Cyclide

A nondegenerate cyclide, also called a Dupin cyclide or cyclid, is a surface that is the common envelope of two distinct one-parameter families of spheres. It is therefore a canal surface in two different ways. The centers of the spheres trace two conic sections, possibly degenerate (Alcázar et al. 2018). The lines of curvature are circles or straight lines (Pinkall 1986).

The standard tori with positive major and minor radii and their inversions in an inversion sphere centered at a point and of radius are cyclides (Pinkall 1986). The inversion sends a point to

Such an image is a quartic surface when the inversion center is off the torus. When the inversion center is a point in the smooth locus of the torus, the image is instead a cubic surface, called a parabolic cyclide. Inversion at a point of the torus outside its smooth locus can reduce the polynomial degree further. For example, a horn torus inverted at its point of self-contact becomes a circular cylinder. Planes, spheres, circular cones, and circular cylinders are included as degenerate cyclides (Pratt 1990).

Illustrated above are ring cyclides, horn cyclides, and spindle cyclides. The figures on the right use an inversion center at a point in the smooth locus of the corresponding torus, producing a parabolic ring cyclide, parabolic horn cyclide, and parabolic spindle cyclide, respectively.

The croissant surface has a similar pinched appearance, but comparison of its catalog equation with the canonical equations for Dupin cyclides shows that it does not belong to this family (Real Sociedad Matemática Española 2011, p. 43; Alcázar et al. 2018).