The invariant functions of the rational Bi-cubic Bézier Surfaces

Abstract

Patterson's work [1] on the invariants of the rational Bézier paths may be extended to permit weight vectors of mixed-sign [2]. In this more general situation, in addition to Patterson's continuous invariants, a discrete sign-pattern invariant is required to distinguish path geometry. The author's derivation of the invariants differs from that of Patterson's and extends naturally to the rational Bézier surfaces. In this paper it is shown that 13 continuous invariant functions and a discrete, sign-pattern, invariant exist for the bi-cubic surfaces. Explicit functional forms of the invariant functions for the bi-cubics are obtained. The results are viewed from the perspective of the Fundamental Theorem on invariants for Lie groups. © 2009 Springer Berlin Heidelberg.