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Courbes de Bézier en géométrie algorithmique : approximation et cohérence topologique

Abstract : In this thesis, we propose a method for solving computational geometry problems posed for curve objects (in contrast with "linear" objects : sets of points, segments, polygons...). The objects we study are composite Bézier curves, chosen, on one hand, for the realism they assure in geometric modeling, and on another hand, for the ease of algorithmic processing that their properties offer. Our approach emphasizes the topological aspects of the addressed problems, avoiding the inconsistencies that floating point arithmetics solving of high degree algebraic equations (generated the direct processing of curves) often induces. This aim is reached through the use of converging polygonal approximations, that, in the case of Bézier curves, are naturally provided by the control polygons via the de Casteljau subdivision. Two of the computational geometry fundamental problems are addressed here, the convex hull and the arrangements, both of them in dimension 2. In the arrangements case, the notion of topology (combinatoric) is well known ; in the convex hull case, we define it rigorously. For the two problems, we show that one can obtain all the topological information defining (implicitely, but correctly and completely) the exact solution dealing exclusively with the polygonal approximations of the given objects. The theoretical results that we have obtained are made concrete by the design of algorithms proven correct and convergent and for which cost studies have been done. Some examples illustrate the functioning of these algorithms, demonstratind the validity of the proposed method.
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Manuela Neagu. Courbes de Bézier en géométrie algorithmique : approximation et cohérence topologique. Modélisation et simulation. Université Joseph-Fourier - Grenoble I, 1998. Français. ⟨tel-00004897⟩

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