In
mathematics, a Dupin cyclide or cyclide of Dupin is any
geometric inversion of a
standard torus,
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
or
double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by (and named after)
Charles Dupin
Baron Pierre Charles François Dupin (6 October 1784, Varzy, Nièvre – 18 January 1873, Paris, France) was a French Catholic mathematician, engineer, economist and politician, particularly known for work in the field of mathematics, where the ...
in his 1803 dissertation under
Gaspard Monge
Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. During ...
. 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 mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \n ...
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
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a 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
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by d ...
s are all circles (possibly through the
point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
). Equivalently, the
curvature sphere
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canoni ...
s, which are the spheres
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to the surface with radii equal to the
reciprocals of the
principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends b ...
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 circles. Equivalently again, both sheets of the
focal surface
For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at th ...
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
Lie sphere geometry is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. The main idea which leads to Lie sphere geometry is th ...
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
In geometry, Soddy's hexlet is a chain of six spheres (shown in grey in Figure 1), each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, the three spheres are the red inner sphere and tw ...
configurations of spheres.
Parametric and implicit representation
: (CS): A Dupin cyclide can be represented in two ways as the
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
of a one parametric pencil of spheres, i.e. it is a
canal surface
In geometry and topology, a channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its '' directrix''. If the radii of the generating spheres are constant, the canal surface is ca ...
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
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by d ...
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 surface
For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at th ...
s 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:
:
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 (see section
Cyclide as channel surface):
:
:
:
: