Multimeridional refraction: dependence of the measurement accuracy on the number of meridians refracted.

A Monte Carlo simulation of multimeridional refraction measurements was used to investigate the dependence of the accuracy of the measurement on the number of meridians refracted, N, and on the standard deviation of a measurement in a single meridian, sigma. For the description of the measurement errors, the residual refraction values were used, i.e., the parameters of the refraction remaining after application of the measured correction. The distributions of the residual refraction values were found to be independent of the "true" refraction values; in addition, by means of a factor square root of N/sigma, reduced residual refraction values could be defined which also were independent of N and sigma. A vector space proposed by Lakshminarayanan and Varadharajan (based on Long's power matrix) was used to represent the joint distribution of the residual refraction values in three-dimensional space. It was found to be a three-variate Gaussian distribution with zero mean and diagonal covariance matrix. It could further be shown that the vector space proposed by Harris is identical to the one used, up to a linear transformation. Several criteria, based on the one- and three-dimensional distributions and corresponding to different levels of accuracy, are discussed resulting in a wide range of answers about the number of meridians to be refracted.