Planar intersection and blending of natural quadrics

In general, two quadric surfaces intersect in a space quartic. However, the intersection may degenerate to a collection of plane curves. In this thesis, new geometric algorithms and characterization theorems are presented for problems related to two axial natural quadrics (i.e., a cylinder or cone) that have planar intersection. These problems include computing the plane intersection of a natural quadric, detecting and calculating the planar intersection curve of two natural quadrics, and constructing a blending Dupin cyclide of two natural quadrics with planar intersection. An important discovery is that all reasonings and computations can be reduced to a plane (the axial plane). Using the Dandelin sphere concept, the first part of this thesis presents a simple geometric method for computing the intersection of a plane and a natural quadric (i.e., a sphere, cylinder and cone). The characteristics of the intersection conic are expressed by simple formulae. Therefore, our algorithm has the advantage that all parallel intersection conics can be determined without extra cost. A planar intersection of two axial natural quadrics consists of two conics, one double line and a conic, or at most four lines. In the second part, for each type of planar intersection, a fast geometric algorithm is presented to compute the intersection. Only line intersections, point/line incidence and vector cross product are used; no lengthy algebraic manipulations are involved. Since these operations can be implemented precisely, our algorithms are robust. In addition to the algorithmic study, characterization results are also obtained. The existence of a conic intersection (i.e., at least one component of the intersection curve is a conic) is characterized by focal lengths, perpendicular focal conics, the common inscribed sphere (for the intersecting axes case), and congruence of cones (for the parallel axes case). With the insights uncovered by these characterization theorems, another conic intersection algorithm is developed. Using the common inscribed sphere, we also present an algorithm to extract the types of the intersection conics without actually computing them. In the third part, we address the problem of blending two axial natural quadrics with planar intersection by Dupin cyclides. The main contributions are a new construction algorithm with more control of the blending cyclides, a complete and new proof that (with one exception) two axial natural quadrics have planar intersection if and only if they have a blending cyclide, and a thorough study of how all of these blending cyclides are organized into families.