Search results for "Dupin"
showing 10 items of 20 documents
G1-Blend between a Differentiable Superquadric of Revolution and a Plane or a Sphere Using Dupin Cyclides
2008
In this article, we present a method to perform G1-continuous blends between a differentiable superquadric of revolution and a plane or a sphere using Dupin cyclides. These blends are patches delimited by four lines of curvature. They allow to avoid parameterization problems that may occur when parametric surfaces are used. Rational quadratic Bezier curves are used to approximate the principal circles of the Dupin cyclide blends and thus a complex 3D problem is now reduced to a simpler 2D problem. We present the necessary conditions to be satisfied to create the blending patches and illustrate our approach by a number of superellipsoid/plane and superellipsoid/sphere blending examples.
Blending Planes and Canal Surfaces Using Dupin Cyclides
2011
We develop two different new algorithms of G1-blending between planes and canal surfaces using Dupin cyclides. It is a generalization of existing algorithms that blend revolution surfaces and planes using a plane called construction plane. Spatial constraints were necessary to do that. Our work consist in building three spheres to determine the Dupin cyclide of the blending. The first algorithm is based on one of the definitions of Dupin cyclides taking into account three spheres of the same family enveloping the cyclide. The second one uses only geometric properties of Dupin cyclide. The blending is fixed by a circle of curvature onto the canal surface. Thanks to this one, we can determine…
Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points
2021
Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. A Dupin cyclide can be defined as the envelope of a one-parameter family of oriented spheres, in two different ways. R. Martin is the first author who thought to use these surfaces in CAD/CAM and geometric modeling. The Minkowski-Lorentz space is a generalization of the space-time used in Einstein’s theory, equipped of the non-degenerate indefinite quadratic form $$Q_{M} ( \vec{u} ) = x^{2} + y^{2} + z^{2} - c^{2} t^{2}$$ where (x, y, z) are the spacial components of the vector $$ \vec{u}$$ and t is the time component of $$ \vec{u}$$ and c is the constant of the spee…
Foliations of $\mathbb{S}^3$ by Cyclides
2018
Throughout the last 2–3 decades, there has been great interest in the extrinsic geometry of foliated Riemannian manifolds (see [2], [4] and [22]). ¶One approach is to build examples of foliations with reasonably simple singularities with leaves admitting some very restrictive geometric condition. For example (see [22], [23] and [17]), consider in particular foliations of $\mathbb{S}^{3}$ by totally geodesic or totally umbilical leaves with isolated singularities. ¶The article [14] provides families of foliations of $\mathbb{S}^{3}$ by Dupin cyclides with only one smooth curve of singularities. Quadrics and other families of cyclides like Darboux cyclides provide other examples. These foliat…
The non-degenerate Dupin cyclides in the space of spheres using Geometric Algebra
2012
International audience; Dupin cyclides are algebraic surfaces of degree 4 discovered by the French mathematician Pierre-Charles Dupin early in the 19th century and \textcolor{black}{were} introduced in CAD by R. Martin in 1982. A Dupin cyclide can be defined, in two different ways, as the envelope of a one-parameter family of oriented spheres. So, it is very interesting to model the Dupin cyclides in the space of spheres, space wherein each family of spheres can be seen as a conic curve. In this paper, we model the non-degenerate Dupin cyclides and the space of spheres using Conformal Geometric Algebra. This new approach permits us to benefit from the advantages of the use of Geometric Alge…
"Nous ne sommes ni vos élèves ni vos amis". Charles Dupin face à la presse ouvrière sous la Monarchie de juillet (1830-1848)
2009
International audience
Points massiques, espace des sphères et « hyperbole »
2015
The use of massic points permits to define a branch of a hyperbola in the Euclidean plane using a Rational Quadratic Bézier Curve. In the space of spheres, a circular cone, a circular cylinder, a torus, a pencil of spheres or a Dupin cyclide is represented by a conic. If the kind of the pencil is Poncelet or if the canal surface is a circular cone, a spindle torus, a spindle or a horned Dupin cyclide, the curve is a circle which is seen as a hyperbole. The limit points of the pencil or the singular points of the Dupin cyclide can be determined using the asymptotes of this circle. In this article, we show that the use of massic points simplifies the modelization of these pencils or these Dup…
Nouveaux modèles géométriques pour la CAO et la synthèse d'images
2017
La géométrie du 21ème siècle est indissociable de l'ordinateur. La performance des logiciels de géométrie qu'ils soient gratuits ou sous licence d'exploitation s'accroit de jour en jour. La plupart repose sur la représentation de courbes à l'aide de points de contrôle communément appelés courbes Bézier ou courbes splines. L'ouvrage traite de ses courbes représentées par des points massiques c'est à dire des points pondérés ou des vecteurs. Plongées dans un espace non euclidien dit espace de Minkovski- Lorentz, elles servent à la représentation de surfaces canal, enveloppes de sphères orientées. En particulier, les coniques planes représentées par des points massiques de contrôle, placées da…
From Dupin cyclides to scaled cyclides
2003
Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. They have a low algebraic degree and have been proposed as a solution to a variety of geometric modeling problems. The circular curvature line’s property facilitates the construction of the cyclide (or the portion of a cyclide) that blends two circular quadric primitives. In this context of blending, the only drawback of cyclides is that they are not suitable for the blending of elliptic quadric primitives. This problem requires the use of non circular curvature blending surfaces. In this paper, we present another formulation of cyclides: Scaled cyclides. A scaled cy…
Dupin et le Duc d’Orléans : des relations complexes
2014
International audience