G1-Blend between a Differentiable Superquadric of Revolution and a Plane or a Sphere Using Dupin Cyclides

In this article, we present a method to perform G¹-continuous blends between a differentiable superquadric of revolution and a plane or a sphere using Dupin cyclides. These blends are patches delimited by four lines of curvature. They allow to avoid parameterization problems that may occur when parametric surfaces are used. Rational quadratic Bezier curves are used to approximate the principal circles of the Dupin cyclide blends and thus a complex 3D problem is now reduced to a simpler 2D problem. We present the necessary conditions to be satisfied to create the blending patches and illustrate our approach by a number of superellipsoid/plane and superellipsoid/sphere blending examples.