A Dupin cyclide
, a Dupin cyclide or cyclide of Dupin is any geometric inversion
of a standard torus
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 surface
s 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 line
s are all circles (possibly through the point at infinity
). Equivalently, the curvature sphere
s, which are the spheres tangent
to the surface with radii equal to the reciprocals
of the principal curvature
s at the point of tangency, are constant along the corresponding curvature lines: they are the tangent spheres containing the corresponding curvature lines as great circle
s. 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 transformation
s; 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.
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''.
An elliptic cyclide can be represented parametrically by the following formulas (see section Cyclide as channel surface
are the semi major and semi minor axes and
the linear eccentricity of the ellipse:
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 surface
s of the cyclide.
can be considered as the average radius of the generating spheres.
respectively one gets the curvature lines (circles) of the surface.
The corresponding implicit representation
In case of
, 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
A parabolic cyclide can be represented by the following parametric representation (see section Cyclide as channel surface