On the complexity of a single cell in certain arrangements of surfaces related to motion planning

Abstract

We obtain near-quadratic upper bounds on the maximum combinatorial complexity of a single cell in certain arrangements of n surfaces in 3-space where the lower bound for this quantity is Ω(n2) or slightly larger. We prove a theorem that identifies a collection of topological and combinatorial conditions for a set of surface patches in space, which make the complexity of a single cell in an arrangement induced by these surface patches near-quadratic. We apply this result to arrangements related to motion-planning problems of two types of robot systems with three degrees of freedom and also to a special type of arrangements of triangles in space. The complexity of the entire arrangement in each case that we study can be Θ(n3) in the worst case, and our single-cell bounds are of the form O(n2 α(n)), O(n2 log n), or O(n2 α(n)log n). The only previously known similar bounds are for the considerably simpler arrangements of planes or of spheres in space, where the bounds are Θ(n) and Θ(n2), respectively. For some of the arrangements that we study we derive near-quadratic-time algorithms to compute a single cell. © 1994 Springer-Verlag New York Inc.