A general fractal distribution function for rough surface profiles

Starting with a very general functional description, involving fractal parameters, of the height along a given line on a rough surface, a distribution function for the corresponding surface profile is derived. This distribution is found to differ from Gaussian form by a convergent power series and to be directly dependent on two fractal parameters: the fractal dimension and topothesy. It is shown how the distribution function can be used to determine the effects of varying the fractal parameters on the height of the bearing-area curve (a standard measure of surface roughness). By truncating the series representation for the distribution function for the surface profiles, two approximate models for the height distribution are obtained. These models are shown to compare favorably with experimentally obtained distributions.