Yvon-Villarceau Circle Equivalents on Dupin Cyclides

International audienceA torus contains four families of circles: parallels, meridians and two sets of Yvon-Villarceau circles. Craftworks and artworks based on Yvon-Villarceau circles can be very attractive. Dupin cyclides are images of tori under sphere inversion, so they contain the images of the torus circles families. I applied operations that are known to create effective artworks on tori to Dupin cyclides, and proved them to be feasible. The regularity and the hidden complexity of the objects I obtained make them very attractive. Reviving the 19th century's tradition of mathematical models making, I printed several models, which can help in understanding their geometry. The tools I developed can be generalized to explore transformations of other mathematical objects under sphere inversion. This exploration is just at its beginning, but has already produced interesting new objects

Re-parameterization reduces irreducible geometric constraint systems

International audienceYou recklessly told your boss that solving a non-linear system of size n (n unknowns and n equations) requires a time proportional to n, as you were not very attentive during algorithmic complexity lectures. So now, you have only one night to solve a problem of big size (e.g., 1000 equations/unknowns), otherwise you will be fired in the next morning. The system is well-constrained and structurally irreducible: it does not contain any strictly smaller well-constrained subsystems. Its size is big, so the Newton–Raphson method is too slow and impractical. The most frustrating thing is that if you knew the values of a small number k<<n of key unknowns, then the system would be reducible to small square subsystems and easily solved. You wonder if it would be possible to exploit this reducibility, even without knowing the values of these few key unknowns. This article shows that it is indeed possible. This is done at the lowest level, at the linear algebra routines level, so that numerous solvers (Newton–Raphson, homotopy, and also p-adic methods relying on Hensel lifting) widely involved in geometric constraint solving and CAD applications can benefit from this decomposition with minor modifications. For instance, with k<<n key unknowns, the cost of a Newton iteration becomes O(kn^2) instead of O(n^3). Several experiments showing a significant performance gain of our re-parameterization technique are reported in this paper to consolidate our theoretical findings and to motivate its practical usage for bigger systems

Disclinations, dislocations and continuous defects: a reappraisal

Disclinations, first observed in mesomorphic phases, are relevant to a number of ill-ordered condensed matter media, with continuous symmetries or frustrated order. They also appear in polycrystals at the edges of grain boundaries. They are of limited interest in solid single crystals, where, owing to their large elastic stresses, they mostly appear in close pairs of opposite signs. The relaxation mechanisms associated with a disclination in its creation, motion, change of shape, involve an interplay with continuous or quantized dislocations and/or continuous disclinations. These are attached to the disclinations or are akin to Nye's dislocation densities, well suited here. The notion of 'extended Volterra process' takes these relaxation processes into account and covers different situations where this interplay takes place. These concepts are illustrated by applications in amorphous solids, mesomorphic phases and frustrated media in their curved habit space. The powerful topological theory of line defects only considers defects stable against relaxation processes compatible with the structure considered. It can be seen as a simplified case of the approach considered here, well suited for media of high plasticity or/and complex structures. Topological stability cannot guarantee energetic stability and sometimes cannot distinguish finer details of structure of defects.Comment: 72 pages, 36 figure

On organizing principles of Discrete Differential Geometry. Geometry of spheres

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse

Tangent-ball techniques for shape processing

Shape processing defines a set of theoretical and algorithmic tools for creating, measuring and modifying digital representations of shapes.  Such tools are of paramount importance to many disciplines of computer graphics, including modeling, animation, visualization, and image processing.  Many applications of shape processing can be found in the entertainment and medical industries. In an attempt to improve upon many previous shape processing techniques, the present thesis explores the theoretical and algorithmic aspects of a difference measure, which involves fitting a ball (disk in 2D and sphere in 3D) so that it has at least one tangential contact with each shape and the ball interior is disjoint from both shapes. We propose a set of ball-based operators and discuss their properties, implementations, and applications.  We divide the group of ball-based operations into unary and binary as follows: Unary operators include: * Identifying details (sharp, salient features, constrictions) * Smoothing shapes by removing such details, replacing them by fillets and roundings * Segmentation (recognition, abstract modelization via centerline and radius variation) of tubular structures Binary operators include: * Measuring the local discrepancy between two shapes * Computing the average of two shapes * Computing point-to-point correspondence between two shapes * Computing circular trajectories between corresponding points that meet both shapes at right angles * Using these trajectories to support smooth morphing (inbetweening) * Using a curve morph to construct surfaces that interpolate between contours on consecutive slices The technical contributions of this thesis focus on the implementation of these tangent-ball operators and their usefulness in applications of shape processing. We show specific applications in the areas of animation and computer-aided medical diagnosis.  These algorithms are simple to implement, mathematically elegant, and fast to execute.Ph.D.Committee Chair: Jarek Rossignac; Committee Member: Greg Slabaugh; Committee Member: Greg Turk; Committee Member: Karen Liu; Committee Member: Maryann Simmon

Le contrôle des inflexions et des extremums de courbure portés par les courbes et les surfaces B-Splines

RÉSUMÉ Le contrôle des propriétés différentielles des courbes et des surfaces B-splines est un enjeu important, en particulier pour le domaine de la conception géométrique assistée par ordinateur. Un enjeu qui sollicite autant les méthodes analytiques que numériques dans le but de permettre au concepteur de manipuler les formes avec une aisance toujours croissante. Ce texte explore les possibilités offertes lorsqu'on combine des méthodes numériques de pointe aux travaux de grands géomètres du 19e siècle. Ainsi, de nouveaux algorithmes pour l’optimisation sous contraintes des B-splines ont été développés. Ensuite, ces algorithmes ont été combinés à la théorie des groupes de transformations comme elle a été développée à l’origine par des pionniers comme Sophus Lie, Gaston Darboux et Felix Klein. Ceci permet d’ouvrir des portes vers de nouveaux horizons. Il devient possible de générer de larges espaces de formes sur lesquels on contrôle les propriétés différentielles. Il devient également possible d’éliminer des oscillations de façon sélective ou de manipuler les formes sans introduire d’oscillations indésirables. Avant de progresser vers cet objectif ambitieux, il faut d’abord être en mesure de bien comprendre et de bien visualiser ces propriétés différentielles que l’on souhaite contrôler. L’histoire de la géométrie différentielle classique des courbes et des surfaces est très riche. Cette histoire est revisitée avec une perspective nouvelle. Soit la perspective du contrôle des inflexions et des extremums de courbure. Ceci permet de faire émerger des liens importants entre la géométrie différentielle, la théorie des singularités, les groupes de transformations et l’optique géométrique. Ensuite viennent les algorithmes d’optimisation des B-splines sous contraintes. Les variables indépendantes sont les positions des points de contrôle de la B-spline alors que les contraintes portent sur la position des points de contrôle d’une fonction qui représente les propriétés différentielles de la B-spline. Les algorithmes sont d’abord développés pour les fonctions B-splines à une et deux variables. Une fois ces algorithmes développés, plusieurs possibilités nouvelles s’offrent à nous. Il devient possible, par exemple d’obtenir la courbe qui s’approche le plus d’une autre courbe quelconque sous la contrainte de posséder certaines propriétés différentielles. De cette manière, il devient possible de travailler avec un plus grand nombre de points de contrôle et ainsi dans un espace de forme plus riche sans avoir à se soucier d’oscillations arbitraires. Ceci permet en particulier d’éliminer de façon sélective des oscillations indésirables sur des profils aérodynamiques.----------ABSTRACT Control of B-spline differential properties is an important stake, especially for the field of computer-aided geometric design. An issue that calls for analytical and numerical skills to allow the designer to manipulate shapes in an increasingly efficient way. This text explores possibilities offered by combining new numerical methods with works of 19th century great geometers. Thus, new algorithms for constrained optimization of B-splines are selected and then grafted to the group theory of transformations as it was originally developed by pioneers such as Sophus Lie, Gaston Darboux and Felix Klein. This opens doors to new horizons. It becomes possible to generate large spaces of shapes with a control over their differential properties. This also gives us a selective eraser of curvature extrema and the option to manipulate shapes without introducing undesirables oscillations. Before progressing towards this ambitious goal, one must first be able to understand and visualize these differential properties that one wishes to control. This rich history of the classical differential geometry is revisited with a new perspective. This new perspective is the one of the control of inflections and extrema of curvature. A perspective that allows to establish important links between differential geometry, the theory of singularities, groups of transformations and geometric optics. Next comes the B-splines optimization algorithms with constraints. The independent variables are the B-spline position of the control points, while the constraints are applied to the control points of a function which represents the differential properties of the B-spline. The algorithms are first developed for B-spline functions. Once these algorithms have been developed, several new possibilities open up to us. It becomes possible, for example, to find the closest curve to another one under specified differential properties. This particular algorithm is introduced as an extension to the standard B-spline least squares method to approximate a series of points. The extension consists in adding constraints to produce curve segments with monotonously increasing or decreasing curvature. The interior point method is used to solve the constrained optimization problem. The method requires gradients and those are provided by symbolic B-spline operators. Therefore, the algorithm relies on the arithmetic, differential and variation diminishing properties of the Bsplines to apply the constraints. Thereby, it becomes possible to work with a greater number of control points and thus in a richer shape space without having to manage undesired oscillations

Geometric algorithms for algebraic curves and surfaces

This work presents novel geometric algorithms dealing with algebraic curves and surfaces of arbitrary degree. These algorithms are exact and complete — they return the mathematically true result for all input instances. Efficiency is achieved by cutting back expensive symbolic computation and favoring combinatorial and adaptive numerical methods instead, without spoiling exactness in the overall result. We present an algorithm for computing planar arrangements induced by real algebraic curves. We show its efficiency both in theory by a complexity analysis, as well as in practice by experimental comparison with related methods. For the latter, our solution has been implemented in the context of the Cgal library. The results show that it constitutes the best current exact implementation available for arrangements as well as for the related problem of computing the topology of one algebraic curve. The algorithm is also applied to related problems, such as arrangements of rotated curves, and arrangments embedded on a parameterized surface. In R3, we propose a new method to compute an isotopic triangulation of an algebraic surface. This triangulation is based on a stratification of the surface, which reveals topological and geometric information. Our implementation is the first for this problem that makes consequent use of numerical methods, and still yields the exact topology of the surface.Diese Arbeit stellt neue Algorithmen für algebraische Kurven und Flächen von beliebigem Grad vor. Diese Algorithmen liefern für alle Eingaben das mathematisch korrekte Ergebnis. Wir erreichen Effizienz, indem wir aufwendige symbolische Berechnungen weitesgehend vermeiden, und stattdessen kombinatorische und adaptive numerische Methoden einsetzen, ohne die Exaktheit des Resultats zu zerstören. Der Hauptbeitrag ist ein Algorithmus zur Berechnung von planaren Arrangements, die durch reelle algebraische Kurven induziert sind. Wir weisen die Effizienz des Verfahrens sowohl theoretisch durch eine Komplexitätsanalyse, als auch praktisch durch experimentelle Vergleiche nach. Dazu haben wir unser Verfahren im Rahmen der Softwarebibliothek Cgal implementiert. Die Resultate belegen, dass wir die zur Zeit beste verfügbare exakte Software bereitstellen. Der Algorithmus wird zur Arrangementberechnung rotierter Kurven, oder für Arrangements auf parametrisierten Oberflächen eingesetzt. Im R3 geben wir ein neues Verfahren zur Berechnung einer isotopen Triangulierung einer algebraischen Oberfläche an. Diese Triangulierung basiert auf einer Stratifizierung der Oberfläche, die topologische und geometrische Informationen berechnet. Unsere Implementierung ist die erste für dieses Problem, welche numerische Methoden konsequent einsetzt, und dennoch die exakte Topologie der Oberfläche liefert

Transform domain texture synthesis on surfaces

In the recent past application areas such as virtual reality experiences, digital cinema and computer gamings have resulted in a renewed interest in advanced research topics in computer graphics. Although many research challenges in computer graphics have been met due to worldwide efforts, many more are yet to be met. Two key challenges which still remain open research problems are, the lack of perfect realism in animated/virtually-created objects when represented in graphical format and the need for the transmissiim/storage/exchange of a massive amount of information in between remote locations, when 3D computer generated objects are used in remote visualisations. These challenges call for further research to be focused in the above directions. Though a significant amount of ideas have been proposed by the international research community in their effort to meet the above challenges, the ideas still suffer from excessive complexity related issues resulting in high processing times and their practical inapplicability when bandwidth constraint transmission mediums are used or when the storage space or computational power of the display device is limited. In the proposed work we investigate the appropriate use of geometric representations of 3D structure (e.g. Bezier surface, NURBS, polygons) and multi-resolution, progressive representation of texture on such surfaces. This joint approach to texture synthesis has not been considered before and has significant potential in resolving current challenges in virtual realism, digital cinema and computer gaming industry. The main focus of the novel approaches that are proposed in this thesis is performing photo-realistic texture synthesis on surfaces. We have provided experimental results and detailed analysis to prove that the proposed algorithms allow fast, progressive building of texture on arbitrarily shaped 3D surfaces. In particular we investigate the above ideas in association with Bezier patch representation of 3D objects, an approach which has not been considered so far by any published world wide research effort, yet has flexibility of utmost practical importance. Further we have discussed the novel application domains that can be served by the inclusion of additional functionality within the proposed algorithms.EThOS - Electronic Theses Online ServiceGBUnited Kingdo