Differential equation-based shape interpolation for surface blending and facial blendshapes.

Differential equation-based shape interpolation has been widely applied in geometric modelling and computer animation. It has the advantages of physics-based, good realism, easy obtaining of high- order continuity, strong ability in describing complicated shapes, and small data of geometric models. Among various applications of differential equation-based shape interpolation, surface blending and facial blendshapes are two active and important topics. Differential equation-based surface blending can be time-independent and time-dependent. Existing differential equation-based surface blending only tackles time-dependen

Utilisation des cyclides de Dupin quartiques et des supercyclides quartiques en modélisation géométrique

Dans cette thèse, nous proposons de nouvelles primitives pour la modélisation géométrique : les cyclides de Dupin quartiques et les supercyclides quartiques. Les cyclides de Dupin ont été inventées en 1822 par le mathématicien français Pierre-Charles Dupin. Ce sont des surfaces algébriques de degré 4, à lignes de courbure circulaires, possédant une équation paramétrique et deux équations implicites équivalentes. nous montrons l'apport de ces surfaces pour la modélisation géométrique à travers l'utilisation de ces cyclides pour la jointure de surfaces algébriques. L'utilisation des cyclides de Dupin permet de remplacer un problème de jointure en 3D par un problème de jointure plus simple en 2D en construisant deux arcs de cercles modélisés par des courbes de Bézier rationnelles quadratiques. De plus, la paramétrisation initiale des surfaces n'intervient pas. Cependant, les propriétés géométriques des cyclides de Dupin impliquent que ces surfaces soient de révolution. Ainsi, l'utilisation des supercyclides permet de généraliser les jointures aux surfaces elliptiques. Pour faire le lien entre ces nouvelles primitives et les surfaces paramétriques, qui sont largement utilisées en modélisation géométrique, nous avons étudié la conversion des cyclides en surfaces de Bézier rationnelles biquadratiques. Afin de représenter une cyclide de Dupin entièrement, nous avons proposé deux variantes de l'algorithme de Mike Pratt. Puis nous en avons proposé deux autres en utilisant les propriétés barycentriques des surfaces de Bézier. Nous avons également donné des critères nécessaires et non suffisants afin de construire une surface de Bézier rationnelle biquadratique convertible en un carreau de cyclide de Dupin. Trois algorithmes de conversion ont été alors proposés.In this thesis, we propose new primitives for geometric modeling : quartics Dupin cyclides and quartics supercyclides. Dupin cyclides were invented in 1822 by the french mathematician Pierre-Charles Dupin. These are algebraic surfaces of degree 4 with circular lines of curvature. They have a parametric equation and two equivalent implicit equations. We show the contributions of these surfaces for the geometrical modeling through their use for the blending of algebraic surfaces. The use of the Dupin cyclides permit to replace a 3D blend problem by a simpler 2D blend problem : building two arcs of circles modelled by quadratic rational Bézier curves. Moreover, the initial parametrization of surfaces does not intervene. However, the geometric properties of the Dupin cyclides require that the blended surfaces are revolution surfaces. Thus, the use of the supercyclides permits to generalize the blending to elliptic surfaces. To establish a link between these new primitives and parametric surfaces, which are widely used in geometric modeling, we studied the conversion of cyclides to biquadratic rational Bézier surfaces. In order to entirely represent a Dupin cyclide, we proposed two alternatives of Mike Pratt's algorithm. Then, we proposed two new algorithms based on the barycentric properties of Bézier surfaces. We also give the necessary and nonsufficient conditions to build a biquadratic rational Bézier surfaces convertible to a patch of Dupin cyclide. Three conversion algorithms are then proposed.DIJON-BU Sciences Economie (212312102) / SudocSudocFranceF

On nonsingular, cyclide transition surfaces.

The natural quadrics (the plane, sphere, right circular cylinder, and right circular cone) are an important subset of the primitives used in computer modeling systems. The (Dupin) cyclide is a degree 4 generalization of a torus that appears to be a natural choice for a new primitive in these modelers. This thesis studies nonsingular, cyclide transition surfaces between any two natural quadrics. These surfaces are tubes that are tangent to the natural quadrics they connect along circles. Necessary and sufficient conditions (with constructive proofs) for the existence of cyclide transition surfaces are given. Except for the cone/cone case, none of the previously published constructions indicate the exact conditions of their validity. Further, in previous work, no effort is made to ensure the construction of nonsingular transition surfaces, even though a designer seldom desires singular surfaces. Each quadric/quadric case is analyzed to determine exactly when the nonsingular, cyclide transition surface is a blend and when it is a join. This distinction, previously ignored, is important since the term blend usually has a very specific meaning. (When visualizing a blend between two intersecting surfaces, a designer typically imagines the surface formed by pressing putty with one's thumb along the curve of intersection between the surfaces to remove the sharp crease.) A further result states that except in the case of intersecting spheres, blends and joins cannot simultaneously exist. Affine transformations of cyclides, a subset of the supercyclides, and their role in blending is studied. This analysis leads to the following original result. A plane and an axial natural quadric can be blended by a supercyclide along any given ellipse on the axial natural quadric if and only if the plane and the axial natural quadric intersect in an ellipse. Also shown is that similar extensions to the remaining cases, that build upon the work presented here, cannot be made. This topic is beyond the scope of the thesis and is left to future investigations.Ph.D.Applied SciencesComputer scienceMathematicsMechanical engineeringPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/130188/2/9721935.pd