Line Transversals of Convex Polyhedra in $\reals^3$

TL;DR

This paper establishes new bounds on the combinatorial complexity of line transversals of convex polyhedra in three-dimensional space and provides randomized algorithms for their computation, improving previous bounds especially when the number of polyhedra is small.

Contribution

The paper introduces nearly tight bounds on the complexity of line transversals and algorithms for their computation, advancing understanding of geometric transversal structures.

Findings

Bound of O(n^2 k^{1+ε}) on the complexity of line transversals

Bound of O(n k^{1+ε}) on the complexity of transversals emanating from a fixed line

Improved bounds for disjoint polyhedra cases

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 $\reals^3$ 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 $\ell_0$.