Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error

Optom Vis Sci. 1997 Jun;74(6):367-75. doi: 10.1097/00006324-199706000-00019.


The description of sphero-cylinder lenses is approached from the viewpoint of Fourier analysis of the power profile. It is shown that the familiar sine-squared law leads naturally to a Fourier series representation with exactly three Fourier coefficients, representing the natural parameters of a thin lens. The constant term corresponds to the mean spherical equivalent (MSE) power, whereas the amplitude and phase of the harmonic correspond to the power and axis of a Jackson cross-cylinder (JCC) lens, respectively. Expressing the Fourier series in rectangular form leads to the representation of an arbitrary sphero-cylinder lens as the sum of a spherical lens and two cross-cylinders, one at axis 0 degree and the other at axis 45 degrees. The power of these three component lenses may be interpreted as (x,y,z) coordinates of a vector representation of the power profile. Advantages of this power vector representation of a sphero-cylinder lens for numerical and graphical analysis of optometric data are described for problems involving lens combinations, comparison of different lenses, and the statistical distribution of refractive errors.

Publication types

  • Research Support, U.S. Gov't, P.H.S.
  • Review

MeSH terms

  • Computer Graphics
  • Data Interpretation, Statistical*
  • Eyeglasses* / standards
  • Eyeglasses* / statistics & numerical data
  • Fourier Analysis*
  • Humans
  • Optics and Photonics
  • Optometry / methods
  • Refraction, Ocular
  • Refractive Errors / therapy*