Comments on the Cover Illustration, Moon

Abstract

Pr(v n)= n Pr(v-/'P> n-l/P) n Pr(v-P > R) R -P. Hence, for the crater radius r, we put r = v -/P. Note that the procedure for forming an appropriate function of v leads to an inverse function. Mandelbrot suggests that p = 2 gives a good approximation to the actual lunar surface, and to those of other known planetary satellites. It is also the value of p which gives statistical self-similarity. To achieve randomly even scattering of points (x, y) within a plane rectangle is simply a matter of obtaining x and y as independent and linear functions of v. To achieve the analogous scattering on a spherical surface-isotropic distribution-requires a little more subtlety. A solution is obtained by using (longitude, latitude) coordinates for the location of each center, with longitude = 2 rrv, latitude = arcos(l - 2 v). Making one of the coordinates a linear function of v forces the other coordinate to be a nonlinear function of v. The reasoning behind the above solution is left to the reader to figure out. (Clue: what is the area of the polar cap bounded by latitude 0 on a globe of unit radius?)