Combinatoire des droites et segments pour la visibilité 3D

Marc Glisse 1
1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : This thesis mainly presents results on combinatorial problems on lines and segments that appear naturally in the study of visibility in three dimensions. We first show results on the size of the silhouette of an object seen from a viewpoint, that is the complexity of the set of lines tangent to the object that go through the point. In particular, we give the first non-trivial theoretical bounds for non-convex polyhedra, that is that under some reasonable hypotheses, the average complexity of the silhouette is at most the square root of the complexity of the polyhedron, a phenomenon widely observed in graphics. We also provide bounds, on average and in the worst case, on the number of lines tangent to four objects where the objects are either polytopes or balls. These bounds give us hope that the size of global structures like the visibility complex might not be prohibitive. The bounds on the polytopes are also the first that take advantage of the structure of scenes where triangles are organized in realistic polytopes, that is polytopes that need not be disjoint. These bounds eventually induce the first non-trivial bounds on the complexity of the umbras created by area light sources. The results presented in this thesis significantly enhance the state of the art on the combinatorial properties of visibility structures in three dimensions and should help the future algorithmic developments for these problems.
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Submitted on : Tuesday, November 27, 2007 - 4:17:08 PM
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  • HAL Id : tel-00192337, version 1

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Marc Glisse. Combinatoire des droites et segments pour la visibilité 3D. Modélisation et simulation. Université Nancy II, 2007. Français. ⟨tel-00192337⟩

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