Mapping of error cells in clinical measure to symmetric power space

Abstract

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.

Bounds and intervals around nonzero cylinder powers in symmetric dioptric power space

“…It is interesting to note the number of cells halved after mapping as a result of invariance of cells under spherocylindrical transposition. 1,6 Numerical examples illustrate how to determine upper and lower bounds. Bounds and intervals of zero cylinder powers at all axis angles complement bounds and intervals about nonzero cylinder powers in an antistigmatic plane.…”

“…Further, 6 symmetric intervals estimated for powers in sphere, cylinder, and axis lead to parallelepipedal error cells. Every power on or in the cell is transformed into a typical power on or in cells in symmetric power space.…”

“…Previous work 6 shows that no common independent variables exist among proper values and vectors of the power matrices for spherocylindrical powers, so that proper values and vectors of the dioptric power matrix are functionally in- dependent. Thus an interval of cylinder powers does not determine an interval of corresponding cylinder axes or vice versa.…”

“…The best way to represent axes and corresponding powers of the cylinder lens is in the plane with a rectangular set of axes. 6 Any hypothetical cylindrical component within the interval of the rounded power has a corresponding hypothetical axis located anywhere within the axis interval. Intervals of axis angles and corresponding cylinder power intervals combine to form error cells around each reported cylinder power.…”