Gluing Dupin cyclides along circles, finding a cyclide given three contact conditions (original) (raw)
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Gluing Dupin cyclides along circles, . . . given three contact conditions
2012
Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential geometry, the simplest after planes and spheres. We prove here that, given three oriented contact conditions, there is in general no Dupin cyclide satisfying them, but if the contact conditions belongs to a codimension one subset, then there is a one-parameter family of solutions, which are all tangent along a curve determined by the three contact conditions.
Finding a cyclide given three contact conditions
Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential geometry, the simplest after planes and spheres. We prove here that, given three oriented contact conditions, there is in general no Dupin cyclide satisfying them, but if the contact conditions belongs to a codimension one subset, then there is a one-parameter family of solutions.
On the Geometry of Dupin Cyclides
1988
In the 19th century, the French geometer Charles Pierre Dupin discovered a nonspherical surface with circular lines of curvature. He called it a cyclide in his book, Applicarions de Geometrie published in 1822. Recently, cyclides have been revived for use as surface patches in computer aided geometric design (CAGD). Other applications of eyelides in CAGD are possible (e.g., variable radius blending) and require a deep understanding of the geometry of the cyclide. We resurrect the geometric descriptions of the cyclide found in the classical papers of James Clerk Maxwell and Arthur Cayley. We present a unified perspective of their results and use them to devise effective algorithms for synthesizing cyclides. We also discuss the morphology of cyclides and present a new classification scheme. Central Cyclides Revolute Cyclides Parabolic Cyclides Degenerate Cyclides Horn
Cyclides, their tangency and deformation
In this paper we propose a new tool for the geometric control of Dupin cyclides. We define Dupin cyclides as envelopes of 1-parameter families of spheres, tangent to three fixed spheres in euclidian 3D space. Using the Moebius model, we identify these 1- parameter families of spheres with certain B´ezier conics, which lie on cones in R5. In fact, the vertex of such a cone corresponds to a sphere or plane tangent to the cyclide. The latter sphere is utilized as a shape parameter to deform the Dupin cyclide in a controlled way.
Symmetries of Canal Surfaces and Dupin Cyclides
2018
We develop a characterization for the existence of symmetries of canal surfaces defined by a rational spine curve and rational radius function. In turn, this characterization inspires an algorithm for computing the symmetries of such canal surfaces. For Dupin cyclides in canonical form, we apply the characterization to derive an intrinsic description of their symmetries and symmetry groups, which gives rise to a method for computing the symmetries of a Dupin cyclide not necessarily in canonical form. As a final application, we discuss the construction of patches and blends of rational canal surfaces with a prescribed symmetry.
In this paper we consider the problem of joining circles with prescribed tangencies, i.e., along given cones or spheres. The joining surfaces are envelopes of quadratic 1-parameter families of spheres which generalize the family of Dupin cyclides, but still their algebraic degree is four. They are represented with conics in 4-space. Given a conic in Bezier form the implicit equation of the corresponding surface is exhibited
Dupin cyclides as conics in extended four dimensional space
We look at Dupin cyclides as they arise naturally in the environment of the Moebius hypersphere in extended 4−space as a subfamily of the envelopes of quadratic families of spheres. The latter correspond to conics in extended 4−space, which are thought of as second degree polynomial Bèzier curves. We present necessary and sufficient conditions for the conic to produce a Dupin cyclide, in terms of the geometry of extended 4−space and also in terms of the control points of the Bèzier curve that represents the quadratic family. We offer remarks on the construction of tensor product patches lying on Dupin cyclides and on quadric cones in terms of the Bèzier conic and de Casteljau’s algorithm.
Dupin Cyclides are Not of L_{1}$$L1-Finite Type
Bulletin of the Iranian Mathematical Society, 2018
In this paper, we prove that the Dupin cyclides are not of L 1-finite type. An isometric immersed surface ψ : M → E 3 is said to be of L 1-finite type if ψ = k i=0 ψ i for some positive integer k, ψ i : M → E 3 is smooth and L
Cyclides and the guiding circle
Using the fact that spheres of 3D correspond to points in 4-dimensional space which lie outside a certain quadric, we introduce a new shape handle that permits the control of general cyclides: the guiding circle. A general cyclide is the envelope of a quadratic family of spheres which in this context is a conic in 4-dimensional space. The guiding circle is associated to the cyclide via the plane of the conic that corresponds to the cyclide.
Computers & Mathematics with Applications, 2014
Ring Dupin cyclides are non-spherical algebraic surfaces of degree four that can be defined as the image by inversion of a ring torus. They are interesting in geometric modeling because: (1) they have several families of circles embedded on them: parallel, meridian, and Yvon-Villarceau circles, and (2) they are characterized by one parametric equation and two equivalent implicit ones, allowing for better flexibility and easiness of use by adopting one representation or the other, according to the best suitability for a particular application. These facts motivate the construction of circular edge triangles lying on Dupin cyclides and exhibiting the aforementioned properties. Our first contribution consists in an analytic method for the computation of Yvon-Villarceau circles on a given ring Dupin cyclide, by computing an adequate Dupin cyclide-torus inversion and applying it to the torusbased equations of Yvon-Villarceau circles. Our second contribution is an algorithm which, starting from three arbitrary 3D points, constructs a triangle on a ring torus such that each of its edges belongs to one of the three families of circles on a ring torus: meridian, parallel, and Yvon-Villarceau circles. Since the same task of constructing right triangles is far from being easy to accomplish when directly dealing with cyclides, our third contribution is an indirect algorithm which proceeds in two steps and relies on the previous one. As the image of a circle by a carefully chosen inversion is a circle, and by constructing different images of a right triangle on a ring torus, the indirect algorithm constructs a one-parameter family of 3D circular edge triangles lying on Dupin cyclides.