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Jackson Cross Cylinder is a lens of fundamental importance inoptometry with a key role in the refraction routine. And yet it appears not tobe as well understood as perhaps it should be. The purpose of this paper is toexamine the linear optical character of the Jackson cross-cylinder and, inparticular, those properties associated with the operations performed on thelens in the refraction routine, namely flipping and turning. Corresponding tothese operations in physical space are steps in an abstract space, symmetricdioptric power space. The powers of all Jackson cross-cylinders lie in theplane of anastigmatic powers in the abstract space. In particular the powers ofan F Jackson cross-cylinder (for example, a 0.5-D Jackson cross-cylinder has F= 0.5 D) lie on a circle of radius F centered on null power. Flipping the lenstakes one diametrically across the circle; turning the lens takes one aroundthe circle at twice the rate. A subsequent paper shows how these operationswork in defining the cylinder in the refraction routine.
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