![]() Also, place the inversion center inside the Torus. To find "interesting" shapes, it is helpful to have minor radius of the generating Torus larger than the major. ![]() I have also made an inc to generate PoV "poly" code. (x^2 y^2 z^2) 4*ri^4(dx*x dy*y)*(-ri^2 dy*y dx*x) In the 19th century, the French geometer Charles Pierre Dupin discovered a non-spherical surface with circular lines of curvature. Every Dupin cyclide can be obtained from the following three examples by inversion in a suitable sphere: a torus of revolution, a circular cylinder and a circular cone. (x^4 y^4 z^4) 2*((r1^2 - dy^2 - (dx r0)^2 )* Duplin Cyclide: a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. This formula is the inversion of a torus in the x-z plane, 1) First definition: the Dupin cyclides (in the strict sense) are the surfaces, different from the tori, the curvature lines of which are circles (as an. The Dupin Cyclide family of surfaces is a member of a larger group of surfaces referred to as Canal surfaces or Swept surfaces, all of which are envelopes of sweeping objects. 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. This formulation, and the program, is based upon the fact that every Dupin Cyclide is the inverse of a Torus. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. All natural quadrics (cone, cylinder, sphere) and the torus are special cases of the cyclide. They are also the envelope of spheres with centres The Dupin cyclide is a quartic surface with useful properties such as circular lines of curvature, rational parametric representations and closure under offsetting. They are the envelope of spheres kissing three other fixed spheres. Blaschke, K.The Dupin Cyclides can be looked at in various ways. Gostiaux, "Differential geometry: manifolds, curves, and surfaces", Springer (1988) (Translated from French) Berger, "Geometry", II, Springer (1987) Ryan, "Tight and taut immersions of manifolds", Pitman (1985) The natural generalization of the Dupin cyclides to higher dimensions are the so-called Dupin-hypersurfaces (see ). have analyzed and generalized Dupin'scyclide in various ways. Mathematicians, including C.,ey 1871J and Darboux 1887). A remarkable property of cyclides is the fact that they carry four families of circles. Principal circles of ring Dupin cyclides in the plane P: y0 (left), and in the plane P: z0 Figure 3. Applications de Geomdrie published in Paris in 1822, he called this surface a cyclide. The Dupin cyclides are surfaces of the second order in pentaspherical coordinates and have, moreover, two equal axis. The first to show this was Maxwell in 1868. Originally, a Dupin cycle was defined more geometrically as the envelope of a family of spheres tangent to three fixed spheres. ly, Dupin cyclides and concentric spheres (the latter can be considered as a particular Dupin cyclide). Cohn-Vossen, "Anschauliche Geometrie", Springer (1932) Klein, "Vorlesungen über höhere Geometrie", Springer (1926)ĭ. When is a circle on S 2, the stereographic projection of the corresponding Hopf torus highly looks like a. Then H 1 ( ) is called the Hopf cylinder or the Hopf torus when is closed, with profile curve. Let H: S 3 S 2 be the Hopf map and let be a curve on S 2. Thus Soddy's construction shows that a cyclide of Dupin is the envelope of a 1-parameter family of spheres in two different ways, and each sphere in either family is tangent to two spheres in same family and three spheres in the other family. Dupin cyclide as the stereographic projection of a Hopf torus. Dupin, "Développements de géométrie", Paris (1813)į. The envelope of Soddy's hexlets is a Dupin cyclide, an inversion of the torus. Both sheets of the focal set of a Dupin cyclide degenerate to curves, $ \Gamma _ ģ) The evolutes are a circle and a straight line the corresponding Dupin cyclide is a torus.ĭupin cyclides are algebraic surfaces of order four in the cases 1) and 3) above, and of order three in the case 2).Ĭh. An algorithm and implementation is presented to compute the exact arrangement induced by arbitrary algebraic surfaces on a parametrized ring Dupin cyclide. A surface for which both families of curvature lines consist of circles, so that it is a special case of a canal surface.
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