5 research outputs found
Rational rolling ball blending of natural quadrics
We construct a blending surface of two natural quadrics using rational variable rolling ball approach, i.e. as a canal surface with a rational spine curve and a rational radius. All general positions of the given quadric surfaces are considered. The proposed construction is Laguerre invariant. In particular, the blending surface has rational offset of the same degree.
NatĆ«raliĆł kvadrikiĆł jungimas racionalaus apriedanÄio rutuliuko metodu
Santrauka
NatĆ«ralios kvadrikos (sferos, apskritiminiai cilindrai ir kĆ«giai) daĆŸnai naudojamos geometriniame modeliavime. Ć iame darbe siĆ«lomas naujas dvieju natĆ«raliu kvadrikiu glodaus jungimo metodas, naudojant kintamo racionalaus spindulio apriedanÄio rutuliuko metoda, t.y. jungiamasis pavirĆĄius â tai kanalinis pavirĆĄius, kuris turi racionalia aĆĄine kreive ir racionalu spinduli. Metodas tinka visiems dvieju kvadrikiu bendru poziciju atvejams. Konstrukcija yra invariantiĆĄka Laguerre geometrijos atĆŸvilgiu: pavyzdĆŸiui, jungiamasis pavirĆĄius turi to paties laipsnio racionalu ofseta.
First Published Online: 14 Oct 201
Geometry of tubular surfaces and their focal surfaces in Euclidean 3-space
In this study, we examined the focal surfaces of tubular surfaces in Euclidean 3-space . We achieved some significant results for these surfaces in accordance with the modified orthogonal frame. Additionally, we proposed a few geometric invariants that illustrated the geometric characteristics of these surfaces, such as flat, minimal, Weingarten, and linear-Weingarten surfaces, using the traditional methods of differential geometry. Additionally, the asymptotic and geodesic curves of these surfaces have been researched. At last, we presented an example as an instance of use to validate our theoretical findings
Generalized plane offsets and rational parameterizations
In the first part of the paper a planar generalization of offset curves is introduced and some properties are derived. In particular, it is seen that these curves exhibit good regularity properties and a study on self-intersection avoidance is performed. The representation of a rational curve as the envelope of its tangent lines, following the approach of Pottmann, is revisited to give the explicit expression of all rational generalized offsets. Other famous shapes, such as constant width curves, bicycle tire-tracks curves and Zindler curves are related to these generalized offsets. This gives rise to the second part of the paper, where the particular case of rational parameterizations by a support function is considered and explicit families of rational constant width curves, rational bicycle tire-track curves and rational Zindler curves are generated and some examples are shown
Le contrÎle des inflexions et des extremums de courbure portés par les courbes et les surfaces B-Splines
RĂSUMĂ
Le contrÎle 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 possibilitĂ©s offertes lorsqu'on combine des mĂ©thodes numĂ©riques de pointe aux travaux de grands gĂ©omĂštres du 19e siĂšcle. Ainsi, de nouveaux algorithmes pour lâoptimisation sous contraintes des B-splines ont Ă©tĂ© dĂ©veloppĂ©s. Ensuite, ces algorithmes ont Ă©tĂ© combinĂ©s Ă la thĂ©orie des groupes de transformations comme elle a Ă©tĂ© dĂ©veloppĂ©e Ă lâorigine par des pionniers comme Sophus Lie, Gaston Darboux et Felix Klein. Ceci permet dâouvrir des portes vers de nouveaux horizons. Il devient possible de gĂ©nĂ©rer de larges espaces de formes
sur lesquels on contrĂŽle les propriĂ©tĂ©s diffĂ©rentielles. Il devient Ă©galement possible dâĂ©liminer des oscillations de façon sĂ©lective ou de manipuler les formes sans introduire dâoscillations indĂ©sirables.
Avant de progresser vers cet objectif ambitieux, il faut dâabord ĂȘtre en mesure de bien comprendre et de bien visualiser ces propriĂ©tĂ©s diffĂ©rentielles que lâon souhaite contrĂŽler. Lâhistoire de la gĂ©omĂ©trie diffĂ©rentielle classique des courbes et des surfaces est trĂšs riche. Cette histoire est revisitĂ©e avec une perspective nouvelle. Soit la perspective du contrĂŽle des inflexions et des extremums de courbure. Ceci permet de faire Ă©merger des liens importants entre la gĂ©omĂ©trie
diffĂ©rentielle, la thĂ©orie des singularitĂ©s, les groupes de transformations et lâoptique gĂ©omĂ©trique.
Ensuite viennent les algorithmes dâoptimisation des B-splines sous contraintes. Les variables indĂ©pendantes sont les positions des points de contrĂŽle de la B-spline alors que les contraintes portent sur la position des points de contrĂŽle dâune fonction qui reprĂ©sente les propriĂ©tĂ©s diffĂ©rentielles de la B-spline.
Les algorithmes sont dâabord dĂ©veloppĂ©s pour les fonctions B-splines Ă une et deux variables. Une fois ces algorithmes dĂ©veloppĂ©s, plusieurs possibilitĂ©s nouvelles sâoffrent Ă nous. Il devient possible, par exemple dâobtenir la courbe qui sâapproche le plus dâune autre courbe quelconque sous la contrainte de possĂ©der certaines propriĂ©tĂ©s diffĂ©rentielles. De cette maniĂšre, il devient possible de travailler avec un plus grand nombre de points de contrĂŽle et ainsi dans un espace
de forme plus riche sans avoir Ă se soucier dâoscillations arbitraires. Ceci permet en particulier
dâĂ©liminer de façon sĂ©lective des oscillations indĂ©sirables sur des profils aĂ©rodynamiques.----------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