Creating a more accurate flat map of the Earth

Projecting the surface of a globe on to a flat surface always introduces distortions. Richard Gott, Dave Goldberg, and Bob Vanderbei claim to have created created a projection that minimizes the errors more than any projection before.

Previously, Goldberg and I identified six critical error types a flat map can have: local shapes, areas, distances, flexion (bending), skewness (lopsidedness) and boundary cuts. These are illustrated by the famous Mercator projection, the base template for Google maps. It has perfect local shapes but is bad at depicting areas. Greenland appears as large as South America even though it covers only one seventh the area on the globe.

One can’t make everything perfect. The Mercator map has a boundary cut error: one makes a cut of 180 degrees along the meridian of the international date line from pole to pole and unrolls the Earth’s surface, thus putting Hawaii on the far-left side of the map and Japan on the far-right side of the map creating an additional distance error in the process. A pilot flying a great circle route straight from New York to Tokyo passes over northern Alaska. His route looks bent on a Mercator map—a flexion error. North America is lopsided to the north: Canada is bigger than it should be, and Mexico is too small. All these errors are important. Ignoring one of them can lead you to bad-looking maps no one would prefer.

The object here is to find map projections that minimize the sum of the squares of the errors—a technique that dates back to the mathematician Carl Friedrich Gauss. The Goldberg-Gott error score (sum of squares of the six normalized individual error terms) for the Mercator projection is 8.296. The lower the score, the smaller the errors and the better the map. A globe of the Earth would have an error score of 0.0. We found that the best previously known flat map projection for the globe is the Winkel tripel used by the National Geographic Society, with an error score of 4.563. It has straight pole lines top and bottom with bulging left and right margins marking its 180 degree boundary cut in the middle of the Pacific.

These three people have created a projection that has a very small Goldberg-Gott error score of just 0.881.

It has zero boundary cut error since continents and oceans are continuous over the circular edge. It has a remarkable property no single-sided flat map possesses: distance errors between pairs of points (such as cities) are bounded, being off by only at most plus or minus 22.2 percent. In the Mercator and Winkel tripel projections, distance errors blow up as one approaches the poles and boundary cuts.

You can see their map by following the link.

1. flex says

Hmm. Interesting, although as it stands I suspect it is of limited usefulness. Now maybe if a interactive, computer-generated map using this mapping technique was developed, where two points of interest were defined, a straight line drawn between them, and that becomes the line orthogonal to an arbitrary equator. Now that may be interesting.

But having enjoyed maps, without really spending the time to understand the details of mapmaking, I think it’s probably worthwhile to drop a couple of my favorite map bits from some of my favorite authors here:

From Louis Carroll’s Slyvia and Bruno (not one of his better works, but it has it’s moments):

“What a useful thing a pocket-map is!” I remarked.
“That’s another thing we’ve learned from your Nation,” said Mein Herr, “map-making. But we’ve carried it much further than you. What do you consider the largest map that would be really useful?”

“About six inches to the mile.”

“Only six inches!” exclaimed Mein Herr. “We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all ! We actually made a map of the country, on the scale of a mile to the mile!”

“Have you used it much?” I enquired.

“It has never been spread out, yet,” said Mein Herr: “the farmers objected: they said it would cover the whole country, and shut out the sunlight ! So we now use the country itself, as its own map, and I assure you it does nearly as well.”

Which was then expanded by Jose Louis Borges in his one-paragraph short story, On Exactitude in Science, as a fictional quotation from the fictional book Viajes de Varones Prudemtes by the fictional author Suarez Miranda. The money quote is:

The Guild of Cartographers created a Map of the Empire, which perfectly coincided with the Empire itself.

And finally, Umberto Eco, in How to Travel with a Salmon and other Essays” has an essay about the Impossibility of creating a 1:1 scale map, in which I find my notes scribbled in the margin about how Eco was really talking about the impossibility of fully defining consciousness. Not because consciousness is undefinable, but because the act of mapping a specific consciousness changes the consciousness sufficiently to make the map either inaccurate or incomplete. I guess I thought about deeper things when I was younger.

2. OverlappingMagisteria says

Seems to be kinda cheating if the map is two sided. One nice thing about a world map is that it lets you see the whole globe at once. And this one would be very inconvenient if looking at any area near the equator -- you’d have to keep flipping it over back and forth.Technically, it might be more accurate, but I’m not sure if it’s more useful.

I wonder how much creating an accurate map even matters anymore. A globe is the most accurate, but they’re inconvenient to carry around while a map folds up and fits in your pocket. But nowadays, map apps like Google maps will show a globe if you zoom out far enough. So you can carry a globe around in your pocket!

3. Holms says

Um, a two sided map? Pretty sure I’ve seen those before. Which of course brings in the new problem of inconvenience whenever the area or two locations being looked at are split by the boundary, but the authors dismiss this as it is not accounted for in the error algorithm.

Why can’t people just the flat Earthers how they get their amazing maps? Surely they have it all figured out by now…

4. blf says

@2, “I wonder how much creating an accurate map even matters anymore.”

Some professions — sea & air navigation, as an example — require very accurate maps.

I’ve yet to find any sort of map app which “works” at even a medium-sized city level. I’m constantly zooming in-and-out, sliding around, trying to match up what very small limited-space inconsistent display (usually with irrelevant & obnoxious annotations like ads) shows with reality. I massively prefer an origami Falkplan. (I am aware there is an on-line Falksomething, at least, but haven’t used it sufficiently to know if it “works”.) Of course, Falkplan was notorious for “not working” in large cities (the NYC version was diabolical), they didn’t — as far as I know — cover anything but select cities, and measuring distances (estimating time) could be tricky, but they were usefully informative, accurate, and didn’t require batteries.

5. mnb0 says

“You can see their map by following the link.”
Unfortunately it’s not possible to compare Saudi-Arabia with Greenland. The two are about as large, so it’s an easy and quick way to evaluate how accurate the map is.