Computing separating planes for pairs of disjoint polytopes

Abstract

A method for computing extreme separating planes for a pair of disjoint polytopes is presented. A projective transformation, determined by a strict separating plane, is applied to each of the original polytopes to produce a transformed polytope with the same face structure as the original. Computing the convex hull of the transformed pair of polytopes is shown to be equivalent to the original problem. General convex hull algorithms which work in all dimensions are well established, but in dimensions 2 and 3 the merge step of Preparata and Hong uses the face structure of the transformed polytopes to make the computation of their convex hull more efficient. An application of extreme separating planes to hidden surface elimination is also presented.