Abstract

We have used the Hartmann–Shack technique previously to measure ocular aberrations along the horizontal meridian of the visual field. This requires considerable modifications from the technique for measuring the aberrations corresponding to the fovea. We now further develop the technique so that it can be used for any meridian of the visual field. Allowance is made for any auxiliary optics placed in front of the eye to compensate for the limited range of the Hartmann–Shack technique and for the case where aberrations are estimated at a wavelength other than the measuring wavelength. Zernike wave aberrations are converted to peripheral refractions. Examples are presented showing the developments, and we discuss change in wave aberrations when converting from a circular to an elliptical pupil.

© 2007 Optical Society of America

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References

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  1. D. A. Atchison, "Recent advances in measurement of monochromatic aberrations of human eyes," Clin. Exp. Optom. 88, 5-27 (2005).
    [CrossRef] [PubMed]
  2. R. Navarro, E. Moreno, and 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]
  3. A. Guirao and P. Artal, "Off-axis monochromatic aberrations estimated from double pass measurements in the human eye," Vision Res. 39, 207-217 (1999).
    [CrossRef] [PubMed]
  4. D. A. Atchison and D. H. Scott, "Monochromatic aberrations of human eyes in the horizontal visual field," J. Opt. Soc. Am. A 19, 2180-2184 (2002).
    [CrossRef]
  5. L. Lundström, P. Unsbo, and J. Gustafsson, "Off-axis wave front measurements for optical correction in eccentric viewing," J. Biomed. Opt. 10, 034002 (2005).
    [CrossRef] [PubMed]
  6. L. Lundström, J. Gustafsson, I. Svensson, and P. Unsbo, "Assessment of objective and subjective eccentric refraction," Optom. Vision Sci. 82, 298-306 (2005).
    [CrossRef]
  7. D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, 2000), p. 269.
  8. D. A. Atchison, D. H. Scott, and W. N. Charman, "Hartmann-Shack technique and refraction across the horizontal visual field," J. Opt. Soc. Am. A 20, 965-973 (2003).
    [CrossRef]
  9. L. Lundström and P. Unsbo, "Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils," J. Opt. Soc. Am. A 24, 569-577 (2007).
    [CrossRef]
  10. "Ophthalmics--methods for reporting optical aberrations of eyes," Doc. Z80.28-2004 (American National Standards Institute, 2004).
  11. T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, "Measurement of refractive errors in young myopes using the COAS Shack-Hartmann aberrometer," Optom. Vision Sci. 80, 6-14 (2003).
    [CrossRef]
  12. L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, "The chromatic eye: a new reduced-eye model of ocular chromatic aberration in humans," Appl. Opt. 31, 3594-3600 (1992).
    [CrossRef] [PubMed]
  13. S. Marcos, S. A. Burns, E. Moreno-Barriuso, and R. Navarro, "A new approach to the study of ocular chromatic aberrations," Vision Res. 39, 4309-4323 (1999).
    [CrossRef]
  14. L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, "Aberrations of the human eye in visible and near infrared illumination," Optom. Vision Sci. 80, 26-35 (2003).
    [CrossRef]
  15. T. O. Salmon and 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]
  16. Y.-Z. Wang and L. N. Thibos, "Oblique (off-axis) astigmatism of the reduced schematic eye with elliptical refracting surface," Optom. Vision Sci. 74, 557-562 (1997).
    [CrossRef]
  17. R. Navarro, J. Santamaría, and J. Bescós, "Accommodation-dependent model of the human eye with aspherics," J. Opt. Soc. Am. A 2, 1273-1281 (1985).
    [CrossRef] [PubMed]
  18. I. Escudero-Sanz and R. Navarro, "Off-axis aberrations of a wide-angle schematic eye model," J. Opt. Soc. Am. A 16, 1881-1891 (1999).
    [CrossRef]

2007 (1)

2005 (3)

L. Lundström, P. Unsbo, and J. Gustafsson, "Off-axis wave front measurements for optical correction in eccentric viewing," J. Biomed. Opt. 10, 034002 (2005).
[CrossRef] [PubMed]

L. Lundström, J. Gustafsson, I. Svensson, and P. Unsbo, "Assessment of objective and subjective eccentric refraction," Optom. Vision Sci. 82, 298-306 (2005).
[CrossRef]

D. A. Atchison, "Recent advances in measurement of monochromatic aberrations of human eyes," Clin. Exp. Optom. 88, 5-27 (2005).
[CrossRef] [PubMed]

2003 (3)

D. A. Atchison, D. H. Scott, and W. N. Charman, "Hartmann-Shack technique and refraction across the horizontal visual field," J. Opt. Soc. Am. A 20, 965-973 (2003).
[CrossRef]

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, "Measurement of refractive errors in young myopes using the COAS Shack-Hartmann aberrometer," Optom. Vision Sci. 80, 6-14 (2003).
[CrossRef]

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, "Aberrations of the human eye in visible and near infrared illumination," Optom. Vision Sci. 80, 26-35 (2003).
[CrossRef]

2002 (2)

1999 (3)

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

S. Marcos, S. A. Burns, E. Moreno-Barriuso, and R. Navarro, "A new approach to the study of ocular chromatic aberrations," Vision Res. 39, 4309-4323 (1999).
[CrossRef]

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

1998 (1)

1997 (1)

Y.-Z. Wang and L. N. Thibos, "Oblique (off-axis) astigmatism of the reduced schematic eye with elliptical refracting surface," Optom. Vision Sci. 74, 557-562 (1997).
[CrossRef]

1992 (1)

1985 (1)

Artal, P.

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

Atchison, D. A.

Bescós, J.

Bradley, A.

Burns, S. A.

S. Marcos, S. A. Burns, E. Moreno-Barriuso, and R. Navarro, "A new approach to the study of ocular chromatic aberrations," Vision Res. 39, 4309-4323 (1999).
[CrossRef]

Charman, W. N.

Diaz-Santana, L.

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, "Aberrations of the human eye in visible and near infrared illumination," Optom. Vision Sci. 80, 26-35 (2003).
[CrossRef]

Dorronsoro, C.

Escudero-Sanz, I.

Gasser, W.

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, "Measurement of refractive errors in young myopes using the COAS Shack-Hartmann aberrometer," Optom. Vision Sci. 80, 6-14 (2003).
[CrossRef]

Guirao, A.

A. Guirao and 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.

L. Lundström, P. Unsbo, and J. Gustafsson, "Off-axis wave front measurements for optical correction in eccentric viewing," J. Biomed. Opt. 10, 034002 (2005).
[CrossRef] [PubMed]

L. Lundström, J. Gustafsson, I. Svensson, and P. Unsbo, "Assessment of objective and subjective eccentric refraction," Optom. Vision Sci. 82, 298-306 (2005).
[CrossRef]

Kenmore, T.

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, "Measurement of refractive errors in young myopes using the COAS Shack-Hartmann aberrometer," Optom. Vision Sci. 80, 6-14 (2003).
[CrossRef]

Lara-Saucedo, D.

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, "Aberrations of the human eye in visible and near infrared illumination," Optom. Vision Sci. 80, 26-35 (2003).
[CrossRef]

Llorente, L.

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, "Aberrations of the human eye in visible and near infrared illumination," Optom. Vision Sci. 80, 26-35 (2003).
[CrossRef]

Lundström, L.

L. Lundström and P. Unsbo, "Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils," J. Opt. Soc. Am. A 24, 569-577 (2007).
[CrossRef]

L. Lundström, J. Gustafsson, I. Svensson, and P. Unsbo, "Assessment of objective and subjective eccentric refraction," Optom. Vision Sci. 82, 298-306 (2005).
[CrossRef]

L. Lundström, P. Unsbo, and J. Gustafsson, "Off-axis wave front measurements for optical correction in eccentric viewing," J. Biomed. Opt. 10, 034002 (2005).
[CrossRef] [PubMed]

Marcos, S.

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, "Aberrations of the human eye in visible and near infrared illumination," Optom. Vision Sci. 80, 26-35 (2003).
[CrossRef]

S. Marcos, S. A. Burns, E. Moreno-Barriuso, and R. Navarro, "A new approach to the study of ocular chromatic aberrations," Vision Res. 39, 4309-4323 (1999).
[CrossRef]

Moreno, E.

Moreno-Barriuso, E.

S. Marcos, S. A. Burns, E. Moreno-Barriuso, and R. Navarro, "A new approach to the study of ocular chromatic aberrations," Vision Res. 39, 4309-4323 (1999).
[CrossRef]

Navarro, R.

Salmon, T. O.

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, "Measurement of refractive errors in young myopes using the COAS Shack-Hartmann aberrometer," Optom. Vision Sci. 80, 6-14 (2003).
[CrossRef]

T. O. Salmon and 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]

Santamaría, J.

Scott, D. H.

Smith, G.

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, 2000), p. 269.

Svensson, I.

L. Lundström, J. Gustafsson, I. Svensson, and P. Unsbo, "Assessment of objective and subjective eccentric refraction," Optom. Vision Sci. 82, 298-306 (2005).
[CrossRef]

Thibos, L. N.

Unsbo, P.

L. Lundström and P. Unsbo, "Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils," J. Opt. Soc. Am. A 24, 569-577 (2007).
[CrossRef]

L. Lundström, J. Gustafsson, I. Svensson, and P. Unsbo, "Assessment of objective and subjective eccentric refraction," Optom. Vision Sci. 82, 298-306 (2005).
[CrossRef]

L. Lundström, P. Unsbo, and J. Gustafsson, "Off-axis wave front measurements for optical correction in eccentric viewing," J. Biomed. Opt. 10, 034002 (2005).
[CrossRef] [PubMed]

Wang, Y.-Z.

Y.-Z. Wang and L. N. Thibos, "Oblique (off-axis) astigmatism of the reduced schematic eye with elliptical refracting surface," Optom. Vision Sci. 74, 557-562 (1997).
[CrossRef]

West, R. W.

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, "Measurement of refractive errors in young myopes using the COAS Shack-Hartmann aberrometer," Optom. Vision Sci. 80, 6-14 (2003).
[CrossRef]

Ye, M.

Zhang, X.

Appl. Opt. (1)

Clin. Exp. Optom. (1)

D. A. Atchison, "Recent advances in measurement of monochromatic aberrations of human eyes," Clin. Exp. Optom. 88, 5-27 (2005).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

L. Lundström, P. Unsbo, and J. Gustafsson, "Off-axis wave front measurements for optical correction in eccentric viewing," J. Biomed. Opt. 10, 034002 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (7)

Optom. Vision Sci. (4)

L. Llorente, L. Diaz-Santana, D. Lara-Saucedo, and S. Marcos, "Aberrations of the human eye in visible and near infrared illumination," Optom. Vision Sci. 80, 26-35 (2003).
[CrossRef]

Y.-Z. Wang and L. N. Thibos, "Oblique (off-axis) astigmatism of the reduced schematic eye with elliptical refracting surface," Optom. Vision Sci. 74, 557-562 (1997).
[CrossRef]

T. O. Salmon, R. W. West, W. Gasser, and T. Kenmore, "Measurement of refractive errors in young myopes using the COAS Shack-Hartmann aberrometer," Optom. Vision Sci. 80, 6-14 (2003).
[CrossRef]

L. Lundström, J. Gustafsson, I. Svensson, and P. Unsbo, "Assessment of objective and subjective eccentric refraction," Optom. Vision Sci. 82, 298-306 (2005).
[CrossRef]

Vision Res. (2)

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

S. Marcos, S. A. Burns, E. Moreno-Barriuso, and R. Navarro, "A new approach to the study of ocular chromatic aberrations," Vision Res. 39, 4309-4323 (1999).
[CrossRef]

Other (2)

D. A. Atchison and G. Smith, Optics of the Human Eye (Butterworth-Heinemann, 2000), p. 269.

"Ophthalmics--methods for reporting optical aberrations of eyes," Doc. Z80.28-2004 (American National Standards Institute, 2004).

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Figures (3)

Fig. 1
Fig. 1

(a) Transformation of a rotationally symmetrical wavefront on an elliptical pupil to a wavefront on a circular pupil; (b) transformation of a rotationally symmetrical wavefront on a circular pupil to a wavefront on an elliptical pupil.

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

Relationship between α , ϕ , κ , and μ, shown from the perspective of an observer looking at a subject’s eye.

Tables (1)

Tables Icon

Table 1 Off-Axis Aberrations and Refractions for Model Eyes

Equations (105)

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κ = tan 1 ( tan ϕ cos α ) , μ = tan 1 ( tan ϕ sin α ) .
x = x cos κ , x a = x a cos κ ,
y = y cos μ , y a = y a cos μ .
W ( x , y ) x = Δ x f , W ( x , y ) y = Δ y f ,
Δ x = x x a , Δ y = y y a ;
Δ x = ( x x a ) cos 2 κ , Δ y = ( y y a ) cos 2 μ ,
W ( x , y ) x = Δ x f , W ( x , y ) y = Δ y f .
R 2 = R 2 ( sin 2 β + cos 2 β cos 2 ϕ ) .
Δ M R 2 R 2 = Δ M + Δ J 45 sin 2 θ + Δ J 180 cos 2 θ ,
R 2 = R 2 [ sin 2 ( θ α ) + cos 2 ( θ α ) cos 2 ϕ ] .
R 2 = R 2 ( sin 2 α + cos 2 α cos 2 ϕ ) .
Δ M ( sin 2 α + cos 2 α cos 2 ϕ ) = Δ M + Δ J 180 .
R 2 = R 2 ( cos 2 α + sin 2 α cos 2 ϕ ) .
Δ M ( cos 2 α + sin 2 α cos 2 ϕ ) = Δ M Δ J 180 .
Δ M ( 1 + cos 2 ϕ ) = 2 Δ M
Δ M = Δ M ( 1 + cos 2 ϕ ) 2 .
Δ J 180 = Δ M cos 2 α sin 2 ϕ 2 .
R 2 = R 2 [ ( 1 2 cos α 1 2 sin α ) 2 + ( 1 2 cos α + 1 2 sin α ) 2 cos 2 ϕ ] .
Δ M 2 [ ( cos α sin α ) 2 + ( cos α + sin α ) 2 cos 2 ϕ ] = Δ M + Δ J 45 .
R 2 = R 2 [ ( 1 2 cos α + 1 2 sin α ) 2 + ( 1 2 cos α + 1 2 sin α ) 2 cos 2 ϕ ] .
Δ M 2 [ ( cos α + sin α ) 2 + ( sin α cos α ) 2 cos 2 ϕ ] = Δ M Δ J 45 .
Δ M sin 2 α sin 2 ϕ = 2 Δ J 45
Δ J 45 = Δ M sin 2 α sin 2 ϕ 2 .
1 r = 2 W R 2 .
C 2 2 Z 2 2 + C 2 0 Z 2 0 + C 2 2 Z 2 2 ,
Z 2 2 = 6 ρ 2 sin 2 θ ,
Z 2 0 = 3 ( 2 ρ 2 1 ) ,
Z 2 2 = 6 ρ 2 cos 2 θ ,
Z 4 2 = 10 ( 4 ρ 4 3 ρ 2 ) sin 2 θ ,
Z 4 0 = 5 ( 6 ρ 4 6 ρ 2 + 1 ) ,
Z 4 2 = 10 ( 4 ρ 4 3 ρ 2 ) cos 2 θ .
W m = 2 3 C 2 0 6 5 C 4 0 + 12 7 C 6 0 20 9 C 8 0 + 30 11 C 10 0 ,
W 45 sin 2 θ = ( 6 C 2 2 3 10 C 4 2 + 6 14 C 6 2 10 18 C 8 2 + 15 22 C 10 2 ) sin 2 θ ,
W 180 cos 2 θ = ( 6 C 2 2 3 10 C 4 2 + 6 14 C 6 2 10 18 C 8 2 + 15 22 C 10 2 ) cos 2 θ .
M = 2 W m R 2 = ( 4 3 C 2 0 12 5 C 4 0 + 24 7 C 6 0 40 9 C 8 0 + 60 11 C 10 0 ) R 2 ,
J 45 sin 2 θ = 2 W 45 sin 2 θ R 2 = ( 2 6 C 2 2 6 10 C 4 2 + 12 14 C 6 2 20 18 C 8 2 + 30 22 C 10 2 ) sin 2 θ R 2 ,
J 180 cos 2 θ = 2 W 180 cos 2 θ R 2 = ( 2 6 C 2 2 6 10 C 4 2 + 12 14 C 6 2 20 18 C 8 2 + 30 22 C 10 2 ) cos 2 θ R 2 .
Δ M = 4 3 Δ C 2 0 R 2 ;
Δ C 2 0 = Δ M R 2 ( 4 3 ) .
Δ C 2 0 = Δ M R 2 ( 1 + cos 2 ϕ ) ( 8 3 ) .
Δ C 2 2 = Δ J 45 R 2 ( 2 6 ) .
Δ C 2 2 = Δ M R 2 sin 2 α sin 2 ϕ ( 4 6 ) .
Δ C 2 2 = Δ J 180 R 2 ( 2 6 ) ,
Δ C 2 2 = Δ M R 2 cos 2 α sin 2 ϕ ( 4 6 ) .
n λ = 1.320535 + 4.685 ( λ 214.102 ) .
Δ M λ = [ Δ M λ ¯ 1 L + D ] n λ 1 n λ ¯ 1 + P e f f .
C 2 λ 0 = C 2 λ ¯ 0 + Δ C 2 λ 0 = C 2 λ ¯ 0 Δ M λ R 2 ( 1 + cos 2 ϕ ) 8 3 ,
C 2 λ 2 = C 2 λ ¯ 2 n λ 1 n λ ¯ 1 + Δ C 2 λ 2
= C 2 λ ¯ 2 n λ 1 n λ ¯ 1 + Δ M λ R 2 sin 2 α sin 2 ϕ 4 6 ,
C 2 λ 2 = C 2 λ ¯ 2 n λ 1 n λ ¯ 1 + Δ C 2 λ 2
= C 2 λ ¯ 2 n λ 1 n λ ¯ 1 + Δ M λ R 2 cos 2 α sin 2 ϕ 4 6 .
C n λ m = C n λ ¯ m n λ 1 n λ ¯ 1 .
tan γ = sin β ( cos β cos ϕ ) .
tan β = tan γ cos ϕ ,
sin β = tan γ cos ϕ 1 + tan 2 γ cos 2 ϕ ,
cos β = 1 1 + tan 2 γ cos 2 ϕ .
R 2 = R 2 cos 2 ϕ ( 1 + tan 2 γ ) 1 + tan 2 γ cos 2 ϕ .
R 2 = R 2 cos 2 ϕ ( 1 + tan 2 α ) 1 + tan 2 α cos 2 ϕ .
tan γ = ( tan 45 tan α ) ( 1 + tan 45 tan α ) = ( 1 tan α ) ( 1 + tan α ) ,
R 2 = 2 R 2 cos 2 ϕ ( 1 + tan 2 α ) ( 1 + tan α ) 2 + cos 2 ϕ ( 1 tan α ) 2 .
tan γ = 1 tan α ,
R 2 = R 2 cos 2 ϕ ( 1 + tan 2 α ) tan 2 α + cos 2 ϕ .
tan γ = ( tan 135 tan α ) ( 1 + tan 135 tan α ) = ( 1 + tan α ) ( 1 tan α ) ,
R 2 = 2 R 2 cos 2 ϕ ( 1 + tan 2 α ) ( 1 tan α ) 2 + cos 2 ϕ ( 1 + tan α ) 2 .
sin 2 θ = sin ( 2 α + β ) = sin 2 β cos 2 α + cos 2 β sin 2 α = cos 2 α ( 2 sin β cos β ) + sin 2 α ( cos 2 β sin 2 β ) .
sin 2 θ = 2 cos 2 α tan γ cos ϕ + sin 2 α ( 1 tan 2 γ cos 2 ϕ ) 1 + tan 2 γ cos 2 ϕ .
cos 2 θ = cos ( 2 α + β ) = cos 2 α ( cos 2 β sin 2 β ) sin 2 α ( 2 sin β cos β ) .
cos 2 θ = cos 2 α ( 1 tan 2 γ cos 2 ϕ ) 2 sin 2 α tan γ cos ϕ 1 + tan 2 γ cos 2 ϕ .
sin 2 θ = 2 cos 2 α tan α cos ϕ + sin 2 α ( 1 tan 2 α cos 2 ϕ ) 1 + tan 2 α cos 2 ϕ ,
cos 2 θ = cos 2 α ( 1 tan 2 α cos 2 ϕ ) + 2 sin 2 α tan α cos ϕ 1 + tan 2 α cos 2 ϕ .
( tan 45 tan α ) ( 1 + tan 45 tan α ) = ( 1 tan α ) ( 1 + tan α )
sin 2 θ = 2 cos 2 α ( 1 tan 2 α ) cos ϕ + sin 2 α [ ( 1 + tan α ) 2 ( 1 tan α ) 2 cos 2 ϕ ] ( 1 + tan α ) 2 + ( 1 tan α ) 2 cos 2 ϕ ,
cos 2 θ = cos 2 α [ ( 1 + tan α ) 2 ( 1 tan α ) 2 cos 2 ϕ ] 2 sin 2 α ( 1 tan 2 α ) cos ϕ ( 1 + tan α ) 2 + ( 1 tan α ) 2 cos 2 ϕ .
sin 2 θ = 2 cos 2 α tan α cos ϕ + sin 2 α ( tan 2 α cos 2 ϕ ) tan 2 α + cos 2 ϕ ,
cos 2 θ = cos 2 α ( tan 2 α cos 2 ϕ ) 2 sin 2 α tan α cos ϕ tan 2 α + cos 2 ϕ .
tan γ = ( tan 135 tan α ) ( 1 + tan 135 tan α ) = ( 1 + tan α ) ( 1 tan α ) ,
sin 2 θ = 2 cos 2 α ( 1 tan 2 α ) cos ϕ + sin 2 α [ ( 1 tan α ) 2 ( 1 + tan α ) 2 cos 2 ϕ ] ( 1 tan α ) 2 + ( 1 + tan α ) 2 cos 2 ϕ ,
cos 2 θ = cos 2 α [ ( 1 tan α ) 2 ( 1 + tan α ) 2 cos 2 ϕ ] + 2 sin 2 α ( 1 tan 2 α ) cos ϕ ( 1 tan α ) 2 + ( 1 + tan α ) 2 cos 2 ϕ .
M + J 45 sin 2 ε + J 180 cos 2 ε = 2 ( W m + W 45 sin 2 θ + W 180 cos 2 θ ) R 2 ,
M + J 180 = 2 [ W m ( 1 + tan 2 α cos 2 ϕ ) + W 45 [ 2 cos 2 α tan α cos ϕ + sin 2 α ( 1 tan 2 α cos 2 ϕ ) ] + W 180 [ cos 2 α ( 1 tan 2 α cos 2 ϕ ) + 2 sin 2 α tan α cos ϕ ] ] R 2 cos 2 ϕ ( 1 + tan 2 α ) .
M J 180 = 2 [ W m ( tan 2 α + cos 2 ϕ ) + W 45 [ 2 cos 2 α tan α cos ϕ + sin 2 α ( tan 2 α cos 2 ϕ ) ] + W 180 [ cos 2 α ( tan 2 α cos 2 ϕ ) 2 sin 2 α tan α cos ϕ ] ] R 2 cos 2 ϕ ( 1 + tan 2 α ) .
2 M = 2 [ W m ( 1 + cos 2 ϕ ) + W 45 sin 2 α sin 2 ϕ + W 180 cos 2 α sin 2 ϕ ] R 2 cos 2 ϕ .
M = M ( 1 + cos 2 ϕ ) + J 45 sin 2 α sin 2 ϕ + J 180 cos 2 α sin 2 ϕ 2 cos 2 ϕ .
2 J 180 = 2 { W m cos 2 α sin 2 ϕ + W 45 cos 2 α sin 2 α ( 1 + cos ϕ ) 2 + W 180 [ cos 2 2 α ( 1 + cos 2 ϕ ) + 2 sin 2 2 α cos ϕ ] } R 2 cos 2 ϕ .
J 180 = { M cos 2 α sin 2 ϕ + J 45 cos 2 α sin 2 α ( 1 + cos ϕ ) 2 + J 180 [ cos 2 2 α ( 1 + cos 2 ϕ ) + 2 sin 2 2 α cos ϕ ] } 2 cos 2 ϕ .
M + J 45 = [ W m [ ( 1 + tan α ) 2 + cos 2 ϕ ( 1 tan α ) 2 ] + W 45 { 2 cos 2 α ( 1 tan 2 α ) cos ϕ + sin 2 α [ ( 1 + tan α ) 2 ( 1 tan α ) 2 cos 2 ϕ ] } + W 180 { cos 2 α [ ( 1 + tan α ) 2 ( 1 tan α ) 2 cos 2 ϕ ] 2 sin 2 α ( 1 tan 2 α ) cos ϕ } ] R 2 cos 2 ϕ ( 1 + tan 2 α ) .
M J 45 = [ W m [ ( 1 tan α ) 2 + ( 1 + tan α ) 2 cos 2 ϕ ] + W 45 { 2 cos 2 α ( 1 tan 2 α ) cos ϕ + sin 2 α [ ( 1 tan α ) 2 ( 1 + tan α ) 2 cos 2 ϕ ] } + W 180 { cos 2 α [ ( 1 tan α ) 2 ( 1 + tan α ) 2 cos 2 ϕ ] + 2 sin 2 α ( 1 tan 2 α ) cos ϕ } ] R 2 cos 2 ϕ ( 1 + tan 2 α ) .
2 J 45 = 2 { W m sin 2 α sin 2 ϕ + W 45 cos 2 α sin 2 α ( 1 + cos ϕ ) 2 + W 180 [ cos 2 α sin 2 α ( 1 + cos 2 ϕ ) 2 sin 2 α cos 2 α cos ϕ ] } R 2 cos 2 ϕ .
J 45 = { M sin 2 α sin 2 ϕ + J 45 cos 2 α sin 2 α ( 1 + cos ϕ ) 2 + J 180 [ cos 2 α sin 2 α ( 1 + cos 2 ϕ ) 2 sin 2 α cos 2 α cos ϕ ] } 2 cos 2 ϕ .
M = [ ( 2 3 C 2 0 6 5 C 4 0 + 12 7 C 6 0 20 9 C 8 0 + 30 11 C 10 0 ) ( 1 + cos 2 ϕ ) + ( 6 C 2 2 3 10 C 4 2 + 6 14 C 6 2 10 18 C 8 2 + 15 22 C 10 2 ) sin 2 α sin 2 ϕ + ( 6 C 2 2 3 10 C 4 2 + 6 14 C 6 2 10 18 C 8 2 + 15 22 C 10 2 ) cos 2 α sin 2 ϕ ] R 2 cos 2 ϕ ,
J 45 = [ ( 2 3 C 2 0 6 5 C 4 0 + 12 7 C 6 0 20 9 C 8 0 + 30 11 C 10 0 ) sin 2 α sin 2 ϕ + ( 6 C 2 2 3 10 C 4 2 + 6 14 C 6 2 10 18 C 8 2 + 15 22 C 10 2 ) [ 2 cos 2 2 α cos ϕ + sin 2 2 α ( 1 + cos 2 ϕ ) ] + ( 6 C 2 2 3 10 C 4 2 + 6 14 C 6 2 10 18 C 8 2 + 15 22 C 10 2 ) cos 2 α sin 2 α ( 1 + cos ϕ ) 2 ] R 2 cos 2 ϕ ,
J 180 = [ ( 2 3 C 2 0 6 5 C 4 0 + 12 7 C 6 0 20 9 C 8 0 + 30 11 C 10 0 ) cos 2 α sin 2 ϕ + ( 6 C 2 2 3 10 C 4 2 + 6 14 C 6 2 10 18 C 8 2 + 15 22 C 10 2 ) cos 2 α sin 2 α ( 1 + cos ϕ ) 2 + ( 6 C 2 2 3 10 C 4 2 + 6 14 C 6 2 10 18 C 8 2 + 15 22 C 10 2 ) [ cos 2 2 α ( 1 + cos 2 ϕ ) + 2 sin 2 2 α cos ϕ ] ] R 2 cos 2 ϕ .
C = 2 J 180 2 + J 45 2 ,
S = M C 2 ,
α = tan 1 ( J 45 J 180 ) 2 .
α = α + 90 .
α = α + 180 .
W = B ( X 2 + Y 2 ) 2 = B R 4 ,
X = X cos ϕ , Y = Y .
W = [ B ( X cos ϕ ) 2 + Y 2 ] 2 ,
W = B cos 4 ϕ ( X 4 + 2 X 2 Y 2 cos 2 ϕ + Y 4 cos 4 ϕ ) .
W = B cos 4 ϕ [ ( X 2 + Y 2 ) 2 2 X 2 Y 2 sin 2 ϕ + Y 4 ( sin 4 ϕ 2 sin 2 ϕ ) ] .
W = C ( X 2 + Y 2 ) 2 ,
W = C ( X 4 cos 4 ϕ + 2 X 2 Y 2 cos 2 ϕ + Y 4 ) .
W = C [ ( X 2 + Y 2 ) 2 2 X 2 Y 2 sin 2 ϕ + X 4 ( sin 4 ϕ sin 2 ϕ ) ] .

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