Central Conics on Parabolic Dupin Cyclides

Hyperbolas, ellipses and degenerate conics on parabolic Dupin cyclides are investigated. These central conics are obtained as the intersections of parabolic cyclides and the planes perpendicular to the two planes of symmetry of the cyclides. They are also the images of central conics in the parametric space. Since the conics are planar curves, they are transformed into planar or spherical curves on Dupin cyclides via inversion. Lemniscates of Bernoulli on Dupin cyclides and Vivianis curves on right-circular cylinders are included in the inverted conics. Two intersecting lines on a parabolic ring cyclide, which are degenerate conics, are inverted into Villarceau circles on a ring cyclides.