special rays and structures in general optical systems: generalized magnifications associated with the fundamental properties

In aprevious study special rays and structures, including nodal, principal and focal rays and structures, were defined in terms of generalized magnification of reduced inclination of rays across thesystem.  The mathematical structure of the transference, however, suggests that there are another three possible approaches.  These involve generalized magnification of transverse position of rays across the system and two cross-magnifications, one of reduced inclination to transverse position and the other of transverse position to reduced inclination. The purpose of this paper is to develop these otherapproaches.  Nodal, principal and focal rays and structures, and, indeed, all special rays and structures, are defined in each of these approaches.  Equations relating the four types of generalized magnification are derived.  The approaches described here may have application in contexts where transverse position is of interest.  In an appendix the generalized magnifications are calculated for nodal rays across a particular model eye. (S Afr Optom 2010 69(2) 51-57)