Lines avoiding balls in three dimensions revisited

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 the discrete representation of the boundary of F(B) in O(n3+ε ) time, for any ε > 0.

• Type: Conference
• Year: 2010
• Where: COMPGEOM
• Authors: Natan Rubin
• Keywords: Almost-tight Upper Bound, Arbitrary Balls, COMPGEOM 2010, Free Isolated Tangents