On quadratic two-parameter families of spheres and their envelopes

In the present paper we investigate rational two-parameter families of spheres and their envelope surfaces in Euclidean R3. The four dimensional cyclographic model of the set of spheres in R3 is an appropriate framework to show that a quadratic triangular Bézier patch in R4 corresponds to a two-parameter family of spheres with rational envelope surface. The construction shows also that the envelope has rational offsets. Further we outline how to generalize the construction to obtain a much larger class of surfaces with similar properties. © 2007 Published by Elsevier B.V.