Finding stabbing lines in 3-space

Abstract

A line intersecting all polyhedra in a set ℬ is called a "stabber" for the set ℬ. This paper addresses some combinatorial and algorithmic questions about the set ℒ(ℬ) of all lines stabbing ℬ. We prove that the combinatorial complexity of ℒ(ℬ) has an {Mathematical expression} upper bound, where n is the total number of facets in ℬ, and c is a suitable constant. This bound is almost tight. Within the same time bound it is possible to determine if a stabbing line exists and to find one. © 1992 Springer-Verlag New York Inc.