Abstract

Measuring flats in the horizontal posture with interferometers is analyzed in detail, taking into account the sag produced by gravity. A mathematical expression of the bending is provided for a plate supported at three unevenly spaced locations along the edge. It is shown that the azimuthal terms of the deformation can be recovered from a three-flat measuring procedure, while the pure radial terms can only be estimated. The effectiveness of the iterative algorithm for data processing is also demonstrated. Experimental comparison on a set of three flats in horizontal and upright posture is provided.

© 2008 Optical Society of America

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References

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  1. L. Rayleigh, “Interference bands and their application,” Nature 48, 212-214 (1893).
    [CrossRef]
  2. H. Barrell and R. Marriner, “Liquid surface interferometry,” Nature 162, 529-530 (1948).
    [CrossRef]
  3. G. D. Dew, “The measurement of optical flatness,” J. Sci. Instrum. 43, 409-415 (1966).
    [CrossRef] [PubMed]
  4. R. Brünnagel, H.-A. Oehring, and K. Steiner, “Fizeau interferometer for measuring the flatness of optical surfaces,” Appl. Opt. 7, 331-335 (1967).
    [CrossRef]
  5. J. P. Marioge, B. Bonino, and M. Mullot, “Standard of flatness: its application to Fabry-Perot interferometers,” Appl. Opt. 14, 2283-2285 (1975).
    [CrossRef] [PubMed]
  6. K.-E. Elssner, A. Vogel, J. Grzanna, and G. Schulz, “Establishing a flatness standard,” Appl. Opt. 33, 2437-2446 (1994).
    [CrossRef] [PubMed]
  7. J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936-1942 (1996).
    [CrossRef]
  8. I. Powell and E. Goulet, “Absolute figure measurements with a liquid-flat reference,” Appl. Opt. 37, 2579-2588 (1998).
    [CrossRef]
  9. M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42, 389-393(2005).
    [CrossRef]
  10. G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077-1084 (1967).
    [CrossRef] [PubMed]
  11. G. Schulz, J. Schwider, C. Hiller, and B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10, 929-934 (1971).
    [CrossRef] [PubMed]
  12. J. Grzanna and G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107-112(1990).
    [CrossRef]
  13. G. Schulz and J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31, 3767-3780 (1992).
    [CrossRef] [PubMed]
  14. G. Schulz, “Absolute flatness testing by an extended rotation method using two angles of rotation,” Appl. Opt. 32, 1055-1059 (1993).
    [CrossRef] [PubMed]
  15. J. Grzanna, “Absolute testing of optical flats at points on a square grid: error propagation,” Appl. Opt. 33, 6654-6661(1994).
    [CrossRef] [PubMed]
  16. B. B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, “Calibration of a 300 mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39, 5161-5171 (2000).
    [CrossRef]
  17. S. Sonozaki, K. Iwata, and Y. Iwahashi, “Measurement of profiles along a circle on two flat surfaces by use of a Fizeau interferometer with no standard,” Appl. Opt. 42, 6853-6858(2003).
    [CrossRef] [PubMed]
  18. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 33, 379-383 (1984).
  19. C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698-4705 (1993).
    [CrossRef] [PubMed]
  20. C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015-1021 (1996).
    [CrossRef] [PubMed]
  21. P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 36, 2478-2481(1997).
    [CrossRef]
  22. C. J. Evans, “Comment on the paper 'Interferometric testing of optical surfaces: absolute measurement of flatness',” Opt. Eng. 37, 1880-1882 (1998).
    [CrossRef]
  23. R. E. Parks, L.-Z. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951-5956(1998).
    [CrossRef]
  24. V. Greco, R. Tronconi, C. Del Vecchio, M. Trivi, and G. Molesini, “Absolute measurement of planarity with Fritz's method: uncertainty evaluation,” Appl. Opt. 38, 2018-2027 (1999).
    [CrossRef]
  25. K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40, 1637-1648(2001).
    [CrossRef]
  26. M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Jena) 112, 381-391 (2001).
    [CrossRef]
  27. U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856-5865 (2006).
    [CrossRef] [PubMed]
  28. W. Gao, P. S. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396-404 (2002).
    [CrossRef]
  29. M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45, 060503 (2006).
    [CrossRef]
  30. R. D. Geckeler, “Optimal use of pentaprism in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18, 115-125(2007).
    [CrossRef]
  31. P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2 m flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).
    [CrossRef]
  32. J. Yellowhair and J. H. Burge, “Analysis of a scanning pentaprism system for measurements of large flat mirrors,” Appl. Opt. 46, 8466-8474 (2007).
    [CrossRef] [PubMed]
  33. U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601(2007).
    [CrossRef]
  34. E. E. Bloemhof, J. C. Lam, V. A. Feria, and Z. Chang, “Precise determination of the zero-gravity surface figure of a mirror without gravity-sag modelling,” Appl. Opt. 46, 7670-7678(2007).
    [CrossRef] [PubMed]
  35. M. Vannoni and G. Molesini, “Iterative algorithm for three flat test,” Opt. Express 15, 6809-6816 (2007).
    [CrossRef] [PubMed]
  36. M. Vannoni and G. Molesini, “Absolute planarity with three flat test: an iterative approach with Zernike polynomials,” Opt. Express 16, 340-354 (2008).
    [CrossRef] [PubMed]
  37. A. Nádai, “Die verbiegungen in einzelnen punkten unterstützter kreisförmiger platten,” Physik Zeitschr. XXIII, 366-376 (1922).
  38. A. E. H. Love, Treatise on the Mathematical Theory of Elasticity (Dover, 1944), p. 481.
  39. W. B. Emerson, “Determination of planeness and bending of optical flats,” J. Res. Natl. Bur. Stand. 49, 241-247 (1952).
  40. W. A Bassali, “The transverse flexure of thin elastic plates supported at several points,” Proc. Cambridge Philos. Soc. 53, 728-742 (1957).
    [CrossRef]
  41. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, 1959), pp. 56-58, 72-74, and 293-295.
  42. L. Landau and E. Lifchitz, Théorie de l'Élasticité (MIR, 1967), p. 67.
  43. L. A. Selke, “Theoretical elastic deflections of a thick horizontal circular mirror on a ring support,” Appl. Opt. 9, 149-153 (1970).
    [CrossRef] [PubMed]
  44. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), pp. 464-466 and 767-772.
  45. V. B. Gubin and V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147-148 (1990).

2008 (1)

2007 (6)

R. D. Geckeler, “Optimal use of pentaprism in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18, 115-125(2007).
[CrossRef]

P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2 m flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).
[CrossRef]

J. Yellowhair and J. H. Burge, “Analysis of a scanning pentaprism system for measurements of large flat mirrors,” Appl. Opt. 46, 8466-8474 (2007).
[CrossRef] [PubMed]

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601(2007).
[CrossRef]

E. E. Bloemhof, J. C. Lam, V. A. Feria, and Z. Chang, “Precise determination of the zero-gravity surface figure of a mirror without gravity-sag modelling,” Appl. Opt. 46, 7670-7678(2007).
[CrossRef] [PubMed]

M. Vannoni and G. Molesini, “Iterative algorithm for three flat test,” Opt. Express 15, 6809-6816 (2007).
[CrossRef] [PubMed]

2006 (2)

U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856-5865 (2006).
[CrossRef] [PubMed]

M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45, 060503 (2006).
[CrossRef]

2005 (1)

M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42, 389-393(2005).
[CrossRef]

2003 (1)

2002 (1)

W. Gao, P. S. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396-404 (2002).
[CrossRef]

2001 (2)

K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40, 1637-1648(2001).
[CrossRef]

M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Jena) 112, 381-391 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (3)

1997 (1)

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 36, 2478-2481(1997).
[CrossRef]

1996 (2)

C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015-1021 (1996).
[CrossRef] [PubMed]

J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936-1942 (1996).
[CrossRef]

1994 (2)

1993 (2)

1992 (1)

1990 (2)

J. Grzanna and G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107-112(1990).
[CrossRef]

V. B. Gubin and V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147-148 (1990).

1984 (1)

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 33, 379-383 (1984).

1975 (1)

1971 (1)

1970 (1)

1967 (2)

1966 (1)

G. D. Dew, “The measurement of optical flatness,” J. Sci. Instrum. 43, 409-415 (1966).
[CrossRef] [PubMed]

1957 (1)

W. A Bassali, “The transverse flexure of thin elastic plates supported at several points,” Proc. Cambridge Philos. Soc. 53, 728-742 (1957).
[CrossRef]

1952 (1)

W. B. Emerson, “Determination of planeness and bending of optical flats,” J. Res. Natl. Bur. Stand. 49, 241-247 (1952).

1948 (1)

H. Barrell and R. Marriner, “Liquid surface interferometry,” Nature 162, 529-530 (1948).
[CrossRef]

1922 (1)

A. Nádai, “Die verbiegungen in einzelnen punkten unterstützter kreisförmiger platten,” Physik Zeitschr. XXIII, 366-376 (1922).

1893 (1)

L. Rayleigh, “Interference bands and their application,” Nature 48, 212-214 (1893).
[CrossRef]

Appl. Opt. (20)

R. Brünnagel, H.-A. Oehring, and K. Steiner, “Fizeau interferometer for measuring the flatness of optical surfaces,” Appl. Opt. 7, 331-335 (1967).
[CrossRef]

J. P. Marioge, B. Bonino, and M. Mullot, “Standard of flatness: its application to Fabry-Perot interferometers,” Appl. Opt. 14, 2283-2285 (1975).
[CrossRef] [PubMed]

K.-E. Elssner, A. Vogel, J. Grzanna, and G. Schulz, “Establishing a flatness standard,” Appl. Opt. 33, 2437-2446 (1994).
[CrossRef] [PubMed]

I. Powell and E. Goulet, “Absolute figure measurements with a liquid-flat reference,” Appl. Opt. 37, 2579-2588 (1998).
[CrossRef]

G. Schulz and J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31, 3767-3780 (1992).
[CrossRef] [PubMed]

G. Schulz, “Absolute flatness testing by an extended rotation method using two angles of rotation,” Appl. Opt. 32, 1055-1059 (1993).
[CrossRef] [PubMed]

J. Grzanna, “Absolute testing of optical flats at points on a square grid: error propagation,” Appl. Opt. 33, 6654-6661(1994).
[CrossRef] [PubMed]

B. B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, “Calibration of a 300 mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. 39, 5161-5171 (2000).
[CrossRef]

S. Sonozaki, K. Iwata, and Y. Iwahashi, “Measurement of profiles along a circle on two flat surfaces by use of a Fizeau interferometer with no standard,” Appl. Opt. 42, 6853-6858(2003).
[CrossRef] [PubMed]

C. Ai and J. C. Wyant, “Absolute testing of flats by using even and odd functions,” Appl. Opt. 32, 4698-4705 (1993).
[CrossRef] [PubMed]

C. J. Evans and R. N. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015-1021 (1996).
[CrossRef] [PubMed]

R. E. Parks, L.-Z. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951-5956(1998).
[CrossRef]

V. Greco, R. Tronconi, C. Del Vecchio, M. Trivi, and G. Molesini, “Absolute measurement of planarity with Fritz's method: uncertainty evaluation,” Appl. Opt. 38, 2018-2027 (1999).
[CrossRef]

K. R. Freischlad, “Absolute interferometric testing based on reconstruction of rotational shear,” Appl. Opt. 40, 1637-1648(2001).
[CrossRef]

G. Schulz and J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077-1084 (1967).
[CrossRef] [PubMed]

G. Schulz, J. Schwider, C. Hiller, and B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10, 929-934 (1971).
[CrossRef] [PubMed]

U. Griesmann, “Three-flat test solutions based on simple mirror symmetry,” Appl. Opt. 45, 5856-5865 (2006).
[CrossRef] [PubMed]

E. E. Bloemhof, J. C. Lam, V. A. Feria, and Z. Chang, “Precise determination of the zero-gravity surface figure of a mirror without gravity-sag modelling,” Appl. Opt. 46, 7670-7678(2007).
[CrossRef] [PubMed]

J. Yellowhair and J. H. Burge, “Analysis of a scanning pentaprism system for measurements of large flat mirrors,” Appl. Opt. 46, 8466-8474 (2007).
[CrossRef] [PubMed]

L. A. Selke, “Theoretical elastic deflections of a thick horizontal circular mirror on a ring support,” Appl. Opt. 9, 149-153 (1970).
[CrossRef] [PubMed]

J. Res. Natl. Bur. Stand. (1)

W. B. Emerson, “Determination of planeness and bending of optical flats,” J. Res. Natl. Bur. Stand. 49, 241-247 (1952).

J. Sci. Instrum. (1)

G. D. Dew, “The measurement of optical flatness,” J. Sci. Instrum. 43, 409-415 (1966).
[CrossRef] [PubMed]

Meas. Sci. Technol. (1)

R. D. Geckeler, “Optimal use of pentaprism in highly accurate deflectometric scanning,” Meas. Sci. Technol. 18, 115-125(2007).
[CrossRef]

Metrologia (1)

M. Vannoni and G. Molesini, “Validation of absolute planarity reference plates with a liquid mirror,” Metrologia 42, 389-393(2005).
[CrossRef]

Nature (2)

L. Rayleigh, “Interference bands and their application,” Nature 48, 212-214 (1893).
[CrossRef]

H. Barrell and R. Marriner, “Liquid surface interferometry,” Nature 162, 529-530 (1948).
[CrossRef]

Opt. Commun. (1)

J. Grzanna and G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107-112(1990).
[CrossRef]

Opt. Eng. (7)

P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2 m flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007).
[CrossRef]

M. Schulz and C. Elster, “Traceable multiple sensor system for measuring curved surface profiles with high accuracy and high lateral resolution,” Opt. Eng. 45, 060503 (2006).
[CrossRef]

J. Chen, D. Song, R. Zhu, Q. Wang, and L. Chen, “Large-aperture high-accuracy phase-shifting digital flat interferometer,” Opt. Eng. 35, 1936-1942 (1996).
[CrossRef]

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 36, 2478-2481(1997).
[CrossRef]

C. J. Evans, “Comment on the paper 'Interferometric testing of optical surfaces: absolute measurement of flatness',” Opt. Eng. 37, 1880-1882 (1998).
[CrossRef]

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 33, 379-383 (1984).

U. Griesmann, Q. Wang, and J. Soons, “Three-flat tests including mounting-induced deformations,” Opt. Eng. 46, 093601(2007).
[CrossRef]

Opt. Express (2)

Optik (Jena) (1)

M. F. Küchel, “A new approach to solve the three flat problem,” Optik (Jena) 112, 381-391 (2001).
[CrossRef]

Physik Zeitschr. (1)

A. Nádai, “Die verbiegungen in einzelnen punkten unterstützter kreisförmiger platten,” Physik Zeitschr. XXIII, 366-376 (1922).

Precis. Eng. (1)

W. Gao, P. S. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396-404 (2002).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

W. A Bassali, “The transverse flexure of thin elastic plates supported at several points,” Proc. Cambridge Philos. Soc. 53, 728-742 (1957).
[CrossRef]

Sov. J. Opt. Technol. (1)

V. B. Gubin and V. N. Sharonov, “Algorithm for reconstructing the shape of optical surfaces from the results of experimental data,” Sov. J. Opt. Technol. 57, 147-148 (1990).

Other (4)

S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, 1959), pp. 56-58, 72-74, and 293-295.

L. Landau and E. Lifchitz, Théorie de l'Élasticité (MIR, 1967), p. 67.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), pp. 464-466 and 767-772.

A. E. H. Love, Treatise on the Mathematical Theory of Elasticity (Dover, 1944), p. 481.

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Figures (11)

Fig. 1
Fig. 1

Location of the pins and the pads in a plate mounting for interferometry.

Fig. 2
Fig. 2

Gravity deformation of a fused silica flat supported at three points, as in Fig. 1, after finite element analysis.

Fig. 3
Fig. 3

Laboratory setup to measure the deformation of a flat under gravity.

Fig. 4
Fig. 4

Map of the deformation induced by gravity on a fused silica flat supported, as in Fig. 1, after laboratory experiment (average of 20 measurements).

Fig. 5
Fig. 5

Map of the standard deviation of the 20 measurements whose average is shown in Fig. 4.

Fig. 6
Fig. 6

Three-flat test with plates in the horizontal posture. (a) Interferometric setup and (b) sequence of measurements with flats in pairs, producing the source data for analysis.

Fig. 7
Fig. 7

Scheme of the iterative algorithm used to analyze the three-flat test data.

Fig. 8
Fig. 8

Rotationally variant component of the deformation induced by gravity. (a) Resulting map from three-flat algebra and (b) map produced by the iterative algorithm.

Fig. 9
Fig. 9

Comparison between measurements of flat M. Upper: map obtained with M in the horizontal posture. Middle: map obtained with M in the upright posture. Lower: difference map.

Fig. 10
Fig. 10

Comparison between measurements of flat L. Upper: map obtained with L in the horizontal posture. Middle: map obtained with L in the upright posture. Lower: difference map.

Fig. 11
Fig. 11

Comparison between measurements of flat K. Upper: map obtained with K in the horizontal posture. Middle: map obtained with K in the upright posture. Lower: difference map.

Tables (4)

Tables Icon

Table 1 List of Zernike Polynomials Satisfying the Biharmonic Homogeneous Equation a

Tables Icon

Table 2 Geometric and Physical Characteristics of Fused Silica Plates Studied and Data Used for Finite Element Analysis

Tables Icon

Table 3 Gravity Deformation (Peak-to-Valley and Power) of the Fused Silica Plate According to Finite Element Analysis, Gravity Sagging Measurement, Three-Flat Algebra, and Iterative Algorithm with Flats in the Horizontal Posture (See Text)

Tables Icon

Table 4 Zernike Coefficients (in Wave Units) of the Deformation Induced by Gravity

Equations (21)

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

q = P π a 2 ,
4 w = q D ;
D = E h 3 12 ( 1 ν 2 ) ,
4 w = 1 r r { r r [ 1 r r ( r w r ) + 1 r 2 2 w θ 2 ] } + 1 r 2 2 w θ 2 [ 1 r r ( r w r ) + 1 r 2 2 w θ 2 ] .
w = w 0 + w 1 ,
a r r r r b + 2 r 2 a r r b θ θ + 2 r a r r r b 2 r 3 a r b θ θ 1 r 2 a r r b + 4 r 4 a b θ θ + 1 r 3 a r b + 1 r 4 a b θ θ θ θ = 0 ,
w 0 = 1 64 q D r 4 ,
w = c 3 Z 3 + c 4 Z 4 + c 5 Z 5 + c 6 Z 6 + c 7 Z 7 + c 8 Z 8 + ... + c 26 Z 26
w = c 3 Z 3 + c 4 Z 4 + c 10 Z 10 + c 26 Z 26 = c 3 ( 2 r 2 1 ) + c 4 r 2 cos 2 θ + c 10 r 3 sin 3 θ + c 26 r 5 sin 5 θ .
D = K F + M , E = L F + M , F = L F + M R , G = L F + K ,
Δ = 8 ( 7 9 cos n θ + 2 cos 2 n θ ) ,
D = K F + M + w F , E = L F + M , F = L F + M R , G = L F + K w .
D = d j Z j , E = e j Z j , F = f j Z j , G = g j Z j ,
K = k j Z j , L = l j Z j , M = m j Z j , w = c j Z j .
d 3 = k 3 + m 3 + c 3 , e 3 = l 3 + m 3 , g 3 = k 3 + l 3 c 3 .
n   odd , sine   term k F j = k j , n   odd , cosine   term k F j = k j , n   even , sine   term k F j = k j , n   even , cosine   term k F j = k j .
m R , p = cos n φ m p + sin n φ m q , m R , q = cos n φ m q sin n φ m p .
d 4 = k 4 + m 4 + c 4 , e 4 = l 4 + m 4 , f 4 = l 4 + ρ m 4 + σ m 5 , g 4 = l 4 + k 4 c 4 , d 5 = k 5 + m 5 c 5 , e 5 = l 5 + m 5 , f 5 = l 5 + ρ m 5 σ m 4 , g 5 = l 5 + k 5 c 5 .
k 4 = d 4 e 4 + g 4 2 , l 4 = ( ρ 1 ) ( e 4 + f 4 ) + σ ( e 5 f 5 ) 2 ( ρ 1 ) , m 4 = ( ρ 1 ) ( e 4 f 4 ) σ ( e 5 f 5 ) 2 ( ρ 1 ) , c 4 = ( ρ 1 ) ( d 4 + f 4 g 4 ) + σ ( e 5 f 5 ) 2 ( ρ 1 ) , k 5 = ( ρ + 1 ) ( f 4 e 4 ) + σ ( g 5 d 5 f 5 ) 2 σ , l 5 = ( ρ + 1 ) ( f 4 e 4 ) σ ( e 5 + f 5 ) 2 σ , m 5 = ( ρ + 1 ) ( f 4 e 4 ) + σ ( e 5 f 5 ) 2 σ , c 5 = e 5 d 5 g 5 2 .
d 9 = k 9 + m 9 c 9 , e 9 = l 9 + m 9 , f 9 = l 9 + ρ m 9 + σ m 10 , g 9 = l 9 + k 9 c 9 , d 10 = k 10 + m 10 + c 10 , e 10 = l 10 + m 10 , f 10 = l 10 + ρ m 10 σ m 9 , g 10 = l 10 + k 10 c 10 .
k 9 = ( ρ + 1 ) ( e 10 f 10 ) + σ ( g 9 d 9 f 9 ) 2 σ , l 9 = ( ρ + 1 ) ( e 10 f 10 ) σ ( e 9 + f 9 ) 2 σ , m 9 = ( ρ + 1 ) ( e 10 f 10 ) + σ ( e 9 f 9 ) 2 σ , c 9 = e 9 d 9 g 9 2 , k 10 = d 10 e 10 + g 10 2 , l 10 = ( ρ 1 ) ( e 10 + f 10 ) σ ( e 9 f 9 ) 2 ( ρ 1 ) , m 10 = ( ρ 1 ) ( e 10 f 10 ) + σ ( e 9 f 9 ) 2 ( ρ 1 ) , c 10 = ( ρ 1 ) ( d 10 + f 10 g 10 ) σ ( e 9 f 9 ) 2 ( ρ 1 ) .

Metrics