A new analytical approximation of corneal topography

Abstract

It is shown that a rotationally symmetric surface, which is the border between two optical media, can focus a parallel light beam to a point only when the profile of the surface is an ellipse or a hyperbola with eccentricity equal to the ratio of refractive indices of both media. In fact, as is well known, the conic sections do not describe the corneal profile well, so a new analytical approximation for the corneal contour is given. This simple approximation in the case of a rotationally symmetric surface is based on the hyperbolic cosine function with two parameters, the radius of curvature at the apex of the cornea and a parameter describing the stability of the central curvature. The hyperbolic cosine function can also be used to approximate a 3-D non-rotational surface in the case of a cornea with axial astigmatism. The 3-D contour map of deviation of the hyperbolic cosine approximation from a reference sphere is compared with an interferogram of a cornea in vivo, recorded in a Twyman-Green interferometer.