3.2 Generating extraterrestrial terrain

Techniques such as Fourier filtering [1, 9], random midpoint displacement [4], Poisson faulting [6] often creates more or less realistic fractal forgeries of nature. They do not consider over that the real terrain is not self-similar or identical from top to bottom. In thus generated terrains we do not find craters, streaks, and other features typical for "extraterrestrial terrain".

Pickover [10] uses a logistic function (2) for simulating these features.


This function was originally introduced by Belgian sociologist and mathematician P.F. Verhulst to model the growth of population limited by finite resources. If tex2html_wrap_inline683 , the tex2html_wrap_inline559 are "random" numbers from the interval (0,1) with U-shaped probability distribution (see Figure 7 a) ). Using transformation of the tex2html_wrap_inline561 according to


we can thought tex2html_wrap_inline561 as random number generator with the numbers uniformly distributed over the interval (0,1) (see Figure 7 a)). Pickover [10] recommended another transformation given by


Figure 7: a) Distribution of tex2html_wrap_inline559 and tex2html_wrap_inline561 b) Distribution of tex2html_wrap_inline563

This transformation may operate on the logistic variable and Collins et al. [2] shows that the transformation (4) generates a sequence of "random" numbers with a near-Gaussian distribution (see Figure 7 b) ). In the next step we extend equations (3) and (4) to 3 dimensions. We assume tex2html_wrap_inline563 distribution to be a radius and tex2html_wrap_inline561 denotes an angle


Variables u and v in equation (5) are thus coordinates of tex2html_wrap_inline563 and tex2html_wrap_inline561 in polar coordinate system. In every iteration the u and v coordinates are evaluated and an element P(u,v) in 2D array is incremented by one. After many iterations array P(u,v) viewed as 3D surface resembles a circular ridge with center at tex2html_wrap_inline723 . Each ridge has a near Gaussian cross-section. The diameter of the crater is controlled by variable tex2html_wrap_inline725 i.e. the distance from center to the ridge.

Advantage of this method is that it can be used for tuning of already created terrains. The near-Gaussian numbers are easy to incorporate into software for stochastic modeling and produce structures that look like a crater with self-similar details.

Ivo Marak - marak@sgi.felk.cvut.cz