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Courbes rationnelles (BR) à grandes régularités : cas du cercle

Abstract : Many shapes and objects in CAD mathematics are defined from the circle using elementary geometric transformations or deformations. Hence the interest of proposing a parametrization of the circle as a curve (BR), image of 0.1, well adapted to CAD. In chapters 2 and 3, we determine several control mass polygons respectively with 5, 7, 9, 11 mass vectors, allowing to obtain the closure Ck (k = 1 to 5) of the complete circle. In each case, we give necessary and sufficient conditions to obtain only strictly positive mass vectors. More important is to provide as uniform a parameterization of the full circle as possible. The previously mentioned representations have two degrees of freedom which we take advantage of in Chapter 4 to obtain a quasi-uniform representation of the circle: the images of the i/n values of 0.1 are as evenly spaced as possible on the circle. We propose solutions with 5 and 7 mass vectors corresponding to a closed circle C1 and C3. The quality of these representations, highlighted by the error estimation, is confirmed by figures. In Chapter 5, we addressed another problem frequently encountered in CAD geometry: for a given rational curve, can we obtain by successive length elevations, a control polygon whose mass vectors are all positive mass? For some types of mass polygons we answer by some necessary or sufficient conditions, and by an evaluation of the number of necessary iterations.
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Submitted on : Friday, December 3, 2021 - 10:35:48 AM
Last modification on : Tuesday, December 21, 2021 - 3:46:58 AM
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Isabelle Cattiaux-Huillard. Courbes rationnelles (BR) à grandes régularités : cas du cercle. Mathématiques [math]. Université de Valenciennes et du Hainaut-Cambrésis, 1997. Français. ⟨NNT : 1997VALE0022⟩. ⟨tel-03464312⟩



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