General or local variations of the refractive elements in the eye, called irregular astigmatism, may manifest as non-orthogonal meridians when each principal meridian on the front surface of a cornea is independently aligned with the focussed mires of a keratometer. These are examples of astigmatic systems that are thick. The representation of power along the principal meridians is not suitable for quantitative work. The purpose of this research note is to convert power along principal meridians that can be non-orthogonal, to the coordinates of the power matrix that are suitable for quantitative analysis.
Lensometers and keratometers yield powers along perpendicular meridians even if the principal meridians of the lens and the cornea are oblique. From each such instrument, multiple raw data represented on optical crosses require conversion to determine elementary statistics. Calculations for research decisions need to be authentic. Principles common to meridians generalize formulaic methods for oblique meridians. Like a lens or a cornea, matrix latent quantities are represented on a matrix cross. Our problem is to determine the matrix whose cross represents quantities on the optical cross. All measurements on an optical cross that include corneal and lens powers and oblique meridians can be considered. Once determined, a portfolio of matrix calculations applies and is justified for ophthalmic calculation. Matrices can be unique and, like a cornea before it is measured, contain latent observations. Asymmetric power component matrices quantify a deviation of a corneal surface from smoothness and toricity. Entries may identify those measurements causing irregular astigmatism that may stem from surgical or other external intervention. Irregular astigmatism is detected primarily from significant measurements in the paraxial range. Measurements are assimilated with matrix factors in a holistic way in order to support choices with calculations and statistics.
Although error cells in clinical measure are not invariant under spherocylindrical transposition, cells represented in positive cylinder form have the same shape as cells for which the power is expressed in negative cylinder form. Error cells in symmetric power space about powers in negative cylinders can be rotated about the axis of scalar powers to coincide perfectly with cells about powers in positive cylinder form for near spherical and astigmatic powers. The error regions in symmetric power space do not depend on the spherocylindrical form in which the original measurements are made and their isomorphism is conserved in the mapping.
During the refraction procedure, the power of the nearest equivalent sphere lens, known as the scalar power, is conserved within upper and lower bounds in the sphere (and cylinder) lens powers. Bounds are brought closer together while keeping the circle of least confusion on the retina. The sphere and cylinder powers and changes in these powers are thus dependent. Changes are depicted in the cylinder-sphere plane by error cells with one pair of parallel sides of negative gradient and the other pair aligned with the graph axis of cylinder power. Scalar power constitutes a vector space, is a meaningful ophthalmic quantity and is represented by the semi-trace of the dioptric power matrix. The purpose of this article is to map to error cells for the following: coordinates of the dioptric power matrix, its principal powers and meridians and its entries from error cells surrounding powers in sphere, cylinder and axis. Error cells in clinical measure for conserved scalar power now contain more compensatory lens powers. Such cells and their respective mappings in terms of most scientific and alternate clinical quantities now image consistently not only to the cells from where they originate but also to each other.
A single approximate matrix expression is derived that gives the thickness at any point on any lens. The lens may have dioptric power or prism power or both. The expression gives, amongst other things, the equation for lines of constant thickness.
We seek to analyze the geometry and explain how bounds and intervals of nonzero purely cylindrical powers are obtained and applied in symmetric dioptric power space and envisaged in the clinic. The principal powers at zero and at the focus at the cylinder power of a lens are subject to the same uncertainty when measured. Accompanying these uncertainties is an error in axis position. Error cells are constructed for typical cylinder axes and an associated power. The geometry contains an elegant clinical determination for cross-cylinder compensation of astigmatism in terms of calculation friendly quantities. The extreme positions in the error cells define bounds for the cross-cylinder powers and their meridians. When clinical powers in a chosen error cell are transposed, the new powers are within a different cell. This ambiguous cell pair maps to a single cell in an antistigmatic plane around cross-cylinder powers.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.