Pipes, Cigars, and Kreplach: The Union of Minkowski Sums in Three Dimensions

Abstract

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.