Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Ray Shooting Amidst Spheres in Three Dimensions and Related Problems
SIAM Journal on Computing
Lines Avoiding Unit Balls in Three Dimensions
Discrete & Computational Geometry
On the Union of Cylinders in Three Dimensions
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Line transversals of convex polyhedra in R3
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Linear Data Structures for Fast Ray-Shooting amidst Convex Polyhedra
Algorithmica - Special Issue: European Symposium on Algorithms 2007, Guest Editors: Larse Arge and Emo Welzl
Counting and representing intersections among triangles in three dimensions
Computational Geometry: Theory and Applications
On the complexity of sets of free lines and line segments among balls in three dimensions
Proceedings of the twenty-sixth annual symposium on Computational geometry
On the complexity of sets of free lines and line segments among balls in three dimensions
Proceedings of the twenty-sixth annual symposium on Computational geometry
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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(n{3+ε) time, for any ε0.