Rational parametric representation of parabolic cyclide: Formulation and applications

Abstract

The Dupin cyclide is a quartic surface with several interesting properties such as circular lines of curvature and rational parametric representation. Non-degenerate forms of the Dupin cyclide have been used for applications such as surface composition, variable radius blending and free space representation. The parabolic cyclide, a cubic surface, is a degenerate form of the Dupin cyclide with a pair of skew lines of curvature on its surface. The parabolic cyclide patch is attractive for geometric design applications due to its lower algebraic degree and (degenerate) straight lines of curvature, which can be used to provide smooth transition from planar to curved surfaces. The parametric representations available for the general Dupin cyclide cannot, in general, be extended to the parabolic cyclide. In this paper, we derive a new rational biquadratic and corresponding NURBS representation for a parabolic cyclide patch. We also discuss the practical issues involved in using these parametric representations and provide examples of applications of this patch.