A VoMVi Complex for Supporting Optimal Path Queries for UAVs
Let B be a convex polyhedron translating in 3-space amidst k convex polyhedral obstacles A1, . . . , Ak with pairwise disjoint interiors. The free configuration space (space of all collision-free placements) of B can be represented as the complement of the union of the Minkowski sums Pi = Ai ⊕ (−B), for i = 1, . . . , k. We show that the combinatorial complexity of the free configuration space of B is O(nk log k), and that it can be Ω(nkα(k)) in the worst case, where n is the total complexity of the individual Minkowski sums P1, . . . , Pk. We also derive an efficient randomized algorithm that constructs this configuration space in expected time O(nk log k logn).