Minimum distance between a canal surface and a simple surface

The computation of the minimum distance between two objects is an important problem in the applications such as haptic rendering, CAD/CAM, NC veri cation, robotics and computer graphics. This paper presents a method to compute the minimum distance between a canal surface and a simple surface (that is, a plane, a natural quadric, or a torus) by nding roots of a function of a single parameter. We utilize the fact that the normals at the closest points between two surfaces are collinear. Given the spine curve C(t), tmin t tmax, and the radius function r(t) for a canal surface, a point on the spine curve C(t ) uniquely determines a characteristic circle K(t ) on the surface. Normals to the canal surface at points on K(t ) form a cone with a vertex C(t ) and an axis which is parallel to C 0(t ). Then we construct a function of t which expresses the condition that the perpendicular from C(t) to a given simple surface is embedded in the cone of normals to the canal surface at points on K(t). By solving this equation, we nd characteristic circles which contain the points of locally minimum distance from the simple surface. Based on these circles, we can compute the minimum distance between given surfaces.