Some lists of map projections cover them in a way that I think is more haphazard than it needs to be. This is my attempt at a simple introduction to some map projections.

I’ll only look at the usual situation, in which we are mapping the surface of a sphere onto a planar map. The Earth is not a perfect sphere, but I’m going to ignore the slight complications arising from that. For the purposes of this post, it won’t matter; but it may mean that I’m slightly misusing the names of some of the projections.

We start by choosing a special point on the surface of the Earth (or some other spheroidal object), say the North Pole. From there, we slice up the surface into concentric strips, of equal height along the surface, centered on the starting point. Most projections also require us to cut the strips along some chosen meridian, so that we can straighten them out.

The strips will be thin enough that some of the details of what we’ll do with them don’t matter. The projection will be the *limit*, as the height of the strips approaches zero.

A terminology note: With the strips lying on their sides, I’m not sure which dimension is the “width”. So I’ll arbitrarily use “length” for the (longer) east-west (horizontal) dimension, and “height” for the north-south (vertical) dimension.

Also note that the overall scale doesn’t matter. I won’t spell it out, but of course we can always shrink/enlarge both dimensions of the projection simultaneously. We’re not trying to build a map the actual size of the Earth.

## Sinusoidal

If we straighten the strips, without changing their length or height, we get a *sinusoidal* projection. The boundaries of the map are sine waves.

If we instead leave the strips curved in circular arcs, reducing their curvature just enough to accommodate a flat surface, we get a Werner projection.

These are among the most fundamental map projections, but neither is used very much. Werner is too “out there”. Sinusoidal has a lot of distortion of the shapes of the features, and it’s just kind of ugly. You can feel the tension in it.

Some polar-oriented projections like Werner are classified as “conical”, others “azimuthal”, or some variant of those terms. I’m just going to call them all “polar”.

When we don’t change the areas of the strips, the resulting projection is an *equal-area* projection: The relative areas of the regions depicted by the projection are accurate. Sinusoidal and Werner are both equal-area projections.

## Equirectangular

If we straighten the strips, then lengthen or shorten them so that they are all the same length, without changing their height, we get an *equirectangular* projection.

I think the best polar counterpart to the equirectangular projection is the *azimuthal equidistant* projection, in which we lengthen each strip until it forms a complete circle (the technical term is *annulus*), just outside of the previous one.

## Mercator

If we straighten the strips, then enlarge them until they are all the same width (keeping their post-straightening *shape* the same), we get a Mercator projection.

I’m not sure what the polar counterpart of the Mercator is, but I think it’s the *stereographic* projection.

Projections like this are called “conformal”. They try to maintain shapes and directions, at the expense of areas.

## Cylindrical equal-area

If we stretch/shrink the strips to the same width, adjusting each one’s height as needed to keep its *area* the same as it was, we get an equal-area cylindrical projection of some sort. Exactly which one depends on the relative width and height of the final map. The most basic is probably the *Lambert cylindrical equal-area*. Others include Gall-Peters, Hobo-Dyer, and Behrmann.

The polar counterpart, with the strips forming full circles, is the *Lambert azimuthal equal-area* projection.

## Pseudocylindrical equal-area

*Pseudocylindrical equal-area* projections are similar to cylindrical ones, but the strips aren’t all stretched to the same width.

The sinusoidal projection is one example of such a projection, though it’s kind of a degenerate example. Others are Mollweide and Eckert II:

You can invent new pseudocylindrical equal-area projections quite easily. Just draw any shape that has the correct area, then fill it with the strips, each stretched to reach the left and right edges of your shape, with each strip’s height adjusted so that its area doesn’t change. Some shapes won’t work well because they’ll lead to unwanted discontinuities, but it’s not hard to avoid that.

With Mollweide, the shape is an ellipse. With Eckert II, it’s a hexagon. Mollweide is a nice-looking projection, in my opinion, but I don’t think there’s any particular mathematical justification for it.

So, that’s my quick explanation of some map projections.

There are *many* other map projections, and I don’t want to give the idea that cutting the globe into strips is always the best way to understand a projection. It’s not; I just think it works pretty well for some of the basic ones.