Signature Sequence of Intersection Curve of Two Quadrics for Exact Morphological Classification

Computing the intersection curve of two quadric surfaces is an important problem in geometric computation, ranging from shape modeling in computer graphics and CAD/CAM, collision detection in robotics and computational physics, to arrangement computation in computational geometry. We present the solution to the fundamental problem of complete classification of morphologies of the intersection curve of two quadrics (QSIC) in PR, 3D real projective space. We show that there are in total 35 different QSIC morphologies with non-degenerate quadric pencils. For each of these 35 QSIC morphologies, through a detailed study of the eigenvalue curve and the index function jump we establish a characterizing algebraic condition expressed in terms of the Segre characteristics and the signature sequence of a quadric pencil. We show how to compute a signature sequence with rational arithmetic so as to exactly classify the morphology of the intersection curve of any two given quadrics. Two immediate applications of our results are the robust topological classification of QSIC in computing B-rep surface representation in solid modeling and the derivation of algebraic conditions for collision detection of quadric primitives.