Penetration Depth of Two Convex Polytopes in 3D

Let A and B be two convex polytopes in R with m and n facets, respectively. The penetration depth of A and B, denoted as (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 (A;B) in O(mn + m + n) expected time, for any constant " > 0. It also computes a vector t such that ktk = (A;B) and int(A + t) \ B = ;. We show that if the Minkowski sum B ( A) has K facets, then the expected running time of our algorithm is O Kmn +m + n , for any " > 0. We also present an approximation algorithm for computing (A;B). For any > 0, we can compute, in time O(m + n + (log(m + n))= ), a vector t such that ktk (1+ ) (A;B) and int(A+t)\B = ;. Our result also gives a -approximation algorithm for computing the width of A in time O(n + (log n)= ), which is simpler and slightly faster than the recent algorithm by Chan [3]. Work by P.A. was supported by Army Research O ce 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 Army Research O ce MURI grant DAAH04-96-1-0013. Work by M.S. was supported by NSF Grants CCR-97-32101, CCR94-24398, by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F., the German-Israeli Foundation for Scienti c 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. Center for Geometric Computing, Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129, USA. E-mail: [email protected] Computer Graphics Laboratory, Computer Science Department, Stanford University, Stanford CA 94305 E-mail: [email protected] Center for Geometric Computing, Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129, USA. Current address: Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2987. E-mail: [email protected] Synopsys Inc., 154 Crane Meadow Rd, Suite 300, Marlboro, MA 01752, USA. E-mail: [email protected] School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel; and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA. E-mail: [email protected]