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A generalization of Patterson's work on the invariants of the rational Bezier curves, to the case of weight vectors of mixed-sign is presented. It is shown that an additional invariant is required in the general case. The results are viewed from the perspective of the Fundamental Theorem on invariants for Lie groups. [ABSTRACT FROM AUTHOR]

Dupin cyclides, their applications in geometric modelling and their parametrization using bi-quadratic patches bounded by lines of curvature, have been investigated in recent years by a number of authors such as Martin, de Pont and Sharrock in 1986, Boehm in 1990, Pratt in 1990, and Degen in 1994. However, no completely reliable and general algorithm for the determination of bi-quadratic cyclide patches has appeared in the literature. This paper presents a new approach that produces any required bi-quadratic patch, bounded by lines of curvature, without non-intrinsic geometric constraints or restrictions. Specifically, if a bi-quadratic parametrization exists for the specified region of the cyclide, then it is correctly determined. Explicit formulae are given for the Bernstein weights and vectors of the parametrizations. The method is neither cyclide specific nor specific to the construction of bi-quadratic rational parametrizations - it may therefore be applied to other surfaces and to higher degree rational constructions. [ABSTRACT FROM AUTHOR]