# Foreword "Calculus" by Bernard Maitenaz

The following website Calculus, prepared by Werner Köppen, illustrates the importance of mathematics for the research targeting improvement in the visual comfort of prespyobic people corrected with progressive lenses…..

Prior to this it may be of interest to recall the first research work (1951) leading to the Varilux lens, launched on the market in 1959, and to specify what was the mathematical starting point ………..

After a first analysis I was convinced, that the realization of a progressive lens implies the solution of two different problems: the definition of an adequate surface and the practical tools for mass production. As, at that time, the basic lens material was mineral glass, difficult to grind and to polish, it was evident to start with the second problem, the practical realization of such a surface, naturally complex.

So the very first approach was “mechanistic”: surface generation by a diamond grinding wheel executing a plane movement controlled by mechanical cams. In the beginning the grinding wheels were of the “cup type” and “conical” afterwards. The progressive lenses obtained met precisely the specifications along the “umbilical”meridian, but in the lateral parts the aberrations were too strong. So the control of the lateral zones required a different process, enabling modification of their structure….The point-by-point process (PpP 1956) based on the polar coordinate system (with reference to the center of the basic sphere) was adopted as solution, because it allowed to quantify easily, but with high precision, the deviations from the basic sphere and their local optical impact:

- The umbilical meridian, as well as its vicinity, could easily be expressed by discrepancies in relation to the basic sphere.
- As regards the lateral zones it became easy to modify them, to shape them like a sculptor. For example with the help of graphs representing the discrepancies in relation to the basic sphere along judiciously chosen sections. Starting from the selected discrepancies, the general structure of the surface could be calculated as well as the map of the local optical performance and by this the initial aberrations could be reduced ……..
- The practical realization was simple. But as it was very time consuming (4000 points to calculate and to grind) the industrialization had to recourse to punching cards for the calculations and three-dimensional copying machines for the surface generation.

The surfaces obtained comprised initially (1959) a spherical far vision zone in order to be “comparable” to bifocal lenses. In a second step (1972), the quality of the lateral surfaces manufactured according to the PpP method, made possible adoption of totally aspherical surfaces with a distinctly higher visual comfort.

For the editing of the corresponding patents theses surfaces were described by equations of the intersection of the surface with a plane rotating around a horizontal axis passing through the center of the polar coordinate system and by the evolution of the geometric parameters. These sections were conics with their geometrical parameters evolving continuously along the umbilical curve (in fact, after a series of less and less flattened ellipses, a circle and then parabolas and hyperbolas). In this manner the first equation of the total Varilux 2 surface was established.

This first global equation of the surface marks the beginning of the “era of mathematics” for the progressive lens design, an era which is described by Werner Köppen in the following chapter. In it the application of the progress of the mathematical discipline is addressed. It also shows how the wide field of research to improve the progressive lens has been the object of various and complementary scientific projects, and the benefit of the results obtained….. It finally shows to what extent mathematics were an essential tool for the evolution of the progressive lens.

In order to illustrate the difficulties, the complexity but also the precision of these research works, we have only to specify that superposing the Varilux 2 surface (1972) to that of a recent progressive with the same characteristics, Varilux Physio or a competitor, the discrepancy between the two surfaces, in the useful vision zone, is in the order of magnitude of one tenth of a millimeter ….and yet in spite of that the visual comfort is beyond compare!

Bernard Maitenaz

Paris, January 2015