R-mapping based method for designing of form cutting tool for sculptured surface machining

Computer-aided design (CAD) and computer-aided machining (CAM) of sculptured surfaces sometimes present problems that were not envisioned in classical differential geometry. This paper presents mathematical results that pertain to designing a form-cutting tool for machining a given sculptured surface on a multiaxis NC machine. The results obtained below are based on a new kind of surface mapping - on the so-called R-mapping of a sculptured surface P onto the machining surface T of a form-cutting tool. This result is significant, for it substantiates earlier works in computer-aided geometric design and computer-aided machining of sculptured surfaces. It also offers practical means of generating the machining surface of a form-cutting tool for sculptured surface machining. From a theoretical viewpoint, it is remarkable that one can generate the machining surface of a form-cutting tool. This becomes possible under an assumption that there is one-to-one correspondence between principle curvatures of a sculptured surface to be machined and of the machining surface of a form-cutting tool to be designed. The presented geometric criteria may be helpful for designers, since maximal rate of conformity of the machining surface of a form-cutting tool to a sculptured surface is a theoretical requirement in developing highly efficient technology or in solving design problems. Usefulness of the approach is illustrated with examples.