Analytical Expression of the Swept Surface of a Rotary Cutter Using the Envelope Theory of Sphere Congruence

Based on the observation that many surfaces of revolution can be treated as a canal surface, i.e., the envelope surface of a one-parameter family of spheres, the analytical expressions of the envelopes of the swept volumes generated by the commonly used rotary cutters undergoing general spatial motions are derived by using the envelope theory of sphere congruence. For the toroidal cutter, two methods for determining the effective patch of the envelope surface are proposed. With the present model, it is shown that the swept surfaces of a torus and a cylinder can be easily constructed without complicated calculations, and that the minimum distance (between the swept surface and a simple surface) and the signed distance (between the swept surface and a point in space) can be easily computed without constructing the swept surface itself. An example of global tool path optimization for flank milling of ruled surface with a conical tool, which requires to approximate the tool envelope surface to the point cloud on the design surface, is given to confirm the validity of the proposed approach.