## Abstract

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 language | English |
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Pages (from-to) | 65-93 |

Number of pages | 29 |

Journal | Discrete and Computational Geometry |

Volume | 48 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jul 2012 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

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