The predicates for the Voronoi diagram of ellipses
Proceedings of the twenty-second annual symposium on Computational geometry
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Offset Approach to Defining 3D Digital Lines
ISVC '08 Proceedings of the 4th International Symposium on Advances in Visual Computing
The Voronoi Diagram of Circles and Its Application to the Visualization of the Growth of Particles
Transactions on Computational Science III
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
Minimal offsets that guarantee maximal or minimal connectivity of digital curves in nD
DGCI'09 Proceedings of the 15th IAPR international conference on Discrete geometry for computer imagery
A certified delaunay graph conflict locator for semi-algebraic sets
ICCSA'05 Proceedings of the 2005 international conference on Computational Science and its Applications - Volume Part I
The offset to an algebraic curve and an application to conics
ICCSA'05 Proceedings of the 2005 international conference on Computational Science and its Applications - Volume Part I
Convex hull and voronoi diagram of additively weighted points
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Connectedness of offset digitizations in higher dimensions
CompIMAGE'10 Proceedings of the Second international conference on Computational Modeling of Objects Represented in Images
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Most of the curves and surfaces encountered in geometric modelling are defined as the set of solutions of a system of algebraic equations and inequalities (semi-algebraic sets). Many problems from different fields involve proximity queries like finding the (nearest) neighbours or quantifying the neighbourliness of two objects. The Voronoi diagram of a set of sites is a decomposition of space into proximal regions. The proximal region of a site is the locus of points closer to that site than to any other one. Voronoi diagrams allow one to answer proximity queries after locating a query point in the Voronoi zone it belongs to. The dual graph of the Voronoi diagram is called the Delaunay graph. Only approximations by conics can guarantee a proper order of continuity at contact points, which is necessary for guaranteeing the exactness of the Delaunay graph. The theoretical purpose of this thesis is to elucidate the basic algebraic and geometric properties of the offset to an algebraic curve and to reduce the semi-algebraic computation of the Delaunay graph to eigenvalues computations. The practical objective of this thesis is the certified computation of the Delaunay graph for low degree semi-algebraic sets embedded in the Euclidean plane. The methodology combines interval analysis and computational algebraic geometry. The central idea of this thesis is that a (one time) symbolic preprocessing may accelerate the certified numerical evaluation of the Delaunay graph conflict locator. The symbolic preprocessing is the computation of the implicit equation of the generalised offset to conics. The reduction of the Delaunay graph conflict locator for conics from a semi-algebraic problem to a linear algebra problem has been possible through the use of the generalised Voronoi vertex (a concept introduced in this thesis). The certified numerical computation of the Delaunay graph has been possible by using an interval analysis based library for solving zero-dimensional systems of equations and inequalities (ALIAS). The certified computation of the Delaunay graph relies on theorems on the uniqueness of a root in given intervals (Kantorovitch, Moore-Krawczyk). For conics, the computations get much faster by considering only the implicit equations of the generalised offsets.