Third-order smoothness metric to characterize progressive addition lenses

Although second-order surface analyses, mainly mean power and cylinder maps, are commonly used to characterize the progressive addition lens (PAL) surface, recently it has been suggested that third-order variations may also have relevancy in PAL optical and visual performance. This paper proposes a third-order smoothness metric, and its associated Riemannian distance, to further characterize PAL's surface optical performance. These metrics can provide a complementary scoring tool to those classical ones, particularly, to analyze the transition zones between far, near, intermediate, and blending zones. A method to compute these metrics is provided. This third-order smoothness metric also enables a formal definition of the PAL principal curve, namely, the curve embedded in the PAL surface, that minimizes the line path integral joining the far and near reference points weighted by the third-order smoothness metric. Finally, the paper describes a comprehensive methodology to compute such principal curves using a level-set geodesic procedure. All these ideas are put into practice with several real PAL surfaces.