Rational Bézier Formulas with Quaternion and Clifford Algebra Weights

Abstract

We consider Bézier-like formulas with weights in quaternion and geometric (Clifford) algebra for parametrizing rational curves and surfaces. The simplest non-trivial quaternionic case of bilinear formulas for surface patches is studied in detail. Such formulas reproduce well known biquadratic parametrizations of principal Dupin cyclide patches, and are characterized in general as special Darboux cyclide patches. We also survey the most general case of geometric algebra generated by pseudo-Euclidean space. This extends the previous constructions to any dimensions and any signatures of ambient space. Applications include Bézier curves and surfaces in the conformal model of Euclidean space, bilinear Clifford-Bézier patches on isotropic cyclides, and rational offset surface modeling.