FITTING CIRCLES AND SPHERES TO COORDINATE MEASURING MACHINE DATA

This work addresses the problem of enclosing given data points between two concentric circles (spheres) of minimum distance whose associated annulus measures the out-of-roundness (OOR) tolerance. The problem ar ises in analyzing coordinate measuring machine (CMM) data taken agains t circular (spherical) features of manufactured parts. It also can be interpreted as the ''geometric'' Chebychev problem of fitting a circle (sphere) to data so as to minimize the maximum distance deviation. A related formulation, the ''algebraic'' Chebychev formula, determines t he equation of a circle (sphere) to minimize the maximum violation of the equation by the data points. In this paper, we describe a linear-p rogramming approach for the algebraic Chebychev formula that determine s reference circles (spheres) and related annuluses whose widths are v ery close to the widths of the true geometric Chebychev annuluses. We also compare the algebraic Chebychev formula against the popular algeb raic least-squares solutions for various data sets. In most of these e xamples, the algebraic and geometric Chebychev solutions coincide, whi ch appears to be the case for most real applications. Such solutions y ield concentric circles whose separation is less than that of the corr esponding least-squares solution. It is suggested that the linear-prog ramming approach be considered as an alternate solution method for det ermining OOR annuluses for CMM data sets.