Isomorphism and Possible Invariance of Error Cells Under Spherocylindrical Transposition

Abstract

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.

Mapping of error cells in clinical measure to symmetric power space

“…We answer the question: what are the effects on the dioptric power matrix, its elements and characteristics of a symmetric region of uncertainty surrounding powers in sphere, cylinder and axis? The region of uncertainty is mapped algebraically, point for point to error cells about powers in symmetric dioptric power space, confirming that cells are fully compatible with error cells about principal powers and error cells about powers in any plane containing the axis of scalar powers (Abelman and Abelman, 2007). The axis component will allow the principal meridians to be considered as entries of the modal matrices (Abelman, 2006).…”

“…The first columns of the matrices are the position vectors of the centres E + and E − respectively, and corners E 2 to E 4 and E 8 of the error cells with respect to O. Matrix 1 in positive cylinder form is converted by transposition to matrix 2. Cells for positive cylinders have the same shape as cells for negative cylinders (Abelman and Abelman, 2007), but are separated from one another for significant cylinder magnitude and 90° apart on the graph axis A of . The error cell for negative cylinders interchanges points E 2 and E 3 and points E 6 and E 7 with the error cell for positive cylinders. …”

“…In negative cylinder form, the position vector of the centre is transposed to the point E − at [ u 0 ± 90°, − v 0 , v 0 + w 0 ]′. An actual cylinder power v 0 is a value somewhere between v 0 − g and v 0 + g , an actual sphere power is in the range from w 0 − g /2 to w 0 + g /2 (Abelman and Abelman, 2007), and an actual axis is a value somewhere between u 0 − f and u 0 + f , where g > 0 is the error in cylinder power and f > 0 is the error in cylinder angle. Error cell parallelepipeds project lengths 2 f parallel to axis A , 2 g parallel to axis F C and g parallel to axis F S ; all projected lengths are symmetrically placed about the centres E + and E − of the cells.…”

“…Viewed along the graph axis of cylinder axes, error cells about sphere and cylinder powers and axis are isomorphous. Cells are, however, not invariant under spherocylindrical transposition (Abelman and Abelman, 2007). Bounds of sphere, cylinder and axis ultimately shape the level of accuracy or confidence in any conclusions that are reached.…”

“…Cylinder and sphere lens powers have been shown to be compensatory when they are included in error cells with a particular parallelogram shape in the plane defined by cylinder and sphere power (Abelman and Abelman, 2007). Respective extreme powers of cells in clinical measure were converted to the extremes of cells in scientific measure.…”