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

Let be a set of pairwise-disjoint polyhedral obstacles in R 3 with a total of n vertices, and let B be a ball in R 3. We show that the combinatorial complexity of the free conguration space F of B amid, i.e., (the closure of) the set of all placements of B at which B does not intersect any obstacle, is O(n 2+"), for any "> 0; the constant of proportionality depends on ". This upper bound almost matches the known quadratic lower bound on the maximum possible complexity of F. The special case in which 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(