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Villarceau circles

About: Villarceau circles is a research topic. Over the lifetime, 11 publications have been published within this topic receiving 28 citations.

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TL;DR: The properties of these surfaces are used to prove that three families of circles meridian arcs, parallel arcs, and Villarceau circles can be computed on every Dupin cyclide, and a geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupincyclide is presented.
Abstract: This paper considers the conversion of the parametric Bezier surfaces, classically used in CAD-CAM, into patched of a class of non-spherical degree 4 algebraic surfaces called Dupin cyclides, and the definition of 3D triangle with circular edges on Dupin cyclides. Dupin cyclides was discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. The authors use the properties of these surfaces to prove that three families of circles meridian arcs, parallel arcs, and Villarceau circles can be computed on every Dupin cyclide. A geometric algorithm to compute these circles so that they define the edges of a 3D triangle on the Dupin cyclide is presented. Examples of conversions and 3D triangles are also presented to illustrate the proposed algorithms.

11 citations

Journal ArticleDOI
01 Mar 2014
TL;DR: In this paper, the oblique circular torus (OCT) and its main geometric properties are introduced and the coordinate-free approach leads to the algebraic equation of an OCT in a privileged Cartesian reference frame.
Abstract: The oblique circular torus (OCT) and its main geometric properties are introduced. Intrinsic vector calculation is utilized to mathematically describe the OCT. The coordinate-free approach leads to the algebraic equation of an OCT in a privileged Cartesian reference frame. The OCT equation is used to confirm a theorem of Euclidean geometry. In a broad category of OCT, through any point five circles can be drawn on the surface, namely the parallel of latitude and four circular generatrices whose planes pass through the OCT center of symmetry. In the special case of a right circular torus, the Villarceau theorem is verified. Next, consider the four RRS open chains whose S spherical-joint centers move on the same OCT and their possible in-parallel assemblies in single-loop RRRS chains. From a category of the foregoing RRRS chains, a new derivation of the amazing Bennett 4R linkage is proposed. Two kinds of Bennett linkages are further verified and each kind contains two enantiomorphic or symmetric linkages. ...

8 citations

Journal ArticleDOI
TL;DR: This work proposes an indirect algorithm which constructs a one-parameter family of 3D circular edge triangles lying on Dupin cyclides, 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.
Abstract: 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 torus-based 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.

4 citations

01 Jan 2000
TL;DR: In this paper, the authors investigated hyperbolas, ellipses and degenerate conics on parabolic Dupin cyclides and inverted them into planar or spherical conics via inversion.
Abstract: : Hyperbolas, ellipses and degenerate conics on parabolic Dupin cyclides are investigated. These central conics are obtained as the intersections of parabolic cyclides and the planes perpendicular to the two planes of symmetry of the cyclides. They are also the images of central conics in the parametric space. Since the conics are planar curves, they are transformed into planar or spherical curves on Dupin cyclides via inversion. Lemniscates of Bernoulli on Dupin cyclides and Viviani's curves on right-circular cylinders are included in the inverted conics. Two intersecting lines on a parabolic ring cyclide, which are degenerate conics, are inverted into Villarceau circles on a ring cyclides.

3 citations

Journal ArticleDOI
TL;DR: In this article, the authors considered the Villarceau circles as the orbits of specific composite rotors in 3D conformal geometric algebra that generate knots on nested tori and computed the conformal parametrization of these circular orbits by giving an equivalent, position-dependent simple rotor that generates the same parametric track for a given point.
Abstract: We consider Villarceau circles as the orbits of specific composite rotors in 3D conformal geometric algebra that generate knots on nested tori. We compute the conformal parametrization of these circular orbits by giving an equivalent, position-dependent simple rotor that generates the same parametric track for a given point. This allows compact derivation of the quantitative symmetry properties of the Villarceau circles. We briefly derive their role in the Hopf fibration and as stereographic images of isoclinic rotations on a 3-sphere of the 4D Clifford torus. We use the CGA description to generate 3D images of our results, by means of GAviewer. This paper was motivated by the hope that the compact coordinate-free CGA representations can aid in the analysis of Villarceau circles (and torus knots) as occurring in the Maxwell and Dirac equations.

3 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20191
20151
20143
20111
20091
20061