Map Projections

1. Why map projections are needed

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• A map projection is the method of moving the graticule (the net of parallels and meridians) from the globe onto flat paper. Class 11, Map Projections also calls it the study of the methods tried for this transfer.
• The globe is true but impractical. It shows shape, size, direction and distance correctly. But:
• it is expensive and hard to carry
• it cannot show small detail

• comparing two regions on it is hard.

• Detailed study needs large-scale maps on flat paper. Class 11, Introduction to Maps lists projection as one of the essentials of map making.

• The core problem: a sphere is a non-developable surface, which means it cannot be flattened.
• Paper laid on a globe touches it only along one line or at one point.

• If light is thrown from the centre, the image gets more distorted the farther it is from that touching point.

• Class 6's orange-peel test shows the same thing: the peel tears before it lies flat.

• On the globe, meridians are semicircles and parallels are circles. On paper they become straight or curved lines that cross each other.
• Some distortion can never be avoided. That is why so many projections exist.

2. Elements of a projection: reduced earth and the global properties

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• Reduced earth: a model of the earth shrunk to the map's scale. The graticule is drawn on it first. It is roughly a spheroid, with the polar diameter shorter than the equatorial one.
• Its radius: R = earth's radius (taken as 640,000,000 cm) ÷ the RF denominator.

• At 1:250,000,000, R = 2.56 cm. At 1:300,000,000, R = 2.1 cm.

• Parallels are circles at right angles to the earth's axis. They are unequal in length, from a point at each pole to the full circle of the equator.

• Meridians are equal semicircles that meet at the poles. Greenwich is 0° only by agreement (details in latitude-longitude-time).

• Global properties: the four qualities a projection tries to keep: 1. correct distance 2. correct shape 3. correct area 4. correct direction.

• No single projection keeps all four. Over large regions, true area and true shape cannot exist together.

• Beyond NCERT: Tissot's indicatrix is a set of small circles drawn across a map. Where they swell or squash, you can see how the projection distorts area or shape.

3. Classification by drawing technique and by source of light

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By drawing technique

• A perspective projection is drawn by casting the shadow of the globe's graticule from a light source onto a developable surface.
• A non-perspective projection is drawn without light or shadow.
• A conventional projection (also called mathematical) is built only from formulae and has little link to any projected image. Mercator's is the best-known example.

By source of light

Projection	Where the light is	Key property (beyond NCERT)
Gnomonic projection	Centre of the globe	Every great circle is a straight line, so navigators plot the shortest route on it
Stereographic projection	On the globe's surface, directly opposite the touching point	Shapes of small areas are true (conformal); used for polar maps
Orthographic projection	At infinity, so the rays are parallel	Looks like the globe seen from space; used for pictures of hemispheres

• Navigators draw the great-circle route on a gnomonic chart, then copy it onto a Mercator chart as a chain of rhumb lines.
• Exercise trap: light at the centre = gnomonic.

4. Classification by developable surface and by position of the tangent

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• A developable surface can be flattened without stretching, tearing or creasing. A cylinder, a cone and a plane are developable.
• A non-developable surface cannot be flattened this way. A sphere or globe is non-developable.
• The developable surfaces give three families:
• Cylindrical projection: a paper cylinder is wrapped round the globe (usually touching the equator) and then cut open. Parallels and meridians become straight lines crossing at right angles.
• Conical projection: a cone touches the globe along one parallel and is then cut open. Parallels become arcs and meridians become straight lines spreading out from the pole. It is best for mid-latitudes.

• Zenithal projection (also called azimuthal): a flat plane touches the globe at one point, usually a pole. Directions from that centre point are true.

• The tangent is the line or point where the surface touches the globe. By its position:

• Normal projection (also called equatorial): it touches at the equator.
• Oblique projection: it touches between the pole and the equator.

• Polar projection: it touches at a pole.

• Matching rule: cylindrical for the tropics and world maps, conical for mid-latitudes, polar zenithal for the Arctic and Antarctic.

• Beyond NCERT: a cylinder turned on its side to touch along a meridian gives the transverse Mercator. UTM uses it (section 9).

5. Classification by preserved property

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Class	What stays true	Cost or use
Equal area projection (homolographic)	Every graticule cell has the same area as on the globe	Shapes distort; used for maps of population, crops and forests
Orthomorphic projection (true shape, conformal)	Shape of small areas, because scale is equal in all directions at each point	Area is sacrificed, as on Mercator
Azimuthal (true bearing)	Direction of every point from the centre	Only from the centre point
Equidistant projection (true scale)	Distance, but only along chosen parallels or meridians	No map keeps scale true everywhere

• Example of equidistant: the simple (equidistant) cylindrical projection keeps true scale along all meridians and along the equator.
• Azimuthal equidistant projection (beyond NCERT): it shows true distance and true direction from its centre point.
• Centred on the North Pole, it forms the UN emblem.

• Centred on a city, it is used to plan air routes and radio links. For example, the distance from Delhi to any capital can be read straight off it.

• Trap 1: equal area and orthomorphic cannot both hold on one map.

• Trap 2: NCERT's exercise "homolographic vs orthographic" most likely means orthomorphic.

6. Simple conical projection with one standard parallel

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• A cone touches the globe along one parallel, the standard parallel. Scale is true along it, and distortion grows with distance from it.
• NCERT construction example: an area from 10°N to 70°N and 10°E to 130°E, at 1:250,000,000, with a 10° interval.
• R = 640,000,000 ÷ 250,000,000 = 2.56 cm.
• The standard parallel is 40°N and the central meridian is 70°E.
• The radius of the standard parallel is the cone's slant length.
• The arc distance between parallels is marked along the central meridian.

• Meridian spacing is marked along the standard parallel. Meridians are then drawn as straight lines to the pole.

• Properties of the simple conical projection with one standard parallel:

• parallels are equally spaced arcs with one common centre
• meridians are straight lines that meet at the pole and cross the parallels at right angles
• scale is true along all meridians and along the standard parallel only
• the pole is shown as an arc, not a point

• it is neither equal area nor orthomorphic.

• Limitations: it is useless for a world map, because distortion is extreme in the opposite hemisphere. It is also poor for large areas within one hemisphere.

• Uses: mid-latitude areas that are narrow north-south but long east-west. Examples: the Trans-Siberian Railway, the Canadian Pacific Railway, the USA-Canada boundary and the Narmada valley.
• Beyond NCERT: conics with two standard parallels (Lambert conformal conic, Albers equal area) suit mid-latitude countries better.

7. Cylindrical equal area projection and Gall-Peters

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• The cylindrical equal area projection is also called Lambert's projection (J.H. Lambert, 1772).
• The globe is projected with parallel rays onto a cylinder that touches it at the equator.

• Parallels and meridians are straight lines crossing at right angles.

• NCERT construction example: a world map at 1:300,000,000 with a 15° interval.

• R = 2.1 cm, so the equator = 2πR = 13.2 cm.
• The equator is divided into 24 parts of 0.55 cm each.

• Each parallel is drawn at a height of R × sine of its latitude, so the parallels crowd together towards the poles.

• Properties:

• the polar parallel is as long as the equator

• scale is true only along the equator.

• How area stays equal: the map stretches land east-west, and squashes it north-south by the same amount. So shapes get badly distorted poleward, and the projection is not orthomorphic.

• Uses: the belt between 45°N and 45°S, and maps of tropical crops: rice, tea, coffee, rubber and sugarcane.
• Gall-Peters projection (beyond NCERT): a cylindrical equal-area variant.
• James Gall described it in 1855. Arno Peters promoted it from 1973 as fairer to Africa, South America and South Asia.
• Critics point out that its tropical shapes look stretched.
• Boston public schools adopted it in 2017.
• The lesson: every world map carries a point of view.

8. Mercator's projection, rhumb lines and Web Mercator

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• Mercator's projection was made by Gerardus Mercator in 1569. It is a mathematical (conventional), cylindrical and orthomorphic projection.
• How it keeps shape: poleward, the gap between parallels is widened by exactly as much as each parallel is stretched (both by the secant of latitude). So the scale along a meridian equals the scale along a parallel at every point.
• NCERT construction example: at 1:250,000,000 with a 15° interval.
• R = 1 inch, so the equator = 6.28 inches.

• Distances of parallels from the equator come from a table: 15° = 0.265, 30° = 0.549, 45° = 0.881, 60° = 1.317 and 75° = 2.027 inches.

• Properties:

• parallels and meridians are straight lines crossing at right angles
• every parallel is as long as the equator, so the 30° parallel is 1.154 times its length on the globe

• parallels get farther apart towards the poles.

• Limitations:

• area is hugely enlarged at high latitudes, about 4 times at 60°
• Greenland (about 2.2 million sq km) looks as big as Africa (about 30 million) and the USA (about 9.8 million)

• the poles cannot be shown at all.

• NCERT errors to flag:

• NCERT says Greenland is 1/10th of the USA. It is about two-ninths.
• NCERT calls Mercator "Dutch". He was Flemish, born Gerard de Kremer.

• NCERT calls the projection "azimuthal". It is conformal, not azimuthal.

• Uses: navigation, because any straight line on it is a rhumb line (loxodrome).

• A rhumb line crosses every meridian at the same angle, so it keeps one constant compass bearing.

• It is longer than the great-circle route (Fig. 4.6), so long voyages combine the two.

• Mercator is also used for atlas maps of drainage, ocean currents, winds, temperature and rainfall.

• Web Mercator (EPSG:3857) is a simplified, spherical form of Mercator.
• Google Maps adopted it around 2005, and most online maps use it today.
• It keeps local shapes, which makes zooming smooth.
• It repeats Mercator's high-latitude enlargement, which shapes how people picture country sizes.

9. Choosing a projection: world maps and the mapping of India

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• The choice depends on the purpose of the map and on the size and shape of the region.
• World maps:
• Goode's homolosine projection (J. Paul Goode, 1923) is equal area. It is split ("interrupted") over the oceans to keep the continents compact, and is used in atlases for world distribution maps.

• The Robinson projection (Arthur Robinson, 1963) is a compromise. It is neither equal area nor true shape, but keeps overall distortion low. The National Geographic Society used it from 1988 to 1998, then switched to the Winkel Tripel.

• Other matches: navigation charts use Mercator, polar regions use zenithal, and mid-latitude countries use conics.

• India, old system: the Survey of India's older topographical sheets used the polyconic projection on the Everest spheroid.
• Each parallel is drawn as an arc of its own cone, touching the globe at that parallel.

• It is accurate near the central meridian of each sheet.

• India, new system: under the National Map Policy 2005, the Open Series Maps use the Universal Transverse Mercator (UTM) on the WGS-84 datum.

• UTM covers the earth from 80°S to 84°N in 60 zones, each 6° of longitude wide, with each zone on a transverse Mercator projection.
• India falls in zones 42 to 47.
• GPS units and GIS layers use UTM's metric grid (see remote-sensing-gis; sheet details are in topographical-maps).

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Exam angles

Prelims — high-yield facts and traps

• Light source matches: at the centre = gnomonic (great circles are straight lines); on the surface opposite the touching point = stereographic; at infinity = orthographic.
• Developable surfaces: cylinder, cone and plane. The globe is non-developable.
• Surface matches: cylindrical for the tropics, conical for mid-latitudes, zenithal for the poles.
• Tangent positions: normal touches at the equator, oblique between the pole and the equator, polar at a pole.
• Property names: homolographic = equal area; orthomorphic = true shape; azimuthal = true bearing from the centre; equidistant = true scale only along chosen lines.
• Mercator:
• cylindrical and conformal
• rhumb lines are straight
• the poles cannot be shown
• Greenland looks as big as Africa

• used for navigation and atlas maps of winds and currents.

• Cylindrical equal area:

• true area but distorted shape
• the polar parallel equals the equator

• best for 45°N–45°S and for tropical crops.

• Simple conical with one standard parallel:

• true scale along the meridians and the standard parallel
• neither equal area nor orthomorphic

• used for the Trans-Siberian, the CPR, the USA-Canada boundary and the Narmada valley.

• NCERT exercise answers: least suitable for a world map = conical; neither area, shape nor direction correct = simple conical; true shape and direction but area exaggerated = Mercator.

• UTM and India:
• UTM has 60 zones of 6°, from 80°S to 84°N
• India lies in zones 42–47
• the Open Series Maps use UTM on WGS-84

• the old Survey of India sheets were polyconic on the Everest spheroid.

• UN emblem: azimuthal equidistant projection centred on the North Pole.

• Statement traps:
• "Mercator is an azimuthal projection": FALSE.
• "One projection can be both equal area and orthomorphic": FALSE.
• "Greenland is 1/10th of the USA": FALSE (it is about 2/9).

Mains — GS-I themes

1. "No map projection represents the globe truly." Explain using the four global properties, and how the choice of projection follows its purpose.
2. Compare Mercator and the cylindrical equal area projection: their properties, limitations and uses.
3. Maps as worldviews: the Mercator-Peters debate, and how Web Mercator shapes the way people see country sizes.
4. Why did the Survey of India move to UTM and global datums? Discuss what this means for GIS, GPS and navigation.

Current-affairs hooks

• Online map platforms and India's own map services (Bhuvan, MapmyIndia), and the projections they use.
• The Survey of India's digital Open Series maps and new national datums and reference frames (verify current).
• Schools and atlases adopting equal-area world maps, and public debates on the "true size" of Africa.
• Arctic shipping routes, where the choice of projection affects route planning (see world-transport-communication).