Calculus: Progressive Lens Design Calculation

1. Introduction

Calculus is a mathematical analysis of some essential patents, which mark the progress in progressive lens design from the first commercial PPL, Varilux 1, until Varilux Comfort, the most sold progressive lens brand ever. The choice of the patents after the Varilux invention is subjective in the sens, that I selected the patents of products where I had occasion to participate in the development. There are other outstanding concepts by Carl Zeiss, Sola, Hoya or from other manufacturers which characterize this era. All the calculations use only information from the patents or other published material, accessible to any person interested in this topic. Even if all these patents can be linked to a commercial product, we have to be cautious, when comparing the calculated design with the features of the market product. The patent claims cover for evident reasons a broad variety of geometries and parameters and the given examples do not necessarily represent the product chosen for the market launch.

Some chapters of Calculus are dedicated to specific aspects of the progressive surface concept, as the chapters concerning the Minkwitz theorem or Orthoscopy. The first and principal goal writing this scientific/technical history is documenting the evolution of research and technical work in progressive surface design at the end of last century. So it will be of interest to the people, who have worked in the spectacle lens business during this period and generally to any technically minded person looking for information about the history of progressive surface design. The paper will also be useful for the student of optical engineering or physiological optics to understand the basics of progressive lens computation. Calculus depicts the history of surface design between the 1950's and 1990's in successive chapters, each dedicated to a particular product or a specific characteristic. Each chapter is available as individual pdf document and subdivided into paragraphs. This structure of chapters and paragraphs is listed in the menu on the left.

The computations have been executed with the help of rather simple tools as the commercial sofware Mathcad and Matlab installed on a laptop. The computer programs written for the calculations are either integrated in the general text (Mathcad) or given in a separate paragraph (Matlab). Mathcad was used for the computation of surfaces defined by analytical functions. For the construction of more sophisticated designs, as Varilux Comfort, the surface representation required tensor product splines and it was advantageous to have recourse to the Matlab language.

In order to simplify the method, some calculations are based on approximation models. Their use is discussed in the respective context. There are two short chapters treating definitions and notions of differential geometry and ophthalmic optics / progressive lenses. Nevertheless we presuppose the basic scientific knowledge in these domains of mathematics and spectacle lens optics. In order to understand the calculations completely, you need the patents referenced in the different chapters. You will find them, without problem, on the Internet with the help of Google Patent Search.

Acknowledgement
One problem for the author was the fact, that he is no expert in computer programming and calculations. So particularly for the last chapter, the surface optimization applying tensor product B-splines, help was very welcome. My special thanks go to Professor Günther Greiner and Matthias Innmann , M. Sc. (computer science), Friedrich-Alexander-University , Erlangen, for writing the program for the minimization of the merit function.

A heartfelt "thank you" to my wife Bärbel for her patience and understanding accompanying this work, that lasted longer than I thought.