  It is useful to have mathematical criteria for evaluating errors in map
projections. The Chebyshev criterion for minimizing rms (root mean square)
local scale factor errors for conformal maps has been useful in developing
conformal map projections of continents. Any local error criterion will be
minimized ultimately by map projections with multiple interruptions, on which
some pairs of points that are close on the globe are far apart on the map.
Since it is as bad to have two points on the map at two times their proper
separation as to have them at half their proper separation, it is the rms
logarithmic distance, s, 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 s=0.343. For comparison, the Mercator has s=0.444, and the Mollweide
has s=0.390. We present 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 s=0.365, s=0.348, and s=0.341 respectively, which improve on previous
projections of their type. The "Gott-Mugnolo azimuthal" 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 them move so as to minimize s. 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.
