# Computing the penetration depth of two convex polytopes in 3d

Published

Conference Paper

© Springer-Verlag Berlin Heidelberg 2000. 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+ɛn3/4+ɛ + m1+ɛ + n1+ɛ) expected time, for any constant ɛ > 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+ɛm1/4n1/4 + m1+ɛ + n1+ɛ), for any ɛ > 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 + (log2 n)/δ), which is simpler and slightly faster than the recent algorithm by Chan.

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Guibas, LJ; Har-Peled, S; Rabinovitch, A; Sharir, M

### Published Date

- January 1, 2000

### Published In

### Volume / Issue

- 1851 /

### Start / End Page

- 328 - 338

### Electronic International Standard Serial Number (EISSN)

- 1611-3349

### International Standard Serial Number (ISSN)

- 0302-9743

### International Standard Book Number 10 (ISBN-10)

- 3540676902

### International Standard Book Number 13 (ISBN-13)

- 9783540676904

### Digital Object Identifier (DOI)

- 10.1007/3-540-44985-X

### Citation Source

- Scopus