Theory and Computation of Median Straight Lines and Planes in Coordinate Metrology

This paper presents the theoretical foundation and computational techniques for establishing median straight lines and planes for point-clouds in coordinate metrology. The theoretical foundation is based on an optimization formulation that exploits the invariant and equivariant properties of the point-clouds that reside in a Euclidean metric space. The optimization formulation leads directly to versatile computational approaches. The paper shows that the computational challenges can be easily overcome with readily available, free, and high-quality software packages. Such median fits are advantageous in many areas including when one-sided anomalies (which arise in some advanced manufacturing contexts and in some advanced measurement systems) should be ignored. Additional application of median straight lines and planes involves a standardized approach to handling outliers in coordinate metrology. Further applications involve standardized specification of tolerance zones using quartile and percentile zones, thus introducing rank-order statistics in the field of geometrical tolerancing and related metrological practices.