Mathematics of Planet Earth

The shape of the Earth can be approximated by a sphere. If we want to be more precise, we use a geoid, which is an ellipsoid of revolution, since the radius of the Earth is larger at the equator than at the poles. It is Gauss who proved that there exists no isometry between a surface of nonzero curvature and a surface of zero curvature like a plane, a cone, or a cylinder. Hence, it is impossible to draw maps of the Earth or of regions of the Earth that preserve ratios of distances, and each process of cartography is a compromise. Most maps are obtained through projections of the Earth on a plane, a cylinder, or a cone. Equivalent projections preserve the ratio of areas. The horizontal projection of a sphere onto a cyclinder tangent to the sphere at the equator is equivalent: this was already known to Archimedes. Conformal projections preserve angles. The stereographic projection of the sphere from one pole to a plane tangent to the other pole; This was already known to Hipparque. Composing with biholomorphic transformations on the plane yields other conformal projections. When the biholomorphic transformation is the function log z, this is the Mercator projection on a cylinder. Other natural functions are the functions z^a, with a positive real, which yield conformal projections on a cone. The geoid is a Riemann surface and there are conformal transformations from it to the plane, but the formulas are more complex. Most of what we presented so far is several centuries old. The mathematician John Milnor, Fields medalist in 1962, got interested in determining which projection minimizes the distortion of distances in a neighborhood of a given point. But what means distortion? Given two points P and Q in a region of the sphere, we consider sharp constants A and B such that the ratios of the distance of P and Q to the distance of their images on the map lies between A and B. The distortion is defined as the logarithm of the ratio B/A. John Milnor showed that a projection minimizing the distortion over a given region always exists. In the particular case where the region is bounded by a circle on the sphere, he could show that this optimal projection is the azimuthal equidistant projection.