# Pipes, cigars, and kreplach: The union of Minkowski sums in three dimensions

Published

Journal Article

Let Ω be a set of pairwise-disjoint polyhedral obstacles in R3 with a total of n vertices, and let B be a ball. We show that the combinatorial complexity of the free configuration space F of B amid Ω, i.e., the set of all placements of B at which B does not intersect any obstacle, is O(n2+ε), for any ε>0; the constant of proportionality depends on ε. This upper bound almost matches the known quadratic lower bound on the maximum possible complexity of F. We also present a randomized algorithm to compute the boundary of F whose expected running time is O(n2+ε), for any ε>0.

### Duke Authors

### Cited Authors

- Agarwal, PK; Sharir, M

### Published Date

- January 1, 1999

### Published In

- Proceedings of the Annual Symposium on Computational Geometry

### Start / End Page

- 143 - 153

### Citation Source

- Scopus