Bounded domain, bi-quadratic rational parametrisations of Dupin cyclides

Dupin cyclides, their applications in geometric modeling and their parametrisation using bi-quadratic patches bounded by lines of curvature, have been investigated in recent years by a number of authors - see, for example, Martin et al (1986); Boehm (1990); Pratt (1990); Degen (1994, 1996). 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 parametrisation exists for the specified region of the cyclide, then it is correctly determined. Explicit formulæ are given for the Bernstein weights and vectors of the parametrisations. The method is neither cyclide specific nor specific to the construction of bi-quadratic rational parametrisations - it may therefore be applied to other surfaces and to higher-degree rational constructions.