title: Thickness extrema at the edge of astigmatic lenses including those with straight, circular and elliptical edges.
abstract: The problem of locating the points of maximum and minimum thickness at the edge of a lens, and of calculating the thickness at those points, is examined for lens powers and for edge shapes in general. The edge extremal problem, as the problem is called, is solved explicitly for general powers along straight cut edges. The extremal problem is also analysed for lenses with circular and elliptical edges but explicit solutions are obtained only for centrally cut lenses, that is, lenses with coincident optical and geometrical centres. For edges that are neither straight nor centrally cut ellipses (including circles), it appears that explicit solutions cannot be obtained for the edge extrema: one has to resort to numerical solution of implicit equations or to the calculation of thickness at sufficiently many points around the edge. For the centrally cut ellipse the edge extremal problem turns out to be the eigenvalue problem of linear algebra. In general the thickness extrema at the edge do not lie on the principal meridians of the lens, nor do they lie on meridians that are mutually perpendicular. With minor modification the results apply equally well to the edge extrema for sagitta of a surface.
authors: Department of Optometry, Rand Afrikaans University, South Africa.
authors: W F
authors: WF
authors: Harris
mesh_terms: D001251:Astigmatism / Q000628:therapy*; D005139:Eyeglasses / Q000592:standards*; D008962:Models, Theoretical; D055095:Optics and Photonics; D009907:Optometry / Q000379:methods*