The union of moving polygonal pseudodiscs - Combinatorial bounds and applications

Abstract Let P be a set of polygonal pseudodiscs in the plane with n edges in total translating with fixed velocities in fixed directions. We prove that the maximum number of combinatorial changes in the union of the pseudodiscs in P is Θ ( n 2 α ( n )). In general, if the pseudodiscs move along curved trajectories, then the maximum number of changes in the union is Θ ( nλ 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 to prove 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 2 λ 4 ( n )). This bound is almost tight in the worst case.