Purpose: The purpose of this article is to investigate the invariance and isomorphism of error cells with cylinder power measured positively and negatively in the cylinder-sphere plane. Cells with cylinder power measured positively and negatively are mapped to symmetric power space to see whether isomorphism and the invariance are conserved under the mapping and spherocylindrical transposition.
Method: Principal powers of refraction are measured and are rounded off to multiples of 0.25 D and 1 degrees or 5 degrees , say, for the principal meridians. Geometric regions in the cylinder-sphere plane with positive and negative cylinder represent uncertainty surrounding the powers and are respectively transformed to a plane containing the axis of scalar powers in symmetric power space.
Results: Cells from principal powers become error cells surrounding readings of cylinder and sphere as well as error regions about astigmatic powers. These are presented in the plane for scalar powers and semipowers of positive and negative cylinders. In symmetric power space, planes containing the axis of scalar powers are distinguished from one another by the cylinder axis of the lens power.
Conclusion: 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.