On stabbling lines for convex polyhedra in 3D

Abstract

A line ℓ is called a stabbling line for a set B of convex polyhedra in R 3 if it intersects every polyhedron of B . This paper presents an upper bound of O( n 3log n) on the complexity of the space of stabbling lines for B , where n is the number of edges in the polyhedra of B . We solve a more general problem that counts the number of faces in a set of convex polyhedra, which are implicitly defined by a set of half-spaces and a set of hyperplanes. We show that the former problem is a special case of the latter problem. We also apply this technique to obtain an upper bound on the number of distinct faces that ever appear on the intersection of a set of half-spaces as we insert or delete half-spaces dynamically.