Convex Decomposition of Polyhedra and Robustness

We present a simple algorithm to compute a convex decomposition of a non-convex, non-manifold polyhedron of arbitrary genus (handles). The algorithm takes a non-convex polyhedron with n edges and r notches (features causing non-convexity in the polyhedra) and produces a worst-case optimal O(r2 ) number of convex polyhedra Si, with U;S; = S, in O(nr2 ) time and O(nr) space. Recenlly, Chazelle and Patios have given a fast O(n r + r2 logr) time algorithm to tetrahedraljze a non-convex simple polyhedron. Their algorithm, however, works for a simple polyhedron of genus 0 and with no shells (inner boundaries). The input polyhedron of our algorithm may have arbitrary genus and inner boundaries and may be a non-manifold. We also present an algorithm for the same problem while doing only finite precision a.rithmetic computations.