What is a map projection?
The network of parallels and meridians on a globe is called the graticule. A map projection is a way of drawing this graticule on a flat sheet.
A globe is a curved surface. It cannot be flattened without stretching or tearing. So every flat map has some errors.
A globe keeps four things correct: area, shape, direction and distance. A projection can keep one or two of them correct, never all four.
Why not just use a globe? A globe is hard to carry, cannot show small areas in detail, and you cannot see the whole world at once.
Types of projection
By the surface used (developable surface)
- Cylindrical: a paper cylinder wraps the globe, touching the Equator.
- Conical: a paper cone sits on the globe, touching one parallel.
- Zenithal (azimuthal): a flat plane touches the globe at one point, often a pole.
A developable surface can be unrolled flat without stretching (cylinder, cone). A sphere is non-developable.
By method
- Perspective: drawn by imagining a light throwing the grid onto the surface.
- Non-perspective: no light; worked out by geometry.
- Mathematical (conventional): made by formulas.
By position of the light source (perspective types)
Gnomonic (light at the centre), stereographic (light at the opposite end), orthographic (light far away, parallel rays).
By property kept
- Homolographic (equal-area): areas correct.
- Orthomorphic (true shape): shapes of small areas correct.
- Azimuthal (true bearing): directions from the centre correct.
- Equidistant: distances correct along certain lines.
Conical projection with one standard parallel
Imagine a paper cone placed on the globe so that it touches the globe along one parallel. That parallel is the standard parallel. Light from the centre throws the graticule onto the cone. Unroll the cone: you get a fan shape.
How it is drawn (outline of the method)
- Draw a circle for the globe (radius from the chosen scale: R = Earth radius × RF).
- Draw the Equator, the polar axis and the standard parallel at its latitude from the centre.
- Draw a tangent at the point where the standard parallel meets the circle; extend it to meet the polar axis. This gives the apex of the cone and the radius of the standard parallel on the map.
- Find the gap between parallels along the circle (arc of the chosen interval) and the gap between meridians along the standard parallel (its length ÷ number of intervals).
- With the apex as centre, draw the standard parallel as an arc. Draw the other parallels as arcs at equal spacing.
- Mark the meridian spacing along the standard parallel and draw straight lines from the apex through these marks.
Properties
- All parallels are arcs of concentric circles; spacing between them is equal.
- All meridians are straight lines meeting at the apex, cutting parallels at right angles.
- Scale is correct along the standard parallel and along the meridians.
- The pole is shown as an arc, not a point.
- Away from the standard parallel, parallels are stretched, so areas and shapes get distorted.
Limitations and uses
- Not suitable for a world map or a whole hemisphere; distortion becomes very large.
- Best for areas in mid-latitudes that stretch east–west but are narrow north–south, e.g. a river valley, a railway line, or a belt of countries between 30° and 60°.
Mercator's projection
Made by Gerardus Mercator in 1569. It is a cylindrical, orthomorphic projection worked out by mathematics.
How it is drawn (outline)
- Find the length of the Equator on the map (2πR, with R from the scale). Draw it as a straight line and divide it for the meridian interval.
- Draw meridians as straight, parallel, vertical lines at equal spacing.
- Draw each parallel at a height taken from a Mercator table (height grows with latitude: roughly R × ln tan(45° + φ/2)).
Properties
- Parallels and meridians are straight lines cutting at right angles.
- All parallels are the same length as the Equator, so they are stretched more and more towards the poles.
- Spacing between parallels increases towards the poles by the same amount the parallels are stretched (by sec φ = 1 ÷ cos φ). So shape is kept for small areas (orthomorphic).
- A straight line drawn anywhere is a line of constant bearing (rhumb line or loxodrome).
- The poles cannot be shown: they would be infinitely far away.
Limitations and uses
- Areas are greatly exaggerated in high latitudes: Greenland looks as big as South America.
- Used for navigation charts at sea and in the air, and for world maps of ocean currents, winds and drainage where direction matters.
Comparing the two
| Feature | Conical, one standard parallel | Mercator |
|---|---|---|
| Surface | Cone | Cylinder |
| Parallels | Concentric arcs | Straight, equal-length lines |
| Meridians | Straight lines from apex | Straight, parallel lines |
| True property | Scale along the standard parallel | Shape and direction |
| Best for | Narrow mid-latitude belts | Navigation, world directions |
Try it
Peel an orange in one piece and try to press it flat: see the gaps and cracks. Then roll a sheet of paper into a cone and a tube around a ball. Finally predict: on a Mercator map, how much is a square at 60° stretched? (1 ÷ cos 60° = 2 times each way, 4 times in area.) Check with the slider in the last 3D step.
Key formulas and definitions
- Graticule = network of parallels and meridians
- Mercator stretch factor at latitude φ = sec φ = 1 ÷ cos φ
- Area exaggeration on Mercator = sec² φ
- Radius of reduced globe R = Earth radius × RF
Worked examples
1. Which projection would you choose for a map of a river valley lying between 40°N and 50°N, stretching east–west?
The conical projection with one standard parallel (standard parallel about 45°N). The area is narrow north–south, mid-latitude, and wide east–west.
2. Why does Greenland look larger than it is on a Mercator map?
Greenland is at high latitude. Mercator stretches parallels by 1 ÷ cos φ both ways, so area grows by 1 ÷ cos² φ. At 70° this is about 8.5 times.
3. By how much is length stretched at 60° latitude on Mercator?
1 ÷ cos 60° = 1 ÷ 0.5 = 2. Lengths are doubled; areas are 4 times.
4. A sailor draws a straight line between two ports on a Mercator chart. What does this line give?
A rhumb line: a constant compass bearing. The sailor can keep the same direction all the way.
Common mistakes
- Saying the conical projection is good for world maps. It is only good for narrow mid-latitude belts.
- Thinking Mercator keeps area correct. It keeps shape and direction; area is badly exaggerated near the poles.
- Drawing the pole as a point on a conical projection. It appears as an arc.
- Drawing Mercator parallels at equal spacing. The spacing increases towards the poles.