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 in ℝ3. We show that the combinatorial complexity of the free configuration space ℱ of B amid Ω, i.e., (the closure of) 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. The special case in which Ω is a set of lines is studied separately. We also present a few extensions of this result, including a randomized algorithm for computing the boundary of ℱ whose expected running time is O(n2+ε), for any ε > 0.

Full Text

Duke Authors

Cited Authors

  • Agarwal, PK; Sharir, M

Published Date

  • January 1, 2000

Published In

Volume / Issue

  • 24 / 4

Start / End Page

  • 645 - 685

International Standard Serial Number (ISSN)

  • 0179-5376

Digital Object Identifier (DOI)

  • 10.1007/s4540010064

Citation Source

  • Scopus