Line transversals to disjoint balls

Ciprian Borcea (1) , Xavier Goaoc (2) , Sylvain Petitjean (2)

1 Departement of mathematics
2 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility

We prove that the set of directions of lines intersecting three disjoint balls in $\mathbb{R}^3$ in a given order is a strictly convex subset of $\mathbb{S}^2$. We then generalize this result to $n$ disjoint balls in $\mathbb{R}^d$. As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.

Mots clés: Hadwiger-type theorem; Helly-type theorem; Hessian; convexity; disjoint balls; geometric transversal theory; lines

Domaines: Géométrie algorithmique [cs.CG]

Ciprian Borcea, Xavier Goaoc, Sylvain Petitjean. Line transversals to disjoint balls. 23rd Annual ACM Symposium on Computational Geometry 2007 - SoCG'07, 2007, Gyeongju, South Korea. pp.245-254, ⟨10.1145/1247069.1247115⟩. ⟨inria-00176201⟩