Abstract
Because dioptric power matrices of thin systems constitute a (three-dimensional) inner-product space, it is possible to define distances and angles in the space and so do quantitative analyses on dioptric power for thin systems. That includes astigmatic corneal powers and refractive errors. The purpose of this study is to generalise to thick systems. The paper begins with the ray transference of a system. Two 10-dimensional inner-product spaces are devised for the holistic quantitative analysis of the linear optical character of optical systems. One is based on the point characteristic and the other on the angle characteristic; the first has distances with the physical dimension L−1 and the second has the physical dimension L. A numerical example calculates the locations, distances from the origin and angles subtended at the origin in the 10-dimensional space for two arbitrary astigmatic eyes.
Highlights
The optical character of a thin system in linear optics can be represented by a symmetric 2×2 matrix F, the symmetric dioptric power matrix
Because symmetric dioptric power space is an inner-product space, we have been able to define distances, angles, orthonormal axes, confidence ellipsoids, etc. in the space. This has provided the basis for the quantitative analysis we have done on powers including refractive errors and corneal powers (e.g. Ref 3)
The optical character of a system that is thick or thin is completely characterised by the ray transference of the system.[5]
Summary
The optical character of a thin system in linear optics can be represented by a symmetric 2×2 matrix F, the symmetric dioptric power matrix. The set of all such powers defines a threedimensional linear (or vector) space, known as symmetric dioptric power space.[1] Because the matrix has uniform physical dimensionality[2] (each entry has the dimension L−1 and is usually measured in dioptres), one can define an inner-product on the space and the space becomes an inner-product space. This has provided the basis for the quantitative analysis we have done on powers including refractive errors and corneal powers (e.g. Ref 3). The optical character of a system that is thick or thin is completely characterised by the ray transference (a real 4×4 matrix)
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