Systematic and random errors in least squares estimation for circular contours

It is assumed that stochastically independent measurements are made of the distance from a reference centre to n points spaced at equi-angular intervals along an arc of length θ on the circular contour of a part. The least squares estimates of the radius of the circular contour and the coordinates of the true centre relative to the reference centre are derived assuming that the distance between the true and reference centres is small, relative to the radius of the part. These estimates are shown to be expressible as linear combinations of the measured distances with coefficients that are explicit functions of n and θ. The estimates are shown to be systematically in error (biased) unless the true and reference centres coincide. Formulas are given for the calculation of the bias error in any situation. The random error of the estimated quantities is expressed in terms of the random error of an individual measurement and the values of n and θ. The correlation coefficient between the estimated quantities is also found. The results enable the determination, in advance of making measurements, of whether the likely random and systematic errors will be tolerable for a given choice of n and θ and a given centring accuracy.