Dupin Cyclide Blends Between Quadric Surfaces for Shape Modeling

We introduce a novel method to define Dupin cyclide blends between quadric primitives. Dupin cyclides are nonspherical algebraic surfaces discovered by French mathematician Pierre-Charles Dupin at the beginning of the 19th century. As a Dupin cyclide can be fully characterized by its principal circles, we have focussed our study on how to determine principal circles tangent to both quadrics being blended. This ensures that the Dupin cyclide we are constructing constitutes a G1 blend. We use the Rational Quadratic Bézier Curve (RQBC) representation of circular arcs to model the principal circles, so the construction of each circle is reduced to the determination of the three control points of the RQBC representing the circle. In this work, we regard the blending of two quadric primitives A and B as two complementary blending operations: primitive A-cylinder and cylinder-primitive B; two Dupin cyclides and a cylinder are then defined for each blending operation. In general the cylinder is not useful and may be reduced to a simple circle. A complete shape design example is presented to illustrate the modeling of Eurographics’04 Hugo using a limited number of quadrics combined using Dupin cyclide blends.