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

S. D. Marhebula
African Vision and Eye Health | South African Optometrist: Vol 66, No 2 | a226 | DOI: | © 2007 S. D. Marhebula | This work is licensed under CC Attribution 4.0
Submitted: 19 December 2007 | Published: 19 December 2007

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S. D. Marhebula, Department of Optometry, University of Limpopo, South Africa

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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.


dioptric power; Hamiltonian matrices; Hamiltonian space; linear optics; ray transference; symplecticity; Scheimpflug photography; Pentacam


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