Map Projections Minimizing Distance Errors

Maps convey important information about distances between pairs of points. It is therefore desirable to minimize the errors made in representing distances between pairs of points on maps. Since it is just as bad to have two points on the map at twice their proper separation as to have them at half their proper separation, it is the root-mean-square (rms) logarithmic distance between random points in the mapped region that we will minimize. The best previously known projection of the entire sphere for distances is the Lambert equal-area azimuthal, with an rms logarithmic distance error of σ = 0.343. By way of comparison, the Mercator projection has σ = 0.444 and the Mollweide, σ = 0.390. We present three new projections – the Gott equal-area elliptical, with perfect shapes on the central meridian; the Gott-Mugnolo equal-area elliptical; and the Gott-Mugnolo azimuthal, with rms logarithmic distance errors of σ = 0.365, σ = 0.348, and σ = 0.341 respectively – that improve on previous projections of their type. The Gott-Mugnolo azimuthal projection has the lowest distance errors of any map and is produced by a new technique using “forces” between pairs of points on a map, which make the points move so as to minimize σ. The Gott equal-area elliptical projection produces a particularly attractive map of Mars, and the Gott-Mugnolo azimuthal projection produces an interesting map of the Moon, both of which we also show.