Classification by drawing technique and by source of light

Detail

Basic terms used in this section

• Map projection: a way of moving the network of latitudes and longitudes from the round globe onto a flat sheet.
• Graticule: the network of parallels (lines of latitude) and meridians (lines of longitude) drawn on a globe or map.
• Developable surface: a surface that can be unrolled flat without stretching or tearing. There are three:
• cylinder, which gives cylindrical projections
• cone, which gives conical projections

• plane, which gives zenithal projections. A zenithal (also called azimuthal) projection uses a flat sheet that touches the globe at one point.

• Tangent point: the single point where the flat sheet touches the globe in a zenithal projection.

A. Classification by drawing technique

Type	How it is drawn	Link to a real "image" of the globe	Example
Perspective projection	A light source shines through the globe and casts the shadow of the graticule onto a developable surface	Strong. The map is a real shadow picture	Gnomonic, stereographic, orthographic
Non-perspective projection	Drawn without any light or shadow. The network is built so that it keeps a chosen property	Weak	Many cylindrical and conical nets
Conventional (mathematical) projection	Worked out only from mathematical formulae and calculation	Very little link to any projected image	Mercator's projection

• Perspective projection: how it works, step by step
• Picture a glass globe with the graticule painted on it.
• A light is placed at a fixed point.
• The shadow of the lines falls on a sheet of paper (or a paper cylinder or cone).

• The shadow is traced, and the traced lines become the map.

• Non-perspective projection

• The net is drawn on paper using geometry.

• No shadow-casting is imagined.

• Conventional (mathematical) projection

• The spacing of parallels and meridians is calculated with formulae.
• The aim is to keep one chosen property, such as true shape, true area or a straight-line compass course.

• Mercator's projection (1569) is the best-known example. It was made for sailors.

  • On it, a rhumb line (also called a loxodrome: a course that cuts every meridian at the same angle, so the ship keeps one constant compass bearing) is drawn as a straight line [3].
  • In 1537, Pedro Nunes had shown in Tratado da sphera that a ship holding a constant compass bearing does not travel in a straight line or along a great circle [3].

• Modern formula-based projections

• Today almost all projections are calculated, not traced. John P. Snyder's Map Projections: A Working Manual (USGS Professional Paper 1395, 1987) gives forward and inverse equations for each projection, so computers can draw them [2].
• The USGS uses conformal projections for its larger-scale topographic maps. A conformal projection keeps the true shape of small areas. The two used are the Transverse Mercator and the Lambert Conformal Conic [2].
• The USGS built the Space Oblique Mercator. It was the first projection designed to map the Earth continuously from a satellite with low distortion [2].

B. Classification by source of light

All three are perspective zenithal projections. The flat sheet touches the globe at one point. Only the position of the light changes.

Projection	Where the light is	What happens to the rays	Key property	Main use
Gnomonic	Centre of the globe	Rays spread out from the centre	Every great circle is a straight line	Planning the shortest sea or air route
Stereographic	On the globe's surface, at the point exactly opposite the tangent point	Rays start from the far side of the globe	Conformal: true shape of small areas	Polar maps, navigation maps
Orthographic	At infinity (very far away)	Rays are parallel	Looks like the globe seen from space	Pictures of hemispheres

Gnomonic projection: light at the centre

• Great circle: any circle on the globe whose centre is the centre of the Earth, such as the equator or any meridian. The shortest path between two points on Earth always runs along a great circle.
• Why great circles come out straight
• Every great circle lies in a flat plane that passes through the Earth's centre.
• The light is also at the centre, so it sits inside that plane.

• The shadow of the great circle therefore falls on the map sheet as a straight line.

• Use

• A navigator draws a straight line between two ports on a gnomonic chart. That line is the shortest route.

• Great-circle charts are usually small-scale charts drawn on the gnomonic projection [4].

• Limit

• Rays from the centre aimed at points 90° away from the tangent point run parallel to the sheet and never meet it.
• So a gnomonic map can never show a full hemisphere.
• Shapes and areas stretch sharply as you move away from the centre of the map.

Stereographic projection: light on the opposite surface

• Light position: on the globe's surface, diametrically opposite the tangent point. If the sheet touches the North Pole, the light is at the South Pole.
• Key property: it is conformal, so angles and the shapes of small areas stay true.
• Use: polar maps (the Arctic and Antarctica) and navigation maps.
• Compared with the gnomonic: the light sits farther back, so the stereographic can show a larger area with less stretching.

Orthographic projection: light at infinity

• Light position: at infinity, so all the rays are parallel to each other.
• Result:
• The map looks like a photograph of the Earth taken from deep space.
• It shows one hemisphere.

• Areas near the outer edge look squeezed (foreshortened).

• Use: pictorial views of hemispheres, and pictures of the Moon or planets.

• Limit: it is neither conformal nor equal-area.

C. Gnomonic and Mercator together in navigation

• Step 1: find the shortest route. Draw the great-circle route as a straight line on a gnomonic chart.
• Step 2: break it into short courses. Mark points along that line and note their latitude and longitude.
• Step 3: move it to Mercator. Plot the points on a Mercator chart and join them with short straight lines. Each straight line on a Mercator chart is a rhumb line.
• Result: the ship sails a chain of rhumb lines. Each leg has one fixed compass bearing, and the whole path stays close to the shortest great-circle route.
• Why both charts are needed
• A great-circle route is the shortest, but the compass bearing keeps changing along it.
• A rhumb line is easy to steer, but it is longer, except along the equator or a meridian.

D. Exercise trap

• Light at the centre of the globe = gnomonic.
• Light on the opposite surface = stereographic.
• Light at infinity with parallel rays = orthographic.
• Cylindrical, conical and zenithal are classes based on the developable surface, not on the light source. They are wrong answers to a "source of light" question.

Prelims Hooks

• The light source is at the centre of the globe → gnomonic projection.
• The light source is on the globe's surface, opposite the tangent point → stereographic projection.
• The light source is at infinity and the rays are parallel → orthographic projection.
• Gnomonic: every great circle is a straight line, so it is used to find the shortest route. It cannot show a full hemisphere.
• Stereographic is conformal (true shape of small areas) and is used for polar maps.
• Mercator's projection (1569) is a conventional (mathematical) projection, not a perspective one. Rhumb lines are straight on it [3].
• Pedro Nunes described the rhumb line in Tratado da sphera, 1537 [3].
• The Space Oblique Mercator (USGS) was the first projection designed to map the Earth continuously from a satellite with low distortion [2].
• Trap: "Mercator is a perspective cylindrical projection." This is wrong. It is built from formulae.

Mains Points

• Choosing a projection for a purpose
• The gnomonic is used for shortest-route planning, the Mercator for constant-bearing steering, the stereographic for polar charts and the orthographic for pictorial globes.

• No single projection keeps shape, area, distance and direction all at once. The mapmaker has to trade one property against another.

• Navigation logic for GS-I geography

• The route is plotted as a great circle on a gnomonic chart, then sailed as a chain of rhumb lines on a Mercator chart.

• This explains why long sea and air routes, such as Indian Ocean–Pacific shipping and polar air routes, look curved on ordinary wall maps.

• From shadow to formula

• Early perspective methods are now mostly replaced by mathematical projections calculated by computer.
• Examples are the conformal Transverse Mercator and Lambert Conformal Conic used for topographic maps, and the Space Oblique Mercator used for satellite mapping. These support modern remote sensing and GIS [2].