Abstract

In wavefront-driven vision correction, ocular aberrations are often measured on the pupil plane and the correction is applied on a different plane. The problem with this practice is that any changes undergone by the wavefront as it propagates between planes are not currently included in devising customized vision correction. With some valid approximations, we have developed an analytical foundation based on geometric optics in which Zernike polynomials are used to characterize the propagation of the wavefront from one plane to another. Both the boundary and the magnitude of the wavefront change after the propagation. Taylor monomials were used to realize the propagation because of their simple form for this purpose. The method we developed to identify changes in low-order aberrations was verified with the classical vertex correction formula. The method we developed to identify changes in high-order aberrations was verified with ZEMAX ray-tracing software. Although the method may not be valid for highly irregular wavefronts and it was only proven for wavefronts with low-order or high-order aberrations, our analysis showed that changes in the propagating wavefront are significant and should, therefore, be included in calculating vision correction. This new approach could be of major significance in calculating wavefront-driven vision correction whether by refractive surgery, contact lenses, intraocular lenses, or spectacles.

© 2009 Optical Society of America

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References

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  1. J. Liang, W. Grimm, S. Goelz, and 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. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eyes,” J. Opt. Soc. Am. A 14, 2873-2883 (1997).
    [CrossRef]
  3. J. Liang, D. R. Williams, and D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884-2892 (1997).
    [CrossRef]
  4. A. Roorda and D. R. Williams, “The arrangement of the three cone classes in the living human eye,” Nature 397, 520-522 (1999).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  8. D. A. Chernyak, “Cyclotorsional eye motion occurring between wavefront measurement and refractive surgery,” J. Cataract Refract. Surg. 30, 633-638 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. H. Shu, L. Luo, and G. Han, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960-1968 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. 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]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  24. G.-M. Dai, “Wavefront expansion basis functions and their relationships,” J. Opt. Soc. Am. A 23, 1657-1666 (2006).
    [CrossRef]
  25. G.-M. Dai, “Wavefront expansion basis functions and their relationships: errata,” J. Opt. Soc. Am. A 23, 2970-2971(2006).
    [CrossRef]
  26. American National Standards Institute, “Methods for reporting optical aberrations of eyes,” ANSI Z80.28-2004 (Optical Laboratories Association, 2004), Annex B, pp. 19-28.
  27. 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]

2007 (1)

2006 (6)

2004 (2)

E. Donnenfeld, “The pupil is a moving target: centration, repeatability, and registration,” J. Refract. Surg. 20, 593-596 (2004).

D. A. Chernyak, “Cyclotorsional eye motion occurring between wavefront measurement and refractive surgery,” J. Cataract Refract. Surg. 30, 633-638 (2004).
[CrossRef] [PubMed]

2003 (3)

2002 (2)

P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” Proc. SPIE 4769, 130-144 (2002).
[CrossRef]

J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937-1945 (2002).
[CrossRef]

2001 (2)

2000 (1)

1999 (1)

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

1997 (2)

1996 (1)

W. F. Harris, “Wavefronts and their propagation in astigmatic systems,” Optom. Vis. Sci. 73, 606-612 (1996).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

M. A. Wilson, M. C. W. Campbell, and P. Simonet, “Change of pupil centration with change of illumination and pupil size,” Optom. Vis. Sci. 69, 129-136 (1992).
[CrossRef] [PubMed]

1988 (1)

G. Walsh, “The effect of mydriasis on the pupillary centration of the human eye,” Ophthal. Physiol. Opt. 8, 178-182 (1988).
[CrossRef]

1976 (1)

Ares, J.

Arines, J.

Atchison, D. A.

Bará, S.

Bille, J. F.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Campbell, C. E.

Charman, W. N.

Chernyak, D. A.

D. A. Chernyak, “Cyclotorsional eye motion occurring between wavefront measurement and refractive surgery,” J. Cataract Refract. Surg. 30, 633-638 (2004).
[CrossRef] [PubMed]

Cox, I.

Dai, G.-M.

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, “A concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

Donnenfeld, E.

E. Donnenfeld, “The pupil is a moving target: centration, repeatability, and registration,” J. Refract. Surg. 20, 593-596 (2004).

E. M. Janssen, A. J.

A. J. E. M. Janssen and P. Dirksen, “A concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

Geary, K.

Goelz, S.

Goldberg, K. A.

Grimm, W.

Guirao, A.

Han, G.

Harris, W. F.

W. F. Harris, “Wavefronts and their propagation in astigmatic systems,” Optom. Vis. Sci. 73, 606-612 (1996).
[CrossRef] [PubMed]

Liang, J.

Lundström, L.

Luo, L.

Mancebo, T.

Miller, D. T.

Moreno-Barriuso, E.

Noll, R. J.

Pankretz, G. S.

P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” Proc. SPIE 4769, 130-144 (2002).
[CrossRef]

Prado, P.

Riera, P. R.

P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” Proc. SPIE 4769, 130-144 (2002).
[CrossRef]

Roorda, A.

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

Schwiegerling, J.

Scott, D. H.

Shu, H.

Simonet, P.

M. A. Wilson, M. C. W. Campbell, and P. Simonet, “Change of pupil centration with change of illumination and pupil size,” Optom. Vis. Sci. 69, 129-136 (1992).
[CrossRef] [PubMed]

Thibos, L. N.

L. N. Thibos, “Propagation of astigmatic wavefronts using power vectors,” S. Afr. Optom. 62, 111-113 (2003).

Topa, D. M.

P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” Proc. SPIE 4769, 130-144 (2002).
[CrossRef]

Unsbo, P.

W. Campbell, M. C.

M. A. Wilson, M. C. W. Campbell, and P. Simonet, “Change of pupil centration with change of illumination and pupil size,” Optom. Vis. Sci. 69, 129-136 (1992).
[CrossRef] [PubMed]

Walsh, G.

G. Walsh, “The effect of mydriasis on the pupillary centration of the human eye,” Ophthal. Physiol. Opt. 8, 178-182 (1988).
[CrossRef]

Williams, D.

Williams, D. R.

Wilson, M. A.

M. A. Wilson, M. C. W. Campbell, and P. Simonet, “Change of pupil centration with change of illumination and pupil size,” Optom. Vis. Sci. 69, 129-136 (1992).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Appl. Opt. (1)

J. Cataract Refract. Surg. (1)

D. A. Chernyak, “Cyclotorsional eye motion occurring between wavefront measurement and refractive surgery,” J. Cataract Refract. Surg. 30, 633-638 (2004).
[CrossRef] [PubMed]

J. Microlith. Microfab. Microsyst. (1)

A. J. E. M. Janssen and P. Dirksen, “A concise formula for the Zernike coefficients of scaled pupils,” J. Microlith. Microfab. Microsyst. 5, 030501 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Liang, W. Grimm, S. Goelz, and 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. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eyes,” J. Opt. Soc. Am. A 14, 2873-2883 (1997).
[CrossRef]

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

A. Guirao, D. Williams, and I. Cox, “Effect of the rotation and translation on the expected benefit of an ideal method to correct the eye's high-order aberrations,” J. Opt. Soc. Am. A 18, 1003-1015 (2001).
[CrossRef]

K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146-2152 (2001).
[CrossRef]

J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937-1945 (2002).
[CrossRef]

C. E. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209-217 (2003).
[CrossRef]

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]

G.-M. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23, 539-543(2006).
[CrossRef]

G.-M. Dai, “Wavefront expansion basis functions and their relationships,” J. Opt. Soc. Am. A 23, 1657-1666 (2006).
[CrossRef]

H. Shu, L. Luo, and G. Han, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960-1968 (2006).
[CrossRef]

S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061-2066 (2006).
[CrossRef]

G.-M. Dai, “Wavefront expansion basis functions and their relationships: errata,” J. Opt. Soc. Am. A 23, 2970-2971(2006).
[CrossRef]

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]

J. Refract. Surg. (1)

E. Donnenfeld, “The pupil is a moving target: centration, repeatability, and registration,” J. Refract. Surg. 20, 593-596 (2004).

Nature (1)

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

Ophthal. Physiol. Opt. (1)

G. Walsh, “The effect of mydriasis on the pupillary centration of the human eye,” Ophthal. Physiol. Opt. 8, 178-182 (1988).
[CrossRef]

Optom. Vis. Sci. (2)

M. A. Wilson, M. C. W. Campbell, and P. Simonet, “Change of pupil centration with change of illumination and pupil size,” Optom. Vis. Sci. 69, 129-136 (1992).
[CrossRef] [PubMed]

W. F. Harris, “Wavefronts and their propagation in astigmatic systems,” Optom. Vis. Sci. 73, 606-612 (1996).
[CrossRef] [PubMed]

Proc. SPIE (1)

P. R. Riera, G. S. Pankretz, and D. M. Topa, “Efficient computation with special functions like the circle polynomials of Zernike,” Proc. SPIE 4769, 130-144 (2002).
[CrossRef]

S. Afr. Optom. (1)

L. N. Thibos, “Propagation of astigmatic wavefronts using power vectors,” S. Afr. Optom. 62, 111-113 (2003).

Other (2)

American National Standards Institute, “Methods for reporting optical aberrations of eyes,” ANSI Z80.28-2004 (Optical Laboratories Association, 2004), Annex B, pp. 19-28.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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Figures (5)

Fig. 1
Fig. 1

Geometry for the vertex correction (the eye is on the right hand side). (a) Myopic case, (b) hyperopic case. Note the focal length is f before the vertex correction and the vertex distance is d, both in meters.

Fig. 2
Fig. 2

Examples of wavefront propagation according to the Huygens principle for (a) a diverging defocus and (b) a spherical aberration.

Fig. 3
Fig. 3

Geometry for (a) myopic and (b) hyperopic wavefront with a pupil radius R propagated a distance d toward the eye to a new wavefront with a pupil radius R .

Fig. 4
Fig. 4

A low-order wavefront ( c 2 2 = 1 μm , c 2 0 = 3 μm , c 2 2 = 2 μm , and R = 3 mm ) propagates to become an elliptical wavefront. (a)  d = 3.5 mm , (b)  d = 12.5 mm , (c)  d = 120 mm . The aspect ratios of the ellipses are 0.9914, 0.9691, and 0.6638, respectively.

Fig. 5
Fig. 5

Contour plots of wavefronts involving a propagation. (a) The original wavefront, (b) the propagated wavefront using the analytical formulas presented in this paper, (c) the propagated wavefront using the ZEMAX ray-tracing software.

Tables (3)

Tables Icon

Table 1 Zernike Coefficients g i of the Direction Factor Expressed as Those in the Original Wavefront c i , up to the Fourth Order

Tables Icon

Table 2 Comparison of Zernike Coefficients, in μm , After a Random Wavefront Propagated by 12.5 mm Using the Analytical Method and ZEMAX

Tables Icon

Table 3 Coordinate Error in μm in x and y for a Normal Eye and a Highly Irregular Eye When the Propagation Distance d is 3.5 mm and 12.5 mm , Respectively

Equations (63)

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

S = 1 f ,
S = 1 f d ,
S = S 1 S d .
S = S 1 S d ,
S + C = S + C 1 ( S + C ) d .
cos ψ = 1 1 + [ W ( x , y ) x ] 2 + [ W ( x , y ) y ] 2 .
cos ψ = 1 1 2 [ W ( x , y ) x ] 2 1 2 [ W ( x , y ) y ] 2 .
cos ψ = 1 1 2 R 2 a ( u , υ ) ,
a ( u , υ ) = [ W ( u , υ ) u ] 2 + [ W ( u , υ ) υ ] 2 .
d = d cos ψ .
W W = d d = d ( 1 cos ψ ) = d 2 R 2 a ( u , υ ) .
W W = d d = d ( 1 cos ψ ) = d 2 R 2 a ( u , υ ) ,
[ W ( u , υ ) u ] 2 = p , q p , q α p q α p q q q T p + p 2 q + q 2 ( u , υ ) ,
[ W ( u , υ ) υ ] 2 = p , q p , q α p q α p q ( p q ) ( p q ) T p + p 2 q + q ( u , υ ) ,
W ( ρ , θ ) = i = 1 J α i T i ( ρ , θ ) = p , q α p q ρ p cos q θ sin p q θ = p , q α p q u q υ p q ,
p = int [ ( 8 i + 1 1 ) / 2 ] ,
q = ( 2 i p 2 p ) / 2 ,
i = p ( p + 1 ) 2 + q .
n = int [ ( 8 i + 1 1 ) / 2 ] ,
m = 2 i n ( n + 2 ) .
i = n 2 + 2 n + m 2 .
a ( u , υ ) = p , q p , q α p q α p q q q T p + p 2 q + q 2 ( u , υ ) + p , q p , q α p q α p q ( p q ) ( p q ) T p + p 2 q + q ( u , υ ) = i = 1 J β i T i ( u , υ ) ,
W ( u , υ ) = W ( u , υ ) + | d | 2 R 2 a ( u , υ ) = i = 1 J α i T i ( u , υ ) + | d | 2 R 2 i = 1 J β i T i ( u , υ ) = i = 1 J ( α i + | d | 2 R 2 β i ) T i ( u , υ ) ,
W ( ρ , θ ) = i = 1 J ( c i + | d | 2 R 2 g i ) Z i ( ρ , θ ) ,
x = x + d [ W ( x , y ) x ] ,
y = y + d [ W ( x , y ) y ] .
r = r + d W ( ρ ) r = r + d R W ( ρ ) ρ .
W ( ρ ) = i = 1 N c 2 i 0 Z 2 i 0 ( ρ ) ,
R = R + d R W ( ρ ) ρ | ρ = 1 .
W ( ρ ) ρ = i = 1 N c 2 i 0 R 2 i 0 ( ρ ) ρ = i = 1 N 2 c 2 i 0 2 i + 1 s = 0 i 1 ( 1 ) s ( i s ) ( 2 i s ) ! s ! [ ( i s ) ! ] 2 ρ 2 i 2 s 1 .
b = i = 1 N 2 c 2 i 0 2 i + 1 s = 0 i 1 ( 1 ) s ( i s ) ( 2 i s ) ! s ! [ ( i s ) ! ] 2 ,
R = R + d R b .
W ( R ρ , θ ) = 6 c 2 2 ρ 2 sin 2 θ + 3 c 2 0 ( 2 ρ 2 1 ) + 6 c 2 2 ρ 2 cos 2 θ = 3 c 2 0 ( 2 ρ 2 1 ) + 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 ρ 2 cos 2 ( θ ϕ ) ,
ϕ = 1 2 tan 1 ( c 2 2 c 2 2 ) .
S = 4 3 c 2 0 R 2 2 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 R 2 ,
C = 4 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 R 2 .
W ( u , υ ) = 2 6 c 2 2 u υ + 3 c 2 0 ( 2 u 2 + 2 υ 2 1 ) + 6 c 2 2 ( u 2 υ 2 ) .
a ( u , υ ) = [ W ( u , υ ) u ] 2 + [ W ( u , υ ) υ ] 2 = ( 2 6 c 2 2 υ + 4 3 c 2 0 u + 2 6 c 2 2 u ) 2 + ( 2 6 c 2 2 u + 4 3 c 2 0 υ 2 6 c 2 2 υ ) 2 = 24 [ ( c 2 2 ) 2 + 2 ( c 2 0 ) 2 + ( c 2 2 ) 2 ] ρ 2 + 48 2 c 2 0 ( c 2 2 ) 2 + ( c 2 2 ) 2 ρ 2 cos 2 ( θ ϕ ) = 16 3 c 2 0 c 2 2 Z 2 2 + 4 3 [ ( c 2 2 ) 2 + 2 ( c 2 0 ) 2 + ( c 2 2 ) 2 ] Z 2 0 + 16 3 c 2 0 c 2 2 Z 2 2 + 12 [ ( c 2 2 ) 2 + 2 ( c 2 0 ) 2 + ( c 2 2 ) 2 ] Z 0 0 .
W ( ρ , θ ) = c 2 2 Z 2 2 + c 2 0 Z 2 0 + c 2 2 Z 2 2 + d + d 2 R 2 { 16 3 c 2 0 c 2 2 Z 2 2 + 4 3 [ ( c 2 2 ) 2 + 2 ( c 2 0 ) 2 + ( c 2 2 ) 2 ] Z 2 0 + 16 3 c 2 0 c 2 2 Z 2 2 + 12 [ ( c 2 2 ) 2 + 2 ( c 2 0 ) 2 + ( c 2 2 ) 2 ] Z 0 0 } = b 2 2 Z 2 2 + b 2 0 Z 2 0 + b 2 2 Z 2 2 + b 0 0 Z 0 0 ,
b 2 2 = ( 1 + d 8 3 c 2 0 R 2 ) c 2 2 ,
b 2 0 = { 1 + d 2 3 c 2 0 R 2 [ ( c 2 2 ) 2 + 2 ( c 2 0 ) 2 + ( c 2 0 ) 2 ] } c 2 0 ,
b 2 2 = ( 1 + d 8 3 c 2 0 R 2 ) c 2 2 ,
b 0 0 = d + 12 [ ( c 2 2 ) 2 + 2 ( c 2 0 ) 2 + ( c 2 2 ) 2 ] .
S = 4 3 b 2 0 R 2 2 6 ( b 2 2 ) 2 + ( b 2 2 ) 2 R 2 ,
C = 4 6 ( b 2 2 ) 2 + ( b 2 2 ) 2 R 2 .
b = 2 6 { [ ( c 2 2 ) 2 + 2 ( c 2 0 ) 2 + ( c 2 2 ) 2 ] + 2 2 c 2 0 ( c 2 2 ) 2 + ( c 2 2 ) 2 cos 2 ( θ ϕ ) } 1 / 2 .
b minp = 4 3 c 2 0 + 2 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 .
R minp = R { 1 + d R 2 [ 4 3 c 2 0 + 2 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 ] } = R ( 1 d S ) .
R R = ( f ) + d ( f ) ,
R = ( 1 d f ) R .
S = 1 R minp 2 [ 4 3 b 2 0 + 2 6 ( b 2 2 ) 2 + ( b 2 2 ) 2 ] = 1 R minp 2 [ 4 3 c 2 0 + 2 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 + d R 2 ( 4 3 c 2 0 + 2 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 ) 2 ] = 1 R 2 ( 1 S d ) 2 [ R 2 S ( 1 S d ) ] = S 1 S d .
b maxp = 4 3 c 2 0 2 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 .
R maxp = R { 1 + d R 2 [ 4 3 c 2 0 2 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 ] } = R [ 1 d ( S + C ) ] .
S + C = 1 R maxp 2 [ 4 3 b 2 0 2 6 ( b 2 2 ) 2 + ( b 2 2 ) 2 ] = 1 R maxp 2 [ 4 3 c 2 0 2 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 + d R 2 ( 4 3 c 2 0 2 6 ( c 2 2 ) 2 + ( c 2 2 ) 2 ) 2 ] = 1 R 2 [ 1 d ( S + C ) ] 2 { R 2 ( S + C ) [ 1 d ( S + C ) ] } = S + C 1 d ( S + C ) .
i = p ( p + 1 ) / 2 1 + ( q + 1 ) = p ( p + 1 ) / 2 + q .
p = ( 8 i + 1 1 ) / 2.
p = int [ ( 8 i + 1 1 ) / 2 ] .
q = ( 2 i p 2 p ) / 2.
i = n ( n + 1 ) / 2 1 + k .
k = ( m + n ) / 2 + 1.
i = ( n 2 + 2 n + m ) / 2.
n = int [ ( 8 i + 1 1 ) / 2 ] .
m = 2 i n ( n + 2 ) .

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