Choosing the Right Way to Flatten the Earth

The art and science of drawing our 3D Earth on to a 2D sheet of paper or computer monitor is worthy of a book. Unfortunately, I've only got a few hundred words. As a result, I'm going to largely concentrate on a single, controversial choice: the Mercator projection.

When evaluating map projections we're not just talking about finding a suitable representation for a sphere in Flatland, because the Earth isn't a sphere. It's an oblate spheroid (i.e. flatter at the poles than the equator). Or, at least, it's more an oblate spheroid than it is a sphere. But it's a bit bumpy too. Still, even if it was a perfect sphere, the task of representing the whole Earth completely accurately on a sheet of flat paper is entirely impossible. A flat map cannot simultaneously display area, shape, direction, bearing, distance and scale perfectly all at once. So we have different map projections which do represent one or two of these things accurately and we pick the most appropriate or go for a compromise projection. At least that's how things probably should work.

You're no doubt familiar with the Mercator projection (even if you didn't know the name). It's long been derided for making places near the poles (like Greenland) look very big and places near the equator (like much of Africa) look comparatively small. It has been suggested that this has led to misunderstanding in the United States about travel to and from Africa in relation to the recent ebola outbreak. It's even been tied to racism by some!

Interestingly, there have been several recent attempts to redress apparent misperceptions about the relative size of Africa by super imposing other countries, like the United States, China and India on top (see, for example, here and here). However, Google Maps, OpenStreetMap, Bing Maps and others use a variant of the Mercator projection termed Web Mercator. Not only do we have the same area issues as with "ordinary" Mercator maps, the Earth is assumed to be a perfect sphere.

So, given all this, why do all these popular "slippy map" applications use (a variant of) the Mercator projection? Should we be using something else?

Bing Maps software architect Joe Schwartz gives an excellent and detailed answer to the first question here. In short, the projection used means north is always up and east to the right regardless of where you may be zoomed in to. Moreover, the projection is (almost) conformal, meaning "small" objects (like buildings) have the right shape. This is critical for street maps.

Using OpenStreeMap and IgniteUI I've created two maps with location markers. For simplicity, I'll show one screenshot for each at an appropriate scale. The first shows the whole Earth with the locations of all urban agglomerations with populations of 300,000 or more as of 2014.

The second map shows a small portion of central London. The locations of seven London Underground stations in this relatively small area are already marked, but I've added my own markers for them too. The magenta-colored circles in the bottom right (i.e. south east) corner are the locations of stations that are on the Piccadilly Line. The black hexagons mark stations that are not on the Piccadilly line. (I used the geolocations given here which obviously don't align perfectly with the markers already on the map.)

The points made by Schwartz and illustrated by these two examples highlight an important issue that I've previously noted in relation to more general data visualization: context is key. The Mercator projection may not be completely ideal for showing your global dataset. But slippy maps have other uses too. Using one to walk a few blocks from A to B (e.g. to get from a station on one train line to a station on a different line) would be a lot more difficult if we had a non-conformal projection where small-scale objects were distorted and angles were all wrong.

None of this is to say that the Mercator projection is the best all-round solution for every scenario. It isn't. In an ideal world we'd be using equal area maps to show data related to areas. Having said that, if we're talking small areas, a Mercator projection should work just fine. On the other hand, we could never use Google Maps, OpenStreetMap or Bing Maps to plot data around the South Pole because the South Pole can't be shown, such is the nature of the projection. All projected maps have limitations, some you may just need to be aware of, others are absolute.

If you're planning to stick a map of the world up on a classroom wall, there are better options than Mercator. Personally I like the compromise of the Robinson projection. National Geographic abandoned Robinson in 1998 in favor of the Winkel-Tripel projection, having previously also used the Van der Grintern projection. There's even some pretty complicated math to back up National Geographic's choice. And when they wanted to focus on the oceans, they decided on something entirely different. The important thing to realize is that all 2D maps are wrong, but some are useful for specific purposes.

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