Paraxial Ocular Measurements and Entries in Spectral and Modal Matrices: Analogy and Application

Abstract

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.

Tolerance and Nature of Residual Refraction in Symmetric Power Space as Principal Lens Powers and Meridians Change

“…In Figure 1(a), nonzero powers λ , μ are scalars formatted as eigenvalue entries [6] in the matrix [7] $\begin{array}{c}\mathrm{\Lambda }=\left(\begin{array}{cc}\lambda & \mathrm{normal0}\\ \mathrm{normal0}& \mu \end{array}\right)=\frac{\lambda +\mu }{\mathrm{2}}\mathbf{I}+\frac{\lambda -\mu }{\mathrm{2}}\mathbf{J}\end{array}$ and, along lines at polar angles α and α + 90° in Figure 1(b), corresponding meridians are formatted as vectors and column entries in the matrix $\begin{array}{c}\mathbf{Q}=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right)=\cos \alpha \mathbf{I}+\sin \alpha \mathbf{L}\end{array}$ of eigenvectors [6] of matrix A whose structure follows in (6). …”

“…As matrices of eigenvalues and eigenvectors, these matrices are multiplied [6] (not scalar products) $\begin{array}{c}\mathbf{Q}\mathrm{\Lambda }{\mathbf{Q}}^{-\mathrm{1}}=\left(\cos \alpha \mathbf{I}+\sin \alpha \mathbf{L}\right)\\ \times \left(\frac{\lambda +\mu }{\mathrm{2}}\mathbf{I}+\frac{\lambda -\mu }{\mathrm{2}}\mathbf{J}\right){\left(\cos \alpha \mathbf{I}+\sin \alpha \mathbf{L}\right)}^{-\mathrm{1}}\end{array}$ to give the symmetric (independent of L ) dioptric power matrix $\begin{array}{c}\mathbf{A}=\frac{\mathrm{1}}{2}(\lambda +\mu )\mathbf{I}+\frac{\mathrm{1}}{2}(\lambda -\mu )\cos \mathrm{2}\alpha \mathbf{J}+\frac{\mathrm{1}}{2}(\lambda -\mu )\sin \mathrm{2}\alpha \mathbf{K}\end{array}$ as a linear combination. The units of A and those of the coefficients I , J , and K are dioptres.…”

“…A practitioner adjusts this (spherical) power as the cylinder power goes from P to S in Figure 3(b) in order to keep the position of the circle of least confusion invariant in the eye. The symmetric residual power matrix G 1 has the eigenfactors [6] $\begin{array}{c}\mathrm{\Lambda }=\left(\begin{array}{cc}{\lambda }_P-{\lambda }_S& \mathrm{normal0}\\ \mathrm{normal0}& {\mu }_P-{\mu }_S\end{array}\right),\\ {\mathbf{Q}}_P=\left(\begin{array}{cc}\cos {\alpha }_P& \cos \left({\mathrm{normal90}}^{°}+{\alpha }_P\right)\\ \sin {\alpha }_P& \sin \left({\mathrm{normal90}}^{°}+{\alpha }_P\right)\end{array}\right)\end{array}$ that, as in Figure 1, can be represented on a power cross where the principal powers are λ P − λ S along α P and μ P − μ S along 90° + α P . The strength or magnitude of the residual power G 1 is obtained from the eigenfactor Λ of G 1 and (8): $\begin{array}{c}‖{\mathbf{G}}_{\mathrm{1}}‖=\frac{\sqrt{\mathrm{normal2}}}{\mathrm{2}}\sqrt{{\left({\lambda }_P-{\lambda }_S\right)}^{\mathrm{normal2}}+{\left({\mu }_P-{\mu }_S\right)}^{\mathrm{normal2}}}.\end{array}$ This square root of the scalar product 〈 G 1 , G 1 〉 [8] is independent of the meridians in Q P .…”

“…Distinct lens principal powers λ and μ in (a) define eigenvalues of matrix A in (6) in columns of $\mathrm{normal\Lambda }=\left(\begin{array}{cc}\lambda & \mathrm{0}\\ \mathrm{0}& \mu \end{array}\right)$. These powers are along meridians at α and α + 90° in (b) that define eigenvectors of matrix A in (6) in the corresponding columns of $\mathrm{boldQ}=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right)$ [6]. …”

Components of Lens Power That Regulate Surface Principal Powers and Relative Meridians Independently

“…Matrices in commutators have coincident eigenvectors and coincident eigenvectors are those of commuting matrices. Surface principal powers along oblique meridians [5] or perpendicular meridians that are not aligned are reasons for lenses to have antisymmetric dioptric power matrices. Symmetry of matrices serves as a frame of reference and leads to knowledge about the problem that can be identified with the eigenvectors often measured by instruments in the consulting area.…”