Sunday Function

Here is a picture of the earth, with latitude and longitude lines in ten-degree increments:

Despite the fact that each vaguely rectangular area formed by the intersection of those lines forms a 10 degree by 10 degree region, the actual square-mile area of each rectangle varies. The ones near the poles are smaller than the ones near the equator. It turns out that the area of a little square patch of land with sides given by some fixed interval of latitude and longitude will be proportional to cos(φ), where φ is the latitude. (NB: for those who're used to seeing sin(φ) in their spherical area elements, remember that in the standard spherical coordinates φ is the colatitude. Don't blame me, I'd have defined the north pole latitude to be 0.)

Mapakers know about this, but it's a minor concern next to the fact that the earth is basically a sphere, while maps are 2D squares. If you've ever tried to peel and orange and lay the peel down as a single flat rectangle, you can understand the cartographer's eternal struggle. Fundamentally, what the cartographers are doing is trying to find two functions x = f(φ, λ) and y = g(φ, λ) that take a point (φ, λ) on the glope and send it to (x, y) on a 2D piece of paper. Fundamental considerations of topology force certain tradeoffs, so cartographers try to develop functions whose disadvantages don't outweigh their practical navigational, pedagogical, or other advantages.

One famous such set of functions is that defining the Mercator projection:

The Mercator projection has some nice features. It's a conformal projection, meaning that it preserves angles. An angle on the globe (say, between roads or rivers) will be the same angle on the Mercator map. The map as a whole is a rectangle, which means you won't waste paper when you print it.

But it also has some bad features. The most obvious is that it doesn't preserve areas. While in reality the latitude/longitude squares get smaller at high latitudes, in this projection they actually get bigger. This blows up northern regions like Alaska and Greenland to preposterously huge size. We can calculate the area of each lat/lon square by computing the Jacobian determinant of this set of functions. It's a little involved, but effectively we just calculate how much change in x and y we get for a change in φ and λ and multiply them together. It turns out that the result is sec(φ) when as we determined earlier it was supposed to be cos(φ). We can plot these as a function of latitude (sec is the one that blows up, cos is the one that doesn't):

Obviously near the poles they disagree badly. On the real globe the lat/lon slices get small near the poles, while on the Mercator map they become infinitely huge. You might prefer to have a map that preserves area, in which case you could use a function like that of the Lambert cylindrical projection:

You can take my word for it that the Jacobian determinant of this transformation is cos(φ), which means it preserves area. Unfortunately preserving area is not the same as preserving shape:

Being rigidly set on preservation of angle or area (or distance, or direction) tends to produce some particularly nasty kind of distortion in some other features. These days it's probably most common to see projection schemes that don't preserve any one feature perfectly but instead try to compromise in a way that doesn't distort any given feature too badly. National Geographic uses the Winkel tripel projection, for instance:

There are tons and tons of other projection functions, which you can find in Wikipedia's listing. Students of calculus should have some fun calculating their Jacobian matrices and proving the area/distance/direction/angle properties that they may or may not preserve. Well, for certain values of "fun" anyway.