Using low-discrepancy sequences and the Crofton formula to compute surface areas of geometric models

Abstract

The surface area of a geometric model, like its volume, is an important integral property that needs to be evaluated frequently and accurately in practice. In this paper, we present a new quasi-Monte Carlo method using low-discrepancy sequences for computing the surface area of a 3D object. We show that the new method is more efficient than a Monte Carlo method using pseudo-random numbers. This method is based on the Cauchy–Crofton formula from integral geometry, and it computes the surface area of a 3D body B by counting the number of intersection points between the boundary surface of B and a set of straight lines in E 3. Low discrepancy sequences are used to generate the set of lines in E 3 to reduce the estimation errors that would be caused by using statistically uniformly distributed lines. We study and compare two different methods for generating 3D random lines, and demonstrate their validity theoretically and experimentally. Experiments suggest that the new quasi-Monte Carlo method is also more efficient than the conventional approach based on surface tessellation.