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In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by (and named after) Charles Dupin in his 1803 dissertation under Gaspard Monge. 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. This property means that Dupin cyclides are natural objects in Lie sphere geometry. Dupin 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. Dupin cyclides were investigated not only by Dupin, but also by A. Cayley und J.C. Maxwell. Today, Dupin cylides are used in computer-aided design (CAD), because cyclide patches have rational representations and are suitable for blending canal surfaces (cylinder, cones, tori, and others). ==Definitions and properties== There are several equivalent definitions of Dupin cyclides. In , they can be defined as the images under any inversion of tori, cylinders and double cones. This shows that the class of Dupin cyclides is invariant under Möbius (or conformal) transformations. In complex space 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). 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. A third property which characterizes Dupin cyclides is that their curvature lines are all circles (possibly through the point at infinity). 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. Equivalently again, both sheets of the focal surface degenerate to conics. 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. The definition in terms of spheres shows that the class of Dupin cyclides is invariant under the larger group of all Lie sphere transformations; any two Dupin cyclides are Lie-equivalent. They form (in some sense) the simplest class of Lie-invariant surfaces after the spheres, and are therefore particularly significant in Lie sphere geometry. 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. It follows that it is tangent to infinitely many Soddy's hexlet configurations of spheres. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dupin cyclide」の詳細全文を読む スポンサード リンク
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