A procedure is described and discussed, by which the requirements imposed on an optical system can be formulated in terms of the unknown surfaces, so that conditional (differential) equations can be written as a basis for the design. This is made possible through a projective transformation by which any ray is represented as a point of a mapping plane, referred by duality to the physical plane of the optical trajectory.
In the first part of the paper, after a general statement of the problem, a summary is given of the principal results of projective geometry that are utilized in the discussion; next, the differential equation of a refracting or reflecting surface is deduced and discussed in dual terms.
The second part of the paper will concern the explicit formulation of a design and the discussion of the principal aberrations, by effecting the ray tracing in the dual plane.
ENZO CAMBI, "Projective Formulation of the Problems of Geometrical Optics. I. Theoretical Foundations," J. Opt. Soc. Am. 49, 2-11 (1959)