Sunday Function

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

i-18b726c7d1b4746da0d320d1a37389a9-earth.png

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:

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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):

i-09e81bd6c64c1e2b182df3948e06f5f0-sec.png

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:

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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:

i-c35bfd8f123a8a2065ba2a0cb4aae06b-lambert.png

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:

i-6b893cf4f786ae481cee03964a6bef32-natgeo.png

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.

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Topology is not really the issue. Remove a single point
from a sphere and the resulting topological space is homeo-
morphic to the plane with stereographic projection for
example being a homeomorphism. Sn and Rn are very little
different topologically - Sn being the one-point compactification of Rn.
The important difference is metric. With it's standard
Riemannian metric Rn has constant zero curvature while
Sn with it's standard metric has constant positive curvature. So even a local isometry is impossible.

By Annonymous (not verified) on 27 Feb 2011 #permalink

The fact that the projection of a sphere from it's axis
onto a cylinder with the same axis and tangent to the sphere
preserves area is one of the most famous theorems of Archimedes.

By Annonymous (not verified) on 27 Feb 2011 #permalink

The revolutionary (at the time) thing about the mercator projection stems from its conformality: if you plot a straight line on the map, you can follow that as a single bearing. You trace out a loxodrome on the sphere (which is not a geodesic, in general), but all-in-all it's pretty good, and makes long distance navigation far easier.

Stereographic projection is also conformal but does not take
loxodromes to straight lines.

By Annonymous (not verified) on 27 Feb 2011 #permalink

I once saw a map that was upside-down and equal area, to say 'look, Africa's massive and a big deal, stop being so North/Euro-centric'. I thought, sure, but a better way to make the point would be to define the "North Pole" as being in Africa, and use a Mercator projection. Then Africa will be infinite in size.

So I wrote a script to move the poles around and see what happens. Here are some examples of alternate Mercator maps: http://www.apathysketchpad.com/blog/2010/09/02/fun-with-the-mercator-pr… I thought Mercator was pretty good because it shows the true shapes of continents, but it turns out that isn't really true. I generated one map with the poles in the oceans and it looked lovely. I generated another with a pole in Africa and one in South America and it was the weirdest map I've seen.

It's more correct to refer to the Mercator-Wright projection because although Mercator published the first map with this projection he did not explain how it was constructed, the first person to do so was the English mathematician and cartographer Edward Wright (1561 - 1615) in his Certaine errors in Navigation published in 1599.

The revolutionary (at the time) thing about the mercator projection stems from its conformality: if you plot a straight line on the map, you can follow that as a single bearing. You trace out a loxodrome on the sphere (which is not a geodesic, in general), but all-in-all it's pretty good, and makes long distance navigation far easier.

True story: The derivative of the y coordinate of the Mercantor projection is sec(phi). The opposite problem comes up in the solution to the shape of a hanging chain(the catenery).

By ObsessiveMathsFreak (not verified) on 04 Mar 2011 #permalink

Ben okyanuslarda kutuplu harita oluÅturulan birini ve güzel görünüyordu. Ben Amerika Güney ile bir baÅka oluÅturulan kutuplu bir Afrika ve de görmüÅtüm harita I've oldu garip.