Lines avoiding balls in three dimensions revisited

Research output: Contribution to journalArticlepeer-review


Let B be a collection of n arbitrary balls in ℝ 3. We establish an almost-tight upper bound of O(n 3+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(n 3+ε) free isolated tangents, for any ε>0. This generalizes the result of Agarwal et al. (Discrete Comput. Geom. 34:231-250, 2005), 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(n 3) of Glisse and Lazard (Proc. 26th Annu. Symp. Comput. Geom., pp. 48-57, 2010). Our approach is constructive and yields an algorithm that computes the discrete representation of the boundary of F(B) in O(n 3+ε) time, for any ε>0.

Original languageEnglish
Pages (from-to)65-93
Number of pages29
JournalDiscrete and Computational Geometry
Issue number1
StatePublished - 1 Jul 2012
Externally publishedYes


  • Combinatorial complexity
  • Free space
  • Geometric arrangements
  • Lines in space
  • Tangents to spheres
  • Union of simply-shaped bodies

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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