The union of moving polygonal pseudodiscs - Combinatorial bounds and applications

Let P be a set of polygonal pseudodiscs in the plane with n edges in total translating with xed velocities in xed directions. We prove that the maximumnumber of combinatorial changes in the union of P is (n (n)). In general, if the pseudodiscs move along curved trajectories, then the maximum number of changes in the union is (n s+2(n)), where s is the maximum number of times any triple of polygon edges meet in a common point. We apply this result in two di erent settings. First, we prove that the complexity of the free space of a constant-complexity polygon translating amidst convex polyhedral obstacles with n edges in total is O(n (n)). Second, we show that the complexity of the space of lines missing a set of n convex homothetic polytopes of constant complexity in 3-space is O(n 4(n)). Both bounds are almost tight in the worst case.