Obtaining implicit equations of supercyclides and definition of elliptic supercyclides

The use of Dupin cyclides and supercyclides in CAGD applications has been the subject of many publications in the last decade. Dupin cyclides are low degree algebraic surfaces having both parametric and implicit representations. In this paper, we aim to give the necessary expansions to derive implicit equations of supercyclides in the affine as well as in the projective space, starting from equations of the Dupin cyclide and the transformation matrix. We introduce a particular subfamily of supercyclides, called elliptic supercyclides, and show how to use them for the blending of elliptic quadratic primitives. We also show how one can convert an elliptic supercyclide into a set of rational biquadratic Bézier patches.