Accuracy and Time in Surface Measurement, Part 1: Mathematical Foundations

Abstract

Accuracy and time are known to be conflicting factors in measurement. This paper re-evaluates the two-dimensional sampling problem for measuring the surface roughness and establishes that an optimal sampling strategy can be obtained by utilizing the point sequences developed in Number Theory. By modeling a machined surfaces as a Wiener process, the root-mean-square (RMS ) error of measurement is shown to be equivalent to the L2 -discrepancy of the complement of the sampling points. It is further shown that this relationship holds for more general surfaces, thus firmly linking the minimum RMS error of the measurement to the celebrated lower bound on L2 -discrepancy asserted by Roth (1954), a 1958 Fields medalist.