Map Projections

The world is not flat (citation needed). This is a very important aspect of our planet Earth; indeed, were our world flat it would not rotate on its axis the same way, would have to have an edge, and would probably crumble into a non-flat ball of rubble from its own sheer weight. We should all, therefore, be grateful that the world we live on is the 3D almost-perfect-sphere that it is. Cartographers tend to be a little less happy about our world’s roundness than others because it presents them with an irksome problem: how do we model a non-flat 3D world on a flat 2D surface? Initially this may seem like a straightforward issue. We could, for example, just draw the earth by what it looks like from space! Well this doesn’t quite work because, among other things, you would only have a 2D projection of the side of the earth facing you. Angles, sizes, and shapes get distorted, especially as we approach the edges of our disc. Perhaps we could imagine we have a globe that we cut a slice through from North Pole to South Pole, which we can spread out flat on a table! This method is unfortunately flawed as well, since the sphere will never lie flat regardless of how many cuts are made. Clearly this problem isn’t quite as straightforward as we initially hoped! Well luckily for us, mathematicians and cartographers love these types of problems, and many have offered many possible solutions, of which we will discuss a few.
The first and probably most familiar solution (called a projection) to this problem is the Mercator projection. This is probably the map that you had hanging on the wall of your elementary school classroom. Gerardus Mercator developed it in 1569. The goal of this particular projection is to maintain direction of rhumb lines (aka paths of constant bearing), which are lines that meet each meridian (lines between the two poles) at the same angle. It was particularly useful for navigation because of these lines. The Mercator projection’s biggest failure is generally that it distorts sizes more and more as we venture away from the equator, causing the poles to have infinite size. For example, Greenland and Africa take up roughly the same size on the Mercator projection when in actuality Africa is nearly fourteen times larger!

The Mercator Projection with red dots showing size distortion. Image: Stefan Kühn, via Wikimedia Commons.

The Mercator can be created by projecting the Earth onto a vertical cylinder with circumference equal to the circumference of the Earth. The next projection, called the Transverse Mercator, is obtained using a horizontal cylinder instead. This projection does not maintain straight rhumb lines like its counterpart and distorts scale, distance, and direction away from the central meridian used.

Transverse Mercator projection. Image: Public domain, via Wikimedia Commons.

Next among the more famous projections is the Robinson. This map features a flat top and bulging sides, with meridians starting and ending equidistant to each other but spreading out as they approach the equator. This projection can be seen as a compromising projection: it loosely preserves size, shape, and distance by not being exact in any one of them in particular. The Robinson, like the Mercator, is frequently used in classroom maps due to providing good guesses for relative shapes and positions and being very easy to understand.

Robinson projection. Image: Strebe, via Wikimedia Commons.

Next up are the Stereographic and Orthographic projections. These projections have existed for thousands of years. They were even used by the ancient Greeks! These two methods are projections of the sphere onto a plane, resembling what it would look like if you were to view the earth from space. The Orthographic projection maps along straight lines perpendicular to the tangent plane of the sphere (think looking through a window from space) while the Stereographic projection maps each point by constructing a line through a predefined point (like the north pole) and drawing where it intersects the tangent plane (think the image in a mirror that the earth is placed on). Thus, the main difference is that Orthographic takes the projection from infinity while the Stereographic takes the projection from a point on the sphere. This means that the Orthographic projection only shows one hemisphere, where the Stereographic can show the entire sphere (except the pole) but in a more distorted way. For both of these projections, directions are true from the center point. With the Orthographic projection, any line going through the center is a great circle.

Orthographic projection. Image: Strebe, via Wikimedia Commons.

Stereographic projection. Image: Strebe, via Wikimedia Commons.

There are many more projections to be found online and in books. Indeed, Wikipedia has a stellar list of some of the many different types. These projections all have different uses and are able to convey information in their own clever way, but they share at least one thing in common: they take quite a bit of ingenuity and creativity to come up with and they reflect a deep love and understanding of math in their creators.

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