Differential geometric approach to atmospheric refraction

Differential geometric techniques are presented and used to model the optical properties of the atmosphere under conditions that produce superior mirages. Optical path length replaces the usual Euclidean metric as a distance-measuring function and is used to construct a surface on which the paths of light rays are geodesics. The geodesic equations are shown to be equivalent to the ray equation in the plane. A differential equation that relates the Gaussian curvature of the surface and the refractive index of the atmosphere is derived. This equation is solved for the cases in which the curvature vanishes or is constant. Illustrative examples based on observation demonstrate the use of geometric techniques in the analysis of mirage images.