Corneal topography and the hirschberg test

A simple trigonometric analysis of the Hirschberg test with the assumption that the corneal surface is spherical predicts a sinusoidal dependence of the corneal reflex displacement on the angle of ocular rotation. A comparison with corneal reflex photographs demonstrates that at angles larger than 50 prism diopters (26 deg) the reflex displacements are larger than predicted by the spherical model. This discrepancy may be accounted for by incorporating a more general description of the corneal topography into the geometric analysis. The linear Hirschberg relation that is seen in typical data is accounted for by a relative flattening of the peripheral cornea by ~ 20% of the apical curvature. This geometric analysis of the functional dependence of the Hirschberg relation on the corneal topography can be expressed as an integral equation. Differentiation yields a second-order differential equation for the corneal topography in terms of the Hirschberg data. If the Hirschberg relation is assumed to be linear, a quadratic dependence is found for the corneal curvature. A similar differential approach can be formulated for the Placido disk. In this sense the corneal topography problem given in terms of Placido disk data is shown to be wellformulated. The relative simplicity of the Hirschberg geometry is seen to stem from the alignment of the light source with the eye of the observer.