Generalized sine condition

The classic Abbe sine condition relates pupil distortion to aberrations with linear field dependence such as coma. This paper provides a fully generalized form of the sine condition that does not use any symmetry. It accurately predicts the change in aberration in the presence of field independent and pupil aberrations. The definitions of the image, object, and coordinate system are completely arbitrary. The relationship is verified using ray trace simulations of a number of systems that have varying degrees of complexity. The potential applications are discussed.