Completely Dioptrically Equivalent Systems

The paraxial optics of astigmatic human eyes is covered by the optics of systems with obliquely crossed toric surfaces. For these systems, the dioptric parameters in terms of the 2 X 2 dioptric matrices are: the equivalent power matrix P e (which may be asymmetric), the back vertex power matrix P v , and the neutralizing or front vertex power matrix P n .1,2 For statistical considerations where thickness is taken into account (as opposed to a thin lens model), one might compute a mean P e , P v , and P n (10 known parameters).3,4 Since the means are statistical calculations and not optical calculations, the question then arises as to whether any real system of toric surfaces has the same dioptric parameters as these means. I'll refer to such matching systems as completely dioptrically equivalent (CDE) systems. These systems might be models of astigmatic eyes or general systems consisting of toric surfaces.