Line transversals of convex polyhedra in ℝ3

Haim Kaplan*, Natan Rubin, Micha Sharir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We establish a bound of O(n2k1+ε), for any ε > 0, on the combinatorial complexity of the set T of line transversals of a collection P of k convex polyhedra in ℝ3 with a total of n facets, and we present a randomized algorithm which computes the boundary of T in comparable expected time. Thus, when k ≪ 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 Tℓ0 of line transversals which emanate from a fixed line ℓ0, establish an almost tight bound of O(nk1+ε) on the complexity of Tℓ0, and provide a randomized algorithm which computes Tℓ0, in comparable expected time. Slightly improved combinatorial bounds for the complexity of Tℓ0, 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 ℓ0 is disjoint from the polyhedra of P, and the case where the polyhedra of P are unbounded in a direction parallel to ℓ0. Our result is related to the problem of bounding the number of geometric permutations of a collection C of k pairwise-disjoint convex sets in ℝ3, namely, the number of distinct orders in which the line transversals of C visit its members. We obtain a new partial result on this problem.

Original languageEnglish
Pages (from-to)3283-3310
Number of pages28
JournalSIAM Journal on Computing
Volume39
Issue number7
DOIs
StatePublished - 2010

Keywords

  • Combinatorial complexity
  • Convex polyhedra
  • Extremal stabbing lines
  • Line transversals
  • Lines in space

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