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

Abstract

We establish a bound of $O(n^2k^{1+\varepsilon})$, for any $\varepsilon>0$, on the combinatorial complexity of the set $\mathcal{T}$ of line transversals of a collection $\mathcal{P}$ of k convex polyhedra in $\mathbb{R}^3$ with a total of n facets, and we present a randomized algorithm which computes the boundary of $\mathcal{T}$ in comparable expected time. Thus, when $k\ll n$, the new bounds on the complexity (and construction cost) of $\mathcal{T}$ improve upon the previously best known bounds, which are nearly cubic in n. To obtain the above result, we study the set $\mathcal{T}_{\ell_0}$ of line transversals which emanate from a fixed line $\ell_0$, establish an almost tight bound of $O(nk^{1+\varepsilon})$ on the complexity of $\mathcal{T}_{\ell_0}$, and provide a randomized algorithm which computes $\mathcal{T}_{\ell_0}$ in comparable expected time. Slightly improved combinatorial bounds for the complexity of $\mathcal{T}_{\ell_0}$ and comparable improvements in the cost of constructing this set are established for two special cases, both assuming that the polyhedra of $\mathcal{P}$ are pairwise disjoint: the case where $\ell_0$ is disjoint from the polyhedra of $\mathcal{P}$, and the case where the polyhedra of $\mathcal{P}$ are unbounded in a direction parallel to $\ell_0$. Our result is related to the problem of bounding the number of geometric permutations of a collection $\mathcal{C}$ of k pairwise-disjoint convex sets in $\mathbb{R}^3$, namely, the number of distinct orders in which the line transversals of $\mathcal{C}$ visit its members. We obtain a new partial result on this problem.