Problèmes de Géométrie Algorithmique sur les Droites et les Quadriques en Trois Dimensions

Sylvain Lazard 1
1 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : This Habilitation thesis presents my main work over the last years in non-linear computational geometry. I present two bodies of work, one on the design and implementation of certified and efficient geometric algorithms for non-linear primitives, in particular quadrics, the other on the properties of lines in space in the context of 3D visibility problems.

Concerning quadrics, my main achievement has been the completion of the first-ever exact, complete, near-optimal and efficient algorithm and implementation for parameterizing the intersection of two quadrics in three-dimensional real projective space. This contribution is a considerable breakthrough on a long-standing open problem and it is the first complete and robust solution to one of the most basic problems of solid modeling by implicit curved surfaces. I also present some very nice results on Voronoi diagrams of three lines, a partition of the space in cells bounded by quadric patches. We show that the topology of such diagrams is invariant for lines in general position and we obtain a monotonicity property on the arcs of the diagram. We deduce a simple algorithm for sorting points along such an arc, which is presumably of great interest for future efficient algorithms for computing the medial axis of a polyhedron. The proof technique, which relies heavily upon modern tools of computer algebra, is also interesting in its own right.

Concerning the properties of lines in space in the context of 3D visibility problems, I present a body of results that address several issues. I first present results on the structural properties of lines that are tangent or transversal to four primitives. In particular, I present a characterization of the degenerate configurations of four spheres that admit infinitely many common tangent, a characterization of the set of line transversals to four segments, and a study of the worst-case number of tangents to four triangles. Second, I present several results on the combinatorial properties of geometric structures in the context of 3D visibility. In particular, I present several important results on the complexity of the silhouette of a polyhedron from a random viewpoint and on the worst-case and expected complexity of the visibility complex, a data structure that encode visibility information. I also present some new surprising bounds on the worst-case complexity of the umbra cast on a plane by polygonal light sources in the presence of convex polyhedral obstacles. Finally, I present the first non-trivial implementable algorithm for computing the set of segments tangent to four among k possibly intersecting convex polyhedra, that is, roughly speaking, the vertices of the visibility complex.
Document type :
Habilitation à diriger des recherches
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Sylvain Lazard. Problèmes de Géométrie Algorithmique sur les Droites et les Quadriques en Trois Dimensions. Génie logiciel [cs.SE]. Université Nancy II, 2007. ⟨tel-00189033⟩

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