INSTITUTO DE OPTICA. CSIC

Progressive lenses have gained worldwide acceptance as an optimal solution for presbyopia compensation in the presence of other ametropia (myopia or hyperopia), due to the possibility of focusing the view at different distances.

Progressive Addition Lenses (PALs) contain a variable curvature surface, which provides variable optical power for different areas of vision over the lens.
Full compatibility equations are derived in the paper that provide the exact magnitude of a cylinder along the curvature lines on any arbitrary smooth PAL surface. These equations reveal that, contrary to current knowledge, the cylinder and its derivative not only depend on the main curvature and its derivatives along the main line, but also on the geodesic curvature and its derivatives along the line orthogonal to the main line.
In this paper the relevance of geodesic curvature is quantified through numerical calculations. It also derives an extended and exact Minkwitz theorem, restricted only to be applied along the curvature lines, but excluding the umbilical points.

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The work is a collaboration between the Institute of Optics and the Department of Mathematics and the ICMAT of the Universidad Autónoma de Madrid.