title: Progressive powered lenses: the Minkwitz theorem.
abstract: OBJECTIVE
  The Minkwitz theorem, which can be proven to apply to the immediate surface surrounding a line of umbilics, states that astigmatism perpendicular to the line changes twice as quickly as the rate of change of power along the line. Our objective is to test how the Minkwitz theorem applies to the design of progressive addition lenses (PALs).
  
  
  METHODS
  Our primary investigation of the astigmatism/power rate relationship used Hoya Tact lenses because they have a relatively large central region with horizontal spherical equivalent power contours and vertical astigmatism power contours. Other PALs were used for subsequent analysis. Lenses were measured with a Rotlex Class Plus lens analyzer.
  
  
  RESULTS
  Zone widths in the central region of the Tact lenses exceeded those predicted by the Minkwitz theorem. Above and below this region, zone widths were narrower than predicted. When averaged along the entire corridor, zone widths approximated the Minkwitz theorem. For other PALs, the measured zone widths exceed Minkwitz theorem in the top (distance) and middle (intermediate) corridor but fell short in the lower (near) corridor. Likewise, on average along the entire corridor, they approximate the Minkwitz theorem.
  
  
  CONCLUSIONS
  Although the Minkwitz theorem must apply exactly to the immediate locale of an umbilic, deviations from Minkwitz can occur within 2 mm of the corridor. Several factors enable enough local deviation from the Minkwitz theorem to "steer" the astigmatism and affect its magnitude in the peripheral portions of a lens. Although the Minkwitz relationship may be altered in some regions of the corridor, there is a global component to the Minkwitz prediction that applies to PALs. On a global level, the gains and losses of astigmatism along the corridor with respect to the Minkwitz prediction have strong tendency to cancel one another. In the end, it appears the unwanted astigmatism associated with a given power change along a given distance can be redistributed but probably not reduced.
authors.affiliation: Center for Ophthalmic Optics Research, The Ohio State University College of Optometry, Columbus, Ohio 43210, USA. jsheedy@optometry.osu.edu
authors.forename: James E
authors.initials: JE
authors.lastname: Sheedy
authors.forename: Charles
authors.initials: C
authors.lastname: Campbell
authors.forename: Ewen
authors.initials: E
authors.lastname: King-Smith
authors.forename: John R
authors.initials: JR
authors.lastname: Hayes
mesh_terms: D001251:Astigmatism / Q000209:etiology*; D004867:Equipment Design / Q000009:adverse effects; D005139:Eyeglasses / Q000009:adverse effects*; D006801:Humans; D008962:Models, Theoretical; D055095:Optics and Photonics; D011305:Presbyopia / Q000628:therapy*