Natural quadrics: projections and intersections

Abstract

Geometrical modelers usually strive to support at least solids bounded by the results of Boolean operations on planes, spheres, cylinders, and cones, that is, the natural quadrics. Most often this set is treated as a subset of the set of quadric surfaces. Although the intersection of two quadrics is a mathematically tractable problem, in implementation it leads to complexity and stability problems. Even in the restriction to the natural quadrics these problems can persist. This paper presents a method which, by using the projections of natural quadrics onto planes and spheres, reduces the intersection of two natural quadrics to the calculation of the intersections of lines and circles on planes and spheres. In order to make the claims of the method easily verifiable and provide the tools necessary for implementation, explicit descriptions of the projections are also included.