Minimum centroid neighborhood for minimum zone sphericity

The minimum zone sphericity tolerance is derived from the ANSI and ISO standards for roundness and has extensive applications in the tribology of ball bearings, hip joints and other lubricated pairs. The worst-case proposed in this paper provides theoretical evidence that the minimum zone center of the two (circumscribed and inscribed reference) spheres with minimum radial separation containing the sampled spherical surface is included in a spherical neighborhood centered in the centroid of radius 2π-2EC, where EC is the sphericity error related to the centroid, which can be determined in closed form. Such linear estimating (about 20% of EC from the centroid, i.e., about one order of magnitude lower than the sphericity tolerance to be assessed) can be used to locate the sphere center with a given tolerance and as a search neighborhood for minimum zone center-based algorithms, such as metaheuristics (genetic algorithms, particle swarm optimization, etc.). The proposed upper bound has been experimentally assessed, using a genetic algorithm (GA) with parameters previously optimized for roundness and extended to three dimensions, which has overcome most of all available datasets from the literature that have been tested with center-based minimum zone algorithms by different authors. The optimum dataset size on artificially generated datasets is also discussed and it is speculated to allow the extension of the proposed upper bound to partial (or incomplete) spherical features.