Generating cutter swept envelopes in five-axis milling by two-parameter families of spheres

This paper presents an efficient parametric approach of determining the shape of the envelope surface by a generalized cutter that follows five-axis tool path during NC machining. In this approach the cutter is modeled as a canal surface. By considering the tool motions the cutter is decomposed into a set of characteristic and great circles which are generated by two-parameter families of spheres. The center of a sphere from these families is described by two parameters which represent the spine curve and the tool path, and the radius of the sphere is described by one parameter representing the spine curve. Considering the relationship between characteristic and great circles the grazing points on the tool surface are identified. Analytically it is proven for the NC cutter geometries that any point on the envelope surface is located at the intersection of the characteristic and great circles. Then based on the proofs a closed-form solution for computing the grazing points generated by a surface of revolution is presented. The presented methodology is reduced to a simpler parametric form when the NC cutters are described by pipe surfaces.