Rational Maximal Parametrisations of Dupin Cyclides

Abstract

Ring Dupin cyclides are algebraic surfaces of low degree that admit rational parametrisation. Their properties and applications in geometric modeling have been investigated in recent years by a number of authors. In particular their parametrisation using bi-quadratic Bézier patches is well documented. It is known, for example, that a minimum of four bi-quadratic Bézier patches is required to parametrise the entire manifold. However, neither the geometry nor the topology of the cyclide impedes the construction of single patch rational parametrisations of the whole surface. This paper constructs and discusses a number of single patch Bézier parametrisations of ring Dupin cyclides. Specifically: bi-quartic rational parametrisations of entire ring cyclides, optimized bi-quartic rational parametrisations of entire ring cyclides, and bi-sextic rational parametrisations of entire ring cyclides for which the parametrisation of iso-parametric lines is ’almost’ identical to those of the familiar trigonometric parametrisation, are presented. The method of construction may be applied to the determination of rational patches of sub-maximal coverage, avoiding all the problems of other methods, such as: the complement of the intended region being parametrised, prohibited parametric values and other geometric constraints, and restriction of angular displacements to < π.