Classification by developable surface and by position of the tangent

Detail

1. Why we need a "developable" surface

• The problem: the Earth is close to a sphere. A sphere cannot be laid flat without stretching, tearing or folding it, so every flat map has some error (distortion).
• Map projection means drawing the network of parallels and meridians (the graticule) from the globe onto a flat sheet.
• Developable surface is a surface that can be flattened without stretching, tearing or creasing. There are three: a cylinder, a cone and a plane (a flat sheet).
• Cut a paper tube along its length and it lies flat. A paper cone does the same.

• The USGS sorts many map projections by these same three surfaces [2].

• Non-developable surface is a surface that cannot be flattened this way. A sphere or globe is non-developable. Peel an orange and try to press the skin flat: it splits.

• How the method works (cause → process → result):
• First, the globe's grid is projected onto a developable surface that is wrapped round it or touching it.
• Then that surface is cut open and laid flat.
• The result is a flat map whose shape depends on which surface was used.

2. Three families by developable surface

Family	Surface and how it touches the globe	Parallels	Meridians	Strength	Best for
Cylindrical	Paper cylinder wrapped round the globe, usually touching the equator, then cut open along its length	Straight lines	Straight lines, crossing parallels at right angles	Low error near the equator	Tropics, world maps
Conical	Cone placed over the globe, touching along one parallel, then cut open (Fig. 4.2)	Arcs of circles	Straight lines spreading out from the pole	Low error along the touching parallel	Mid-latitudes
Zenithal (azimuthal)	Flat plane touching the globe at one point, usually a pole	Circles round the centre (polar case)	Straight lines spreading out from the centre	Directions from the centre point are true	Polar regions

Cylindrical projection

• Cylindrical projection is a projection made by wrapping a paper cylinder round the globe and then cutting it open.
• In the normal case the cylinder touches the globe along the equator, so the map is most correct near 0° latitude.
• Parallels and meridians become straight lines that cross at 90°. This gives a rectangular grid.
• The USGS describes it as a cylinder that is tangent (touching along one line) or secant (cutting into the globe along two lines), which is then "cut lengthwise and laid flat" [2].
• Mercator projection: made by the Flemish map-maker Gerardus Mercator in 1569 [4].
• It is usually called cylindrical, but its spacing is worked out by mathematics, not by simply projecting light onto a cylinder [4].
• It is conformal, which means shapes are correct over small areas [4].
• Any straight line on it is a line of constant compass bearing (a rhumb line). So a sailor can draw a straight course on it. This is why it is used for navigation charts [4].

Conical projection

• Conical projection is a projection made by placing a paper cone over the globe so that it touches along one parallel, then cutting it open (NCERT Fig. 4.2).
• The parallel the cone touches is the standard parallel. Along this line the map scale is true.
• Parallels become arcs that share one centre. Meridians become straight lines spreading out from the pole.
• When the cone's axis lines up with the Earth's polar axis, all meridians are straight, equally spaced lines from the centre of these arcs [2].
• Why it suits mid-latitudes: a cone touching around 30°–60° latitude fits closely over a belt that runs a long way east–west. Examples are the USA, Europe and the main body of India north of the Tropic of Cancer.
• A cone can also be secant, cutting the globe along two standard parallels. This lowers error across a wider belt [2].

Zenithal (azimuthal) projection

• Zenithal projection (also called azimuthal) is a projection made by touching a flat plane to the globe at one point, usually a pole.
• Key property: directions (azimuths) from the centre point to every other point are true [2].
• Error is lowest at the centre point and grows as you move away from it.
• Its main use is mapping the Arctic and Antarctica, and showing routes and directions from one central place.

3. Classification by position of the tangent

• Tangent here means the line or point where the developable surface touches the globe. Moving this contact line or point changes which part of the map is most correct.
• The USGS calls these positions aspects, for example the polar aspect (tangent at the pole) and the equatorial aspect (tangent at the equator) [2].

Type	Where the surface touches	Typical example	Most accurate zone
Normal (equatorial)	At the equator	Cylinder touching the equator	Tropics, about 0°–23½° N/S
Oblique	At a point between the pole and the equator	Plane centred on, say, 30° N	Area round the chosen point
Polar	At a pole (90° N or 90° S)	Zenithal plane on the North Pole	Arctic and Antarctic

• Normal projection (also called equatorial) is one where the surface touches the globe at the equator.
• Oblique projection is one where the surface touches at a point between the pole and the equator. It is used for a region centred on a mid-latitude place.
• Polar projection is one where the surface touches at a pole.
• NCERT's exercise asks you to tell normal and oblique projections apart. Learn this pair well.

4. Matching rule of thumb

• Cylindrical → tropics and world maps. It touches the equator, so it is most correct at low latitudes.
• Conical → middle latitudes. It touches along a mid-latitude parallel.
• Polar zenithal → Arctic and Antarctic. It touches at the pole.
• The logic behind the rule: pick the surface whose contact line or point lies closest to the area you are mapping. Distortion is lowest there.

5. Transverse cylinder and UTM

• How it works: turn the cylinder on its side so that it touches the globe along a meridian instead of the equator. The result is the transverse Mercator projection.
• Error is now lowest along that central meridian.

• This suits areas that are long from north to south.

• UTM (Universal Transverse Mercator) is a plane grid system named after the transverse Mercator projection it is built on [3]:

• It splits the world into 60 north–south zones, each 6° of longitude wide [3].
• Zone 1 covers 180°–174° W. The numbers rise eastward to Zone 60 at 174°–180° E [3].
• The central meridian of each zone gets an easting value of 500,000 metres. Points to its west have smaller values and points to its east have larger ones [3].
• Northings (distances north, in metres) are counted from zero at the equator [3].

• Each zone uses its own transverse cylinder. So the projection "touches" the Earth 60 times, and error stays low within each strip.

• UTM is covered in detail in section 9.

Prelims Hooks

• Developable surfaces are the cylinder, cone and plane. The sphere or globe is non-developable. A favourite trap is to list "sphere" as developable.
• Zenithal = azimuthal. Directions from the centre point are true.
• In the normal case, a cylindrical projection has parallels and meridians as straight lines crossing at right angles.
• In a conical projection, parallels are arcs and meridians are straight lines spreading out from the pole. It is best for mid-latitudes.
• Tangent at the equator gives a normal (equatorial) projection. Between the pole and the equator gives oblique. At a pole gives polar.
• Region–projection match: tropics → cylindrical, mid-latitudes → conical, Arctic and Antarctica → polar zenithal.
• Mercator projection: made by Gerardus Mercator in 1569. It is conformal, and straight lines on it are constant-bearing rhumb lines, so it is used for navigation [4].
• Transverse Mercator: the cylinder touches along a meridian, not the equator. It is the base of UTM: 60 zones, each 6° wide, with the central meridian easting set at 500,000 m [3].
• Tangent means the surface touches the globe along one line. Secant means it cuts the globe along two lines [2].

Mains Points

• Why no flat map is perfect:
• A globe is non-developable, so flattening it must cause distortion in area, shape, distance or direction.

• Map-makers choose a developable surface and a tangent position that keep the distortion small in the area they care about.

• Choosing the right projection for the region:

• Use cylindrical for tropical and world maps, conical for mid-latitude countries that stretch east–west, and polar zenithal for Antarctic research and Arctic routes.

• This matters for India's Antarctic stations, such as Maitri and Bharati, and for planning Arctic shipping routes.

• Navigation and defence uses:

• Mercator's 1569 conformal grid lets ships keep a constant compass bearing [4].

• The transverse Mercator and UTM grid (60 zones of 6°) give accurate local coordinates in metres for military, survey and GIS work [3].

• Limits of world maps:

• A normal cylindrical world map like Mercator makes high-latitude lands look too big.
• Atlas and classroom maps should therefore be chosen with care.