On nonsingular, cyclide transition surfaces
Seth Allen · Deep Blue (University of Michigan) · 1997
The natural quadrics (the plane, sphere, right circular cylinder, and right circular cone) are an important subset of the primitives used in computer modeling systems. The (Dupin) cyclide is a degree 4 generalization of a torus that appears to be a natural choice for a new primitive in these modelers. This thesis studies nonsingular, cyclide transition surfaces between any two natural quadrics. These surfaces are tubes that are tangent to the natural quadrics they connect along circles. Necessary and sufficient conditions (with constructive proofs) for the existence of cyclide transition surfaces are given. Except for the cone/cone case, none of the previously published constructions indicate the exact conditions of their validity. Further, in previous work, no effort is made to ensure the construction of nonsingular transition surfaces, even though a designer seldom desires singular surfaces. Each quadric/quadric case is analyzed to determine exactly when the nonsingular, cyclide transition surface is a blend and when it is a join. This distinction, previously ignored, is important since the term blend usually has a very specific meaning. (When visualizing a blend between two intersecting surfaces, a designer typically imagines the surface formed by pressing putty with one's thumb along the curve of intersection between the surfaces to remove the sharp crease.) A further result states that except in the case of intersecting spheres, blends and joins cannot simultaneously exist. Affine transformations of cyclides, a subset of the supercyclides, and their role in blending is studied. This analysis leads to the following original result. A plane and an axial natural quadric can be blended by a supercyclide along any given ellipse on the axial natural quadric if and only if the plane and the axial natural quadric intersect in an ellipse. Also shown is that similar extensions to the remaining cases, that build upon the work presented here, cannot be made. This topic is beyond the scope of the thesis and is left to future investigations.