Line Transversals of Convex Polyhedra in ${\mathbb{R}}^3$

Abstract

We establish a bound of $O(n^2k^{1+\eps})$, for any $\eps>0$, on the combinatorial complexity of the set $\T$ of line transversals of a collection $\P$ of $k$ convex polyhedra in ${\mathbb{R}}^3$ with a total of $n$ facets, and present a randomized algorithm which computes the boundary of $\T$ in comparable expected time. Thus, when $k\ll n$, the new bounds on the complexity (and construction cost) of $\T$ improve upon the previously best known bounds, which are nearly cubic in $n$. To obtain the above result, we study the set $\TL$ of line transversals which emanate from a fixed line ${\ell }_0$, establish an almost tight bound of $O(nk^{1+\eps})$ on the complexity of $\TL$, and provide a randomized algorithm which computes $\TL$ in comparable expected time. Slightly improved combinatorial bounds for the complexity of $\TL$, and comparable improvements in the cost of constructing this set, are established for two special cases, both assuming that the polyhedra of $\P$ are pairwise disjoint: the case where ${\ell }_0$ is disjoint from the polyhedra of $\P$, and the case where the polyhedra of $\P$ are unbounded in a direction parallel to ${\ell }_0$.

10 pages+ 15 page appendix