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


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