Computing the Penetration Depth of Two Convex Polytopes in 3D

Let A and B be two convex polytopes in R 3 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(m 3=4+" n 3=4+" + m 1+" + n 1+") 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 ? K 1=2+" m 1=4 n 1=4 + m 1+" + n 1+" , 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 2 (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 2 n)==), which is simpler and slightly faster than the recent algorithm by Chan 4].