The concept of geodesic curvature applied to optical surfaces

Abstract

Purpose To propose geodesic curvature as a metric to characterise how an optical surface locally differs from axial symmetry. To derive equations to evaluate geodesic curvatures of arbitrary surfaces expressed in polar coordinates. Methods The concept of geodesic curvature is explained in detail as compared to other curvature‐based metrics. Starting with the formula representing a surface as function of polar coordinates, an equation for the geodesic curvature is obtained depending only on first and second radial and first order angular derivatives of the surface function. The potential of the geodesic curvature is illustrated using different surface tests. Results Geodesic curvature reveals local axial asymmetries more sharply than other types of curvatures such as normal curvatures. Conclusion Geodesic curvature maps could be used to characterise local axial asymmetries for relevant optometry applications such as corneal topography anomalies (keratoconus) or ophthalmic lens metrology.

Admissible surfaces in progressive addition lenses

“…The dependence with curvature properties along both lines of curvatures is an expected result considering that the cylinder is a function of the two principal curvatures, at a specific point of the surface. Particularly, choosing the principal line to be one of the lines of curvatures we can extract a very important property; contrary to what is predicted by Alonso version of Minkwitz theorem [13], (7) reveals that there is one, and only one, extra independent variable for controlling cylinder besides the ratio of change in the principal curvature along the principal line: the geodesic curvature.…”

“…This property is not meaningless from a practical point of view, since the geodesic curvature is a measure of how the curve within the surface twists out locally [13], which is related to the manufacturing process of that surface. Given a ratio of change of one the principal curvatures, the only way to minimize cylinder is at the expense of twisting the surface locally, which makes manufacturing more complex.…”

Optical methods for measuring corneal topography: A review

“…Some of the algorithms used to obtain this type of maps assume that the normal at each point on the corneal surface lies in the meridional plane, so that not necessarily a point of interest will lie in this plane. A recent study proposes the use of geodesic curvature maps [28], which is a measure of how far a part of a specific curve is from the beginning of a curvature line and its corresponding geodesic line. These maps can be used for the local axial characterization of asymmetries in corneal topography offering a better sensitivity than the radial curvature to locate axial asymmetries.…”