Topic

# Villarceau circles

About: Villarceau circles is a(n) research topic. Over the lifetime, 11 publication(s) have been published within this topic receiving 28 citation(s).

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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

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01 Mar 2014Abstract: 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. ...

6 citations

01 Jan 2000

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

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TL;DR: Celestials in 3-space that are obtained from translating a circle along a circle, in either Euclidean or elliptic space are considered, and a classically avored theorem in elliptic geometry is obtained: if the authors translate a line along acircle but not along a line then exactly 2 translated lines will coincide.

Abstract: The sphere in 3-space has an innite number of circles through any closed point. The torus has 4 circles through any closed point. Two of these circles are known as Villarceau circles ([0]). We dene a \celestial" to be a real surface with at least 2 real circles through a generic closed point. Equivalently, a celestial is a surface with at least 2 families of real circles. In 1980 Blum [1] conjectured that a real surface has either at most 6 families of circles or an innite number. For compact surfaces this conjecture has been proven by Takeuchi [2] in 1987 using topological methods. In 2001 Schicho [3] classied complex surfaces with at least 2 families of conics. This result together with Moebius geometry led to a classication of celestials in 3-space [4]. In 2012 Pottmann et al. [5] conjectured that a surface in 3-space with exactly 3 circles through a closed point is a Darboux Cyclide. We conrm this conjecture as a corollary from our classication in [4]. We recall that a translation is an isometry where every point moves with the same distance. In this talk we consider celestials in 3-space that are obtained from translating a circle along a circle, in either Euclidean or elliptic space. This is a natural extension of classical work by William Kingdon Cliord and Felix Klein on the Cliord torus. Krasauskas, Pottmann and Skopenkov conjectured, that celestials in 3-space of Moebius degree 8 are Moebius equivalent to an Euclidean or Elliptic translational celestial. This conjecture is true if its Moebius model has a family of great circles ([6]). Moreover, its real singular locus consist of a great circle. As a corollary we obtain a classically avored theorem in elliptic geometry: if we translate a line along a circle but not along a line then exactly 2 translated lines will coincide ([6]).

3 citations

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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.

2 citations