Dupin cyclide

They were discovered c. 1802 by (and named after) Charles Dupin, while he was still a student at the École polytechnique following Gaspard Monge's lectures.

[1] 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 are used in computer-aided design because cyclide patches have rational representations and are suitable for blending canal surfaces (cylinder, cones, tori, and others).

, 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.

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 again, both sheets of the focal surface degenerate to conics.

[2] 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.

It follows that it is tangent to infinitely many Soddy's hexlet configurations of spheres.

The two conics form the two degenerated focal surfaces of the cyclide.

More intuitive design parameters are the intersections of the cyclide with the x-axis.

A corresponding implicit representation is Remark: By displaying the circles there appear gaps which are caused by the necessary restriction of the parameters

There are two ways to generate an elliptic Dupin cyclide as a channel surface.

The Maxwell-property gives reason for determining a ring cyclide by prescribing its intersections with the x-axis: Given: Four points

yields The foci (on the x-axis) are The center of the focal conics (ellipse and hyperbola) has the x-coordinate If one wants to display the cyclide with help of the parametric representation above one has to consider the shift

The second way to generate the ring cyclide as channel surface uses the focal hyperbola as directrix.

It has the equation In this case the spheres touch the cyclide from outside at the second family of circles (curvature lines).

The spheres of one family enclose the cyclide (in diagram: purple).

Parametric representation of the hyperbola: The radii of the corresponding spheres are In case of a torus (

Calculation in detail leads to the parametric representation of the elliptic cyclide given above.

An advantage for investigations of cyclides is the property: The inversion at the sphere with equation

The images of the tangent planes of the cylinder become the second pencil of spheres touching the cyclide.

and to use the parametric representation above: Given: A torus, which is shifted out of the standard position along the x-axis.

Thus it is a quartic surface in Cartesian coordinates, with an equation of the form: where Q is a 3x3 matrix, P and R are a 3-dimensional vectors, and A and B are constants.

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.

Many other cyclidic geometries can be obtained by studying R-separation of variables for the Laplace equation.