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 n2 n. In general, if the pseudodiscs move along curved trajectories, then the maximum number of changes in the union is ns+2n, where s is the maximumnumber 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 On2 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 On24n. Both bounds are almost tight in the worst case.

• Type: Journal
• Year: 1998
• Where: COMGEO
• Authors: Mark de Berg, Hazel Everett, Leonidas J. Guibas
• Keywords: Applied Computing, Combinatorial Changes, COMGEO 1998, Polygon Edges, Polygonal Pseudodiscs