# 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)