Dupin cyclide

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).

There are several equivalent definitions of Dupin cyclides. In ${\displaystyle \mathbb {R} ^{3}}$, 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 ${\displaystyle \mathbb {C} ^{3}}$ 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).

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