Generalized Dupin Cyclides with Rational Lines of Curvature

Dupin cyclides are algebraic surfaces of order three and four whose lines of curvature are circles. These surfaces have a variety of interesting properties and are aesthetic from a geometric and algebraic viewpoint. Besides their special property with respect to lines of curvature they appear as envelopes of one-parameter families of spheres in a twofold way. In the present article we study two families of canal surfaces with rational lines of curvature and rational principal curvatures, which contain the Dupin cyclides of order three and four as special instances in each family. The surfaces are constructed as anticaustics with respect to parallel illumination and reflection at tangent planes of curves on a cylinder of rotation.