# 8 The mathematics of projections

Cartographers have known for centuries that Mercator isn’t an especially accurate way of viewing the world, and they’ve developed a number of mathematical tools for projecting the earth’s spherical surface onto a two-dimensional map. But **there’s no such thing as a perfect map projection!** Every projection has its advantages and disadvantages. To understand why, we have to venture back into mathematics.

## Properties of map projections

Let’s consider what an ideal map would look like.

First, the map should be **contiguous** — there shouldn’t be any interruptions in it. Just as any two points on the earth’s surface can be connected by a line, someone should be able to draw a line between any two points on the map. We don’t, in other words, want something like this:

For the remainder of this discussion we’re going to consider only contiguous maps — maps that preserve the *topological characteristics* we talked about before.

In drawing these maps, there are four properties of the earth’s surface we’d like to preserve.

- Area
- The areas of regions on the map should be proportional to their areas on the earth’s surface. Maps that preserve relative areas of regions are called
**equal-area**maps. - Distance
- Distances between points on the map should be proportional to the distances between those points on the earth’s surface. Maps that preserve distance are called
**equidistant**maps. - Direction
- Directions from the center of the map to various points on the map should be shown correctly. Maps that do this are called
**azimuthal**maps. - Shape
- Shapes of regions should be drawn correctly, which means we have to preserve local angles — all angles at any point are preserved, the scale at any point is the same in every direction, and lines of latitude and longitude intersect at right angles. Maps that do this are called
**conformal**maps.

The problem is that achieving all four of these properties on a two-dimensional map is impossible. Mathematicians have proven this. Preserving any one requires distorting at least one other.

This table shows which properties can be combined in a single map projection:

Equal-Area | Equidistant | Azimuthal | Conformal | |
---|---|---|---|---|

Equal-Area | ||||

Equidistant | ||||

Azimuthal | ||||

Conformal |

## Types of projections

Mapmakers and mathematicians have invented dozens of map projections — the possibilities are almost limitless. As new needs arise — such as the need to project satellite imagery onto a computer screen with minimal distortion — new projections are invented.

Most projections fall into a few basic categories:

### Cylindrical

Cylindrical projections unwrap the earth’s surface and project it onto the surface of a cylinder. The Mercator projection is cylindrical, as are several variations on it.

### Conic

In a conic projection, the earth’s surface is mathematically projected onto a cone.

### Azimuthal

An azimuthal projection geometrically maps the earth’s surface onto a plane.

## Properties of common projections

The U.S. Geological Survey provides an overview of several common map projections, with an explanation of how each is created and what it’s most useful for. The table below summarizes the properties of common map projections.

Projection | Type | Conformal | Equal area | Equidistant | Azimuthal (true direction) |
---|---|---|---|---|---|

Globe | Sphere | ||||

Mercator | Cylindrical | ||||

Transverse Mercator | Cylindrical | ||||

Oblique Mercator | Cylindrical | ||||

Space Oblique Mercator | Cylindrical | ||||

Miller Cylindrical | Cylindrical | ||||

Robinson | Pseudo-cylindrical | ||||

Sinusoidal Equal Area | Pseudo-cylindrical | ||||

Orthographic | Azimuthal | ||||

Stereographic | Azimuthal | ||||

Gnomonic | Azimuthal | ||||

Azimuthal Equidistant | Azimuthal | ||||

Lambert Azimuthal Equal Area | Azimuthal | ||||

Albers Equal Area Conic | Conic | ||||

Lambert Conformal Conic | Conic | ||||

Equidistant Conic | Conic | ||||

Polyconic | Conic | ||||

Bipolar Oblique Conic Conformal | Conic |

## Choosing a projection

Finally, this table gives a summary of what kinds of maps each projection is more and less suitable for.

Projection | Type | World | Hemisphere | Continent/ Ocean |
Region/ sea |
Medium scale | Large scale |
---|---|---|---|---|---|---|---|

Globe | Sphere | ||||||

Mercator | Cylindrical | ||||||

Transverse Mercator | Cylindrical | ||||||

Oblique Mercator | Cylindrical | ||||||

Space Oblique Mercator | Cylindrical | ||||||

Miller Cylindrical | Cylindrical | ||||||

Robinson | Pseudo- cylindrical |
||||||

Sinusoidal Equal Area | Pseudo- cylindrical |
||||||

Orthographic | Azimuthal | ||||||

Stereographic | Azimuthal | ||||||

Gnomonic | Azimuthal | ||||||

Azimuthal Equidistant | Azimuthal | ||||||

Lambert Azimuthal Equal Area | Azimuthal | ||||||

Albers Equal Area Conic | Conic | ||||||

Lambert Conformal Conic | Conic | ||||||

Equidistant Conic | Conic | ||||||

Polyconic | Conic | ||||||

Bipolar Oblique Conic Conformal | Conic |