Lines avoiding balls in three dimensions revisited

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations


Let ℬ be a collection of n arbitrary balls in ℝ3. We establish an almost-tight upper bound of O(n3+ε), for any ε > 0, on the complexity of the space ℱ(ℬ) of all the lines that avoid all the members of ℬ. In particular, we prove that the balls of ℬ admit O(n3+ε) free isolated tangents, for any ε > 0. This generalizes the result of Agarwal et al. [1], who established this bound only for congruent balls, and solves an open problem posed in that paper. Our bound almost meets the recent lower bound of Ω(n3) of Glisse and Lazard [6]. Our approach is constructive and yields an algorithm that computes a discrete representation of the boundary of ℱ(ℬ) in O(n3+ε) time, for any ε > 0.

Original languageEnglish
Title of host publicationProceedings of the 26th Annual Symposium on Computational Geometry, SCG'10
Number of pages10
StatePublished - 30 Jul 2010
Externally publishedYes
Event26th Annual Symposium on Computational Geometry, SoCG 2010 - Snowbird, UT, United States
Duration: 13 Jun 201016 Jun 2010

Publication series

NameProceedings of the Annual Symposium on Computational Geometry


Conference26th Annual Symposium on Computational Geometry, SoCG 2010
Country/TerritoryUnited States
CitySnowbird, UT


  • Arrangements
  • Combinatorial complexity
  • Free lines
  • Lines in space
  • Tangency surfaces

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics


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