title: Sequential tilt and tilted power of thin lenses.
abstract: OBJECTIVE
  The purposes of this article were, for any sequence of tilts applied to a thin lens, to calculate an equivalent turn and single tilt, to show how to use the equivalent turn and tilt to calculate the tilted power of the lens, and, given the desired tilted power, to calculate the power of the untilted lens necessary to compensate for the effects of tilt. The untilted lens may be stigmatic (spherical) or astigmatic (spherocylindrical).
  
  
  METHODS
  The analysis makes use of rotation matrices to represent rotation in space and previous work in third-order optics on oblique central refraction.
  
  
  RESULTS
  Equations are presented for calculating the combination of turn and tilt that is equivalent to any sequence of tilts. They are specialized for the particular case of combinations of faceform and pantoscopic tilts and allow the decomposition of an arbitrary tilt into a combination of turn and pantoscopic and faceform tilts. The equations also lead to a procedure for calculating or compensating for the tilted power of a sequentially tilted thin lens.
  
  
  CONCLUSIONS
  Previous work on the effect of tilt on thin lenses has been generalized to handle combinations of arbitrary tilts.
authors.affiliation: Optometric Science Research Group, Department of Optometry, University of Johannesburg, Johannesburg, South Africa. wfh@na.rau.ac.za
authors.forename: W F
authors.initials: WF
authors.lastname: Harris
mesh_terms: D004867:Equipment Design; D005139:Eyeglasses / Q000592:standards*; D006801:Humans; D066298:In Vitro Techniques; D008962:Models, Theoretical; D012029:Refraction, Ocular; D012030:Refractive Errors / Q000628:therapy*