Torus/Sphere Intersection Algorithm

The torus/sphere intersection problem could be converted into the intersection problem between a sphere and a cluster of circles if a torus is considered as a cluster of circles with centers on an outer circle. No tracing is required at all. With the theory of the minimum distance between a point and a circle, some special cases are directly figured out such as no intersection, one tangent point, one intersecting circle, or two intersecting circles. For other cases, the intersection problem is solved by a quartic equation with respect to the parameter of the central circle of the given torus. The parametric interval 0, 2π is divided and a sign-detection method is presented to find out those intervals that intersection points lie in. The resultant curves are provided in a parametric form.