Admissible surfaces in progressive addition lenses

TL;DR

This paper derives exact compatibility equations for progressive addition lenses, revealing new dependencies of cylinder on geodesic curvature, and extends the Minkwitz theorem for these surfaces.

Contribution

It introduces complete compatibility equations for PAL surfaces, showing the influence of geodesic curvature on optical properties, and extends the Minkwitz theorem.

Findings

Cylinder depends on geodesic curvature and its derivatives.

Numerical computations quantify geodesic curvature's relevance.

Extended Minkwitz theorem derived for PAL surfaces.

Abstract

Progressive addition lenses contain a surface of spatially-varying curvature, which provides variable optical power for different viewing areas over the lens. We derive complete compatibility equations that provide the exact magnitude of cylinder along lines of curvatures on any arbitrary PAL smooth surface. These equations reveal that, contrary to current knowledge, cylinder, and its derivative, does not only depend on principal curvature and its derivatives along the principal line but also on the geodesic curvature and its derivatives along the line orthogonal to the principal line. We quantify the relevance of the geodesic curvature through numerical computations. We also derive an extended and exact Minkwitz theorem only restricted to be applied along lines of curvatures, but excluding umbilical points.