Thickness extrema at the edge of shaped lenses: the general problem and the solution for a straight edge obtained by means of direct elimination and of Lagrange multipliers

Abstract

In general thickness varies along the cut edge of a lens. There is some interest in being able to predict the locations of the thickest and thinnest points on the edge. One purpose of this paper was to formulate the general problem mathematically of finding the locations of thickness extrema on the edge of an arbitrary lens cut into an arbitrary shape. A second purpose was to illustrate how the problem can be solved. In particular, the problem is solved completely and explicitly for what is probably the simplest case, the straight cut edge. A component-free matrix expression for the position of the extremum is derived by employing Lagrange multipliers and the concept of the generalized inverse of a matrix. The equation applies to spheres, cylinders and sphero-cylinders and allows for the presence of prism as well. Along some edges the thickness is constant. Along other edges the thickness varies linearly; there are no extrema except for the practical extrema at the two ends of the edge: thickest at the one and thinnest at the other. The mathematical conditions for these two cases are presented. Equivalent to the matrix equation for the position vector of the thickness extremum is a pair of scalar equations expressed in terms of components of the matrices. The pair is useful when locating extrema using manual calculations. On the other hand, the component-free matrix equation is useful in other circumstances such as for mathematical manipulations and when using computer software that handles matrices. The former is used in a number of numerical examples; the latter was used to check the answers by computer. The mathematical techniques described here are likely to find application in other areas of ophthalmic and visual optics.