Abstract

Orthogonality is exploited for fitting analytically-specified freeform shapes in terms of orthogonal polynomials. The end result is expressed in terms of FFTs coupled to a simple explicit form of Gaussian quadrature. Its efficiency opens the possibilities for proceeding to arbitrary numbers of polynomial terms. This is shown to create promising options for quantifying and filtering the mid-spatial frequency structure within circular domains from measurements of as-built parts.

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  1. A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt.51(15), 3054–3058 (2012), doi:.
    [CrossRef] [PubMed]
  2. I. Kaya and J. P. Rolland, “Hybrid RBF and local ϕ-polynomial freeform surfaces,” Adv. Opt. Technol.2(1), 81–88 (2012), doi:.
    [CrossRef]
  3. P. Jester, C. Menke, and K. Urban, “Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces,” IMA J. Appl. Math.77, 357–363 (2012).
  4. R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:.
    [CrossRef]
  5. G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express20(3), 2483–2499 (2012).
    [CrossRef] [PubMed]
  6. C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2012), doi:.
    [CrossRef]
  7. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 15.4.
  8. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express18(19), 19700–19712 (2010), doi:.
    [CrossRef] [PubMed]
  9. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978). See 25.4.38.
  10. See Section 12.3 of [7]. Or see Sec. 12.4 in http://apps.nrbook.com/empanel/index.html#
  11. J. H. Hannay and J. F. Nye, “Fibonacci numerical integration on a sphere,” J. Phys. Math. Gen.37(48), 11591–11601 (2004), doi:.
    [CrossRef]
  12. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express18(13), 13851–13862 (2010), doi:.
    [CrossRef] [PubMed]
  13. J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
    [CrossRef]

2012 (5)

A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt.51(15), 3054–3058 (2012), doi:.
[CrossRef] [PubMed]

I. Kaya and J. P. Rolland, “Hybrid RBF and local ϕ-polynomial freeform surfaces,” Adv. Opt. Technol.2(1), 81–88 (2012), doi:.
[CrossRef]

P. Jester, C. Menke, and K. Urban, “Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces,” IMA J. Appl. Math.77, 357–363 (2012).

G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express20(3), 2483–2499 (2012).
[CrossRef] [PubMed]

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2012), doi:.
[CrossRef]

2011 (1)

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:.
[CrossRef]

2010 (2)

G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express18(19), 19700–19712 (2010), doi:.
[CrossRef] [PubMed]

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express18(13), 13851–13862 (2010), doi:.
[CrossRef] [PubMed]

2004 (1)

J. H. Hannay and J. F. Nye, “Fibonacci numerical integration on a sphere,” J. Phys. Math. Gen.37(48), 11591–11601 (2004), doi:.
[CrossRef]

1999 (1)

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Auerbach, J. M.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Cotton, C. T.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Dick, L.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:.
[CrossRef]

Eberhardt, R.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:.
[CrossRef]

English, R. E.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Forbes, G. W.

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2012), doi:.
[CrossRef]

G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express20(3), 2483–2499 (2012).
[CrossRef] [PubMed]

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express18(13), 13851–13862 (2010), doi:.
[CrossRef] [PubMed]

G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express18(19), 19700–19712 (2010), doi:.
[CrossRef] [PubMed]

Gebhardt, A.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:.
[CrossRef]

Hannay, J. H.

J. H. Hannay and J. F. Nye, “Fibonacci numerical integration on a sphere,” J. Phys. Math. Gen.37(48), 11591–11601 (2004), doi:.
[CrossRef]

Henesian, M. A.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Hunt, J. T.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Jester, P.

P. Jester, C. Menke, and K. Urban, “Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces,” IMA J. Appl. Math.77, 357–363 (2012).

Kaya, I.

I. Kaya and J. P. Rolland, “Hybrid RBF and local ϕ-polynomial freeform surfaces,” Adv. Opt. Technol.2(1), 81–88 (2012), doi:.
[CrossRef]

Kelly, J. H.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Kopf, T.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:.
[CrossRef]

Lawson, J. K.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Menke, C.

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2012), doi:.
[CrossRef]

P. Jester, C. Menke, and K. Urban, “Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces,” IMA J. Appl. Math.77, 357–363 (2012).

Nye, J. F.

J. H. Hannay and J. F. Nye, “Fibonacci numerical integration on a sphere,” J. Phys. Math. Gen.37(48), 11591–11601 (2004), doi:.
[CrossRef]

Risse, S.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:.
[CrossRef]

Rolland, J. P.

I. Kaya and J. P. Rolland, “Hybrid RBF and local ϕ-polynomial freeform surfaces,” Adv. Opt. Technol.2(1), 81–88 (2012), doi:.
[CrossRef]

Sacks, R. A.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Shoup, M. J.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Steinkopf, R.

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:.
[CrossRef]

Trenholme, J. B.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Urban, K.

P. Jester, C. Menke, and K. Urban, “Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces,” IMA J. Appl. Math.77, 357–363 (2012).

Williams, W. H.

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Yabe, A.

A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt.51(15), 3054–3058 (2012), doi:.
[CrossRef] [PubMed]

Adv. Opt. Technol. (2)

I. Kaya and J. P. Rolland, “Hybrid RBF and local ϕ-polynomial freeform surfaces,” Adv. Opt. Technol.2(1), 81–88 (2012), doi:.
[CrossRef]

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Technol.2(1), 97–109 (2012), doi:.
[CrossRef]

Appl. Opt. (1)

A. Yabe, “Representation of freeform surfaces suitable for optimization,” Appl. Opt.51(15), 3054–3058 (2012), doi:.
[CrossRef] [PubMed]

IMA J. Appl. Math. (1)

P. Jester, C. Menke, and K. Urban, “Wavelet Methods for the Representation, Analysis and Simulation of Optical Surfaces,” IMA J. Appl. Math.77, 357–363 (2012).

J. Phys. Math. Gen. (1)

J. H. Hannay and J. F. Nye, “Fibonacci numerical integration on a sphere,” J. Phys. Math. Gen.37(48), 11591–11601 (2004), doi:.
[CrossRef]

Opt. Express (3)

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express18(13), 13851–13862 (2010), doi:.
[CrossRef] [PubMed]

G. W. Forbes, “Characterizing the shape of freeform optics,” Opt. Express20(3), 2483–2499 (2012).
[CrossRef] [PubMed]

G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express18(19), 19700–19712 (2010), doi:.
[CrossRef] [PubMed]

Proc. SPIE (2)

R. Steinkopf, L. Dick, T. Kopf, A. Gebhardt, S. Risse, and R. Eberhardt, “Data handling and representation of freeform surfaces,” Proc. SPIE8169, 81690X, 81690X-9 (2011), doi:.
[CrossRef]

J. K. Lawson, J. M. Auerbach, R. E. English, M. A. Henesian, J. T. Hunt, R. A. Sacks, J. B. Trenholme, W. H. Williams, M. J. Shoup, J. H. Kelly, and C. T. Cotton, “NIF optical specifications: the importance of the RMS gradient,” Proc. SPIE3492, 336–343 (1999), doi:.
[CrossRef]

Other (3)

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978). See 25.4.38.

See Section 12.3 of [7]. Or see Sec. 12.4 in http://apps.nrbook.com/empanel/index.html#

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 15.4.

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Figures (10)

Fig. 1
Fig. 1

Sample locations for a plane-symmetric conversion with eleven spokes and six rings. Reflection symmetry means that it is sufficient to sample in just one half of the aperture.

Fig. 2
Fig. 2

A sinusoidal sag function is presented in each column. Both correspond to a single full period, i.e. C = 1 in Eq. (6.2), but one is symmetric while the other is antisymmetric. The sags are plotted in the first row with a linear color ramp between the displayed extreme values. The peak-to-valley and standard deviation are also shown. The spectra of fit coefficients are tabulated in the second row as functions of m and n and include all terms up to Cartesian order t = 11. The PSS’s are presented in the third row. The rms gradient is given by the square root of the sum of these values, and the result is shown as “rss” in those plots since it is the root-sum-square of all the tabulated coefficients.

Fig. 3
Fig. 3

Sinusoidal sag functions presented much as in Fig. 2 but now as a combination of the even and odd cases and for C = 5 and 25. The fit coefficient amplitudes are plotted on a logarithmic color scale in the second row as functions of m and n. Their peak value is shown next to the red block of the legend and each colored block below that represents a reduction by a factor of ten down to 10−7 times the maximum. The third row holds the PSS’s.

Fig. 4
Fig. 4

The PSS for C = 100 follows upon fitting over 150,000 coefficients.

Fig. 5
Fig. 5

The data shown at top right in Fig. 3 is presented here after a low-, medium-, and high-pass filter has been applied in terms of the Cartesian order in the Q basis. These plots contain only the components in the data from (a) t = 1 to 30 at left, (b) t = 31 to 60 in the central plot, and (c) t = 61 to 90 at right. The rss (i.e. the rms gradient) for these three bands is 12.1, 24.9, and 48.5, respectively, and the standard deviation of each band appears in the plots.

Fig. 6
Fig. 6

The data in the rightmost plot of Fig. 5 is now passed through a low-, medium-, and high-pass filter that discriminates based on the azimuthal order rather than the Cartesian order. These plots contain only the components in the data from (a) m = 0 to 25 at left, (b) m = 26 to 50 in the central plot, and (c) m = 51 to 90 at right. This division is chosen so that there is roughly an equal number of terms in each of these three sub-bands [two of the domains are parallelograms and one a trapezoid in the (m,n)-plane]. Together, these components exactly comprise the original function.

Fig. 7
Fig. 7

The default sample locations at left for N = 25 (hence M = 50, J = 102, and K = 27) involves 2,754 samples to extract (2M + 1)(N + 1) = 2,626 coefficients. This is appropriate for a general fit of a spectrum like that in the left column of Fig. 3. The Fibonacci-inspired and analytically equivalent option at right avoids unattractive bunching in the outer zones.

Fig. 8
Fig. 8

The peak coefficient in this spectrum is 2.3 and the color legend steps down to purple for an amplitude of 2.3e-7. The PSS is also plotted on a log scale as the green curve at right, while the blue curves give the results when the sampled sag values are corrupted by additive noise uniformly distributed between ± 1.0e-3, ± 1.0e-4, and ± 1.0e-5. Only the highest level of noise can be distinguished from the noise-free case in the left half of the plot (t < 80). The purple curve is found by using interpolation from a 250x250 pixelated grid.

Fig. 9
Fig. 9

(a) Normal departure extracted from a synthetic data set. The components in that data from (b) t = 1 to 10, (c) t = 11 to 24, (d) t = 25 to 64, (e) t = 65 to 84, and (f) t = 85 to 140.

Fig. 10
Fig. 10

Plots of the components of the spectrum in Fig. 8 for the data in Fig. 9(a) with (a) m = 14, 28, 42,… 140 and (b) m = 0 and n > 7. Notice that the multiples of 14 stand out as vertical bands in the plot at left in Fig. 8 and that there are anomalously large values for large n when m = 0. These are the indicators that suggest these two final filtering steps.

Equations (58)

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

f( ρ,θ )= c ρ 2 1+ 1 c 2 ρ 2 + 1 1 c 2 ρ 2 { u 2 (1 u 2 ) n=0 a n 0 Q n 0 ( u 2 ) + m=1 u m n=0 [ a n m cosmθ+ b n m sinmθ] Q n m ( u 2 ) }.
t={ 2n+4,m=0, 2n+m,m>0.
g(θ) θ := 1 2π 0 2π g(θ)dθ ,
S( ρ max ,θ) S 0 θ = c ρ max 2 1+ 1 c 2 ρ max 2 .
c= 2 S( ρ max ,θ) S 0 θ S( ρ max ,θ) S 0 θ 2 + ρ max 2 .
S ¯ ( ρ/ ρ max ,θ ):= 1 c 2 ρ 2 [ S(ρ,θ) S 0 c ρ 2 1+ 1 c 2 ρ 2 ].
S ¯ (u,θ) u 2 (1 u 2 ) n=0 a n 0 Q n 0 ( u 2 )+ m=1 u m n=0 [ a n m cosmθ+ b n m sinmθ] Q n m ( u 2 ).
S ¯ (u,θ) m=0 [ A m (u)cosmθ+ B m (u)sinmθ].
A 0 (u) u 2 (1 u 2 ) n=0 a n 0 Q n 0 ( u 2 ),
A m (u) u m n=0 a n m Q n m ( u 2 ), B m (u) u m n=0 b n m Q n m ( u 2 ),whenm>0.
L(θ)= k= k exp(ikθ).
exp(ikθ)exp(ijθ) θ = 1 2π α α+2π exp[i(jk)θ]dθ= δ jk .
k = 1 2π α α+2π L(θ)exp(ikθ)dθ.
k ˜ k := 1 2π j=1 J L(jΔ)exp[ikjΔ]Δ= 1 J j=1 J L(jΔ)exp[ikjΔ].
˜ k = 1 J j=1 J { k = k exp[i k jΔ] }exp[ikjΔ]= k = k { 1 J j=1 J exp[i(k k )j2π/J] }.
˜ k = n= k+nJ = k +( k+J + kJ )+( k+2J + k2J )+....
L(θ)= 1 2 a 0 + k=1 ( a k coskθ+ b k sinkθ),
a k =2 L(θ)coskθ θ , b k =2 L(θ)sinkθ θ ,
a ˜ k := 2 J j=1 J L(jΔ)cos(kjΔ), b ˜ k := 2 J j=1 J L(jΔ)sin(kjΔ).
f(θ) θ 1 J j=1 J f(α+jΔ),
β n = (1) n A ¯ 0 (cosθ)cos[(2n+1)θ] θ
A ¯ 0 (u):= A 0 (u) / [u(1 u 2 )] .
a n 0 = f n β n + g n β n+1 + h n β n+2 ,
A 0 (u)= S ¯ (u,θ) θ A ˜ 0 (u):= 1 J j=1 J S ¯ (u,jΔ),
β n = (1) n 2 π 0 π/2 A ¯ 0 (cosϕ)cos[(2n+1)ϕ]dϕ (1) n K k=1 K A ¯ 0 (cos ϕ k )cos[(2n+1) ϕ k ] ,
ϕ k :=(2k1) π 4K .
C(u)= u m n=0 N c n Q n m ( u 2 ),
U pq :=[ p 0 m q 0 m 0 ... 0 0 p 1 m q 1 m ... 0 0 0 p 2 m q 2 m ... 0 0 0 ... 0 p N1 m q N1 m 0 ... 0 0 p N m ].
u n = p n m v n + q n m v n+1 ,
v n =( u n q n m v n+1 )/ p n m .
u n = q n1 m v n1 + p n m v n ,
v n =( u n q n1 m v n1 )/ p n m .
c= U fg U st 1 U hk 1 L hk 1 r,
r n = 2 π 0 1 C(u) P n m+1 ( u 2 ) u m 1 u 2 du = 2 π 0 π/2 C(cosϕ) cos m ϕ P n m+1 ( cos 2 ϕ)dϕ 1 K k=1 K C(cos ϕ k ) cos m ϕ k P n m+1 ( cos 2 ϕ k ) .
F( 1 2 ρ max ,0)=7,989,797,240,841F( 1 2 ρ max , 1 2 ρ max )=16,201,346,177,785 F(0, 1 2 ρ max )=7,996,883,312,162F( 1 2 ρ max , 1 2 ρ max )=16,038,915,342,886.
{r,e,d,c}={ ( 40,446,180,687 53,565,404,125 11,587,305,007 328,228,051 32,163,607 366,902 28,017 321 6 0 ),( 687,615,292,982 242,697,231,660 9,224,094,424 817,823,612 8,029,264 777,228 9,268 189 5 0 ),( 924,648,807,539 474,067,029,112 25,486,726,968 2,023,663,245 20,477,869 1,728,692 19,989 399 9 0 ),( 225,290,889,213 223,995,088,263 12,915,787,282 1,568,376,338 19,668,263 2,261,608 31,561 737 20 1 ) }.
{r,e,d,c}={ ( 98,410,809 162,552,300 23,414,045 221,374 2,464 31 1 0 0 0 ),( 2,080,090,082 262,103,956 3,095,825 39,133 556 11 0 0 0 0 ),( 2,145,718,876 328,143,971 4,317,814 58,363 865 18 0 0 0 0 ),( 2,650,626,613 801,139,998 10,459,850 151,133 2,422 55 1 0 0 0 ) }.
c=(70,530,723,064,43,064,584,81,637,144, 10,310,272,546,272,13,714,464,17,1,0).
a n m cosmθ+ b n m sinmθ= α n m cos(mθ ϕ n m ),
z=sin(πCy/ ρ max +δ)=sinδcos(πCy/ ρ max )+cosδsin(πCy/ ρ max ),
t2π ρ max /λ.
g(u) u := 2 π 0 1 g(u) 1 u 2 du ,
[C(u) u m n=0 N e n P n m+1 ( u 2 )] 2 u .
u m P n m+1 ( u 2 ) u m P n m+1 ( u 2 ) u = 2 π 0 1 P n m+1 ( u 2 ) P n m+1 ( u 2 ) u 2m 1 u 2 du = 1 π 0 1 P n m+1 (x) P n m+1 (x) x m1/2 1x dx ,
r n := C(u) u m P n m+1 ( u 2 ) u = 2 π 0 1 C(u) P n m+1 ( u 2 ) u m 1 u 2 du .
K n m ={ (2m+1)!! (2m+2)!!2 , n=0, 2(2m+3)m 3(m+3)(m+2) K 0 m , n=1, (n+1)(m+2n2)(m+2n3)(2m+2n+1) (2n+1)(m+n2)(m+2n+1)(m+2n) K n1 m , n>1.
H n m ={ m+1 2m+1 K 0 m , n=0, 3m+2 m+2 K 0 m , n=1, (m+2n3)[(m+n2)(4n1)+5n] (m+n2)(2n1)(m+2n) K n1 m , n>1.
k n1 m = K n1 m / h n1 m ,and h n m = H n m ( k n1 m ) 2 .
K n 0 =( n 2 1)/(32 n 2 8),and H n 0 =[1+ (12n) 2 ]/16.
n=0 N e n P n m+1 (x) n=0 N d n P n m (x).
P n m (x)={ s 0 m P 0 m+1 (x), n=0, s n m P n m+1 (x)+ t n1 m P n1 m+1 (x), n>0.
s n m ={ 1, n=0, 1/2, m=1andn=1, m+n2 m+2n2 , otherwise,
t n m ={ 1/2, m=1andn=0, (12n)(n+1) (m+2n)(2n+1) otherwise.
( H 0 1 H 0 2 H 0 3 H 0 4 H 1 1 H 1 2 H 1 3 H 1 4 H 2 1 H 2 2 H 2 3 H 2 4 H 3 1 H 3 2 H 3 3 H 3 4 )=( 1 8 3 32 5 64 35 512 5 16 5 16 77 256 147 512 17 144 7 48 31 192 1045 6144 37 400 9 80 649 5120 7007 51200 ),
( K 0 1 K 0 2 K 0 3 K 0 4 K 1 1 K 1 2 K 1 3 K 1 4 K 2 1 K 2 2 K 2 3 K 2 4 )=( 3 16 5 32 35 256 63 512 5 96 7 96 21 256 11 128 7 160 9 160 33 512 143 2048 ),
( h 0 1 h 0 2 h 0 3 h 0 4 h 1 1 h 1 2 h 1 3 h 1 4 h 2 1 h 2 2 h 2 3 h 2 4 h 3 1 h 3 2 h 3 3 h 3 4 )=( 2 4 6 8 5 8 70 32 2 8 30 24 3 7 32 105 40 2 8 70 40 30 24 5 462 448 2 8 3 14 56 770 128 3003 240 ),
( k 0 1 k 0 2 k 0 3 k 0 4 k 1 1 k 1 2 k 1 3 k 1 4 k 2 1 k 2 2 k 2 3 k 2 4 )=( 3 2 8 5 6 24 7 5 32 9 70 160 5 2 24 7 30 120 7 8 11 105 336 7 2 40 9 70 280 33 30 640 13 462 960 ),
( s 0 1 s 0 2 s 0 3 s 0 4 s 1 1 s 1 2 s 1 3 s 1 4 s 2 1 s 2 2 s 2 3 s 2 4 s 3 1 s 3 2 s 3 3 s 3 4 )=( 1 1 1 1 1 2 1 2 2 3 3 4 1 3 1 2 3 5 2 3 2 5 1 2 4 7 5 8 ),( t 0 1 t 0 2 t 0 3 t 0 4 t 1 1 t 1 2 t 1 3 t 1 4 t 2 1 t 2 2 t 2 3 t 2 4 )=( 1 2 1 2 1 3 1 4 2 9 1 6 2 15 1 9 9 25 3 10 9 35 9 40 ).

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