The Jackson Cross-Cylinder. Part 1: Properties

Abstract

The Jackson cross-cylinder is a lens of fundamental importance in optometry with a key role in the refraction routine.  And yet it appears not to be as well understood as perhaps it should be.  The purpose of this paper is to examine the linear optical character of the Jackson cross-cylinder and, in particular, those properties associated with the operations performed on the lens in the refraction routine, namely flipping and turning.  Corresponding to these operations in physical space are steps in an abstract space, symmetric dioptric power space.  The powers of all Jackson cross-cylinders lie in the plane of antistigmatic powers in the abstract space.  In particular the powers of an F Jackson cross-cylinder (for example, a 0.5-D Jackson cross-cylinder has  5 . 0 = F  D) lie on a circle of radius F centred onnull power.  Flipping the lens takes one diametrically across the circle; turning the lens takes one around the circle at twice the rate.  A subsequent paper shows how these operations work in defining the cylinder in the refraction routine.