Computing the Penetration Depth of Two Convex Polytopes in 3D

Abstract

Let A and B be two convex polytopes in ℝ³ with m and n facets, respectively. The penetration depth of A and B, denoted as π(A,B), is the minimum distance by which A has to be translated so that A and B do not intersect. We present a randomized algorithm that computes π(A,B) in O(m^3/4+εn^3/4+ε + m^1+ε + n^1+ε) expected time, for any constant ε > 0. It also computes a vector t such that ‖t‖ = π(A, B) and int(A + t) ∩ B = 0. We show that if the Minkowski sum B⊕(-A) has K facets, then the expected running time of our algorithm is O (K ^1/2+εm^1/4n^1/4 + m^1+ε + n^1+ε), for any ε > 0. We also present an approximation algorithm for computing π(A, B). For any δ > 0, we can compute, in time O(m + n+ (log² (m + n))/δ), a vector t such that ‖t‖ ≤ (1 + δ)π(A, B) and int(A +t) ∩ B = 0. Our result also gives a δ-approximation algorithm for computing the width of A in time O(n + (log²n)/δ), which is simpler and slightly faster than the recent algorithm by Chan [4].

Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by NSF grants EIA-9870724, and CCR-9732787, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by L.G. was supported in part by National Science Foundation grant CCR-9623851 and by US Army MURI grant 5-23542-A. Work by S.H.-P. was supported by the second author was supported by Army Research Office MURI grant DAAH04-96-1-0013. Work by M.S. was supported by NSF Grants CCR-97-32101, CCR-94-24398, by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scientific Research and Development, and the ESPRIT IV LTR project No. 21957 (CGAL), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University.