Generalized ray tracing equations are derived to include special cases in optics such as uncentered spherical interfaces, prisms, mirrors, and centered aspheric surfaces. These equations were constructed for use with the IBM Card Programmed Electronic Calculator. The method has been applied to complicated optical systems involving uncentered lenses in addition to reflecting and refracting prisms. A method is given for the design of an aspheric surface, and a design curve of an aspheric optical system is given as an example.
WILLIAM A. ALLEN and JOHN R. SNYDER, "Ray Tracing through Uncentered and Aspheric Surfaces," J. Opt. Soc. Am. 42, 243-249 (1952)