The metric geometric mean transference and the problem of the average eye

Abstract

An average refractive error is readily obtained as an arithmetic average of refractive errors.  But how does one characterize the first-order optical character of an average eye?  Solutions have been offered including via the exponential-mean-log transference.  The exponential-mean-log transference ap-pears to work well in practice but there is the niggling problem that the method does not work with all optical systems.  Ideally one would like to be able to calculate an average for eyes in exactly the same way for all optical systems. This paper examines the potential of a relatively newly described mean, the metric geometric mean of positive definite (and, therefore, symmetric) matrices.  We extend the definition of the metric geometric mean to matrices that are not symmetric and then apply it to ray transferences of optical systems.  The metric
geometric mean of two transferences is shown to satisfy the requirement that symplecticity be pre-served.  Numerical examples show that the mean seems to give a reasonable average for two eyes.  Unfortunately, however, what seem reasonable generalizations to the mean of more than two eyes turn out not to be satisfactory in general.  These generalizations do work well for thin systems.  One concludes that, unless other generalizations can be found, the metric geometric mean suffers from more disadvantages than the exponential-mean-logarithm and has no advantages over it.