From Dupin Cyclides to Scaled Cyclides

Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. They have a low algebraic degree and have been proposed as a solution to a variety of geometric modeling problems. The circular curvature line’s property facilitates the construction of the cyclide (or the portion of a cyclide) that blends two circular quadric primitives. In this context of blending, the only drawback of cyclides is that they are not suitable for the blending of elliptic quadric primitives. This problem requires the use of non circular curvature blending surfaces. In this paper, we present another formulation of cyclides: Scaled cyclides. A scaled cyclide is the image of a Dupin cyclide under an affine scaling application. These surfaces are well suited for the blending of elliptic quadrics primitives since they have elliptical lines of curvature. We also show how one can convert a scaled cyclide into a set of rational quadric Bézier patches.