įamilies of cyclides give rise to various cyclidic coordinate geometries. Where Q is a 3x3 matrix, P and R are a 3-dimensional vectors, and A and B are constants. Weisstein, Eric W., "Cyclide", MathWorld.There are several equivalent definitions of Dupin cyclides. Wikimedia Commons has media related to Dupin cyclide. Miller, Willard (1977), Symmetry and Separation of Variables. Fischer, Mathematical Models from the Collections of Universities and Museums, Braunschweig, Germany: Vieweg, pp. Pinkall, Ulrich (1986), "§3.3 Cyclides of Dupin", in G. (2000), "Pierre Charles François Dupin", MacTutor History of Mathematics archive. Moon, Parry Spencer, Domina Eberle (1961), Field Theory Handbook: including coordinate systems, differential equations, and their solutions, Springer, ISBN 2-7. Hilbert, David Cohn-Vossen, Stephan (1999), Geometry and the Imagination, American Mathematical Society, ISBN 0-8218-1998-4. (1960), "§133 Cyclides of Dupin", A Treatise on the Differential Geometry of Curves and Surfaces, New York: Dover, pp. (1992), Lie sphere geometry, New York: Universitext, Springer-Verlag, ISBN 978-7-8.Įisenhart, Luther P. Many other cyclidic geometries can be obtained by studying R-separation of variables for the Laplace equation.Ĭecil, Thomas E. ![]() In Maxime Bôcher's 1891 dissertation, Ueber die Reihenentwickelungen der Potentialtheorie, it was shown that the Laplace equation in three variables can be solved using separation of variables in 17 conformally distinct quadric and cyclidic coordinate geometries. Where Q is a 3x3 matrix, P and R are a 3-dimensional vectors, and A and B are constants.įamilies of cyclides give rise to various cyclidic coordinate geometries. Thus it is a quartic surface in Cartesian coordinates, with an equation of the form: Whereas a quadric can be described as the zero-set of second order polynomial in Cartesian coordinates ( x 1, x 2, x 3), a cyclide is given by the zero-set of a second order polynomial in ( x 1, x 2, x 3, r 2), where r 2= x 1 2 x 2 2 x 3 2. It follows that it is tangent to infinitely many Soddy's hexlet configurations of spheres.ĭupin cyclides are a special case of a more general notion of a cyclide, which is a natural extension of the notion of a quadric surface. The definition also means that a Dupin cyclide is the envelope of the one parameter family of spheres tangent to three given mutually tangent spheres. They form (in some sense) the simplest class of Lie invariant surfaces after the spheres, and are therefore particularly significant in Lie sphere geometry. In fact any two Dupin cyclides are Lie equivalent. The definition in terms of spheres shows that the class of Dupin cyclides is invariant under the larger group of all Lie sphere transformations. It follows that any Dupin cyclide is a channel surface (i.e., the envelope of a one parameter family of spheres) in two different ways, and this gives another characterization. ![]() ![]() Equivalently again, both sheets of the focal surface degenerate to conics. Equivalently, the curvature spheres, which are the spheres tangent to the surface with radii equal to the reciprocals of the principal curvatures at the point of tangency, are constant along the corresponding curvature lines: they are the tangent spheres containing the corresponding curvature lines as great circles. Since a standard torus is the orbit of a point under a two dimensional abelian subgroup of the Möbius group, it follows that the cyclides also are, and this provides a second way to define them.Ī third property which characterizes Dupin cyclides is the fact that their curvature lines are all circles (possibly through the point at infinity). In complex space \( \C^3 \) these three latter varieties can be mapped to one another by inversion, so Dupin cyclides can be defined as inversions of the torus (or the cylinder, or the double cone). This shows that the class of Dupin cyclides is invariant under Möbius (or conformal) transformations. ![]() In \( \R^3 \), they can be defined as the images under any inversion of tori, cylinders and double cones. There are several equivalent definitions of Dupin cyclides. This property means that Dupin cyclides are natural objects in Lie sphere geometry.ĭupin cyclides are often simply known as "cyclides", but the latter term is also used to refer to a more general class of quartic surfaces which are important in the theory of separation of variables for the Laplace equation in three dimensions. The key property of a Dupin cyclide is that it is a channel surface (envelope of a one parameter family of spheres) in two different ways. 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. In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone.
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