# Lines and free line segments Tangent to Arbitrary Three-dimensional Convex Polyhedra

2 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée
4 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : Motivated by visibility problems in three dimensions, we investigate the complexity and construction of the set of tangent lines in a scene of three-dimensional polyhedra. We prove that the set of lines tangent to four possibly intersecting convex polyhedra in $\Real^3$ with a total of $n$ edges consists of $\Theta(n^2)$ connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrarily degenerate scenes. More generally, we show that a set of $k$ possibly intersecting convex polyhedra with a total of $n$ edges admits, in the worst case, $\Theta(n^2k^2)$ connected components of maximal free line segments tangent to at least four polytopes. Furthermore, these bounds also hold for possibly occluded lines rather than maximal free line segments. Finally, we present a $O(n^2 k^2 \log n)$ time and $O(nk^2)$ space algorithm that, given a scene of $k$ possibly intersecting convex polyhedra, computes all the \emph{minimal} free line segments that are tangent to any four of the polytopes and are isolated transversals to the set of edges they intersect; in particular, we compute at least one line segment per connected component of tangent lines.
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Article dans une revue
SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 2007, 37 (2), pp.522-551. 〈10.1137/S0097539705447116〉

Littérature citée [22 références]

https://hal.inria.fr/inria-00103916
Contributeur : Sylvain Lazard <>
Soumis le : lundi 19 novembre 2007 - 17:38:14
Dernière modification le : mardi 27 février 2018 - 17:56:49
Document(s) archivé(s) le : mardi 6 avril 2010 - 18:30:23

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Hervé Brönnimann, Olivier Devillers, Vida Dujmovic, Hazel Everett, Marc Glisse, et al.. Lines and free line segments Tangent to Arbitrary Three-dimensional Convex Polyhedra. SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 2007, 37 (2), pp.522-551. 〈10.1137/S0097539705447116〉. 〈inria-00103916〉

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