Quantitative analysis of transformed ray transferences of optical  systems in a space of augmented Hamiltonian matrices*

There is a need for methods for quantitative analysis of the first-order optical character of optical systems including the eye and components of the eye. Because of their symplectic nature ray transferences themselves are not closed under addition and multiplication by ascalar and, hence, are not amenable to conventional quantitative analysis such as the calculation of an arithmetic mean. However transferences can be transformed into augmented Hamiltonian matrices which are amenable to such analysis. This paper provides a general methodology and in particular shows how to calculate means and variance-covariances representing the first-order optical character of optical systems. The systems may be astigmatic and may have decentred elements. An accompanying paper shows application to the cornea of the human eye with allowance for thickness.

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Optical transferences and their application to ray tracing through the human cornea*

African Vision and Eye Health ◽

10.4102/aveh.v70i4.114 ◽

2011 ◽

Vol 70 (4) ◽

Author(s):

S. D. Mathebula ◽

A. Rubin

Keyword(s):

Healthy Subjects ◽

Vector Fields ◽

Surrounding Medium ◽

Optical Systems ◽

Human Cornea ◽

First Order ◽

Optical Character ◽

Light Rays ◽

The Mean ◽

The Right

The purpose of this paper is two fold, firstly to describe aspects of the quantitative analysis of the linear optical character of the corneas of ten young and healthy subjects using the exp-mean-log-transference and secondly to illustrate how mean transference and ray vector fields or diagrams can be used to explain and understand the optical properties of corneas as thick optical systems.An Oculus Pentacam was used to obtain 43 successive measurements of the radii of curvature of the anterior and posterior corneal surfaces and the central corneal thicknesses of the right eyes of ten subjects. From these measurements 4×4 ray transferences were calculated. Mean transferences were obtained via multi-dimensional Hamiltonian space and these mean transferences were used to produce stereo-pairs of ray vector fields. The mean transferences are also important in understanding the behaviour of light through each of the corneas concerned. This paper provides the first order optical characters of corneas from the positions and inclinations of rays entering and leaving such systems. As anticipated, light rays through the cornea are deflected inwards when the refractive index of the cornea is greater than the index of the surrounding medium. The exp-mean-log transference for a specific cornea exists and is the optical transference of the averaged cornea of the sample of measurements for that cornea. Within the limitations of linear or paraxial optics, the corneas of the different eyes in this sample and their averages were found to be close to that of thin optical systems; but they were not truly thin and instead should be considered as being thick optical systems. (S Afr Optom 2011 70(4) 156-167)

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The metric geometric mean transference and the problem of the average eye

African Vision and Eye Health ◽

10.4102/aveh.v67i2.183 ◽

2008 ◽

Vol 67 (2) ◽

Author(s):

W. F. Harris

Keyword(s):

Geometric Mean ◽

Optical Systems ◽

Symmetric Matrices ◽

Refractive Errors ◽

Arithmetic Average ◽

First Order ◽

Optical Character ◽

All Optical ◽

The Mean ◽

Definition Of

An average refractive error is readily obtained as an arithmetic average of refractive errors. But how does one characterize the first-order optical character of an average eye? Solutions have been offered including via the exponential-mean-log transference. The exponential-mean-log transference ap-pears to work well in practice but there is the niggling problem that the method does not work with all optical systems. Ideally one would like to be able to calculate an average for eyes in exactly the same way for all optical systems. This paper examines the potential of a relatively newly described mean, the metric geometric mean of positive definite (and, therefore, symmetric) matrices. We extend the definition of the metric geometric mean to matrices that are not symmetric and then apply it to ray transferences of optical systems. The metric geometric mean of two transferences is shown to satisfy the requirement that symplecticity be pre-served. Numerical examples show that the mean seems to give a reasonable average for two eyes. Unfortunately, however, what seem reasonable generalizations to the mean of more than two eyes turn out not to be satisfactory in general. These generalizations do work well for thin systems. One concludes that, unless other generalizations can be found, the metric geometric mean suffers from more disadvantages than the exponential-mean-logarithm and has no advantages over it.

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Quantitative analysis in Hamiltonian space of the transformed ray transferences of a cornea

African Vision and Eye Health ◽

10.4102/aveh.v66i2.226 ◽

2007 ◽

Vol 66 (2) ◽

Author(s):

S. D. Marhebula

Keyword(s):

Quantitative Analysis ◽

Linear Space ◽

Optical Systems ◽

Single Subject ◽

Linear Optics ◽

A Value ◽

Optical Character ◽

The Matrix ◽

The Mean ◽

The Right

The primary purpose of this paper is to illustrate the quantitative analysis of the linear-optical character of a cornea using transformed ray transferences in a 10-dimensional Hamiltonian linear space. A Pentacam was utilized to obtain 43 successive measurements of the powers of the anterior and posterior corneal surfaces and the central corneal thicknesses of the right eye of a single subject. From these measurements 44 × ray transferences were calculated (and principal matrix logarithms for all the transferences were determined). This produced a set of 43 transformed transferences for the cornea which represent 43 points in a 10- dimensional Hamiltonian space. A 10-component mean and 10 10 × variance-covariance matrix were calculated from the transformed transferences. The mean, and variances and covariances represent, in the 10-space, the average and the spread respectively of the measurements characterising the optical nature of the cornea. The matrix exponential of the mean gives a value for the mean transference of the cornea; it represents the average cornea. The analysis described here can be applied to most optical systems including whole eyes and is complete within linear optics. We believe it to be the first such analysis.

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Inner-product spaces for quantitative analysis of eyes and other optical systems

African Vision and Eye Health ◽

10.4102/aveh.v75i1.348 ◽

2016 ◽

Vol 75 (1) ◽

Author(s):

William F. Harris ◽

Tanya Evans ◽

Radboud D. Van Gool

Keyword(s):

Quantitative Analysis ◽

Dimensional Space ◽

Three Dimensional ◽

Optical Systems ◽

Inner Product ◽

Product Spaces ◽

Inner Product Spaces ◽

Optical Character ◽

Dioptric Power ◽

Quantitative Analyses

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.

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N.M.R. studies of ligand exchange on acetylacetonatotrimethylplatinum(IV)

Australian Journal of Chemistry ◽

10.1071/ch9750759 ◽

1975 ◽

Vol 28 (4) ◽

pp. 759 ◽

Cited By ~ 9

Author(s):

NS Ham ◽

JR Hall ◽

GA Swile

Keyword(s):

Activation Energy ◽

Quantitative Analysis ◽

Ligand Exchange ◽

Variable Temperature ◽

Order Reaction ◽

First Order Reaction ◽

First Order ◽

Cdcl3 Solution ◽

Made In

A quantitative analysis of the variable-temperature 1H N.M.R. spectra of acetylacetonatotrimethyl-platinum(IV) has been made. In CDCl3 solution the exchange of acetylacetonate ligands is a first-order reaction and proceeds predominantly by dissociation of the dimer into two separated five-coordinate activated complexes. The activation energy is 61.5 � 0.8 kJ mol-1.

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