Axial curvature and the skew ray error in corneal topography

Many intuitions about corneal shape are based on corneas with axial symmetry. In that case, the corneal normal intersects the axis of symmetry and there is an unambiguous definition for axial curvature. If, however, the corneal normal is not in the meridional plane, at least three definitions are available. The present paper defines, analyzes, and makes quantitative comparisons among the different definitions of axial curvature. When Placido rings are used as targets, current algorithms incur an error (the skew ray error) because the axial curvature that is measured is different from the axial curvature used for corneal reconstruction.

A cylindrical coordinate system is used that greatly simplifies the visualization and mathematics of skew rays. In the process of handling the axial curvature for different shapes, we develop a number of convenient formulas for handling ellipsoids and bispheres such as occur in keratoconus and photorefractive keratectomy (PRK). We also develop a simple approach based on vector cross-products for calculating axial curvature for an arbitrarily placed axis.

The skew error is calculated for ellipsoidal and keratoconic corneas, and for corneas that have undergone decentered PRK ablation. For reasonable ellipsoidal and PRK corneas, the skew ray error is found to be negligible. However, for keratoconus and for decentered views of PRK corneas the error can be substantial. The implications of the skew ray ambiguity for instantaneous curvature are also discussed.

The formalism developed in this paper facilitates the interpretation of axial curvature maps for corneas that are not axially symmetric. The skew ray error is negligible except for a small class of distorted corneas.