Blending two cones with Dupin cyclides

Abstract

This paper presents a complete theory of blending cones with Dupin cyclides and consists of four major contributions. First, a necessary and sufficient condition for two cones to have a blending Dupin cyclide is established. Second, based on the intersection structure of the cones, finer characterization results are obtained. Third, a new construction algorithm that establishes a correspondence between points on one or two coplanar lines and all constructed blending Dupin cyclides makes the construction easy and well-organized. Fourth, the completeness of the construction algorithm is proved. Consequently, all blending Dupin cyclides are organized into one to four one-parameter families, each of which is “parameterized” by points on a special line. It is also shown that each family contains an infinite number of ring cyclides, ensuring the existence of singularity free blending surfaces.