Penetration Depth of Two Convex Polytopes in 3D

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(m3/4+e n3/4+e +m1+e + n1+e) expected time, for any constant e > 0. It also computes a vector t such that ¶t¶ = π(A, B) and int(A + t) ∩ B = θ. We show that if the Minkowski sum B ⊕ (-A) has K facets, then the expected running time of our algorithm is O (K1/2+e m1/4 n1/4 + m1+e + n1+e), for any e > 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 = θ. Our result also gives a δ-approximation algorithm for computing the width of A in time O(n + (1/δ) log2(1/δ)), which is simpler and faster than the recent algorithm by Chan [3].

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