Quartic supercyclides I: Basic theory

Various classes of algebraic surfaces have been examined as to their suitability for CAGD purposes. This paper contributes further to the study of a class of quartic surfaces recently investigated by Degen, having strong potential for use in blending, and possibly also in free-form surface design. These surfaces are here put in the context of a classi cation of quartic surfaces originally given more than one hundred years ago. An algebraic representation is provided for them, and a simple geometric interpretation given for their rational biquadratic parametric formulation. Their theory is established from an analytic geometry viewpoint which is more straightforward than Degen's original approach and gives further useful geometric insight into their properties. A major subclass of Degen's surfaces consists of projective transforms of the Dupin cyclides; for this reason (and others, explained in the text) the name supercyclides is proposed for them.