 60ºN, 50ºN, 40ºN, 30ºN,
20ºN, and 10ºN parallels. You must always remember that all of these
concentric circles are parallels.

59
Block - 1 Fundamentals of Cartography
…………………………………………………………………………………………………………………………

Step 7: Take a protractor and place it by matching with the center point of
intersection i.e. pole (90N). Then, mark the divisions of required angles for all
the concentric circles. With the help of setsquares or scale, straight lines are
drawn by touching each of these division points joining to the pole point.
These lines are meridians.

60
Unit – 2 Map Projections Exercise 4 - Construction of Zenithal Map Projections
…………………………………………………………………………………………………………………………
Step 8: You have now completed the construction of graticules of Polar Zenithal
Equidistant projection. Finally, you have to mark proper labels as given below.
All the graticules and the scale of projection must be writen on the drawing.

METHOD-2: GRAPHICAL CONSTRUCTION

Step 1: Draw a circle with O as centre and radius equal to 1.28 cm. With the
help of protractor, draw 10 (given interval) angle as YOX. Measure the arc
XY distance, which will be used for drawing parallels 80ºN, 70ºN, 60ºN, 50ºN,
40ºN, 30ºN, 20ºN, and 10ºN.

61
Block - 1 Fundamentals of Cartography
…………………………………………………………………………………………………………………………
Construction of the Projection:

Step 2: Take a drawing sheet and place it on a plane surface. With the help of
pencil and scale, draw a pair of straight lines on centre of the sheet. The lines
must be intersected with 90 angle. The horizontal line represents 90E-90W
and the other vertical line indicates 0-180 meridians. The point of
intersection represents 90N pole.

Step 3: Using the divider, mark the divisions with the measuremnt of XY
distance on these lines. Now, with the help of compass, from intersection point
the concentric circles are drawn connecting through these markings. These
circles will represent parallels. For meridians, draw lines at 10 interval from O
using protractor. All meridians will be straight lines from the common centre of
the concentric circles.

Step 4: You have now completed the construction of graticules of Polar

Zenithal Equidistant projection. Finally, you have to mark proper labels as
given below. All the graticules and the scale of projection must be writen on
the drawing.

62
Unit – 2 Map Projections Exercise 4 - Construction of Zenithal Map Projections
…………………………………………………………………………………………………………………………
4.3 CONSTRUCTION OF POLAR ZENITHAL
ORTHOGRAPHIC PROJECTION
This is a perspective projection in which the source of light is presumed to be
at infinity so that the rays passing through the globe come parallel to each
other. If the plane surface touches the globe at the North pole, the rays of light
are supposed to be coming from the south and vice versa.

Properties:

 The parallels are concentric circles and are not equally-spaced.

 The distances between two parallels decrease away from the centre of
the projection.
 Meridians are straight lines radiating from the common centre and are
uniformly spaced.
 The direction from the centre to any other point on this projection is
correct.
 Scale along the parallels is correct.
 Because of the excessive shortening of scales along the meridians both
the area and shape are excessively distorted particularly away from the
centre and towards the periphery.

Uses:

 Since the source of light being at an infinite distance, the projection

becomes suitable for astronomical purposes. It prevents a site which is
visible when the globe is viewed from a distance exactly above the pole.
Therefore, this projection is used to prepare astronomical charts.
 This projection is used for a limited area near the pole.

Now let us learn the construction of Polar Zenithal Orthographic Projection

with the help of Example 4.2.

Example 4.2: Construct Polar Zenithal Orthographic Projection for Northern

Hemisphere when R.F is 1:750,000,000 and the interval is 15.

Solution:
Step 1: Calculate the radius of the generating globe (R) for the given scale.
1
𝑅𝐹 = 1: 750,000,000 =
750,000,000
1
𝑅𝑎𝑑𝑖𝑢𝑠𝑅 = (𝑐𝑚) × 𝐸𝑎𝑟𝑡ℎ𝑟𝑎𝑑𝑖𝑢𝑠
750,000,000
1 64
= (𝑐𝑚) × 640,000,000 = = 0.85 𝑐𝑚
750,000,000 75
We have now obtained the value of radius i.e., R = 0.85 cm.

Step 2: Note the latitudinal extent i.e. from 0ºN to 90ºN and the interval i.e.,
15º. So, the following parallels are to be drawn.

15ºN, 30ºN, 45ºN, 60ºN, 75ºN

63
Block - 1 Fundamentals of Cartography
…………………………………………………………………………………………………………………………
Step 3: Note the extent of meridians i.e., from 0º to 180º E/W and the interval
is 10º. So the following meridians are to be drawn.

0, 15E, 30E, 45E, 60E, 75E, 90E, 105E, 120E, 135E, 150E, 165E,
180E/180W, 165W, 150W, 140W, 130W, 120W,105W, 90W, 75W,
60W, 45W, 30W, 15W

Step 3: Calculate the length of the radius (rɸ) of a parallel using the standard
equation as follows:
𝑅𝑎𝑑𝑖𝑢𝑠 (𝑟∅ ) = 𝑅 cos ∅
𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 75° 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 (𝑟 ) = 𝑅 cos ∅ = 0.85 × cos 75° = 0.85 × 0.259 = 0.22 𝑐𝑚
𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 60° 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 (𝑟 ) = 𝑅 cos ∅ = 0.85 × cos 60° = 0.85 × 0.5 = 0.42 𝑐𝑚
𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 45° 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 (𝑟 ) = 𝑅 cos ∅ = 0.85 × cos 45° = 0.85 × 0.707 = 0.60 𝑐𝑚
𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 30° 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 (𝑟 ) = 𝑅 cos ∅ = 0.85 × cos 30° = 0.85 × 0.866 = 0.73 𝑐𝑚
𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 15° 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 (𝑟 ) = 𝑅 cos ∅ = 0.85 × cos 15° = 0.85 × 0.965 = 0.82 𝑐𝑚
𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 0° 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 (𝑟 ) = 𝑅 cos ∅ = 0.85 × cos 0° = 0.85 × 1.00 = 0.85 𝑐𝑚

You can use the scientific calculator to get the cos values directly or refer to
annexures provided at the end of this excersice. You have now all the
calculated values for constructing the projection. Let us start the construction
of the projection. Keep ready with you a drawing sheet, pencil, ruler, protractor
and divider etc.

Construction of the Projection:

Step 5: Take a drawing sheet and place it on a plane surface. With the help of
pencil and scale, draw a pair of straight lines on centre of the sheet. The lines
must be intersected with 90º angle. The horizontal line represents 90ºE-90ºW
and the other vertical line indicates 0º-180º meridians. The point of
intersection represents 90ºN pole. With the help of compass, from intersection
point the concentric circles are drawn using the calculated raddi (Step 3).
These circles will represent 75ºN, 60ºN, 45ºN, 30ºN, 15ºN, and 0º parallels.
You must always remember that all of these concentric circles are parallels.

64
Unit – 2 Map Projections Exercise 4 - Construction of Zenithal Map Projections
…………………………………………………………………………………………………………………………
Step 6: Take a protractor and place it by matching with the center point of
intersection i.e. pole (90ºN). Then, mark the divisions of required angles for all
the concentric circles. With the help of setsquares or scale, straight lines are
drawn by touching each of these division points joining to the pole point.
These lines are meridians.

Step 7: You have now completed the construction of graticules of Polar

Zenithal Orthographic projection. Finally, you have to mark proper labels as
given below. All the graticules and the scale of projection must be writen on
the drawing.

65
Block - 1 Fundamentals of Cartography
…………………………………………………………………………………………………………………………
METHOD-2: GRAPHICAL CONSTRUCTION

Step 1: Draw a circle with O as centre and radius equal to 0.85 cm so that it
represents the globe. Now draw ONOS and OEOW as polar and equatorial
diameters. Also NX tangent to the circle at N to represent the plane of the
projection.

Step 2: Draw Oe, Od, Oc, Ob, and Oa at the given interval of 15 with the help
of protractor. Now draw parallel lines at the marked angles and produce them
to meet the tangent plane. So, draw lines parallel to ON through f, e, d, c, b,
and a, to meet NX at f', e', d', c', b' and a'.

Step 3: Thus, Nf', Ne', Nd', Nc', Nb', and Na' are the radii of the circles
showing parallels at 0, 15N, 30N, 45N, 60N band 75N, respectively. So,
you need to measure the distance of Nf', Ne', Nd', Nc', Nb', and Na' that will be
used for drawing parallels of the projection.

Construction of the Projection:

Step 4: Take a drawing sheet and place it on a plane surface. With the help of
pencil and scale, draw a pair of straight lines on centre of the sheet. The lines
must be intersected with 90º angle. The horizontal line represents 90ºE-90ºW
and the other vertical line indicates 0º-180º meridians. The point of
intersection represents 90ºN pole.

Step 5: Now, with the help of compass, from intersection point the concentric
circles are drawn using the measured radii for different intervals (Step 3).
These circles will represent parallels of 0, 15N, 30N, 45N, 60N and 75N.
For meridians, draw lines at 15 interval from O using protractor. All meridians
will be straight lines from the common centre of the concentric circles.

Step 6: You have now completed the construction of graticules of Polar

Zenithal Equidistant projection. Finally, you have to mark proper labels as
given below. All the graticules and the scale of projection must be writen on
the drawing.
66
Unit – 2 Map Projections Exercise 4 - Construction of Zenithal Map Projections
…………………………………………………………………………………………………………………………

Now, you have understood the construction of Polar Zenthial Orthographic

Projection and Polar Zenthial Equidistant Projections.

You should now complete the following laboratory tasks for your practical
record before submitting it. You can also utilize the dataset given at your study
center for doing exercises. After completing the exercises, you need to submit
the practical record to the concerned academic counselor/lab instructor for
evaluation.

4.4 PRACTICAL EXERCISES

Exercise 4.1: Construct polar zenithal equidistant projection for southern
hemisphere on scale 1:250,000,000 for covering an area of 90°S-40S° at
interval of 10°.
Exercise 4.2: Construct polar zenithal orthomorphic projection for Northern
Hemisphere covering 20°N -90°N latitudes on the scale 1:250,000,000.The
projection interval is 10°.

67
 EXERCISE 5

CONSTRUCTION OF CONICAL
MAP PROJECTIONS
Structure
5.1 Introduction
Expected Learning Outcomes
5.2 Construction of Simple Conical Projection with One Standard Parallel
5.3 Construction of Polyconic Projection
5.4 Practical Exercises

5.1 INTRODUCTION
Till now, you have learned the techniques of constructing different types of
cylindrical and zenithal map projections in the previous exercises. In this
Exercise, you will learn about conical projections.

A conical projection is derived by projecting the image of the network of

parallels and meridians of a globe in such a way that its vertex is above on of
the poles and it touches the globe along latitude which is termed as standard
parallel. The cone is unrolled into a flat surface. The light is supposed to be
located at the centre of the globe. The scale along the standard parallel on a
conical projection is always correct. The position and length of other parallels
on either side of the standard parallel are distorted. Its length is also
considerably increased. The distortion of shape and area away from the
standard parallel is progressive in this projection.

There may be one or two standard parallels in conical projections. The axis
along which the cone is flattened, forms the central meridians of the map.
Other meridians are straight lines radiating from the vertex of the cone at
equal intervals, dividing the standard parallels into equal arcs. Other parallels
will be concentric with the standard parallel. So the scale along the meridians
becomes correct.

To minimize the distortion of shape and area a number of non-perspective

conical projections have been developed. A few of the important ones are;
simple conical projection with one standard parallel, simple conical projection
with two standard parallels, Bonne's projection and Polyconic projection.

K. Nageswara Rao 69
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In this Exercise 5, you will learn the construction of Simple Conical
Projection with One Standard Parallel and Polyconic Projections.

Expected Learning Outcomes

After doing this exercise, you should be able to:
 Understand the construction method of conical map projections;
 Construction of simple conical projection with one standard parallel; and
 Construction of Polyconic projection.

5.2 CONSTRUCTION OF SIMPLE CONICAL

PROJECTION WITH ONE STANDARD
PARALLEL
The standard parallel is the line along which the cone touches the globe in the
simple conical projection. Parallel lines of latitude are projected onto the cone
as concentric arcs of circles in this projection. The meridians are projected
onto the conical surface and intersect at the cone's tip. Along the standard
parallel, the distortion is essentially non-existent. It is limited to representing
one hemisphere, either the northern or southern hemisphere.

Properties:

 All parallels are arcs of concentric circles and are equally spaced.
 All meridians are straight lines radiating from the common center.
 The scale is true along the standard parallel.
 The scale is true along all meridians.
 The distance between them decreases towards the poles.
 The area is true in a narrow-belt on both sides of the standard parallel.

Uses:
 The simple conical projection with one standard parallel is appropriate for
small areas in the mid-latitudes with an east-west extent but a small north-
south extent due to distortion further away from the standard parallel.

Let us start to learn the construction of graticules for Simple Conical

Projection with One Standard Parallel with the help of Example 5.1. Here,
you will learn Mathematical Construction and Graphical Construction for the
construction of graticules of the projections.

Example 5.1: Construct simple conical projection with one standard parallel
on R.F. 1:150,000,000 for the extensions 10ºN-50ºN and 60ºE-100ºE. The
interval is 10º.

Solution:
METHOD-1: MATHEMATICAL CONSTRUCTION

In the preliminary stage, we have to calculate the values for constructing the
map projection. Then, the calculated values are to be used for making the
projection in the next stage.
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Unit – 2 Map Projections Exercise 5 - Construction of Conical Map Projections
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Calculations for Constructing the Projection:

Step 1: Calculate the radius (R) of the generating globe for the given scale.
Let us note down the RF.
1
𝑅𝐹 = 1: 150,000,000 =
150,000,000
1
𝑅𝑎𝑑𝑖𝑢𝑠𝑅 = (𝑐𝑚) × 𝐸𝑎𝑟𝑡ℎ𝑟𝑎𝑑𝑖𝑢𝑠
150,000,000
1 64
= (𝑐𝑚) × 640,000,000 = = 4.26 𝑐𝑚
150,000,000 15
We have now obtained the value of radius i.e., R = 4.26 cm
Step 2: Let us choose the standard parallel. Note the latitudinal extent i.e.,
10ºN to 50ºN and the interval between the parallels i.e., 10º. So, the following
parallels are to be drawn.
10ºN, 20ºN, 30ºN, 40ºN, 50ºN
Hence, the standard parallel would be 30ºN.
Step 3: Also record the longitudinal extent i.e., 60ºE to 100ºE and the interval
between the meridians i.e., 10º. Thus, the following meridians have to be drawn.
60ºE, 70ºE, 80ºE, 90ºE, 100ºE
The central meridian would be 80ºE.
Step 4: Calculate the length of the radius (r1) of the standard parallel (30ºN)
using the standard equation as follows:
𝑟 = 𝑅 cot ∅ = 4.26 × cot 30° = 4.26 × 1.732 = 𝟕. 𝟑𝟖 𝒄𝒎
Step 5: Calculate the length of a division (d1) on the central meridian at 10º
interval using the standard equation as follows.
2𝜋𝑅 2 × 3.14 × 4.26
𝑑 = × 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 (°) = × 10 = 0.74 𝑐𝑚
360° 360
22
𝑤ℎ𝑒𝑟𝑒 𝜋 = = 3.14 (𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡); 𝑅 = 4.26 𝑐𝑚 (𝐹𝑟𝑜𝑚 𝑆𝑡𝑒𝑝 1)
7
The length of a division on central meridian for spacing the parallels is
d1 = 0.74 cm.
Step 6: Calculate the length of a division (d2) on the standard parallel (30ºN)
at 10º interval using the standard equation as follows.
2𝜋𝑅 cos ∅
𝑑 = × 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 (°)
360°
2 × 3.14 × 4.26 × cos 30° 26.75 × 0.866 23.16
= × 10 = = = 0.64 𝑐𝑚
360 36 36
The length of division on the standard parallel at 10º interval for spacing
the meridians is d2 = 0.64 cm.
You can use the scientific calculator to get the cos and cot values directly or
refer to annexures provided at the end of this excersice.You have now all the
calculated values for constructing the projection. Let us start the construction
of the projection. Keep ready with you a drawing sheet, pencil, ruler, compass,
protractor and divider etc.
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Construction of the Projection:

Step 7: Take a drawing sheet and place it on plane surface. With the help of
pencil and scale, draw a straight line vertically on centre of the sheet that is to
be represented as central meridian. Set a compass with the calculated value
of radius i.e., r1 = 7.38 cm and draw an arc of circle going through the central
meridian from NP. This arc will be the standard parallel of 30ºN.

Then, Set the divider with a measurement of d1 = 0.74 cm (Step 5) and mark
the divisions on the central meridian starting from the arc of standard parallel
for spacing the parallels. Draw other concentric arcs by connecting each
division on the central meridian. These arcs will represent all other parallels.

Now, take the divider and divide the standard parallel (30ºN) with the
measurement of d2 = 0.64 cm (refer to Step 6) for spacing the meridians.

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Step 8: With the help of scale or setsquares, straight lines are drawn by touching each of
these division points joining to the pole point of central meridian. These lines are meridians.

Step 9: You have now successfully completed the construction of graticules of

simple conical projection with one standard parallel. But, you have to erase
the extended lines of the projection and finish it with proper labelling as given
below. You must also write the scale of projection.

R. F. 1:150,000,000
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METHOD-2: GRAPHICAL CONSTRUCTION

Calculations for Constructing the Projection:

Step 1: The radius is R = 4.26 cm.

The following parallels need to be drawn.

10ºN, 20ºN, 30ºN, 40ºN, 50ºN; hence, the standard parallel would be 30ºN.

The following meridians have also to be drawn.

60ºE, 70ºE, 80ºE, 90ºE, 100ºE; hence, the central meridian would be 80ºE.

Now, we have to calculate the value of the radius of the standard parallel
(30ºN), the length of a division (d1) on the central meridian (80ºE), and the
length of a division (d2) on the standard parallel.

Step 2: Take a drawing sheet and place it on plane surface. With the help of
pencil and scale, draw a pair of straight lines on the sheet. The lines must be
intersected with 90º angle. The interaction point is named as O. Draw a circle
with the measurement of radius R = 4.26 cm from the center (O). With the
help of protractor, mark the angles of 10° (given interval) as OEI and 30°
(standard parallel) as OEL on the circle. Then, draw lines OI for the given
interval and OL for the standard parallel through the markings from O.

Step 3: Draw a tangent line from the standard parallel point (L) to meet the
vertical line at polar point (NP). Now, the length of LNP will be the radius of
standard parallel (30°).

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Now, you will get the the radius of standard parallel (30°) i.e., LNP = 6.4 cm.

Measure the arc distance of IE which will be the value of the division on
central meridian. Hence, IE is the true distance between two parallels at 10°
interval. So, the length of division on central meridian for spacing the
parallels is d1 = 0.74 cm.

Step 4: Now, you have to calculate the value of division on the standard
parallel. Draw an inner circle with the measurement of IE=0.74 from the center
O. Draw a line xy parallel to OE. Now, measure the length of xy and it will be
the value of division of the standard parallel for spacing the meridians. The
calculated value of the length of division on the standard parallel (30ºN) for
spacing the meridians is d2 = 0.64 cm.

Construction of the Projection:

Step 5: Take a drawing sheet and place it on plane surface. With the help of
pencil and scale, draw a straight line vertically on centre of the sheet. Set a
compass with the calculated value of radius i.e., LP = 6.4 cm and draw an arc of
circle going through the central meridian from point NP.

This arc will be the standard parallel of 30ºN. Set the divider with a measurement
of d1 = 0.74 cm (from Step 3) and mark the divisions on the central meridian
starting from the arc of standard parallel for spacing the parallels.

Step 6: Draw other concentric arcs by connecting each division on the central
meridian. These arcs will represent all other parallels. Now, take the divider
and divide the standard parallel (30ºN) with the measurement of d2 = 0.64 cm
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(refer to Step 4) for spacing the meridians. With the help of scale or
setsquares, straight lines are drawn by touching each of these division points
joining to the central meridian pole point. These lines are meridians.

Step 7: You have now successfully completed the construction of graticules of

simple conical projection with one standard parallel. But you have to erase the
extended lines of the projection and finish it with proper labelling as given
below. You must also write the scale of projection.

R.F. 1:150,000,000

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Unit – 2 Map Projections Exercise 5 - Construction of Conical Map Projections
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5.3 CONSTRUCTION OF POLYCONIC
PROJECTION
This projection is a modified form of the simple conical projection in which all
the parallels are drawn as well as divided as standard parallels. The projection
is derived by considering a number of cones placed over a globe. It was
developed by Ferdinand Hassler, from the Coast and Geodetic Survey of the
United States of America. While constructing this projection, it is assumed that
each parallel on the globe is touched by a different cone. Consequently all
these parallels are standard parallels and representation is correct along all of
them. This is why the projection has been given the name of polyconic
projection which means many cones.

Properties:

 Each parallel is projected as standard parallel.

 Parallels are non-concentric arcs of circles with radius of curvature
decreasing toward the Poles.
 The scale is true only along the central meridian and parallels.
 The meridianal scale increased as we proceed away from the central
meridian.
 Meridians are smooth curves.
 East-West distance between two points is correct along the parallels.
 Near the central meridian both area and shape are approximately correct,
but away from it both become inaccurate.

Uses:
 The projection is not suitable for covering large regions. It is excellent for
depicting regions on small sheets with northern and southern ends that fit
together. Countries in the temperate zone with a wide latitudinal but
restricted longitudinal extent can be displayed successfully on this
projection.
 The framework of international map projection is a modified version of this
projection. In reality, the projection is insufficient for a nation that extends
beyond 30° on either side of the centre meridian. The projection is
appropriate for a map of Europe. It is also used for preparing
topographical sheets of small areas.
Let us start to learn the construction of graticule for Polyconic Projection by
doing the following example.

Example 5.2: Draw a graticule at 10 interval on a scale of 1:320,000,000 for

the extensions 20ºN to 60ºN and 20ºE to 100ºE.

Solution:
METHOD-1: MATHEMATICAL CONSTRUCTION
In the preliminary stage, we have to calculate the values for constructing the
map projection. Then, the calculated values are to be used for making the
projection in the next stage.
77
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Calculations for Constructing the Projection:

Step 1: Calculate the radius (R) of the generating globe for the given scale.
Note down the RF.

1
𝑅𝐹 = 1: 320,000,000 =
320,000,000
1
𝑅𝑎𝑑𝑖𝑢𝑠𝑅 = (𝑐𝑚) × 𝐸𝑎𝑟𝑡ℎ𝑟𝑎𝑑𝑖𝑢𝑠
320,000,000
1 64
= (𝑐𝑚) × 640,000,000 = = 2.0 𝑐𝑚
320,000,000 32
We have now obtained the value of radius i.e., R = 2.0 cm

Step 2: Note the latitudinal extent i.e., from 20ºN to 60ºN and the interval
between the parallels i.e., 10º. So, the following parallels are to be drawn.

20ºN, 30ºN, 40ºN, 50ºN, 60ºN

Step 3: Also record the longitudinal extent of the area i.e., from 20ºE to 100ºE
and the interval between the meridians i.e., 10º. Thus, the following meridians
have also to be drawn.

20ºE, 30ºE, 40ºE, 50ºE, 60ºE, 70ºE, 80ºE, 90ºE, 100ºE

The central meridian would be 60ºE.

Step 4: Calculate the length of a division (d) on the central meridian using the
standard equation as follows.
2𝜋𝑅 2 × 3.14 × 2.0 12.56
𝑑= × 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 (°) = × 10 = = 0.35 𝑐𝑚
360° 360 36
22
𝑤ℎ𝑒𝑟𝑒 𝜋 = = 3.14 (𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡); 𝑅 = 2.0 𝑐𝑚 (𝐹𝑟𝑜𝑚 𝑆𝑡𝑒𝑝 1)
7

Now, you got to know the length of division on central meridian (60ºE) for
spacing the parallels at 10º interval i.e.,d = 0.35 cm.

Step 5: Calculate the length of the radius of parallel (r1) using the standard
equation as follows:

𝑟 = 𝑅 cot ∅ = 2.0 × cot 20° = 2.0 × 2.747 = 5.494 𝑐𝑚

= 2.0 × cot 30° = 2.0 × 1.732 = 3.464 𝑐𝑚
= 2.0 × cot 40° = 2.0 × 1.192 = 2.384 𝑐𝑚
= 2.0 × cot 50° = 2.0 × 0.839 = 1.678 𝑐𝑚
= 2.0 × cot 60° = 2.0 × 0.577 = 1.154 𝑐𝑚

Step 6: Calculate the length of a division (d1) on the parallel at the given
interval using the standard equation as follows.

2𝜋𝑅 cos ∅ 2𝜋𝑅

𝑑 = × 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 (°) = × 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 (°) × cos ∅
360° 360°
2 × 3.14 × 2.0
= × 10 × cos ∅ = 𝟎. 𝟑𝟓 × 𝐜𝐨𝐬 ∅
360
Where, R= 2.0 cm; 𝜋 = 3.14; Interval = 10º.

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Now, we can calculate the length of division on each 20ºN, 30ºN, 40ºN, 50ºN,
and 60ºN parallels using the above formula.
𝑑 ° = 0.35 × cos ∅ = 0.35 × cos 20° = 0.35 × 0.940 = 0.33 𝑐𝑚
𝑑 ° = 0.35 × cos ∅ = 0.35 × cos 30° = 0.35 × 0.866 = 0.30 𝑐𝑚
𝑑 ° = 0.35 × cos ∅ = 0.35 × cos 40° = 0.35 × 0.766 = 0.27 𝑐𝑚
𝑑 ° = 0.35 × cos ∅ = 0.35 × cos 50° = 0.35 × 0.643 = 0.22 𝑐𝑚
𝑑 ° = 0.35 × cos ∅ = 0.35 × cos 60° = 0.35 × 0.500 = 0.18 𝑐𝑚

You can use the scientific calculator to get the cos (cosine) and cot
(cotangent) values directly or refer to annexures provided at the end of this
exercise. You have now all the calculated values for constructing the map
projection. Let us start the construction of the projection. Keep ready with you
a drawing sheet, pencil, ruler, compass, protractor and divider, etc.

Construction of the Projection:

Step 7: Take a drawing sheet and place it on a plane surface. With the help of
pencil and scale, draw a straight line vertically on centre of the sheet that is to
be represented as central meridian (60ºE). Using the divider, mark the
divisions on the central meridian with the measurement of d= 0.35 cm (refer
to Step 4) for spacing the parallels. Set compass with the calculated value of
radius i.e., r1 = 5.494 cm for 20° (refer to Step 5) and mark the point on
central meridian from the bottom of marking division. Now draw an arc going
through this division from the north pole point. With respective radii, the other
parallels are also be drawn.

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Step 8: Using the divider, mark the divisions on 20º parallel with the
measuremnt of 0.33 cm (d1) (refer to Step 6) for spacing the meridians.
Similarly, divide the remaining all parallels i.e.,30º, 40º, 50º and 60º with their
corresponding measurements 0.30 cm, 0.27 cm, 0.22 cm, and 0.18 cm,
respectively for spacing the meridians.

Step 9: Now,draw smooth free-hand curves by connecting through

corresponding division points on the parallels to represent the meridians.

Step 10: You have now completed the construction of graticules of

Polyconicprojection. Finally, you have to mark proper labels as given below.
All the graticules and the scale of projection must be writen on the drawing.

R.F. 1:320,000,000

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Unit – 2 Map Projections Exercise 5 - Construction of Conical Map Projections
……………………………………...……………………………………………………………………………………
METHOD-2: GRAPHICAL CONSTRUCTION

In the preliminary stage, we have to calculate the values for constructing the
map projection. Then, the calculated values are to be used for making the
projection in the next stage.

Calculations for Constructing the Projection:

Step 1: Calculate the radius of the generating globe (R) for the given scale.
We have obtained the value of radius is R = 2.0 cm.
Note the latitudinal extent i.e., from 20ºN to 60ºN and longitudinal extent i.e.,
from 20ºE to 100ºE and the interval i.e., 10º.
The central meridian would be 60ºE.
Step 2: Now, we have to calculate the value of the radius of each parallel (r),
division (d1) on the central meridian, and division (d2) on each parallel. Draw a
circle with the measurement of radius R = 2.0 cm from the center (O). With the
help of protractor, mark the angle for given interval (10°) as OIi’. Also mark
other angles of parallels 20ºN, 30ºN, 40ºN, 50ºN, and 60ºN as Oa, Ob, Oc,
Od, and Oe for making lines through these markings. Now, draw tangent lines
for parallel angles which meet the OP line at a’, b’, c’, d’, and e’. Measure the
length of aa’, bb’, cc’, dd’, and ee’, these will be the radii of 20ºN, 30ºN, 40ºN,
50ºN, and 60ºN parallels.

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Step 3: Measure the arc distance of Ii’ which will be the value of the division
on central meridian. Hence, Ii’ is the true distance between two parallels at 10°
interval.

Step 4: Now, you have to calculate the value of division on each parallel.
Draw an inner circle with the measurement of Ii’ from the center O. Draw lines
parallel to OI by connecting each parallel line in the inner circle as z-z’, y-y’, x-
x’, w-w’, and v-v’ corresponding to the parallels of Oa, Ob, Oc, Od, and Oe.
Then, measure each line with the help of divider and scale. These will be
divisions on respective parallel for spacing the meridians.

Construction of the Projection:

You can follow the procedure of construction as explained in the above

METHOD-1.

Step 7: Take a drawing sheet and place it on a plane surface. With the help of
pencil and scale, draw a straight line vertically on centre of the sheet that is to
be represented as central meridian (60ºE). Using the divider, mark the
divisions on the central meridian with the measurement of d for spacing the
parallels. Set compass with the calculated value of radius i.e., r1 for 20° and
mark the point on central meridian from the bottom of marking division. Now

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Unit – 2 Map Projections Exercise 5 - Construction of Conical Map Projections
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draw an arc going through this division from the north pole point. With
respective radii, the other parallels are also be drawn.

Step 8: Using the divider, mark the divisions on 20º parallel with the
measuremnt value d1. Similarly, divide the remaining all parallels i.e.,30º, 40º,
50º and 60º with their corresponding measurements for spacing the meridians.
Now,draw smooth free-hand curves by connecting through corresponding
division points on the parallels to represent the meridians.

Step 9: You have now completed the construction of graticules of Polyconic

projection. Finally, you have to mark proper labels as given below. All the
graticules and the scale of projection must be writen on the drawing.

R.F. 1:320,000,000

Now, you have understood the construction of Polyconic Projection and

Construction of Simple Conical Projection with One Standard Parallel.

You should now complete the following laboratory tasks for your practical
record before submitting it. You can also utilize the dataset given at your study
center for doing exercises. After completing the exercises, you need to submit
the practical record to the concerned academic counselor/lab instructor for
evaluation.

5.5 PRACTICAL EXERCISES

Exercise 5.1: Draw a graticule of Polyconic projection on a scale of
1:125,000,000 for an area extending from 30°S to 75°S and 15° W to 135°E.
Graticule interval is 15°.

Exercise 5.2: Construct graticules at 10º interval on conical projection with

one standard parallel on RF 1:200,000,000 for the extensions, 10ºN-60ºN and
100ºE-160ºE.

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84