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 Charles Dupin in his 1803 dissertation under Gaspard Monge. The key property of a Dupin cyclide is that it is a channel surface 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, J.C. Maxwell and Mabel M. Young.
Today, Dupin cylides are used in computer-aided design, because cyclide patches have rational representations and are suitable for blending canal surfaces.

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 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.
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. 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 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 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.

Parametric and implicit representation

A further special property of a cyclide is:

Elliptic cyclides

An elliptic cyclide can be represented parametrically by the following formulas :
The numbers are the semi major and semi minor axes and the linear eccentricity of the ellipse:
The hyperbola
is the focal conic to the ellipse. That means: The foci/vertices of the ellipse are the vertices/foci of the hyperbola. The two conics form the two degenerated focal surfaces of the cyclide.
can be considered as the average radius of the generating spheres.
For , respectively one gets the curvature lines of the surface.
The corresponding implicit representation is:
In case of one gets, i. e. the ellipse is a circle and the hyperbola degenerates to a line. The corresponding cyclides are tori of revolution.
More intuitive design parameters are the intersections of the cyclide with the x-axis. See section [|Cyclide through 4 points on the x-axis].

Parabolic cyclides

A parabolic cyclide can be represented by the following parametric representation :
The number determines the shape of both the parabolas, which are focal conics:
determines the relation of the diameters of the two holes. means: both diameters are equal. For the diagram is.
A corresponding implicit representation is
Remark: By displaying the circles there appear gaps which are caused by the necessary restriction of the parameters.

Cyclide as channel surface

There are two ways to generate an elliptic Dupin cyclide as a channel surface. The first one uses an ellipse as directrix, the second one a hyperbola:

Ellipse as directrix

In the x-y-plane the directrix is the ellipse with equation
It has the parametric representation
is the semi major and the semi minor axis.
is the linear eccentricity of the ellipse. Hence:.
The radii of the generating spheres are
is a design parameter. It can be seen as the average of the radii of the spheres. In case of the ellipse is a circle and the cyclide a torus of revolution with the radius of the generating circle.
In the diagram:.
of the blue and purple circles.

Maxwell property

The following simple relation between the actual sphere center and the corresponding sphere radius is due to Maxwell:
;Proof
The foci of the ellipse are . If one chooses and calculates the distance, one gets. Together with the radius of the actual sphere one gets.
Choosing the other focus yields:
Hence:
In the x-y-plane the envelopes of the circles of the spheres are two circles with the foci of the ellipse as centers and the radii .

Cyclide through 4 points on the x-axis

The Maxwell-property gives reason for determining a ring cyclide by prescribing its intersections with the x-axis:
Given: Four points on the x-axis.
Wanted: Center, semiaxes, linear eccentricity and foci of the directrix ellipse and the parameter of the corresponding ring cyclide.
From the Maxwell-property one derives
Solving for yields
The foci are
The center of the focal conics has the x-coordinate
If one wants to display the cyclide with help of the parametric representation [|above] one has to consider the shift of the center !
;Meaning of the order of the numbers :
' Swapping generates a horn cyclide.
' Swapping, generates a spindel cyclide.
' For one gets a 1-horn cyclide.
' For one gets a ring cyclide touching itself at the origin.

Parallel surfaces

By increasing or decreasing parameter, such that the type does not change, one gets parallel surfaces of the same type.

Hyperbola as directrix

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. To each arm of the hyperbola belongs a subfamily of circles. The spheres of one family enclose the cyclide. Spheres of the other family are touched from outside by the cyclide.
Parametric representation of the hyperbola:
The radii of the corresponding spheres are
In case of a torus the hyperbola degenerates into the axis of the torus.

Maxwell-property (hyperbola case)

The foci of the hyperbola are. The distance of hyperbola point to the focus is and together with the sphere radius one gets. Analogously one gets . For a point on the second arm of the hyperbola one derives the equations:
Hence:
In the x-z-plane the circles of the spheres with centers and radii have the two circles with centers and radii as envelopes.

Derivation of the parametric representation

Elliptic cyclide

The ellipse and hyperbola are the degenerated focal surfaces of the elliptic cyclide. For any pair of points of the ellipse and hyperbola the following is true :
From the parametric representation of the focal conics and the radii of the spheres
one gets the corresponding point of the cyclide :
Calculation in detail leads to the parametric representation of the elliptic cyclide given above.
If one uses the parametric representation given in the article on channel surfaces, then, in general, only one family of parametric curves consists of circles.

Parabolic cyclide

The derivation of the parametric representation for the parabolic case runs analogously:
With the parametric representations of the focal parabolas and the radii of the spheres:
one gets
which provides the parametric representation above of a parabolic cyclide.

Dupin cyclides and geometric inversions

An advantage for investigations of cyclides is the property:
The inversion at the sphere with equation can be described analytically by:
The most important properties of an inversion at a sphere are:
  1. Spheres and circles are mapped on the same objects.
  2. Planes and lines containing the origin are mapped on themselves.
  3. Planes and lines not containing the origin are mapped on spheres or circles passing the origin.
  4. An inversion is involutory.
  5. An inversion preserves angles.
One can map arbitrary surfaces by an inversion. The formulas above give in any case parametric or implicit representations of the image surface, if the surfaces are given parametrically or implicitly. In case of a parametric surface one gets:
But: Only in case of right circular cylinders and cones and tori of revolution one gets Dupin cyclides and vice versa.

Example cylinder

a) Because lines, which do not contain the origin, are mapped by an inversion at a sphere on circles containing the origin the image of the cylinder is a ring cyclide with mutually touching circles at the origin. As the images of the line segments, shown in the picture, there appear on line circle segments as images. The spheres which touch the cylinder on the inner side are mapped on a first pencil of spheres which generate the cyclide as a canal surface. The images of the tangent planes of the cylinder become the second pencil of spheres touching the cyclide. The latter ones pass through the origin.

b) The second example inverses a cylinder that contains the origin. Lines passing the origin are mapped onto themselves. Hence the surface is unbounded and a parabolic cyclide.

Example cone

The lines generating the cone are mapped on circles, which intersect at the origin and the image of the cone's vertex. The image of the cone is a double horn cyclide. The picture shows the images of the line segments, which are circles segments, actually.

Example torus

Both the pencils of circles on the torus are mapped on the corresponding pencils of circles on the cyclide. In case of a self-intersecting torus one would get a spindle cyclide.
;Villarceau circles
Because Dupin ring-cyclides can be seen as images of tori via suitable inversions and an inversion maps a circle onto a circle or line, the images of the Villarceau circles form further two families of circles on a cyclide.
;Determining the designparameters
The formula of the inversion of a parametric surface provides a parametric representation of a cyclide with circles as parametric curves. But the points of a parametric net are not well distributed. So it is better to calculate the design parameters and to use the parametric representation above:
Given: A torus, which is shifted out of the standard position along the x-axis.
Let be the intersections of the torus with the x-axis. All not zero. Otherwise the inversion of the torus would not be a ring-cyclide.
Wanted: semi-axes and linear eccentricity of the ellipse and parameter of the ring-cyclide, which is the image of the torus under the inversion at the unitsphere.
The inversion maps onto, which are the x-coordinates of 4 points of the ring-cyclide. From section Cyclide through 4 points on the x-axis one gets
The center of the focal conics has the x-ccordinate

Separation of variables

Dupin 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. Whereas a quadric can be described as the zero-set of second order polynomial in Cartesian coordinates, a cyclide is given by the zero-set of a second order polynomial in, where
r2=x12+x22+x32. 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.
Families of cyclides give rise to various cyclidic coordinate geometries.
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.