SWAT 2000: Algorithm Theory - SWAT 2000 pp 328-338

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

• Pankaj K. Agarwal
• Leonidas J. Guibas
• Sariel Har-Peled
• Alexander Rabinovitch
• Micha Sharir
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1851)

## 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(m3/4+εn3/4+ε + m1+ε + n1+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+εm1/4n1/4 + m1+ε + n1+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].

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## Authors and Affiliations

• Pankaj K. Agarwal
• 1
• Leonidas J. Guibas
• 2
• Sariel Har-Peled
• 3
• Alexander Rabinovitch
• 4
• Micha Sharir
• 5
• 6
1. 1.Center for Geometric Computing, Department of Computer ScienceDuke UniversityDurhamUSA
2. 2.Computer Graphics Laboratory, Computer Science DepartmentStanford UniversityStanford
3. 3.Center for Geometric Computing, Department of Computer ScienceDuke UniversityDurhamUSA
4. 4.Synopsys Inc.MarlboroUSA
5. 5.School of Mathematical SciencesTel Aviv UniversityIsrael
6. 6.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA