Algebraic algorithms for computing intersections between Torus and natural Quadrics

We present in this paper efficient and robust algebraic algorithms for computing intersection curves between torus and natural quadrics used in CAD, namely sphere, cylinder and cone. A proper local coordinate system (LCS) is chosen first. Under the chosen LCS, cylinders and cones are parametrized by their line generators, and the problem of computing intersections between tori and cylinders or cones are converted to the line/torus intersection problem. Then, cases that the intersections are conic sections are identified and computed in geometrical and intuitive ways. For all other cases, discrete intersection points are computed and sorted to form intersection curves. For torus/sphere intersection, we present algorithms to compute the intersection curves directly from explicit algebraic representations.