Contribution to the minimax evaluation of circularity error

The measurement of form and profile errors of mechanical parts involves the fitting of continuous surfaces, curves or lines to a set of coordinate points returned by inspection instrumentation. For circularity, both ASME (Y14.5M - 1994 [1995]) and ISO (1R 1001, 1983) prescribe the fitting of a pair of concentric circles with minimum radial separation to a set of discrete coordinate points sampled around the periphery of a surface of revolution at a plane perpendicular to the axis of revolution. This criterion is often referred to as the minimax or minimum zone criterion. The contribution of this paper is to provide a refinement to the basic results of both Ventura and Yeralan and Etesami and Qiao as an improved path to the optimum solution to the minimax roundness problem. The proposed solution applies the Voronoi diagram approach to the P2( k ) problem to show that the optimum solution can only reside at a particular type of vertex of the MAX region. This particular type of vertex corresponds to one of the possible type (ii) solutions of Ventura and Yeralan's P2( k ) problem. In other words, Etesami and Qiao's approach need not examine all vertices of the MAX region and Ventura and Yeralan's P2( k ) solution need not examine any type (i) solutions. This results in a substantial saving in time required to identify the optimum solution and would be useful for applications that return a large number of points such as dedicated roundness testers, axis of rotation analysers or telescopic ball bars.