Abstract

We compared refractions across the horizontal visual field, based on different analyses of wave aberration obtained with a Hartmann–Shack instrument. The wave aberrations had been determined for 6-mm-diameter pupils up to at least the sixth Zernike order in five normal subjects [J. Opt. Soc. Am. A 19, 2180 (2002)]. The polynomials were converted into refractions based on 6-mm pupils and second-order Zernike aberrations (6 mm/2nd order), 3-mm pupils and second-order aberrations (3 mm/2nd order), 1-mm pupils and second-order aberrations (1 mm/2nd order), and 6-mm pupils with both second- and fourth-order aberrations (6 mm/4th order). The 3-mm/2nd-order and 6-mm/2nd-order refractions differed by as much as 0.9 D in mean sphere on axis, but the differences reduced markedly toward the edges of the visual field. The cylindrical differences between these two analyses were small at the center of the visual field (<0.3 D) but increased into the periphery to be greater than 1.0 D for some subjects. Much smaller differences in mean sphere and cylinder were found when 3-mm/2nd-order refractions and either the 1-mm/2nd-order refractions or the 6-mm/4th-order refractions were compared. The results suggest that, for determining refractions based on wave aberration data with large pupils, similar results occur by either restricting the analysis to second-order Zernike aberrations with a smaller pupil such as 3 mm or using both second- and fourth-order Zernike aberrations. Since subjective refraction is largely independent of the pupil size under photopic conditions, objective refractions based on either of these analyses may be the most useful.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Liang, B. Grimm, S. Goelz, J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
    [CrossRef]
  2. J. Porter, A. Guirao, I. G. Cox, D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
    [CrossRef]
  3. T. O. Salmon, L. N. Thibos, A. Bradley, “Comparison of the eye’s wave-front aberration measured psychophysically and with the Shack–Hartmann wave-front sensor,” J. Opt. Soc. Am. A 15, 2457–2465 (1998).
    [CrossRef]
  4. E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
    [CrossRef]
  5. P. Artal, E. Berrio, A. Guirao, P. Piers, “Contribution of the cornea and internal surfaces to the change of ocular aberrations with age,” J. Opt. Soc. Am. A 19, 137–143 (2002).
    [CrossRef]
  6. H. Hofer, P. Artal, B. Singer, J. L. Aragon, D. R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A 18, 497–506 (2001).
    [CrossRef]
  7. J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
    [CrossRef]
  8. D. T. Miller, D. R. Williams, G. M. Morris, J. Liang, “Images of cone photoreceptors in the living human eye,” Vision Res. 36, 1067–1079 (1996).
    [CrossRef] [PubMed]
  9. A. Roorda, D. R. Williams, “The arrangement of the three cone classes in the living human eye,” Nature 397, 520–522 (1999).
    [CrossRef] [PubMed]
  10. G.-Y. Yoon, D. R. Williams, “Visual performance after correcting the monochromatic and chromatic aberrations of the eye,” J. Opt. Soc. Am. A 19, 266–275 (2002).
    [CrossRef]
  11. K. Munson, X. Hong, L. N. Thibos, “Use of a Hartmann aberrometer to assess the optical outcome of corneal transplantation in a keratoconic eye,” Optom. Vision Sci. 78, 866–871 (2001).
    [CrossRef]
  12. X. Hong, N. Himebaugh, L. N. Thibos, “On-eye evaluation of optical performance of rigid and soft contact lenses,” Optom. Vision Sci. 78, 872–880 (2001).
    [CrossRef]
  13. L. N. Thibos, X. Hong, “Clinical applications of the Shack–Hartmann aberrometer,” Optom. Vision Sci. 76, 817–825 (1999).
    [CrossRef]
  14. D. A. Atchison, D. H. Scott, “Monochromatic aberrations of human eyes in the horizontal visual field,” J. Opt. Soc. Am. A 19, 2180–2184 (2002).
    [CrossRef]
  15. C. E. Campbell, W. J. Bemjamin, H. C. Howland, “Objective refraction: retinoscopy, autorefraction, and photorefraction,” in Borish’s Clinical Refraction, W. J. Benjamin, ed. (Saunders, Philadelphia, Pa., 1998), Chap. 15.
  16. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
    [PubMed]
  17. M. C. Mrochen, M. Bueeler, T. Seiler, “Influence of higher-order optical aberrations on refraction,” Invest. Ophthalmol. Visual Sci. Suppl. 43, S82 (2002).
  18. M. Koomen, R. Scolnik, R. Tousey, “A study of night myopia,” J. Opt. Soc. Am. 41, 80–90 (1951).
    [CrossRef]
  19. W. N. Charman, J. A. Jennings, H. Whitefoot, “The refraction of the eye in relation to spherical aberration and pupil size,” Brit. J. Physiol. Opt. 32, 78–93 (1978).
  20. D. A. Atchison, G. Smith, N. Efron, “The effect of pupil size on visual acuity in uncorrected and corrected myopia,” Am. J. Optom. Physiol. Opt. 56, 315–323 (1979).
    [CrossRef] [PubMed]
  21. R. Navarro, E. Moreno, C. Dorronsoro, “Monochromatic aberrations and point-spread functions of the human eye across the visual field,” J. Opt. Soc. Am. A 15, 2522–2529 (1998).
    [CrossRef]
  22. A. Guirao, P. Artal, “Off-axis monochromatic aberrations estimated from double pass measurements in the human eye,” Vision Res. 39, 207–217 (1999).
    [CrossRef] [PubMed]
  23. R. Navarro, J. Santamarı́a, J. Bescós, “Accommodation-dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A 2, 1273–1281 (1985).
    [CrossRef] [PubMed]
  24. I. Escudero-Sanz, R. Navarro, “Off-axis aberrations of a wide-angle schematic eye model,” J. Opt. Soc. Am. A 16, 1881–1891 (1999).
    [CrossRef]
  25. D. A. Atchison, G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Oxford, UK, 2000), pp. 147–149, 173–176.
  26. C. E. Ferree, G. Rand, C. Hardy, “Refraction for the peripheral field of vision,” Arch. Ophthalmol. 9, 925–938 (1931).
    [CrossRef]
  27. C. E. Ferree, G. Rand, C. Hardy, “Refractive asymmetry in the temporal and nasal halves of the visual field,” Am. J. Ophthalmol. 15, 513–522 (1932).
  28. C. E. Ferree, G. Rand, “Interpretation of refractive conditions in the peripheral field of vision,” Arch. Ophthalmol. 5, 717–731 (1933).
    [CrossRef]
  29. F. Rempt, J. Hoogerheide, W. P. H. Hoogenbloom, “Peripheral retinoscopy and the skiagram,” Ophthalmologica 162, 1–10 (1971).
    [CrossRef] [PubMed]
  30. W. Lotmar, T. Lotmar, “Peripheral astigmatism in the human eye: experimental data and theoretical model predictions,” J. Opt. Soc. Am. 64, 510–513 (1974).
    [CrossRef] [PubMed]
  31. M. Millodot, “Effect of ametropia on peripheral refraction,” Am. J. Optom. Physiol. Opt. 58, 691–695 (1981).
    [PubMed]
  32. M. C. M. Dunne, G. P. Mission, E. K. White, D. Barnes, “Peripheral astigmatic asymmetry and angle alpha,” Ophthalmic Physiol. Opt. 13, 303–305 (1993).
    [CrossRef] [PubMed]
  33. J. Gustafsson, E. Terenius, J. Buchheister, P. Unsbo, “Peripheral astigmatism in emmetropic eyes,” Ophthalmic Physiol. Opt. 21, 393–400 (2001).
    [CrossRef] [PubMed]
  34. A. Seidemann, F. Schaeffel, A. Guirao, N. Lopez-Gil, P. Artal, “Peripheral refractive errors in myopic, emmetropic, and hyperopic young subjects,” J. Opt. Soc. Am. A 19, 2363–2373 (2002).
    [CrossRef]
  35. J. Love, B. Gilmartin, M. C. M. Dunne, “Relative peripheral refractive error in adult myopia and emmetropia,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S302 (2000).
  36. L. Thibos, X. Hong, A. Bradley, X. Cheng, “Statistical variation of aberration structure and image quality in a normal population,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
    [CrossRef]
  37. A. Guirao, D. R. Williams, “An objective method to predict refractive errors from wave aberration data,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S98 (2001).
  38. A. Guirao, D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
    [CrossRef]
  39. G. Smith, D. A. Atchison, The Eye and Visual Optical Instruments (Cambridge U. Press, New York, 1997), Chap. 33.
  40. D. A. Atchison, D. H. Scott, M. J. Cox, “Mathematical treatment of ocular aberrations: a user’s guide,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 110–130.
  41. T. O. Salmon, L. N. Thibos, “Videokeratoscope—line-of-sight misalignment and its effect on measurements of corneal and internal ocular aberrations,” J. Opt. Soc. Am. A 19, 657–669 (2002).
    [CrossRef]

2003 (1)

A. Guirao, D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
[CrossRef]

2002 (8)

T. O. Salmon, L. N. Thibos, “Videokeratoscope—line-of-sight misalignment and its effect on measurements of corneal and internal ocular aberrations,” J. Opt. Soc. Am. A 19, 657–669 (2002).
[CrossRef]

L. Thibos, X. Hong, A. Bradley, X. Cheng, “Statistical variation of aberration structure and image quality in a normal population,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

P. Artal, E. Berrio, A. Guirao, P. Piers, “Contribution of the cornea and internal surfaces to the change of ocular aberrations with age,” J. Opt. Soc. Am. A 19, 137–143 (2002).
[CrossRef]

G.-Y. Yoon, D. R. Williams, “Visual performance after correcting the monochromatic and chromatic aberrations of the eye,” J. Opt. Soc. Am. A 19, 266–275 (2002).
[CrossRef]

D. A. Atchison, D. H. Scott, “Monochromatic aberrations of human eyes in the horizontal visual field,” J. Opt. Soc. Am. A 19, 2180–2184 (2002).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

M. C. Mrochen, M. Bueeler, T. Seiler, “Influence of higher-order optical aberrations on refraction,” Invest. Ophthalmol. Visual Sci. Suppl. 43, S82 (2002).

A. Seidemann, F. Schaeffel, A. Guirao, N. Lopez-Gil, P. Artal, “Peripheral refractive errors in myopic, emmetropic, and hyperopic young subjects,” J. Opt. Soc. Am. A 19, 2363–2373 (2002).
[CrossRef]

2001 (7)

K. Munson, X. Hong, L. N. Thibos, “Use of a Hartmann aberrometer to assess the optical outcome of corneal transplantation in a keratoconic eye,” Optom. Vision Sci. 78, 866–871 (2001).
[CrossRef]

X. Hong, N. Himebaugh, L. N. Thibos, “On-eye evaluation of optical performance of rigid and soft contact lenses,” Optom. Vision Sci. 78, 872–880 (2001).
[CrossRef]

H. Hofer, P. Artal, B. Singer, J. L. Aragon, D. R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A 18, 497–506 (2001).
[CrossRef]

J. Porter, A. Guirao, I. G. Cox, D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
[CrossRef]

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

A. Guirao, D. R. Williams, “An objective method to predict refractive errors from wave aberration data,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S98 (2001).

J. Gustafsson, E. Terenius, J. Buchheister, P. Unsbo, “Peripheral astigmatism in emmetropic eyes,” Ophthalmic Physiol. Opt. 21, 393–400 (2001).
[CrossRef] [PubMed]

2000 (1)

J. Love, B. Gilmartin, M. C. M. Dunne, “Relative peripheral refractive error in adult myopia and emmetropia,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S302 (2000).

1999 (4)

A. Roorda, D. R. Williams, “The arrangement of the three cone classes in the living human eye,” Nature 397, 520–522 (1999).
[CrossRef] [PubMed]

A. Guirao, P. Artal, “Off-axis monochromatic aberrations estimated from double pass measurements in the human eye,” Vision Res. 39, 207–217 (1999).
[CrossRef] [PubMed]

I. Escudero-Sanz, R. Navarro, “Off-axis aberrations of a wide-angle schematic eye model,” J. Opt. Soc. Am. A 16, 1881–1891 (1999).
[CrossRef]

L. N. Thibos, X. Hong, “Clinical applications of the Shack–Hartmann aberrometer,” Optom. Vision Sci. 76, 817–825 (1999).
[CrossRef]

1998 (2)

1997 (1)

1996 (1)

D. T. Miller, D. R. Williams, G. M. Morris, J. Liang, “Images of cone photoreceptors in the living human eye,” Vision Res. 36, 1067–1079 (1996).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

M. C. M. Dunne, G. P. Mission, E. K. White, D. Barnes, “Peripheral astigmatic asymmetry and angle alpha,” Ophthalmic Physiol. Opt. 13, 303–305 (1993).
[CrossRef] [PubMed]

1985 (1)

1981 (1)

M. Millodot, “Effect of ametropia on peripheral refraction,” Am. J. Optom. Physiol. Opt. 58, 691–695 (1981).
[PubMed]

1979 (1)

D. A. Atchison, G. Smith, N. Efron, “The effect of pupil size on visual acuity in uncorrected and corrected myopia,” Am. J. Optom. Physiol. Opt. 56, 315–323 (1979).
[CrossRef] [PubMed]

1978 (1)

W. N. Charman, J. A. Jennings, H. Whitefoot, “The refraction of the eye in relation to spherical aberration and pupil size,” Brit. J. Physiol. Opt. 32, 78–93 (1978).

1974 (1)

1971 (1)

F. Rempt, J. Hoogerheide, W. P. H. Hoogenbloom, “Peripheral retinoscopy and the skiagram,” Ophthalmologica 162, 1–10 (1971).
[CrossRef] [PubMed]

1951 (1)

1933 (1)

C. E. Ferree, G. Rand, “Interpretation of refractive conditions in the peripheral field of vision,” Arch. Ophthalmol. 5, 717–731 (1933).
[CrossRef]

1932 (1)

C. E. Ferree, G. Rand, C. Hardy, “Refractive asymmetry in the temporal and nasal halves of the visual field,” Am. J. Ophthalmol. 15, 513–522 (1932).

1931 (1)

C. E. Ferree, G. Rand, C. Hardy, “Refraction for the peripheral field of vision,” Arch. Ophthalmol. 9, 925–938 (1931).
[CrossRef]

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Aragon, J. L.

Artal, P.

Atchison, D. A.

D. A. Atchison, D. H. Scott, “Monochromatic aberrations of human eyes in the horizontal visual field,” J. Opt. Soc. Am. A 19, 2180–2184 (2002).
[CrossRef]

D. A. Atchison, G. Smith, N. Efron, “The effect of pupil size on visual acuity in uncorrected and corrected myopia,” Am. J. Optom. Physiol. Opt. 56, 315–323 (1979).
[CrossRef] [PubMed]

D. A. Atchison, G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Oxford, UK, 2000), pp. 147–149, 173–176.

G. Smith, D. A. Atchison, The Eye and Visual Optical Instruments (Cambridge U. Press, New York, 1997), Chap. 33.

D. A. Atchison, D. H. Scott, M. J. Cox, “Mathematical treatment of ocular aberrations: a user’s guide,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 110–130.

Barnes, D.

M. C. M. Dunne, G. P. Mission, E. K. White, D. Barnes, “Peripheral astigmatic asymmetry and angle alpha,” Ophthalmic Physiol. Opt. 13, 303–305 (1993).
[CrossRef] [PubMed]

Bemjamin, W. J.

C. E. Campbell, W. J. Bemjamin, H. C. Howland, “Objective refraction: retinoscopy, autorefraction, and photorefraction,” in Borish’s Clinical Refraction, W. J. Benjamin, ed. (Saunders, Philadelphia, Pa., 1998), Chap. 15.

Berrio, E.

Bescós, J.

Bille, J. F.

Bradley, A.

Buchheister, J.

J. Gustafsson, E. Terenius, J. Buchheister, P. Unsbo, “Peripheral astigmatism in emmetropic eyes,” Ophthalmic Physiol. Opt. 21, 393–400 (2001).
[CrossRef] [PubMed]

Bueeler, M.

M. C. Mrochen, M. Bueeler, T. Seiler, “Influence of higher-order optical aberrations on refraction,” Invest. Ophthalmol. Visual Sci. Suppl. 43, S82 (2002).

Burns, S. A.

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

Campbell, C. E.

C. E. Campbell, W. J. Bemjamin, H. C. Howland, “Objective refraction: retinoscopy, autorefraction, and photorefraction,” in Borish’s Clinical Refraction, W. J. Benjamin, ed. (Saunders, Philadelphia, Pa., 1998), Chap. 15.

Charman, W. N.

W. N. Charman, J. A. Jennings, H. Whitefoot, “The refraction of the eye in relation to spherical aberration and pupil size,” Brit. J. Physiol. Opt. 32, 78–93 (1978).

Cheng, X.

Cox, I. G.

Cox, M. J.

D. A. Atchison, D. H. Scott, M. J. Cox, “Mathematical treatment of ocular aberrations: a user’s guide,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 110–130.

Dorronsoro, C.

Dunne, M. C. M.

J. Love, B. Gilmartin, M. C. M. Dunne, “Relative peripheral refractive error in adult myopia and emmetropia,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S302 (2000).

M. C. M. Dunne, G. P. Mission, E. K. White, D. Barnes, “Peripheral astigmatic asymmetry and angle alpha,” Ophthalmic Physiol. Opt. 13, 303–305 (1993).
[CrossRef] [PubMed]

Efron, N.

D. A. Atchison, G. Smith, N. Efron, “The effect of pupil size on visual acuity in uncorrected and corrected myopia,” Am. J. Optom. Physiol. Opt. 56, 315–323 (1979).
[CrossRef] [PubMed]

Escudero-Sanz, I.

Ferree, C. E.

C. E. Ferree, G. Rand, “Interpretation of refractive conditions in the peripheral field of vision,” Arch. Ophthalmol. 5, 717–731 (1933).
[CrossRef]

C. E. Ferree, G. Rand, C. Hardy, “Refractive asymmetry in the temporal and nasal halves of the visual field,” Am. J. Ophthalmol. 15, 513–522 (1932).

C. E. Ferree, G. Rand, C. Hardy, “Refraction for the peripheral field of vision,” Arch. Ophthalmol. 9, 925–938 (1931).
[CrossRef]

Gilmartin, B.

J. Love, B. Gilmartin, M. C. M. Dunne, “Relative peripheral refractive error in adult myopia and emmetropia,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S302 (2000).

Goelz, S.

Grimm, B.

Guirao, A.

A. Guirao, D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
[CrossRef]

A. Seidemann, F. Schaeffel, A. Guirao, N. Lopez-Gil, P. Artal, “Peripheral refractive errors in myopic, emmetropic, and hyperopic young subjects,” J. Opt. Soc. Am. A 19, 2363–2373 (2002).
[CrossRef]

P. Artal, E. Berrio, A. Guirao, P. Piers, “Contribution of the cornea and internal surfaces to the change of ocular aberrations with age,” J. Opt. Soc. Am. A 19, 137–143 (2002).
[CrossRef]

J. Porter, A. Guirao, I. G. Cox, D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
[CrossRef]

A. Guirao, D. R. Williams, “An objective method to predict refractive errors from wave aberration data,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S98 (2001).

A. Guirao, P. Artal, “Off-axis monochromatic aberrations estimated from double pass measurements in the human eye,” Vision Res. 39, 207–217 (1999).
[CrossRef] [PubMed]

Gustafsson, J.

J. Gustafsson, E. Terenius, J. Buchheister, P. Unsbo, “Peripheral astigmatism in emmetropic eyes,” Ophthalmic Physiol. Opt. 21, 393–400 (2001).
[CrossRef] [PubMed]

Hardy, C.

C. E. Ferree, G. Rand, C. Hardy, “Refractive asymmetry in the temporal and nasal halves of the visual field,” Am. J. Ophthalmol. 15, 513–522 (1932).

C. E. Ferree, G. Rand, C. Hardy, “Refraction for the peripheral field of vision,” Arch. Ophthalmol. 9, 925–938 (1931).
[CrossRef]

Himebaugh, N.

X. Hong, N. Himebaugh, L. N. Thibos, “On-eye evaluation of optical performance of rigid and soft contact lenses,” Optom. Vision Sci. 78, 872–880 (2001).
[CrossRef]

Hofer, H.

Hong, X.

L. Thibos, X. Hong, A. Bradley, X. Cheng, “Statistical variation of aberration structure and image quality in a normal population,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

X. Hong, N. Himebaugh, L. N. Thibos, “On-eye evaluation of optical performance of rigid and soft contact lenses,” Optom. Vision Sci. 78, 872–880 (2001).
[CrossRef]

K. Munson, X. Hong, L. N. Thibos, “Use of a Hartmann aberrometer to assess the optical outcome of corneal transplantation in a keratoconic eye,” Optom. Vision Sci. 78, 866–871 (2001).
[CrossRef]

L. N. Thibos, X. Hong, “Clinical applications of the Shack–Hartmann aberrometer,” Optom. Vision Sci. 76, 817–825 (1999).
[CrossRef]

Hoogenbloom, W. P. H.

F. Rempt, J. Hoogerheide, W. P. H. Hoogenbloom, “Peripheral retinoscopy and the skiagram,” Ophthalmologica 162, 1–10 (1971).
[CrossRef] [PubMed]

Hoogerheide, J.

F. Rempt, J. Hoogerheide, W. P. H. Hoogenbloom, “Peripheral retinoscopy and the skiagram,” Ophthalmologica 162, 1–10 (1971).
[CrossRef] [PubMed]

Howland, H. C.

C. E. Campbell, W. J. Bemjamin, H. C. Howland, “Objective refraction: retinoscopy, autorefraction, and photorefraction,” in Borish’s Clinical Refraction, W. J. Benjamin, ed. (Saunders, Philadelphia, Pa., 1998), Chap. 15.

Jennings, J. A.

W. N. Charman, J. A. Jennings, H. Whitefoot, “The refraction of the eye in relation to spherical aberration and pupil size,” Brit. J. Physiol. Opt. 32, 78–93 (1978).

Koomen, M.

Liang, J.

Lopez-Gil, N.

Lotmar, T.

Lotmar, W.

Love, J.

J. Love, B. Gilmartin, M. C. M. Dunne, “Relative peripheral refractive error in adult myopia and emmetropia,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S302 (2000).

Marcos, S.

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

Miller, D. T.

J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
[CrossRef]

D. T. Miller, D. R. Williams, G. M. Morris, J. Liang, “Images of cone photoreceptors in the living human eye,” Vision Res. 36, 1067–1079 (1996).
[CrossRef] [PubMed]

Millodot, M.

M. Millodot, “Effect of ametropia on peripheral refraction,” Am. J. Optom. Physiol. Opt. 58, 691–695 (1981).
[PubMed]

Mission, G. P.

M. C. M. Dunne, G. P. Mission, E. K. White, D. Barnes, “Peripheral astigmatic asymmetry and angle alpha,” Ophthalmic Physiol. Opt. 13, 303–305 (1993).
[CrossRef] [PubMed]

Moreno, E.

Moreno-Barriuso, E.

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

Morris, G. M.

D. T. Miller, D. R. Williams, G. M. Morris, J. Liang, “Images of cone photoreceptors in the living human eye,” Vision Res. 36, 1067–1079 (1996).
[CrossRef] [PubMed]

Mrochen, M. C.

M. C. Mrochen, M. Bueeler, T. Seiler, “Influence of higher-order optical aberrations on refraction,” Invest. Ophthalmol. Visual Sci. Suppl. 43, S82 (2002).

Munson, K.

K. Munson, X. Hong, L. N. Thibos, “Use of a Hartmann aberrometer to assess the optical outcome of corneal transplantation in a keratoconic eye,” Optom. Vision Sci. 78, 866–871 (2001).
[CrossRef]

Navarro, R.

Piers, P.

Porter, J.

Rand, G.

C. E. Ferree, G. Rand, “Interpretation of refractive conditions in the peripheral field of vision,” Arch. Ophthalmol. 5, 717–731 (1933).
[CrossRef]

C. E. Ferree, G. Rand, C. Hardy, “Refractive asymmetry in the temporal and nasal halves of the visual field,” Am. J. Ophthalmol. 15, 513–522 (1932).

C. E. Ferree, G. Rand, C. Hardy, “Refraction for the peripheral field of vision,” Arch. Ophthalmol. 9, 925–938 (1931).
[CrossRef]

Rempt, F.

F. Rempt, J. Hoogerheide, W. P. H. Hoogenbloom, “Peripheral retinoscopy and the skiagram,” Ophthalmologica 162, 1–10 (1971).
[CrossRef] [PubMed]

Roorda, A.

A. Roorda, D. R. Williams, “The arrangement of the three cone classes in the living human eye,” Nature 397, 520–522 (1999).
[CrossRef] [PubMed]

Salmon, T. O.

Santamari´a, J.

Schaeffel, F.

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Scolnik, R.

Scott, D. H.

D. A. Atchison, D. H. Scott, “Monochromatic aberrations of human eyes in the horizontal visual field,” J. Opt. Soc. Am. A 19, 2180–2184 (2002).
[CrossRef]

D. A. Atchison, D. H. Scott, M. J. Cox, “Mathematical treatment of ocular aberrations: a user’s guide,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 110–130.

Seidemann, A.

Seiler, T.

M. C. Mrochen, M. Bueeler, T. Seiler, “Influence of higher-order optical aberrations on refraction,” Invest. Ophthalmol. Visual Sci. Suppl. 43, S82 (2002).

Singer, B.

Smith, G.

D. A. Atchison, G. Smith, N. Efron, “The effect of pupil size on visual acuity in uncorrected and corrected myopia,” Am. J. Optom. Physiol. Opt. 56, 315–323 (1979).
[CrossRef] [PubMed]

D. A. Atchison, G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Oxford, UK, 2000), pp. 147–149, 173–176.

G. Smith, D. A. Atchison, The Eye and Visual Optical Instruments (Cambridge U. Press, New York, 1997), Chap. 33.

Terenius, E.

J. Gustafsson, E. Terenius, J. Buchheister, P. Unsbo, “Peripheral astigmatism in emmetropic eyes,” Ophthalmic Physiol. Opt. 21, 393–400 (2001).
[CrossRef] [PubMed]

Thibos, L.

Thibos, L. N.

T. O. Salmon, L. N. Thibos, “Videokeratoscope—line-of-sight misalignment and its effect on measurements of corneal and internal ocular aberrations,” J. Opt. Soc. Am. A 19, 657–669 (2002).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

X. Hong, N. Himebaugh, L. N. Thibos, “On-eye evaluation of optical performance of rigid and soft contact lenses,” Optom. Vision Sci. 78, 872–880 (2001).
[CrossRef]

K. Munson, X. Hong, L. N. Thibos, “Use of a Hartmann aberrometer to assess the optical outcome of corneal transplantation in a keratoconic eye,” Optom. Vision Sci. 78, 866–871 (2001).
[CrossRef]

L. N. Thibos, X. Hong, “Clinical applications of the Shack–Hartmann aberrometer,” Optom. Vision Sci. 76, 817–825 (1999).
[CrossRef]

T. O. Salmon, L. N. Thibos, A. Bradley, “Comparison of the eye’s wave-front aberration measured psychophysically and with the Shack–Hartmann wave-front sensor,” J. Opt. Soc. Am. A 15, 2457–2465 (1998).
[CrossRef]

Tousey, R.

Unsbo, P.

J. Gustafsson, E. Terenius, J. Buchheister, P. Unsbo, “Peripheral astigmatism in emmetropic eyes,” Ophthalmic Physiol. Opt. 21, 393–400 (2001).
[CrossRef] [PubMed]

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

White, E. K.

M. C. M. Dunne, G. P. Mission, E. K. White, D. Barnes, “Peripheral astigmatic asymmetry and angle alpha,” Ophthalmic Physiol. Opt. 13, 303–305 (1993).
[CrossRef] [PubMed]

Whitefoot, H.

W. N. Charman, J. A. Jennings, H. Whitefoot, “The refraction of the eye in relation to spherical aberration and pupil size,” Brit. J. Physiol. Opt. 32, 78–93 (1978).

Williams, D. R.

A. Guirao, D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
[CrossRef]

G.-Y. Yoon, D. R. Williams, “Visual performance after correcting the monochromatic and chromatic aberrations of the eye,” J. Opt. Soc. Am. A 19, 266–275 (2002).
[CrossRef]

J. Porter, A. Guirao, I. G. Cox, D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
[CrossRef]

H. Hofer, P. Artal, B. Singer, J. L. Aragon, D. R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A 18, 497–506 (2001).
[CrossRef]

A. Guirao, D. R. Williams, “An objective method to predict refractive errors from wave aberration data,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S98 (2001).

A. Roorda, D. R. Williams, “The arrangement of the three cone classes in the living human eye,” Nature 397, 520–522 (1999).
[CrossRef] [PubMed]

J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
[CrossRef]

D. T. Miller, D. R. Williams, G. M. Morris, J. Liang, “Images of cone photoreceptors in the living human eye,” Vision Res. 36, 1067–1079 (1996).
[CrossRef] [PubMed]

Yoon, G.-Y.

Am. J. Ophthalmol. (1)

C. E. Ferree, G. Rand, C. Hardy, “Refractive asymmetry in the temporal and nasal halves of the visual field,” Am. J. Ophthalmol. 15, 513–522 (1932).

Am. J. Optom. Physiol. Opt. (2)

M. Millodot, “Effect of ametropia on peripheral refraction,” Am. J. Optom. Physiol. Opt. 58, 691–695 (1981).
[PubMed]

D. A. Atchison, G. Smith, N. Efron, “The effect of pupil size on visual acuity in uncorrected and corrected myopia,” Am. J. Optom. Physiol. Opt. 56, 315–323 (1979).
[CrossRef] [PubMed]

Arch. Ophthalmol. (2)

C. E. Ferree, G. Rand, C. Hardy, “Refraction for the peripheral field of vision,” Arch. Ophthalmol. 9, 925–938 (1931).
[CrossRef]

C. E. Ferree, G. Rand, “Interpretation of refractive conditions in the peripheral field of vision,” Arch. Ophthalmol. 5, 717–731 (1933).
[CrossRef]

Brit. J. Physiol. Opt. (1)

W. N. Charman, J. A. Jennings, H. Whitefoot, “The refraction of the eye in relation to spherical aberration and pupil size,” Brit. J. Physiol. Opt. 32, 78–93 (1978).

Invest. Ophthalmol. Visual Sci. Suppl. (3)

M. C. Mrochen, M. Bueeler, T. Seiler, “Influence of higher-order optical aberrations on refraction,” Invest. Ophthalmol. Visual Sci. Suppl. 43, S82 (2002).

J. Love, B. Gilmartin, M. C. M. Dunne, “Relative peripheral refractive error in adult myopia and emmetropia,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S302 (2000).

A. Guirao, D. R. Williams, “An objective method to predict refractive errors from wave aberration data,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S98 (2001).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (14)

R. Navarro, E. Moreno, C. Dorronsoro, “Monochromatic aberrations and point-spread functions of the human eye across the visual field,” J. Opt. Soc. Am. A 15, 2522–2529 (1998).
[CrossRef]

R. Navarro, J. Santamarı́a, J. Bescós, “Accommodation-dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A 2, 1273–1281 (1985).
[CrossRef] [PubMed]

I. Escudero-Sanz, R. Navarro, “Off-axis aberrations of a wide-angle schematic eye model,” J. Opt. Soc. Am. A 16, 1881–1891 (1999).
[CrossRef]

A. Seidemann, F. Schaeffel, A. Guirao, N. Lopez-Gil, P. Artal, “Peripheral refractive errors in myopic, emmetropic, and hyperopic young subjects,” J. Opt. Soc. Am. A 19, 2363–2373 (2002).
[CrossRef]

L. Thibos, X. Hong, A. Bradley, X. Cheng, “Statistical variation of aberration structure and image quality in a normal population,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

J. Liang, B. Grimm, S. Goelz, J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
[CrossRef]

J. Porter, A. Guirao, I. G. Cox, D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
[CrossRef]

T. O. Salmon, L. N. Thibos, A. Bradley, “Comparison of the eye’s wave-front aberration measured psychophysically and with the Shack–Hartmann wave-front sensor,” J. Opt. Soc. Am. A 15, 2457–2465 (1998).
[CrossRef]

P. Artal, E. Berrio, A. Guirao, P. Piers, “Contribution of the cornea and internal surfaces to the change of ocular aberrations with age,” J. Opt. Soc. Am. A 19, 137–143 (2002).
[CrossRef]

H. Hofer, P. Artal, B. Singer, J. L. Aragon, D. R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A 18, 497–506 (2001).
[CrossRef]

J. Liang, D. R. Williams, D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
[CrossRef]

G.-Y. Yoon, D. R. Williams, “Visual performance after correcting the monochromatic and chromatic aberrations of the eye,” J. Opt. Soc. Am. A 19, 266–275 (2002).
[CrossRef]

D. A. Atchison, D. H. Scott, “Monochromatic aberrations of human eyes in the horizontal visual field,” J. Opt. Soc. Am. A 19, 2180–2184 (2002).
[CrossRef]

T. O. Salmon, L. N. Thibos, “Videokeratoscope—line-of-sight misalignment and its effect on measurements of corneal and internal ocular aberrations,” J. Opt. Soc. Am. A 19, 657–669 (2002).
[CrossRef]

J. Refract. Surg. (1)

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, “Standards for reporting optical aberrations of eyes,” J. Refract. Surg. 18, S652–S660 (2002).
[PubMed]

Nature (1)

A. Roorda, D. R. Williams, “The arrangement of the three cone classes in the living human eye,” Nature 397, 520–522 (1999).
[CrossRef] [PubMed]

Ophthalmic Physiol. Opt. (2)

M. C. M. Dunne, G. P. Mission, E. K. White, D. Barnes, “Peripheral astigmatic asymmetry and angle alpha,” Ophthalmic Physiol. Opt. 13, 303–305 (1993).
[CrossRef] [PubMed]

J. Gustafsson, E. Terenius, J. Buchheister, P. Unsbo, “Peripheral astigmatism in emmetropic eyes,” Ophthalmic Physiol. Opt. 21, 393–400 (2001).
[CrossRef] [PubMed]

Ophthalmologica (1)

F. Rempt, J. Hoogerheide, W. P. H. Hoogenbloom, “Peripheral retinoscopy and the skiagram,” Ophthalmologica 162, 1–10 (1971).
[CrossRef] [PubMed]

Optom. Vision Sci. (5)

K. Munson, X. Hong, L. N. Thibos, “Use of a Hartmann aberrometer to assess the optical outcome of corneal transplantation in a keratoconic eye,” Optom. Vision Sci. 78, 866–871 (2001).
[CrossRef]

X. Hong, N. Himebaugh, L. N. Thibos, “On-eye evaluation of optical performance of rigid and soft contact lenses,” Optom. Vision Sci. 78, 872–880 (2001).
[CrossRef]

L. N. Thibos, X. Hong, “Clinical applications of the Shack–Hartmann aberrometer,” Optom. Vision Sci. 76, 817–825 (1999).
[CrossRef]

E. Moreno-Barriuso, S. Marcos, R. Navarro, S. A. Burns, “Comparing laser ray tracing, spatially resolved refractometer and Hartmann–Shack sensor to measure the ocular wavefront aberration,” Optom. Vision Sci. 78, 152–156 (2001).
[CrossRef]

A. Guirao, D. R. Williams, “A method to predict refractive errors from wave aberration data,” Optom. Vision Sci. 80, 36–42 (2003).
[CrossRef]

Vision Res. (2)

D. T. Miller, D. R. Williams, G. M. Morris, J. Liang, “Images of cone photoreceptors in the living human eye,” Vision Res. 36, 1067–1079 (1996).
[CrossRef] [PubMed]

A. Guirao, P. Artal, “Off-axis monochromatic aberrations estimated from double pass measurements in the human eye,” Vision Res. 39, 207–217 (1999).
[CrossRef] [PubMed]

Other (4)

D. A. Atchison, G. Smith, Optics of the Human Eye (Butterworth-Heinemann, Oxford, UK, 2000), pp. 147–149, 173–176.

C. E. Campbell, W. J. Bemjamin, H. C. Howland, “Objective refraction: retinoscopy, autorefraction, and photorefraction,” in Borish’s Clinical Refraction, W. J. Benjamin, ed. (Saunders, Philadelphia, Pa., 1998), Chap. 15.

G. Smith, D. A. Atchison, The Eye and Visual Optical Instruments (Cambridge U. Press, New York, 1997), Chap. 33.

D. A. Atchison, D. H. Scott, M. J. Cox, “Mathematical treatment of ocular aberrations: a user’s guide,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 110–130.

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 (3)

Fig. 1
Fig. 1

Calculated right-eye refractions between (-)40° temporal and (+)40° nasal visual field angles for five subjects and the Navarro model eye. The data are derived from Zernike second-order aberrations with a 3-mm-diameter pupil. Open squares represent mean spherical corrections. Line lengths and orientations represent negative cylinder corrections and axes. The scale for the cylinders is that of the vertical axis. The lower extremity of CSA’s cylinder at +35° is outside the vertical plot limits.

Fig. 2
Fig. 2

Refraction differences between (-)40° temporal and (+)40° nasal visual field angles for five subjects and the Navarro model eye. The differences are between refractions that use Zernike second-order aberrations with a 3-mm-diameter pupil and (1) refractions that use Zernike second-order aberrations with a 6-mm-diameter pupil (squares), (2) refractions that use Zernike second-order aberrations with a 1-mm-diameter pupil (circles), and (3) refractions that use Zernike second- and fourth-order aberrations with a 6-mm pupil (triangles). The symbols represent the mean spherical correction differences, and line sizes and orientations represent the negative cylinder correction and axes differences. The scale for the cylinders is that of the vertical axis.

Fig. 3
Fig. 3

Differences in pupil dimensions when viewed on axis (left) and off axis (right).

Equations (66)

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

1/r=2W/R2.
C2-2Z2-2+C20Z20+C22Z22,
Z2-2=6 ρ2sin(2θ),
Z20=3(2ρ2-1),
Z22=6 ρ2cos(2θ)
Z4-2=10(4ρ4-3ρ)sin(2θ),
Z40=5(6ρ4-6ρ2+1),
Z42=10(4ρ4-3ρ2)cos(2θ).
Wm=23C20-65C40+127C60-209C80+3011C100- ,
W45sin(2θ)=(6C2-2-310C4-2+614C6-2-1018C8-2+1522C10-2-)×sin(2θ),
W0cos(2θ)=(6C22-310C42+614C62-1018C82+1522C102-)cos(2θ).
M=-2Wm/R2=-(43C20-125C40+247C60-409C80+6011C100-)/R2,
J45sin(2θ)=-2W45sin(2θ)/R2=-(26C2-2-610C4-2+1214C6-2-2018C8-2+3022C10-2-)×sin(2θ)/R2,
J180cos(2θ)=-2W0cos(2θ)/R2=-(26C22-610C42+1214C62-2018C82+3022C102-)×cos(2θ)/R2.
Mmod=M+ΔM.
C=-2J1802+J452,
S=Mmod-C/2,
α=[tan-1(J45/J180)]/2.
α=α+90°.
α=α+180°.
R2=R2(sin2 θ+cos2 θ cos2 ϕ),
tan γ=sin θ/(cos θ cos ϕ)=tan θ/cos ϕ.
tan θ=tan γ cos ϕ,
sin θ=tan θ/(1+tan2 θ)=tan γ cos ϕ/(1+tan2 γ cos2 ϕ),
cos θ=1/(1+tan2 θ)=1/(1+tan2 γ cos2 ϕ).
M+J45sin(2γ)+J180cos(2γ)
=-2[Wm+W45sin(2θ)+W0cos(2θ)]/R2,
R2=R2cos2 ϕ.
M+J180=-2(Wm+W0)/(R2cos2 ϕ).
R2=R2.
M-J180=-2(Wm-W0)/R2.
sin θ=cos ϕ/(1+cos2 ϕ),
cos θ=1/(1+cos2 ϕ).
R2=2R2cos2 ϕ/(1+cos2 ϕ).
M+J45=-[Wm+2W45cos ϕ/(1+cos2 ϕ)+W0sin2 ϕ/(1+cos2 ϕ)]/[R2cos2 ϕ/(1+cos2 ϕ)].
sin θ=-cos ϕ/(1+cos2 ϕ),
cos θ=1/(1+cos2 ϕ).
M-J45=-[Wm-2W45cos ϕ/(1+cos2 ϕ)+W0sin2 ϕ/(1+cos2 ϕ)]/[R2cos2 ϕ/(1+cos2 ϕ)].
(M+J180)+(M-J180)
=-2(Wm+W0)/(R2cos2 ϕ)-2(Wm-W0)/R2,
M=-[Wm(1+cos2 ϕ)+W0sin2 ϕ]/(R2cos2 ϕ).
(M+J180)-(M-J180)
=-2(Wm+W0)/(R2cos2 ϕ)+2(Wm-W0)/R2,
J180=-[W0(1+cos2 ϕ)+Wmsin2 ϕ]/(R2cos2 ϕ).
2J45=-W45cos ϕ/(1+cos2 ϕ)/[R2cos2 ϕ/(1+cos2 ϕ)],
J45=-2W45/(R2cos ϕ).
M=[M(1+cos2 ϕ)+J180sin2 ϕ]/(2 cos2 ϕ),
J180=[J180(1+cos2 ϕ)+M sin2 ϕ]/(2 cos2 ϕ),
J45=J45/cos ϕ.
M=-[(23C20-65C40+127C60-209C80+3011C100-)(1+cos2 ϕ)+(6C22-310C42+614C62-1018C82+1522C102-)sin2 ϕ]/(R2cos2 ϕ),
J45=-(26C2-2-610C4-2+1214C6-2-2018C8-2+3022C10-2-)/(R2cos ϕ),
J180=-[(23C20-65C40+127C60-209C80+3011C100-)sin2 ϕ+(6C22-310C42+614C62-1018C82+1522C102-)(1+cos2 ϕ)]/(R2cos2 ϕ).
ΔMR2/R2=ΔM+ΔJ45sin(2θ)+ΔJ180cos(2θ),
ΔMcos2 ϕ=ΔM+ΔJ180.
ΔM=ΔM-ΔJ180.
R2=R2(1+cos2 ϕ)/2.
ΔM(1+cos2 ϕ)/2=ΔM+ΔJ45.
ΔM(1+cos2 ϕ)/2=ΔM-ΔJ45.
ΔM=ΔM(1+cos2 ϕ)/2.
ΔJ180=-ΔM(sin2 ϕ)/2.
ΔJ45=0.
ΔM=-43ΔC20/R2;
ΔC20=-ΔMR2/(43).
ΔC20=-ΔMR2(1+cos2 ϕ)/(83).
ΔC22=-ΔJ180R2/(26).
ΔC22=ΔMR2sin2 ϕ/(46).

Metrics