Rational Bicubic Formulation of Dupin Cyclide

Abstract

Dupin cyclides are canal surfaces with circular lines of curvature and are widely used in geometric modelling applications such as Computer-Aided Design (CAD), surface fitting, and blending. In this research paper, we present a numerical approximation scheme for Dupin cyclide patches using rational bicubic Bézier surfaces. The boundary and interior control points are determined by enforcing C¹ continuity conditions, while the associated weights are optimized through curvature based variational fairness criterion. The complete Dupin cyclide is constructed by applying the suitable isometries to the constructed patches. The proposed approximation scheme interpolates the given cyclide data and ensures continuity across patch interfaces. The approximation quality is assessed using standard geometric error measures, including root mean square error and maximum relative error. Numerical experiments demonstrate that the maximum relative error attains the values of 0.6306 and 0.3982 for the two primary patches, confirming the computational efficiency of the proposed scheme.