Computing All Conic Sections in Torus and Natural Quadric Intersections

Conic sections embedded in a torus must be circles of special types: (i) proole circles, (ii) cross-sectional circles , and (iii) Yvone-Villarceau circles. Based on this classiication, we present eecient and robust geometric algorithms that detect and compute all degenerate conic sections (circles) in torus/plane and torus/natural-quadric intersections. 2 Introduction Simple surfaces (such as planes, spheres, cylinders, cones, and tori) are important in conventional solid model-ing systems since they can represent a large number of simple mechanical parts. In the boundary evaluation of such mechanical parts, we need to compute the intersection curves of these simple surfaces. There are many previous results on intersecting two natural quadrics. The detection of degenerate conic sections, in particular, has attracted considerable research attention 2, 5, 6]. Conic sections can be represented precisely and eeciently in geometric databases. Therefore, it is important to detect and compute all conic sections in surface/surface intersections 2, 6]. Surprisingly, there are only a few previous results that can detect all conic sections in the intersection curve of a torus with other surfaces. Piegl 3] considered a degenerate case of Torus Plane Intersection (TPI), in which a torus has two tangential intersections with a plane. In this case, the TPI curve consists of two circles (called Yvone-Villarceau circles). Kim et al. 1] observed a similar case in Torus Sphere Intersection (TSI). That is, when a torus has two tangential intersections with a sphere, the TSI curve consists of two Yvone-Villarceau circles (which are non-coplanar, in general). In this paper, we present eecient and robust geometric algorithms that can detect and compute all conic sections in Torus/Plane Intersection (TPI) and Torus/natural-Quadric Intersection (TQI). These algorithms are based on a simple classiication of all conic sections that can be embedded in a torus. That is, they must be circles of special types: (i) proole circles, (ii) cross-sectional circles, and (iii) Yvone-Villarceau circles. Based on this classiication, we formulate all necessary and suucient conditions in algebraic and semi-algebraic constraints. Let ? T and ? Q denote the sets of all circles that can be embedded in a torus T and a natural-quadric Q, respectively. The two surfaces T and Q intersect in circles if and only if the two sets ? T and ? Q of circles have a non-empty intersection: ? T \ ? Q 6 = ;; that is, there are some circles that can be embedded in both T …