A dual approach to detect polyhedral intersections in arbitrary dimensions

Abstract

This paper presents a dual approach to detect intersections of hyperplanes and convex polyhedra in arbitrary dimensions. In d dimensions, the time complexities of the dual algorithms are 0(2 d log n) for the hyperplane-polyhedron intersection problem, and O((2d) d- 1 logd- 1 n) for the polyhedron- polyhedron intersection problem. These results are the first of their kind for d > 3. In two dimensions, these time bounds are achieved with linear space and preprocessing. In three dimensions, the hyperplane-polyhedron intersection problem is also solved with linear space and preprocessing; quadratic space and preprocessing, however, is required for the polyhedron-polyhedron intersection problem. For general d, the dual algorithms require O(n 2d) space and preprocessing. All of these results readily extend to unbounded polyhedra.