Comments on ‘The average eye’ by W.F. Harris, Ophthal. Physiol. Opt. 2004 24: 580–585

The definition of an average eye is a non-trivial problem. Harris proposed a procedure based on ray transferences in the framework of linear optics. Because the symplectic transferences do not form a vector space transferences might be added but the result is not symplectic. In other words, the resulting matrix does not represent an optical system. Thus, the arithmetic average is not at hand. Harris for the first time brought in a non-linear average of transferences, which seems to be a working solution for calculating averages. The starting point of Harris are the transferences of the optical systems to be averaged. In the case of averaging eyes they represent the properties of the considered eyes each taken as a whole. If the eye (or any other optical system) is taken as a whole, i.e. considering only the mapping rendered by the related transference, then the making of the transferences is a non-issue. How and by which components the transference was constructed is a question which is simply superfluous in this holistic approach. Harris coined it as follows: Similarly one can determine the average corneal power and the average of the properties of other components of the eye but such averages are averages for those particular parts of the eye and not of the optical character of the eye taken as a whole. Indeed, as we shall see, an average eye determined naı̈vely by averaging individual properties is, in general, not a possible eye at all . The last claim might be misleading as a detailed knowledge of the construction data (radii, thicknesses and refractive indices) allows for an alternative definition of an average eye as will be shown below. However, only if the supplementary information is available and only if the number of elements is the same for all eyes to be averaged is the proposed procedure feasible. Contrary to this restriction Harris general approach can be applied in the moment the transferences of optical systems are at hand. The mean for a schematic eye with one surface only and, say, a Gullstrand eye with six surfaces, can easily be calculated. Even an average for binoculars, eyes and microscopes could be produced. Clearly, there might be little interest in such exotic combinations. In the following it will be shown that an average eye determined by averaging individual properties is a possible eye. This procedure might be of interest while constructing model eyes from biometric data. Consider a set of biometric data for K eyes. These data comprise information on corneal and lens radii, refractive indices and thicknesses of the components. Ignoring tilts etc. the 4 · 4 system matrix (or transference) S for one of these eyes is given by a product of matrices. Because there are only two events in the life of a light ray each factor in this product is related either to a refraction (R) or to a transfer (T) which are given by