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.