Robust decompositions of polyhedra

Abstract

We present a simple and robust algorithm to compute a convex decomposition of a non-convex polyhedron of arbitrary genus. The algorithm takes a topologically correct representation of a non-convex polyhedron S and produces a worst-case optimal O(N2) number of topologically correct representations of convex polyhedra Si, with ∪iSi = S, in O(nN2 + N4) time and O(n N + N3) space, where n is the number of edges of S, N is the total number of notches or reflex edges. Our algorithm can be made to run in O((nN + N3) logN) time if robustness is not desired. The robustness of the algorithm stems from its ability to handle all degenerate configurations as well as to maintain topological consistency, while doing floating-point numerical computations. The convex decomposition algorithm is independent of the precision used in the numerical calculations. With slight modifications it also yields a triangulation of the polyhedra into a set of tetrahedra.