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.
"Bounds and intervals around nonzero cylinder powers in symmetric dioptric power space," Journal of Biomedical Optics 14(1), 014025 (1 January 2009). https://doi.org/10.1117/1.3079809