Elements of a projection: reduced earth and the global properties

Detail

1. Reduced earth — what it is

• Reduced earth: a model of the earth made smaller to the scale of the map you want to draw. Picture a small globe. The graticule (the network of parallels and meridians) is drawn on this small globe first. It is then moved onto flat paper.
• Map projection means moving the graticule from this curved model onto a flat surface in a fixed, step-by-step way.
• The reduced earth is roughly a spheroid, not a perfect sphere. Its polar diameter is shorter than its equatorial diameter.
• Why the real earth bulges:
• The earth spins on its axis.
• The spin pushes material outward most strongly at the equator.

• Result: an oblate spheroid. This means a ball that is flattened at the poles and bulges at the equator [3].

• How big the bulge is:

• The equatorial radius and the polar radius differ by about 21 km. This flattening is about 1 in 300 [3].
• WGS-84 (World Geodetic System 1984) is the model of the earth that every GPS device uses. In it, the equatorial radius is 6,378,137.0 m, the polar radius is 6,356,752.3 m and the inverse flattening is 298.257223563 [4].
• Flattening f = (a − b) ÷ a. Here a is the equatorial (semi-major) radius and b is the polar (semi-minor) radius [4].

2. Radius of the reduced earth — the calculation

• Formula: R = earth's radius ÷ the denominator of the RF.
• RF (Representative Fraction) is the map scale written as a ratio, such as 1:250,000,000. It means 1 cm on the map stands for 250,000,000 cm on the ground.
• NCERT takes the earth's radius as 640,000,000 cm (6,400 km). This is a rounded figure. The WGS-84 equatorial radius is 6,378 km [4].

RF (scale)	Working	Radius of reduced earth (R)
1:250,000,000	640,000,000 ÷ 250,000,000	2.56 cm
1:300,000,000	640,000,000 ÷ 300,000,000	2.13 cm ≈ 2.1 cm

• Rule to remember: the larger the RF denominator (a smaller-scale map), the smaller the reduced earth.
• Why it is drawn as a sphere in practice:
• The polar and equatorial diameters differ by about 42 km in real life (about 21 km on each radius [3]).
• At 1:250,000,000, 42 km becomes about 0.17 mm on paper. That is thinner than a pencil line.
• So on small-scale world maps, the reduced earth is drawn as a sphere. The spheroid shape matters only for large-scale survey maps.

3. Parallels and meridians on the reduced earth

Feature	Parallels of latitude	Meridians of longitude
Shape	Full circles	Semicircles, from pole to pole
Plane	At right angles to the earth's axis	Pass through the earth's axis
Length	Unequal. The equator is the longest. They shrink towards the poles and become a point at 90° N and 90° S	All equal
Meeting	Never meet. They are parallel to each other	All meet at the two poles
Zero line	Equator (0°). It is fixed by nature, because it lies halfway between the poles	Greenwich (0°). It was chosen only by agreement (convention)

• How long is a parallel? Its length = the equator's length × cos(latitude). So the 60° parallel is half as long as the equator.
• Prime meridian: the 0° meridian. It runs through Greenwich (London, UK). Nothing in nature fixes it, so it was picked by agreement. (More on this in the latitude-longitude-time note.)

4. The four global properties

• Global properties: the four qualities of the real globe that a projection tries to keep when it moves the graticule onto flat paper: 1. Correct distance between any two points (kept on equidistant projections) 2. Correct shape of areas (kept on orthomorphic or conformal projections) 3. Correct size or area (kept on homolographic or equal-area projections) 4. Correct direction of one point from another (kept on azimuthal or true-bearing projections)

• Why all four cannot be kept:

• A sphere's surface is curved in two directions. It cannot be laid flat without stretching or tearing, just as an orange peel cannot.
• Every projection must stretch some parts and squeeze others.

• So no single projection keeps all four properties. The mapmaker picks the property the map's purpose needs most.

• Area and shape cannot both be kept over large regions:

• To keep shape, the map must stretch in all directions by the same amount at each point. Away from the line of true scale, that stretching blows up area.
• To keep area, the map must stretch one way and shrink the other way. That bends shapes out of form.
• No projection can be both conformal and equal-area [2].

5. Example — Mercator (1569) chooses shape over area

• The Mercator projection was introduced by the Flemish mapmaker Gerardus Mercator in 1569 [6].
• It is conformal. It keeps angles correct over small areas [6].
• Why sailors used it:
• Any straight line on it is a rhumb line or loxodrome, a line of constant compass bearing [6].

• A sailor can draw a straight line and steer one fixed course. So it became the standard map for sea navigation [6].

• The price it pays — area:

• Areas far from the equator look much larger than they really are [6].

• Greenland looks bigger than South America on it. In fact, Greenland is smaller than the Arabian Peninsula [7].

• The opposite choice: the cylindrical equal-area projection keeps area correct. But shapes near the poles become badly squashed (NCERT).

6. Tissot's indicatrix — a way to see distortion

• Tissot's indicatrix: a tool to see and measure how a projection distorts the earth. It was introduced by Nicolas Auguste Tissot in the 19th century [2].
• How it works, step by step:
• Draw a very small circle on the globe, about 0.1° of arc in radius [2].
• Enlarge it and project it onto the map at the same spot [2].

• On the map it always comes out as an ellipse [2]. A circle is the special case where both axes of the ellipse are equal.

• How to read it:

• Shape of the ellipse: if every ellipse stays a circle, the map is conformal [2].
• Size of the ellipse: if every ellipse has the same area, the map is equal-area [2].

• Spacing of the ellipses: shows whether distances stay true (equidistance) [2].

• On the Mercator projection the circles stay round but grow bigger towards the poles. This shows that shape is kept and area is not [7].

• It is the most common way to compare distortion within one projection and between different projections [2].

7. Indian link — the reference spheroid used by the Survey of India

• Before the National Map Policy 2005, Survey of India topographical sheets (topo sheets) used the Polyconic projection on the Everest datum (spheroid) [5].
• Datum: the reference spheroid and its fixed position, used as the base for measuring positions on the earth.
• Under NMP 2005, Open Series Maps (OSM) use the UTM projection (Universal Transverse Mercator) on the WGS-84 datum. Defence Series Maps (DSM) use the Everest or WGS-84 datum with the Polyconic or UTM projection [5].
• WGS-84 is the datum for all future ground surveys in India. Old Polyconic/Everest data is being converted to UTM/WGS-84 [5].

Prelims Hooks

• Radius of the reduced earth = earth's radius ÷ RF denominator. At 1:250,000,000 → 2.56 cm. At 1:300,000,000 → 2.1 cm (using 640,000,000 cm).
• The earth is an oblate spheroid: its polar diameter is shorter than its equatorial diameter. The radii differ by about 21 km, and the flattening is about 1/300 [3].
• WGS-84: equatorial radius 6,378,137 m, polar radius about 6,356,752 m [4].
• Parallels: unequal circles, shrinking to a point at the poles. Meridians: equal semicircles that meet at the poles.
• Greenwich is 0° by agreement only. The equator is 0° because of the earth's own geometry.
• Trap: "A projection can be both equal-area and conformal" is false [2].
• Pairs: Conformal/orthomorphic → shape. Homolographic → area. Equidistant → distance. Azimuthal → direction.
• Mercator (1569): conformal; rhumb lines are straight; area is exaggerated near the poles [6]. Greenland is smaller than the Arabian Peninsula [7].
• Tissot's indicatrix: an infinitely small circle on the globe always appears on the map as an ellipse [2].
• Survey of India Open Series Maps: UTM + WGS-84 (National Map Policy 2005) [5].

Mains Points

• Why every map is a compromise: a curved globe cannot be flattened without distortion. So the purpose of the map decides the projection:
• Navigation needs true direction (Mercator) [6].
• Comparing population density or forest cover needs true area (equal-area maps).

• Air routes and radio links need true distance or direction from one centre (azimuthal maps).

• Distortion shapes how people see the world: on Mercator maps, high-latitude lands such as Greenland look too big and tropical lands look too small [7]. Map literacy should therefore teach Tissot-style distortion checks [2].

• India's geospatial policy: India moved from the Polyconic/Everest system to UTM/WGS-84 under the National Map Policy 2005 [5]. This lines up Indian maps with GPS and global datasets (WGS-84 is the GPS datum [4]). It helps disaster mapping, infrastructure planning and sharing of open data.