# Computing the Penetration Depth of Two Convex Polytopes in 3D

## Abstract

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 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+ε}n^{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+ε}m^{1/4}n^{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+* (log^{2} *(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 +* (log^{2}n)/δ), which is simpler and slightly faster than the recent algorithm by Chan [4].

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