The number of lines tangent to arbitrary convex polyhedra in 3D

Abstract

We prove that the lines tangent to four possibly intersecting convex polyhedra in ℝR 3 with n edges in total form θ(n 2) connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrary degenerate scenes. More generally, we show that a set of κ convex polyhedra with a total of n edges admits, in the worst case, θ(n 2κ 2) connected components of (possibly occluded) lines tangent to any four of these polyhedra. We also show a lower bound of Ω(n 2κ 2) on the number of non-occluded maximal line segments tangent to any four of these κ convex polyhedra.