Next term, I’ll be teaching a course in general relativity, and while preparing my notes on the curvature of space, I was reminded of a really fun paper I worked on with Rich Gott a few years ago. As you probably know, the earth is roughly a sphere, and if you try to wrap a piece of flat paper around it to make a map, you’re going to get a lot of crinkles and folds. In short, you can’t make a perfect map of the whole earth.
There are some, like the Gall-Peters, that are area-preserving. In other words, equal areas on the map always represent equal areas of land.
Peters argued that the only way to properly appreciate the importance of places like Africa and South America in the world was to use the Gall-Peters projection. Other maps, like the well-known Mercator projection are not area preserving, and thus make North America and Europe look huge compared to more equatorial continents. The thinking is that if people constantly see South America and Africa as tiny and insignificant, they will think of them that way. This argument even found its way into the West Wing:
Area preserving maps are good in general, but obviously, if you’re going to preserve area, you’re going to sacrifice fidelity in another area: shape. The Gall-Peters (and other area-preserving maps including the Eckert IV, the Lambert, and the Sinusoidal) all grossly distort the shape of the continents. Here’s the Gall-Peters shown another way:
Each of those blobby ellipsoids represent a perfect circle drawn on the earth with a radius of about 12 degrees (about 1300 km). They get very distorted on the map. It doesn’t have to be so. There are maps that do a crappy job with area, but do a great job with shape. The Mercator, bad as it is with area, does shape very well:
So where does the general relativity come into it? Rich and I realized that classic measures of how maps of the whole world perform relied on what happened to small circles when you project them onto the map. This is the origin of the Tissot Indicatrix, and is basically the ellipses you saw in the maps above.
However, there are also huge large scale effects. The northern boundary between the continental United States and Canada at the 49th parallel of latitude is shown as a straight line in the Mercator Map, but really it is a small circle that is concave to the north. If one drove a truck down that border from west to east, one would have to turn the steering wheel slightly to the left so that one was continually changing direction. The great circle route (the straightest route) connecting the Washington State and Minnesota (both at the 49th parallel) is a straight line which goes entirely
through Canada. This straight line on the globe when extended, passes south of the northern part of Maine, so the continental United States is bend downward like a frown in the Mercator Map.
The tools of curved space — the calculation of geodesics — turn out to be invaluable. Graphically, the ellipses/ovals on the maps above represent, as I said, a perfect circle on the ground around 1300km in radius. However, there are also a cross-hair in the middle, representing a radius 1300km North, South, East, and West. Their projection onto the maps represent small sections of great circle routes. The “flexion” represents how much those lines get bent (on the map), and the “skewness” describes how off center the crosshairs are. The variation in sizes of the ovals tells you about area errors, and the variation in shape tells you about shape errors. A non-existent perfect map would have uniform, circular, even, straight crosshairs throughout.
We ultimately combined all 4 of these metrics: area, shape, flexion, and skewness, averaged over the entire map, to figure out the all-around best world map, the Winkel-Tripel:
Interestingly, even before we did this work, National Geographic used this map projection for their all-world maps. How does it do with our little indicatrices?
Very good indeed!
Curious about the other top performers? Check out the maps (with the indicatrices) and their overall scores below. Lower is better with details in our paper.
- Winkel-Tripel (4.5629) [map]
- Kavrayskiy VII (4.8390) [map]
- Gall Stereographic (5.7582)
- Hammer-Wagner (5.7847) [map]
- Eckert IV (5.8519, but the best area-preserving map) [map]
Incidentally, the Gall-Peters (also area preserving) gets a 7.0722. The Mollweide (also area-preserving and used for all-sky projections in astronomy), does a little better. It gets a 6.387. What does this mean? It means that even though you’re probably used to seeing the cosmic microwave background like this:
You’re probably better off seeing it like this:
The advantage of the Eckert IV is that you don’t get much distortion around the edges. Thanks to Wes Colley for producing both of those.