Triangles in space or building (and analyzing) castles in the Air

摘要

We show that the combinatorial complexity of all non-convex cells in an arrangement of n (possibly intersecting) triangles in 3-space is &Ogr;(n7/3+δ), for any δ0, and that this bound is almost tight in the worst case. Our bound significantly improves a previous nearly cubic bound of Pach and Sharir. We also present a (nearly) worst-case optimal randomized algorithm for calculating a single cell of the arrangement, analyze some special cases of the problem where improved bounds (and better algorithms) can be obtained, and describe applications of our results to translational motion planning for polyhedra in 3-space.