Surface-independent, theoretically exact bestfit for arbitrary sculptured, complex, or standard geometries

Abstract In the coordinate measuring technique (CMT), an unambiguous mathematical criterion such as Gaussian least squares sum of departures measured perpendicularly to the calculated substitute feature ensures that the results of a geometry bestfit will be independent of the algorithm used. Mathematical methods solving the bestfit problem strictly according to this precisely defined minimizing criterion are available only for a limited number of commonly used standard elements. The proposed algorithm provides this bestfit on any parametrically represented surface x(u,v,p), even if the normally required, special implicit representation f(x,p) = 0 is missing. On the other hand, when applied to any standard element, it provides the classical results. The possibility of using the same, unmodified algorithm for a whole variety of parametrically describable surfaces, comprising complex/sculptured surfaces as well as standard surfaces, allows obvious simplification and standardization of bestfit procedure. The relevance of the proposed algorithm is also made greater by the fact that parametric description is in practice a supposition for automatic measuring/manufacturing as well as for CAD-directed representations, and for that reason CAD/CAM/CAQ-data-interchange standards are often based on this mathematical form. In this respect, it serves as a link between CAD-directed automatic measuring/manufacturing and the evaluation concepts of the classical coordinate measuring technique.