title: Cardinal and anti-cardinal points, equalities and chromatic dependence.
abstract: Cardinal points are used for ray tracing through Gaussian systems. Anti-principal and anti-nodal points (which we shall refer to as the anti-cardinal points), along with the six familiar cardinal points, belong to a much larger set of special points. The purpose of this paper is to obtain a set of relationships and resulting equalities among the cardinal and anti-cardinal points and to illustrate them using Pascal's ring.
  
  
  
  The methodology used relies on Gaussian optics and the transference T. We make use of two equations, obtained via the transference, which give the locations of the six cardinal and four anti-cardinal points with respect to the system. We obtain equalities among the cardinal and anti-cardinal points. We utilise Pascal's ring to illustrate which points depend on frequency and their displacement with change in frequency.
  
  
  
  Pascal described a memory schema in the shape of a hexagon for remembering equalities among the points and illustrating shifts in these points when an aspect of the system changes. We modify and extend Pascal's ring to include the anti-cardinal points. We make use of Pascal's ring extended to illustrate which points are dependent on the frequency of light and the direction of shift of the equalities with change in frequency. For the reduced eye the principal and nodal points are independent of frequency, but the focal points and the anti-cardinal points depend on frequency. For Le Grand's four-surface model eye all six cardinal and four anti-cardinal points depend on frequency. This has implications for definitions, particularly of chromatic aberrations of the eye, that make use of cardinal points and that themselves depend on frequency.
  
  
  
  Pascal's ring and Pascal's ring extended are novel memory schema for remembering the equalities among the cardinal and anti-cardinal points. The rings are useful for illustrating changes among the equalities and direction of shift of points when an aspect of a system changes. Care should be taken when defining concepts that rely on cardinal points that depend on frequency.
authors.affiliation: Department of Optometry, University of Johannesburg, Johannesburg, South Africa.
authors.forename: Tanya
authors.initials: T
authors.lastname: Evans
authors.affiliation: Department of Optometry, University of Johannesburg, Johannesburg, South Africa.
authors.forename: William F
authors.initials: WF
authors.lastname: Harris
mesh_terms: D003261:Contact Lenses; D005139:Eyeglasses; D006801:Humans; D008027:Light; D008962:Models, Theoretical; D012029:Refraction, Ocular / Q000502:physiology*; D012030:Refractive Errors / Q000503:physiopathology*; D012031:Refractometry / Q000379:methods*
keywords: Pascal's ring; anti-cardinal points; cardinal points; chromatic dependence; equalities