Magnitude, distance, and orthogonality for thin spherocylindrical lenses

This paper shows that for thin spherocylindrical lenses there is only one reasonable way to define the magnitude of a lens, the distance between two lenses, and orthogonality of lenses. Although any selection of a real inner product on the vector space of thin spherocylindrical lenses extends the space to be a Euclidean space and automatically defines these concepts, requiring the inner product to be rotation invariant makes selecting a Euclidean space equivalent to fixing only the magnitudes of two particular lenses. It is shown that among Euclidean spaces having rotation-invariant inner products there is one that is much preferred to the others. For this space, and for all other rotation-invariant spaces, equations are provided to determine magnitude, distance, inner product, and orthogonality, both in terms of the coordinates of lenses and in terms of the parameters of the prescriptions of the lenses. Also introduced are spherical magnitude and distance, Jackson magnitude and distance, and the six-dimensional Euclidean space of pairs of lenses, which is necessary to describe fully the right and left lenses needed by a patient.