![]() Charles Dupin discussed his discovery of cyclides in his 1803 dissertation. These surfaces have a variety of interesting properties and are aesthetic from a geometric and algebraic viewpoint. A Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. Then the implicit equation of this Dupin cyclide is a factor of the polynomial f(0, x, y, z), deg f 4, defined by the determinant det M (P), which can be identically zero only in the spherical or planar cases. Dupin cyclides are algebraic surfaces of order three and four whose lines of curvature are circles. In order to do this we shall construct a Legendre map enveloping s. Let P(s, t) be the bilinear quaternionic parametrization of the principal Dupin cyclide patch in R 3. We now seek a parametrisation of the envelope of this sphere curve. ( 2018), Clarke ( 2012) and Pember ( 2018), we show that Legendre immersions parametrising channel surfaces are \(\Omega _/s\) is positive definite. They were discovered by (and named after) Charles Dupin in his 1803 dissertation under Gaspard Monge. In particular, these latter are themselves examples of Dupin cyclides. Three smectic layers corresponding to a single focal conic domain as a result of the Dupin cyclides equations system. 3, by applying the gauge theoretic approach of Burstall et al. In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. Divide A B with the ratio m: n gives the Dupin cyclide where m e ( 1 cos u) 1 e cos u k n 1 e e cos v k For the circular hole centres at the origin, k, e ( 0, 1) For a torus, e 0 For a parabolic Dupin cyclide, e 1 Please also refer to Wikipedia article here. These surfaces are well suited for the blending of elliptic quadrics. 2016), we exploit the hexaspherical coordinate model introduced by Lie ( 1872). A scaled cyclide is the image of a Dupin cyclide under an affine scaling application. Dupin cyclides can also be represented as the envelope of a moving sphere whose center varies along a conic curve and radius is given by a quadratic function 17. Following the example of Blaschke ( 1929) and more recently (Jensen et al. Dupin cyclides are the envelopes of a one-parameter family of spheres that are tangent to three given mutually tangent spheres. In this paper, we discuss various aspects of channel surfaces in the context of Lie sphere geometry. Channel surfaces are also widely used in Computer Aided Geometric Design and the existence of a rational parametrisation was investigated in Peternell and Pottmann ( 1997). All natural quadrics (cone, cylinder, sphere) and the torus are special cases of the cyclide. Moreover, a novel application of channel surfaces to semi-discrete curvature line nets was explored in Burstall et al. The Dupin cyclide is a quartic surface with useful properties such as circular lines of curvature, rational parametric representations and closure under offsetting. 3d Tutorial Dupin Cyclide Revisited - New Technique Blender 2.8 Luxxeon 3D 26.5K subscribers Subscribe 6.2K views 4 years ago Luxxeon's Blender Tutorials Best viewed in HD 1080p In this. 2015) and Willmore channel surfaces (Musso and Nicolodi 1999) were recently discussed. 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. He found that Dupin cyclides were such surfaces if the two curves were conics in perpendicular planes, with vertices of one passing through the foci of the. ![]() Mathematicians sometimes refer to this generalisation of a circle or line as a 'circline'. Let H: S 3 S 2 be the Hopf map and let be a curve on S 2. A Dupin cyclide is characterised by the property that all of its lines of curvature are (arcs/segments of) either circles or lines. Furthermore, the subclasses of channel linear Weingarten surfaces (Hertrich-Jeromin et al. Dupin cyclide as the stereographic projection of a Hopf torus. ( 2016) and Musso and Nicolodi ( 1999, 2002) channel surfaces were studied in the context of Möbius geometry and in Musso and Nicolodi ( 1995) and Peternell and Pottmann ( 1998) they were given a Laguerre geometric treatment. For example, in Bernstein ( 2001), Hertrich-Jeromin ( 2003), Hertrich-Jeromin et al. They are known to be the images of cylinders, or cones of revolution, or tori under inversions. Although these surfaces are a classical notion (e.g., Blaschke 19 Monge 1850), they are also a subject of interest in recent research. Dupin cyclides are algebraic surfaces of degree three or four 17. (1) where B i(t) are quadratic Bernstein polynomials defined as: B o(t)= (1-t) 2 B 1(t)= 2t(1-t) B 2(t)=t 2and w i,, where bar is an abbreviation for barycentre.Channel surfaces, that is, envelopes of one-parameter families of spheres, have been intensively studied for many years. ![]()
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