Some form of the edge-ray principle is used to design most nonimaging systems. (Only systems based on geometrical optics are considered.) For proving certain statements of this principle, the optical system is considered to be surrounded by an enclosure that any ray emerging from the system must intersect. A phase space, of topology Sphere × Disk, corresponding to the intersection of rays with the enclosure is introduced, and the system gives rise to a mapping f among points of this space. The proofs hold only if f is continuous, which is not the case for all real systems. Discontinuities in f may be caused by (1) tangential incidence of a ray with a surface, (2) incidence where the radius of curvature of a surface is zero, (3) transition from refraction to total internal reflection, and (4) intersection of different types of optical surface. Although the condition of the continuity of f is used in the proofs, even among systems in which continuity is absent it is difficult in practice to find counterexamples to the edge-ray principle formulated.
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