Cyclides in Geometric Modeling: Computational Tools for an Algorithmic Infrastructure

Abstract

The Dupin cyclide is a quartic surface with useful properties such as circular lines of curvature, rational parametric representations and closure under offsetting. All natural quadrics (cone, cylinder, sphere) and the torus are special cases of the cyclide. Applications of cyclides include variable radius blending, piping design and design of tubular geometry, such as mold gates and wire harnesses. While the mathematical treatment of cyclides is well developed, the tools required to incorporate this surface into conventional modeling applications have not received sufficient attention. In this paper, we detail various methods for manipulating and computing with the cyclide. Issues such as auxiliary representations, inverse parameter mapping, point classification, normal projection of a point onto the surface, distance computation, bounding volumes and surface intersection detection are discussed in detail. We also provide implemented examples of design applications that utilize these computational tools.