The case for the Bonne projection

People seem to dislike the Mercator projection because it exaggerates the size of the areas closer to the poles. People have instead proposed various compromise and equal area projections. But what people really seem to care about when it comes to world maps is not areas or angles, but distances. It’s easy to make a projection that consistently preserves distances away from some line or point. For instance, the azimuthal equidistant projection on the emblem of the UN consistently preserves distances away from the North Pole (though does not preserve areas). The equirectangular projection consistently preserves distances along the meridians:

Equirectangular projection

Mercator projection

The Mercator projection, as we all know, stretches both the distances along the parallels closer to the poles (as does the equirectangular projection) as well as the distances between the parallels closer to the poles. But due to this, it is the projection that best preserves shape.

There is, however, a family of projections that preserves both areas and distances -the Bonne family of projections. When the undistorted parallel of this family of projections is at the equator, the result is the sinusoidal projection, which looks exactly as if a person took apart the slices of an orange and squished both the top and bottom ends together:

Sinusoidal projection

Sinusoidal projection in three parts

The shapes along the equator are completely correct. The problem with this projection, of course, is the further one gets away from the equator, the more severely this projection distorts the shapes along the outer meridians. In particular, Korea and the U.S. West Coast are almost unrecognizeable. However, if one moves the standard parallel of this projection family away from the equator to areas people actually care about, these problems are swiftly remedied:

As with the sinusoidal projection, all the distances along the parallels are correct, as are all the distances along the central meridian and distances away from the standard parallel. The shapes along the standard parallel (in this case, 45 degrees North, around the latitude of Turin, Simferopol, Harbin, or Minneapolis) are completely correct. However, while this map projection might be what most people are looking for, the shape is rather unusual, as the parallels are curved, and the shape is not the best for modern rectangular screens.

A logical approach to creating a compromise map projection with the parallels still portrayed as straight lines would then be averaging the equirectangular and sinusoidal projections:

Winkel I/Eckert V projection

Note, however, this creates huge and unnecessary space near the poles. A solution to this would be to discard the idea of conserving distances along the central meridian and averaging the Lambert Cylindrical Equal-Area and the sinusoidal projections:

Foucaut Sinusoidal projection

In the Foucaut Sinusoidal projection, the regions around the poles are compressed in order to make an equal area projection while reducing the sinusoidal projection’s extreme distortion of the outer meridians away from the equator. As you can see, there is much less empty space around the poles in the Foucaut Sinusoidal projection than in the Winkel I/Eckert V projection. The distortion in shape is very similar to that in Winkel I/Eckert V. Given the lack of vast space around the poles and the relative lack of distortion around the outer meridians, the Foucaut Sinusoidal projection might be the best one for displaying countries of the projections that portray the equator as a straight line. The fact that it distorts the meridians makes it a good competitor to the Robinson projection.