Lines Avoiding Balls in Three Dimensions Revisited |
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Authors: | Natan Rubin |
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Institution: | 1. The Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel 2. Department of Mathematics and Computer Science, Freie Universit?t Berlin, Takustr. 9, 14195, Berlin, Germany
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Abstract: | Let $\mathcal{B}$ be a collection of n arbitrary balls in ?3. We establish an almost-tight upper bound of O(n 3+?? ), for any ??>0, on the complexity of the space $\mathcal{F}(\mathcal{B})$ of all the lines that avoid all the members of $\mathcal{B}$ . In particular, we prove that the balls of $\mathcal{B}$ admit O(n 3+?? ) free isolated tangents, for any ??>0. This generalizes the result of Agarwal et al.?(Discrete Comput. Geom. 34:231?C250, 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 ??(n 3) of Glisse and Lazard (Proc. 26th Annu. Symp. Comput. Geom., pp. 48?C57, 2010). Our approach is constructive and yields an algorithm that computes the discrete representation of the boundary of $\mathcal{F}(B)$ in O(n 3+?? ) time, for any ??>0. |
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