Bézier-like Parametrizations of Spheres and Cyclides Using Geometric Algebra

We introduce new Bézier-like formulas with quaternionic weights for rational curves and surfaces in Euclidean 3-space. They are useful for representation of Möbius invariant geometric objects. Any Bézier curves and surfaces on 2-spheres can be converted to the quaternionic Bézier (QB) form of twice lower degree. Our focus is on bilinear QB-surfaces: we derive their implicitization formula and show that in general they are patches on special quartic surfaces usually called Darboux cyclides. In particular we derive QB-formula for principal patches of Dupin cyclides which are important in geometric modeling applications. One can extend this approach to higher dimensions and to pseudo-Euclidean spaces by changing quaternions with elements of the corresponding geometric algebra. In particular all formulas are valid in case of conformal geometric algebra with signature (4,1).