Dupin cyclide

A surface for which both families of curvature lines consist of circles, so that it is a special case of a canal surface. Both sheets of the focal set of a Dupin cyclide degenerate to curves, $ \Gamma _ {1} $ and $ \Gamma _ {2} $, which are curves of the second order. There are three types of Dupin cyclides.

1) The evolutes are an ellipse and a hyperbola; the radius vector of the corresponding Dupin cyclide is

$$ x = \frac{V b \sin u }{U + V } ,\ y = \frac{U b \sinh v }{U + V } , $$

$$ z = \frac{V a \cos u + U c \cosh v }{U + V } , $$

where

$$ U = c \cos u + d ,\ V = - a \cosh v - d ,\ d = \textrm{ const } . $$

2) The evolutes are focal parabolas; the radius vector is

$$ x = \frac{Vu}{U + V } ,\ y = \frac{Uv}{U + V } , $$

$$ z = \frac{V ( 2 u ^ {2} - p ^ {2} ) - U ( 2 v ^ {2} - p ^ {2} ) }{U p ( U + V ) } , $$

where

$$ U = \frac{2 u ^ {2} + p + q }{u p } ,\ V = \frac{2 v ^ {2} + p ^ {2} - q ^ {2} }{u p } ,\ q = \textrm{ const } . $$

3) The evolutes are a circle and a straight line; the corresponding Dupin cyclide is a torus.

Dupin cyclides are algebraic surfaces of order four in the cases 1) and 3) above, and of order three in the case 2).

References

[1]	Ch. Dupin, "Développements de géométrie" , Paris (1813)
[2]	F. Klein, "Vorlesungen über höhere Geometrie" , Springer (1926)
[3]	D. Hilbert, S.E. Cohn-Vossen, "Anschauliche Geometrie" , Springer (1932)

Comments

Originally, a Dupin cycle was defined more geometrically as the envelope of a family of spheres tangent to three fixed spheres. Every Dupin cyclide can be obtained from the following three examples by inversion in a suitable sphere: a torus of revolution, a circular cylinder and a circular cone.

The Dupin cyclides are surfaces of the second order in pentaspherical coordinates and have, moreover, two equal axis. For more on Dupin cyclides see [a2], pp. 359-360 and [a3], pp. 355-356; 441. A remarkable property of cyclides is the fact that they carry four families of circles.

The natural generalization of the Dupin cyclides to higher dimensions are the so-called Dupin-hypersurfaces (see [a1]).

References

[a1]	T.E. Cecil, P.J. Ryan, "Tight and taut immersions of manifolds" , Pitman (1985)
[a2]	M. Berger, "Geometry" , II , Springer (1987)
[a3]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a4]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)