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, while he was still a student at the École polytechnique following Gaspard Monge's lectures. 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, J.C. Maxwell and Mabel M. Young.
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).
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 spheres.
Parametric and implicit representation (CS): A Dupin cyclide can be represented in two ways as the envelope of a one parametric pencil of spheres, i.e. it is a canal surface with two directrices. The pair of directrices are focal conics and consists either of an ellipse and a hyperbola or of two parabolas. In the first case one defines the cyclide as elliptic, in the second case as parabolic. In both cases the conics are contained in two mutually orthogonal planes. In extreme cases (if the ellipse is a circle) the hyperbola degenerates to a line and the cyclide is a torus of revolution.
A further special property of a cyclide is: (CL): Any curvature line of a Dupin cyclide is a circle.
Elliptic cyclides An elliptic cyclide can be represented parametrically by the following formulas (see section Cyclide as channel surface):
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 (circles) of the surface.
The corresponding implicit representation is: provided that
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.
(ellipt.) Dupin cyclides for designparameters a,b,c,d symm. horn cyclidehorn cyclidehorn cyclidering cyclidering cyclidespindle cyclide
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 (see section Cyclide as channel surface):
The number determines the shape of both the parabolas, which are focal conics: and
determines the relation of the diameters of the two holes (see diagram). means: both diameters are equal. For the diagram is .
A corresponding implicit representation is
parabolic Dupin cyclides for designparameters p=1, kring cyclidehorn cyclidehorn cyclide
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:W. Blaschke: Analytische Geometrie, Springer-Verlag, 2013, , S. 115
Ellipse as directrix
In the x-y-plane the directrix is the ellipse with equation and . 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 (generatrix).
In the diagram: .
Maxwell property The following simple relation between the actual sphere center (ellipse point) and the corresponding sphere radius is due to Maxwell:mentioned in W. Boehm: On Cyclides in Geometric Modeling.Computer Aided Geometric Design 7, 1990, p. 243–255.
The difference/sum of the sphere's radius and the distance of the sphere's center (ellipse point) from one (but fixed) of the foci is constant.
Proof The foci of the ellipse are . If one chooses and calculates the distance , one gets . Together with the radius of the actual sphere (see above) 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 (see diagram).
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 (see diagram).
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 (on the x-axis) are and hence
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 of the center !
Meaning of the order of the numbers (The calculation above presumes , see diagram.)
(H) Swapping generates a horn cyclide. (S) Swapping , generates a spindel cyclide. (H1) For one gets a 1-horn cyclide. (R) 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 (similar to parallel curves) of the same type (see diagram).
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 (curvature lines). To each arm of the hyperbola belongs a subfamily of circles. The spheres of one family enclose the cyclide (in diagram: purple). Spheres of the other family are touched from outside by the cyclide (blue).
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 (in diagram grey) with centers and radii as envelopes.
Derivation of the parametric representation Elliptic cyclide
The ellipse and hyperbola (focal conics) are the degenerated focal surfaces of the elliptic cyclide. For any pair of points of the ellipse and hyperbola the following is true (because of the definition of a focal surface): 1) The line is a normal of the cyclide and 2) the corresponding point of the cyclide divides the chord with relation (see diagram).
From the parametric representation of the focal conics and the radii of the spheres Ellipse: Hyperbola:
one gets the corresponding point of the cyclide (see diagram):
(For the unusual but convenient parametric representation of the hyperbola: see hyperbola.)
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 (degenerated focal surfaces) 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: (I): Any Dupin cyclide is the image either of a right circular cylinder or a right circular double cone or a torus of revolution by an inversion (reflection at a sphere).
The inversion at the sphere with equation can be described analytically by:
The most important properties of an inversion at a sphere are: Spheres and circles are mapped on the same objects. Planes and lines containing the origin (center of inversion) are mapped on themselves. Planes and lines not containing the origin are mapped on spheres or circles passing the origin. An inversion is involutory (identical with the inverse mapping). 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