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Let Ω be a set of pairwise-disjoint polyhedral obstacles in ¿3 with a total of n vertices, and let B be a ball in ¿3. We show that the combinatorial complexity of the free configuration space ${cal F}$ 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(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 ${cal F}$. 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 ${cal F}$ whose expected running time is O(n 2+¿),for any ¿ > 0.

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