31 Map Projection: Concept and classification

Prof. Masood Ahsan Siddiqui

Map projections are logical and methodical transformations that permit the orderly representation of the Earth’s spherical graticule on a flat map. Mathematically speaking, map projections are transformations of geographic co-ordinates (latitude, longitude) into the Cartesian (x, y) co-ordinate space of the map. A map projection is the manner in which the three dimensional spherical surface of the Earth is represented on a two-dimensional surface of a piece of paper. The term ‘projection’ comes from the idea of placing a light source within a transparent globe and projecting shadows of the meridians, and parallels onto a sheet of paper placed tangent to the globe. It can be defined as a “systematic arrangement of intersecting lines on a plane that represent and have one to one correspondence with the meridians and parallels on the datum surface”. Each map projection has certain strengths and weaknesses in terms of accuracy of shape, area, distance and direction. It is not possible for any of the projection to retain more than one of these characteristics over a large part of the earth. It is important to identify that because of the curvature of the earth, all map projections distort distance and directional relationships. The effect of curvature on the distances and directions can be safely ignored for a small area but for a larger area distortion cannot be avoided.

The classification should follow a standard pattern so that any regular (non- conventional) can be described by a set of criteria and conversely a set of criteria will define a regular projection. The classification scheme is as follows:

Classification on the basis of plane of projection:

First order:

Planar (azimuthal): Planar or azimuthal projections are accomplished by drawing lines from a given perspective point through the globe on a tangent plane. A tangent plane intersects th global surface at only one point and is perpendicular to a line passing through the center of the sphere. Thus, these projections are symmetrical around a chosen center or central meridian. Choice of the projection center determines the aspect, or orientation of the projection surface. Planar projections may be centered on the poles (polar aspect), at a point on the equator (equatorial aspect), or at any other orientation (oblique aspect).

Figure 1 Zenithal Projection

The origin of the projection lines i.e. the perspective point may also assume various positions. For example, it may be the center of the Earth (gnomonic), an infinite distance away (orthographic), or on the Earth’s surface opposite the projection plane (stereographic). Meridians are projected as straight lines radiating from the center (pole) and spaced at true angels. The parallels are complete circles.

The important features of this category of projection are:

Ø Only a part of the earth surface is visible,

Ø Distortion will occur at all the four corners,

Ø Distance for the most part is preserved.

Conical: Conical projections are accomplished by intersecting, or touching, a cone with the global surface and mathematically projecting lines on this developable surface. A tangent cone intersects the earth’s surface to form a circle. On this line of intersection, termed the standard parallel, the map will be relatively error-free and possess the quality of equidistance. Cones may also be secant, and intersect the global surface forming two circles which will possess equidistance. The word “secant”, in this instance, is only conceptual, not geometrically accurate. As with planar projections, the conical aspect may be polar, equatorial, or oblique. Theses have been shown in the following figure. It has been assumed that the center, the pole and the vertex of the cone are in a straight line. The cone touches the globe at two points. These points are neither the pole the equator. It means the points are lying in between the pole and the equator. The parallels are arcs of circles drawn using the vertex as the center and the meridians are straight lines radiating from the vertex at equal angles.

Figure 2: Conical Projection

The important features are:

Ø Scale for the most part is preserved,

Ø Distance is distorted towards the bottom,

Ø Area is distorted.

Cylindrical: Cylindrical projections are obtained by intersecting, or touching, a cylinder with the global surface. The surface is mathematically projected on the cylinder, which is then “cut”, and “unrolled”. If the cylinder is rotated 90 degrees from the vertical, i.e. the long axis becomes horizontal, the aspect becomes transverse and the central line of the projection becomes a chosen standard meridian. A tangent cylinder will intersect the global surface on only one line to form a circle as with a tangent cone. This central line of the projection is generally the equator and will possess equidistance..

Figure 3: Normal Cylindrical Projections

Because of tangency, the scale is true at or near the central line. The meridians are parallel and equi-spaced vertical straight lines intersecting the equator at right angles and divide the globe into 360 parts. The amount of distortion increases towards the poles.The parallels are horizontal lines drawn from east to west. All the parallels are equal to the length of equator. The important features are:

Ø The position of the countries near the equator is true to a large extent,

Ø Distortion is maximum towards the poles,

Ø Area for the most part is preserved.

Second order:

a) Tangent b) Secantc) Poly superficial (a series of successive projection surfaces).

These three approaches are applied to minimize the area of distortion by intersecting the globe instead of touching it. On the basis of the nature of contact to the generating globe the projections may be called as tangent,secant and polysuperficial.

A flat plane will intersect the globe along a circular line whereas a cone or cylinder along two circular lines. Thus by intersecting the generating globe the extent of distortion free area is increased from a point to a line in case of a flat plane and from one line to two lines for cone and cylinder. Meaning thereby there will be two standard parallels, instead of one. The distortion free area may further be increased if more than one identical shape surfaces are made tangent to the globe. It will result into Polyconic (for successive cones) or polycylindrical (for successive cylinders) or polyhydric (successive plane) projections.

Third order:

a) Normal b) Transverse c) Oblique

Alignment of the generating projection surface with respect to the datum surface results to the three types of projections. It is said to be normal projection, if the axis of symmetry of the plane of projection is coincident with the rotational axis of the generating globe.

Normal Transverse Oblique

Figure 4: Cylindrical Projections

If however the axis of symmetry of the projection surface is at right angles to that of the globe, a transverse case is attained. That is, the axis of the cylinder is at 90 to the polar axis. If the axis of symmetry of the projection surface is in between the two i.e. the two intersect at an acute angle we get an oblique case.

Method of projection:

Perspective projections: The projections in which it has been imagined that a series of rays are emerging from a source of light and they pass through a transparent globe to cast the shadow of the parallels and meridians on the projection surface. The surface may be cone, cylinder, or plane. All of these surfaces are developable meaning, they can be unfolded without distortion. These are also known as geometric projections. Changing the position of the light source alters the pattern of parallels and meridians on the map, resulting in maps that have different geometric properties.

Figure 5: Perspective Projections

Non-perspective: It is a modified perspective projection for achieving certain desired properties such as true area, shape, directions and bearing. If the mutually intersecting lines are drawn on the projection surface without any reference to the source of light i.e. no rays or plane are involved, but are achieved by a mathematical operation. Such projections are called as non-perspective.

Properties of the map projection

The classification of map projection based on the quality gives three mutually exclusive varieties representing the intrinsic geometrical properties are:

1. Homolographic (equal area or equivalency)

2. Orthomorphic (conformality or true shape)

3. Azimuthal (true bearing)

4. Equidistant

Equivalency indicates that the areas of figures are retained on the projection surface as compared to the datum surface. This however takes place at the cost of shapes and angles getting distorted. This is used best for spatial distribution and relative sizes of spatial features, e.g. political units, population distribution, land use and natural resources etc. Albers conical, Lamberts, cylindrical and Sinusoidal are the typical projections of this category. Typically, reference lines such as the equator or a meridian are chosen to have equidistance and are termed standard parallels or standard meridians.

Conformality refers to retention of shapes or forms also retention of angles. This property is limited to close points and not the areas of significant dimensions. The angel of intersection of meridians and parallels is as it is on the globe i.e. the meridians intersect parallels at right angles, just at globe. These are also called as conformal or orthomorphic projections. These are widely used for topographic mapping and navigation purposes. Mercators and its various variants are typical example

True Directionis characterized by a direction line between two points, which crosses reference lines, e.g. meridians, at a constant angle or azimuth. These are termed rhumb lines and this property makes it comparatively easy to chart a navigational course. However, on a spherical surface, the shortest surface distance between two points is a great circle along which azimuths constantly change. Thus, a more desirable property may be where certain great circles are represented by straight lines. This characteristic is most important in aviation. All meridians are great circles, but the only parallel that is a great circle is the equator. This particular property can be retained with some other qualities of the projections. For example, conformal projections retain the true directions. Typical example of this type are the variants of azimuthal and Mercators projections.

Equidistant means there is correct representation of distances between two points on the projection surface as compared to the distance on the datum surface. This property is limited to a certain specified set of points and is not a general property between points on the two surfaces e.g. along central meridians in case of cylindrical projection and along standard parallels in the conical projections. Such projections are generally used in atlas maps. Typical examples are Azimuthal Equidistant and Equidistant conic projections.

Although a great number of projections have been devised, the majority of them are geometric or mathematical variants of the basic direct geometric projection families described above. Choice of the projection to be used will depend upon the true property or combination of properties desired for effective cartographic analysis.

Important Map projections

Universal Transverse Mercator’s Projection(UTM)

The UTM is a modified version of Mercator’s projection. It is probably the most famous projection of all time and the best map projection for navigators because of its unique quality that an earth-bound traveler, following a line of constant compass direction, can chart the course on the Mercator projection with a Straight Line. This quality was not accidental; Gerardus, the developer, used a logarithmic displacement of the parallels of latitude to obtain this effect as a service to navigators. Because the standard Mercator map is tangent to the Earth at the Equator, it represents size and scale only in lower latitudes, and grossly overestimates them near the poles. Consider that the North Pole is actually a point but in cylindrical projections it explodes to a line.

The UTM is a conformal projection and preserves the shape of the features such as coastlines and rivers and largely used in GIS. Lambert, J.H. (1772) modified the equatorial Mercator’s projection by changing the aspect of projection to transverse with a view to minimize the distortion in a narrow strip running from pole to pole. Later Gauss in 1822 and Kurger in 1912 and 1919 modified and adjusted for polar flattening. Thus it is also known as Gauss-Kurger projection. It is a cylindrical projection with the axis of projection lying perpendicular to the axis of symmetry of datum surface i.e. the ellipsoid. That is, instead of using the standard parallel as in the case of Mercator projection, the transverse Mercator projection uses the standard meridian. In order to minimize the distortions that occur in the transverse projection, further modifications were made. The modified transverse projection is called as Universal Transverse Mercators projection. It loses the properties of straight meridians and parallels of the standard Mercator projection. Transverse Mercator also loses the straight rhumb lines, but it is a conformal projection. Scale is true along the central meridian or along two straight lines equidistant from and parallel to the central meridian. It is repeatedly applied by using multiple cylinders that touch the globe at 6 intervals. The cylinder is made to intersect the globe at two standard meridians that are 180 km east and west of the central meridian.

The UTM defines two-dimensional horizontal positions, consists of 60 zones of 6° of longitudes in width, each extending 3° east and 3° west of a central meridian extending between 80° S to 84° N latitude. An additional one-half degree (40-km) on each side provides for overlap into the adjacent zone. The first zone starts at 180° W to 174° W with 177° W ascentral meridian. The final zone, zone 60 starts at 174° E and extends east to the International Date Line.

Figure 6: Transverse Mercator’s Projection

Figure 7: Overlap in UTM

Table: 1

Universal Transverse Mercators zone numbers with central and bounding meridians

Zone
no.

Central
meridian

Bounding
meridians

Zone
no.

Central
meridian

Bounding
meridians

Zone
no.

Central
meridian

Bounding
meridians

177 W

180,174w

57W

60,54 W

63E

60,66E

171 W

174,168w

51 W

54,48 W

69E

66,72E

165 W

168,162w

45 W

48,42 W

75E

72,78E

159 W

162,156w

39 W

42,36 W

81E

78,84E

153 W

156,150w

33 W

36,30 W

87E

84,90E

147 W

150,144w

27 W

30,24 W

93E

90,96E

141 W

144,138w

21 W

24,18 W

99E

96,102E

135 W

138,132w

15 W

18,12 W

105E

102,108E

129 W

132,126w

09 W

12,06 W

111E

108,114E

123 W

126,120w

03 W

06,00 W

117E

114,120E

117 W

120,114w

03E

00,06E

123E

120,126E

111 W

114,108w

09E

06,12E

129E

126,132E

105W

108,102w

15E

12,18E

135E

132,138E

99 W

102,96 W

21E

18,24E

141E

138,144E

93 W

96,90 W

27E

24,30E

147E

144,150E

87 W

90,84 W

33E

30,36E

153E

150,156E

81 W

84,78 W

39E

36,42E

159E

156,162E

75 W

78,72 W

45E

42,48e

165E

162,168E

69 W

72,66 W

51E

48,54E

171E

168,174E

63 W

66,60 W

57E

54,60E

177E

174,180E

Figure 8: Universal Transverse Mercator (UTM) System

The central meridian is a great circle and is the only line of latitude or longitude represented by a straight line in the map projection. Each UTM zone is divided into horizontal bands of 8° latitude. These band are identified by letters, south to north, starting at 80° S with letter C and ending with letter X. letter I and O are omitted whereas letters A, B, Y and Z are reserved for Universal Polar Stereographic coordinate system.

This projection system is largely used for large mapping projects (like the national map series produced by the U.S. Geological Survey and the Ordnance Surveys of Britain and France. The UTM coordinate system is based on the Transverse Mercators projection. The students of geography know that equatorial Mercator’s projection distorts areas so much towards the poles that Greenland appears about seven times bigger than India. However the distortion is minimum laterally along the equator. In the geometrical sense, the transverse cylinder is tangent along the central meridian of each zone.

A scale factor of 0.9996 is applied along the central meridian of each zone. The scale factor is simply the ratio between the actual map scales at a particular location to the nominal map scale. The actual map scale is always the ratio of map distance to ground distance. This results in the transverse cylinder becoming Secant to the datum and the scale distortion is more favorably distributed over the zone. The choice of scale factor of 0.9996 was made so as to limit the scale error to 1: 2500 within the zone. There are special UTM zone between 0° and 36° longitude above 72° latitude and a special zone 32 between 56° and 64° north of latitude.

Figure 9: Universal Transverse Mercator Grid System

(Northern and Southern hemisphere)

For areas beyond the 80° S and 84° N, the Universal Polar Stereographic projection (UPS) is used. This is an azimuthal projection obtained by projecting from the opposite pole. All of either the northern or southern hemispheres can be shown as a map, but not both. This projection produces a circular map with one of the poles at the center. Polar Stereographic stretches areas toward the periphery, and scale increases for areas farther from the central pole. Meridians are straight and radiating; parallels are concentric circles. Even though scale and area are not constant, this projection, like all stereographic projections, possesses the property of conformality.

A grid system is superimposed upon the Polar Stereographic projection with a provision of half-degree overlap. The vertical grid lines are parallel to the meridians 0° and 180° whereas the horizontal grid lines are parallel to meridians 90° W and 90° E longitudes. Though the origin of the grid is the pole, but an arbitrary false easting and false northing of 2000,000 meters is assigned. The scale factor is set arbitrarily as 0.994 at pole. The scale is true to the grid at 810 whereas at 800, the limit of UPS system, the scale is large by a factor of 1.0016. The UTM coordinate system has been widely used for large-scale topographic mapping all over the world and even for mapping of planet Mars.

The Lambert Conformal Conic Projection

A conformal projection is one in which all the angles are preserved and the scale factor is constant in any direction. The Lambert Conformal Conic projection was developed by Johann Heinrich in 1772 to provide a minimum of angular and scalar distortion. It is conformal map projection of the so-called conical type, on which all geographic meridians are represented by straight lines that meet in a common point outside the limits of the map, and the geographic parallels are represented by a series of arcs of circles having this common point for a center. Meridians and parallels intersect at right angles, and angles on the earth are correctly represented on the projection. This projection may have one standard parallel along which the scale is held exact; or there may be two such standard parallels, both maintaining exact scale. At any point on the map, the scale is the same in every direction. It changes along the meridians and is constant along each parallel. Where there are two standard parallels, the scale between those parallels is too small; beyond them, it is too large. The straight lines plotted are approximately great circles. These are used for VFR (visual flight rules) aeronautical charts.

Figure 10: Lamberts conic conformal Projection with one and two standard parallels

It is a secant conical projection. The parallels are chosen to minimize errors over the area of interest. On the standard parallels, arcs of latitude are represented in their true lengths or to an exact scale. The scale is too small between the standards parallels and to large beyond them. This projection is best suited for mid- latitude area especially with east- west dimension.

The major property of this projection is its conformality. At all coordinates, meridians and parallels cross at right angles. The correct angles produce correct shapes. Also, great circles are approximately straight. This projection is useful for landmark flying. Lambert Conformal Conic is the State Plane coordinate system projection for states having an east-west expanse. Since 1962, Lambert Conformal Conic has been used for the International Map of the World between 84° North and 80° South. It possesses true shape of small areas whereas Albers possesses equal area.

Polyconic Projection: Polyconic projection was developed in 1820 by Ferdinand Hassler specifically for mapping the eastern coast of the U.S. Polyconic projections are made up of an infinite number of conic projections tangent to an infinite number of parallels. These conic projections are placed in relation to a central meridian. Polyconic projections compromise properties such as equal area and conformality, although the central meridian is held true to scale.

Figure 11: Conical Projection

All parallels are arcs of circles, but not concentric. All meridians, except the central meridian, are concave toward the central meridian. Parallels cross the central meridian at equal intervals but move farther apart at the east and west peripheries.

The significant properties are:

All the parallels are standard parallels and the scale is correct along all the parallel and the central meridian. All meridians, except the central meridian, are concave toward the central meridian.
Intersection of parallels and meridians are rectangular,
Scale is distorted away from the central meridian,
Polyconic projections compromise properties such as equal area and Conformality
Suitable for areas with large latitudinal and small longitudinal extent.

The survey of India (SoI) uses Polyconic projection for its topographic map series on 1:250, 000, 1:50000 and 1:25000 series. The USGS also uses it for, the topographic map series on 1:24000.

Table: 2

Choice of map projection

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