Approximating ocular surfaces by generalised conic curves

Abstract

Most of the optical models of the human eye use simple conic functions to represent its individual components such as corneal surfaces and the surfaces of the crystalline lens. Although a conic function provides an acceptable approximation for most anatomical eye surfaces, it also leads to a simple optical analysis of the whole eye system. To fill the gap between the classical use of conic surfaces and the use of more sophisticated functions that often invoke numerically expensive procedures in the optical analysis, a functional generalisation of the conic curve is proposed. A detailed derivation of the generalised conic function is presented for a two-dimensional (meridional) case. This is followed by a three-dimensional surface approximation. Examples are given in which the superiority of the proposed approximation over a classical conic function as well as the hyperbolic cosine approximation is evident. In particular, it is shown that for an average total corneal profile, the proposed generalisation results in a residual height error that is of an order smaller than those achieved with the conic and hyperbolic cosine approximations. In conclusion, the proposed generalised conic function can be a useful tool in eye modelling, where the simplicity of expression is often desirable.

Shape analysis of a parametric human lens model based on geometrical constraints

“…Vice versa, more complex mathematical models such as the augmented conic equation [16], figuring conicoid [29], generalized conic [34], conic patch [5,35], polynomial [10], Fourier series [11], modulated hyperbolic cosine [30] and other curves which involve polynomial and trigonometric functions [31], where the coefficients involved are typically derived by fitting techniques, give a continuous junction between the anterior and the posterior surface, but do not provide a geometrical or optical interpretation of the coefficients of the model.…”

Zernike vs. Bessel circular functions in visual optics

“…Many alternative functional representations have been proposed to describe ophthalmic surfaces, ranging from generalised conic functions to more complicated representations such as the fractional Zernike polynomials and spherical harmonics . Other techniques include combinations of modal and zonal approaches .…”

Some Properties of the Fractional Circle Zernike Polynomials

“…There are many types of mathematical models for corneal topography. The most common and simple models are based on conic sections [5], or on generalized conic functions [11]. Currently, the Zernike polynomials are the standard functions for the description of the wave front aberrations of the human eye and have been also used in the modeling of cornea surfaces [4].…”