Abstract

The locus of photokeratoscope target points that yield images lying in a plane perpendicular to the camera axis was evaluated for a 15.8-mm radius spherical convex reflector. The linear transform of an ellipse was fitted to sets of theoretical and experimental points by the method of least squares. The semimajor and semiminor axes for the theoretical and experimental points were 103.1, 40.0, 95.0, and 42.9 mm, respectively. The theoretical points conformed to the linear transform of the ellipse with a coefficient of correlation departing from unity by less than 4 parts in 10 000. For the combined experimental points, the coefficient of correlation was 0.998. The theoretical locus was also calculated for an elliptical reflector simulating the gross dimensions of the cornea and found to be essentially elliptical, closely approximating that for the spherical reflector.

© 1970 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Wittenberg and W. Ludlam, J. Opt. Soc. Am. 56, 1612 (1966).
    [Crossref]
  2. W. Ludlam and S. Wittenberg, Am. J. Optom., Arch. Am. Acad. Optom. 43, 249 (1966).
    [Crossref]
  3. H. Dekking, Graefes Arch. Ophthalmol. 124, 708 (1930).
    [Crossref]
  4. A. Gullstrand, Kgl. Svenska Vetenskaps akad. Handl. 28, 12 (1896), English translation by W. Ludlam, Am. J. Optom., Arch. Am. Acad. Optom. 43, 143 (1966).
    [Crossref]
  5. F. Berg, Acta Ophthalmol. 7, 386 (1929).
    [Crossref]
  6. E. Fincham, Med. Biol. Illust. 3, 87 (1953).
  7. H. Knoll, R. Stimson, and C. Weeks, J. Opt. Soc. Am. 47, 221 (1957).
    [Crossref] [PubMed]
  8. H. Knoll, Am. J. Optom., Arch. Am. Acad. Optom. 38, 399 (1961).
    [Crossref]
  9. J. Stone, Brit. J. Physiological Opt. 19, 205 (1962).
  10. Primed variables are employed here because position was measured with respect to the center of the steel ball, and not with respect to the principal plane, for which the coordinate axes of the theoretical construct is defined.
  11. The goodness of fit of the ellipse to the experimentally or theoretically determined points, as indicated by the correlation coefficient, is somewhat exaggerated by the linear transformation applied.
  12. H. von Helmholtz, Treatise on Physiological Optics, edited by J. P. C. Southall (Optical Society of America, 1924; Dover, New York, 1965), Vol. I, Ch. 2, p. 9.

1966 (2)

W. Ludlam and S. Wittenberg, Am. J. Optom., Arch. Am. Acad. Optom. 43, 249 (1966).
[Crossref]

S. Wittenberg and W. Ludlam, J. Opt. Soc. Am. 56, 1612 (1966).
[Crossref]

1962 (1)

J. Stone, Brit. J. Physiological Opt. 19, 205 (1962).

1961 (1)

H. Knoll, Am. J. Optom., Arch. Am. Acad. Optom. 38, 399 (1961).
[Crossref]

1957 (1)

1953 (1)

E. Fincham, Med. Biol. Illust. 3, 87 (1953).

1930 (1)

H. Dekking, Graefes Arch. Ophthalmol. 124, 708 (1930).
[Crossref]

1929 (1)

F. Berg, Acta Ophthalmol. 7, 386 (1929).
[Crossref]

1896 (1)

A. Gullstrand, Kgl. Svenska Vetenskaps akad. Handl. 28, 12 (1896), English translation by W. Ludlam, Am. J. Optom., Arch. Am. Acad. Optom. 43, 143 (1966).
[Crossref]

Berg, F.

F. Berg, Acta Ophthalmol. 7, 386 (1929).
[Crossref]

Dekking, H.

H. Dekking, Graefes Arch. Ophthalmol. 124, 708 (1930).
[Crossref]

Fincham, E.

E. Fincham, Med. Biol. Illust. 3, 87 (1953).

Gullstrand, A.

A. Gullstrand, Kgl. Svenska Vetenskaps akad. Handl. 28, 12 (1896), English translation by W. Ludlam, Am. J. Optom., Arch. Am. Acad. Optom. 43, 143 (1966).
[Crossref]

Knoll, H.

H. Knoll, Am. J. Optom., Arch. Am. Acad. Optom. 38, 399 (1961).
[Crossref]

H. Knoll, R. Stimson, and C. Weeks, J. Opt. Soc. Am. 47, 221 (1957).
[Crossref] [PubMed]

Ludlam, W.

S. Wittenberg and W. Ludlam, J. Opt. Soc. Am. 56, 1612 (1966).
[Crossref]

W. Ludlam and S. Wittenberg, Am. J. Optom., Arch. Am. Acad. Optom. 43, 249 (1966).
[Crossref]

Stimson, R.

Stone, J.

J. Stone, Brit. J. Physiological Opt. 19, 205 (1962).

von Helmholtz, H.

H. von Helmholtz, Treatise on Physiological Optics, edited by J. P. C. Southall (Optical Society of America, 1924; Dover, New York, 1965), Vol. I, Ch. 2, p. 9.

Weeks, C.

Wittenberg, S.

W. Ludlam and S. Wittenberg, Am. J. Optom., Arch. Am. Acad. Optom. 43, 249 (1966).
[Crossref]

S. Wittenberg and W. Ludlam, J. Opt. Soc. Am. 56, 1612 (1966).
[Crossref]

Acta Ophthalmol. (1)

F. Berg, Acta Ophthalmol. 7, 386 (1929).
[Crossref]

Am. J. Optom., Arch. Am. Acad. Optom. (2)

W. Ludlam and S. Wittenberg, Am. J. Optom., Arch. Am. Acad. Optom. 43, 249 (1966).
[Crossref]

H. Knoll, Am. J. Optom., Arch. Am. Acad. Optom. 38, 399 (1961).
[Crossref]

Brit. J. Physiological Opt. (1)

J. Stone, Brit. J. Physiological Opt. 19, 205 (1962).

Graefes Arch. Ophthalmol. (1)

H. Dekking, Graefes Arch. Ophthalmol. 124, 708 (1930).
[Crossref]

J. Opt. Soc. Am. (2)

Kgl. Svenska Vetenskaps akad. Handl. (1)

A. Gullstrand, Kgl. Svenska Vetenskaps akad. Handl. 28, 12 (1896), English translation by W. Ludlam, Am. J. Optom., Arch. Am. Acad. Optom. 43, 143 (1966).
[Crossref]

Med. Biol. Illust. (1)

E. Fincham, Med. Biol. Illust. 3, 87 (1953).

Other (3)

Primed variables are employed here because position was measured with respect to the center of the steel ball, and not with respect to the principal plane, for which the coordinate axes of the theoretical construct is defined.

The goodness of fit of the ellipse to the experimentally or theoretically determined points, as indicated by the correlation coefficient, is somewhat exaggerated by the linear transformation applied.

H. von Helmholtz, Treatise on Physiological Optics, edited by J. P. C. Southall (Optical Society of America, 1924; Dover, New York, 1965), Vol. I, Ch. 2, p. 9.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Relationship between an object point a, b, an image point xi, yi, and a convex reflector.

Fig. 2
Fig. 2

Schematic of apparatus.

Fig. 3
Fig. 3

A comparison of the object locii that produce planar images, as determined theoretically ● and experimentally ▲ for a spherical convex reflector 7.9375-mm radius. The position of the reflector would lie between 180 and 200. Note the magnification of the ordinate scale.

Fig. 4
Fig. 4

A comparison of the theoretically determined object locii for a spherical convex reflector 7.9375-mm radius ● and an ellipsoidal reflector ▲ approximating the shape of the cornea and producing the same measured image height for the central keratoscopic ring as does the sphere. The position of the reflector would lie between 180 and 200. Note the magnification of the ordinate scale.

Tables (2)

Tables Icon

Table I Target position producing flat reflected imagery expressed in polar and rectangular coordinates. The P values were interpolated from parametric studies of the image plane. Positive and negative rotations differentiate measurements for the two sides of the optic axis. The final set of columns contains the means of measurements obtained on positive and negative rotation. For ϕ=14°50′, there is no value P because this is the position of the lens housing, whose rectangular coordinates are directly determined and not calculated from an experimental value of P.

Tables Icon

Table II The semimajor and semiminor axis values for the ellipses of best fit for both experimentally derived and theoretical values, where r=coefficient of correlation of a″ and b″ in the linearized transform of the ellipse; f=semimajor axis, g=semiminor axis, and x0=center of ellipse.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

b = y - u sin ( β + α ) a = x - u sin ( β + α ) ,
α = tan - 1 d x / d y β = θ + α
u = u r cos β / ( 2 u - r cos β ) ,
r = [ 1 + ( d x / d y ) 2 ] 3 2 ( d 2 x / d y 2 ) u = ( x i - x ) / cos θ .
( a - x 0 ) 2 / f 2 + ( b - y 0 ) 2 / g 2 = 1
b = m a + c ,