Precise bi-arc curve fitting algorithm for machining an aspheric surface

With the rapid development of the information/image system and aero-space industries, high quality optic aspheric surface lenses play an increasingly important role for completion of the functionalities. Aspheric lenses are non-spherical surfaces having rotation symmetry about the lens axis. The aspheric lens has various shapes according to its application and often requires tens nanometer order form accuracy since surface roughness and form accuracy play essential roles in the functional performance of the optical products. Interpolation of the aspherical surface path must precisely meet the allowable tolerance. Linear interpolation of the aspheric surface path for CNC machining generates an enormous amount of NC code to satisfy the extremely small tolerance, and produces scallops on the machined surface due to the acceleration and deceleration of the tool during every linear motion. Alternatively, interpolations with bi-arcs are used. In this paper, in order to minimize the error induced by the cutting tool path and to shorten the calculation time of interpolation, a precise -arc interpolation method is proposed. The developed algorithm of bi-arc interpolation meets the given tolerance precisely. This is guaranteed by an analytical proof and error maps. Another advantage is its ability to calculate about five times faster than the existing arc interpolation, since iterative calculations for the maximum error can be omitted. The developed algorithm has been used for the precise aspheric machining.