Abstract

A method for testing fast aspheric convex surfaces with dynamic null screens using LCDs is shown. A flat null screen is designed and displayed on an LCD monitor with drop-shaped spots in such a way that the image, which is formed by reflection on the test surface, becomes an exactly square array of circular spots if the surface is perfect. Any departure from this geometry is indicative of defects on the surface. Here the whole surface is tested at once. The position of the spots on the LCD can be changed in a dynamic way, to perform point-shifting of the image spots. The proposed procedure improves the dynamic point-shifting method. As has been shown previously, this process reduces the numerical error during the integration procedure, thereby improving the sensitivity of the test. The positioning accuracy for the screen spots is related to the LCD’s spatial resolution. Results of the evaluation of a parabolic convex surface with f/#=0.22 are shown.

© 2011 Optical Society of America

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References

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  1. R. Díaz-Uribe and M. Campos-García, “Null-screen testing of fast convex aspheric surfaces,” Appl. Opt. 39, 2670–2677(2000).
    [CrossRef]
  2. M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing fast aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47, 849–859 (2008).
    [CrossRef] [PubMed]
  3. M. Campos-García, R. Díaz-Uribe, and F. S. Granados-Agustin, “Testing fast aspheric surfaces with a linear array of sources,” Appl. Opt. 43, 6255–6264 (2004).
    [CrossRef] [PubMed]
  4. M. Avendano-Alejo and R. Dıaz-Uribe, “Testing a fast off-axis parabolic mirror using tilted null screens,” Appl. Opt. 45, 2607–2614 (2006).
    [CrossRef] [PubMed]
  5. M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. 48, 1008–1015 (2009).
    [CrossRef] [PubMed]
  6. V. I. Moreno-Oliva, M. Campos-Garcia, F. Granados-Agustin, M. J. Arjona-Pérez, R. Díaz-Uribe, and M. Avendaño-Alejo, “Optical testing of a parabolic trough solar collector by a null screen with stitching,” Proc. SPIE 7390, 739012(2009).
    [CrossRef]
  7. P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412(2010).
    [CrossRef] [PubMed]
  8. P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “SCOTS: A quantitative slope measuring method for optical shop use,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuB3.
  9. L. Wang, P. Su, R. E. Parks, J. M. Sasian, and J. H. Burge, “Low-cost, flexible, high dynamic range test for free-form illumination optics,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuE3.
  10. V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point-shifting in the optical testing of fast aspheric concave surfaces by a cylindrical screen,” Appl. Opt. 47, 644–651 (2008).
    [CrossRef] [PubMed]
  11. V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Improving the quantitative testing of fast aspherics with two-dimensional point-shifting by only rotating a cylindrical null screen,” J. Opt. A 10, 104029–104035(2008).
    [CrossRef]
  12. R. Díaz-Uribe, “Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. 39, 2790–2804(2000).
    [CrossRef]
  13. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1990).
  14. L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).
  15. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161–166.
  16. R. Díaz-Uribe and A. Cornejo-Rodríguez, “Conic constant and paraxial radius of curvature measurement for conic surfaces,” Appl. Opt. 25, 3731–3734 (1986).
    [CrossRef] [PubMed]
  17. W. Rasban, National Institutes of Health, USA. Image J V. 1.312u, Image Processing and Analysis in Javahttp://rsb.info.nih.gov/ij/.
  18. M. F. González-Cardel and R. Díaz-Uribe, “Profile and deformation coefficients measurement of fast optical surfaces,” Opt. Express 14, 9917–9930 (2006).
    [CrossRef] [PubMed]

2010

2009

M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. 48, 1008–1015 (2009).
[CrossRef] [PubMed]

V. I. Moreno-Oliva, M. Campos-Garcia, F. Granados-Agustin, M. J. Arjona-Pérez, R. Díaz-Uribe, and M. Avendaño-Alejo, “Optical testing of a parabolic trough solar collector by a null screen with stitching,” Proc. SPIE 7390, 739012(2009).
[CrossRef]

2008

2007

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).

2006

2004

2000

1986

Angel, R. P.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412(2010).
[CrossRef] [PubMed]

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “SCOTS: A quantitative slope measuring method for optical shop use,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuB3.

Arjona-Pérez, M. J.

V. I. Moreno-Oliva, M. Campos-Garcia, F. Granados-Agustin, M. J. Arjona-Pérez, R. Díaz-Uribe, and M. Avendaño-Alejo, “Optical testing of a parabolic trough solar collector by a null screen with stitching,” Proc. SPIE 7390, 739012(2009).
[CrossRef]

Avendano-Alejo, M.

Avendaño-Alejo, M.

V. I. Moreno-Oliva, M. Campos-Garcia, F. Granados-Agustin, M. J. Arjona-Pérez, R. Díaz-Uribe, and M. Avendaño-Alejo, “Optical testing of a parabolic trough solar collector by a null screen with stitching,” Proc. SPIE 7390, 739012(2009).
[CrossRef]

M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. 48, 1008–1015 (2009).
[CrossRef] [PubMed]

Bevington, P. R.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161–166.

Bolado-Gómez, R.

Burge, J. H.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412(2010).
[CrossRef] [PubMed]

L. Wang, P. Su, R. E. Parks, J. M. Sasian, and J. H. Burge, “Low-cost, flexible, high dynamic range test for free-form illumination optics,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuE3.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “SCOTS: A quantitative slope measuring method for optical shop use,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuB3.

Campos-Garcia, M.

V. I. Moreno-Oliva, M. Campos-Garcia, F. Granados-Agustin, M. J. Arjona-Pérez, R. Díaz-Uribe, and M. Avendaño-Alejo, “Optical testing of a parabolic trough solar collector by a null screen with stitching,” Proc. SPIE 7390, 739012(2009).
[CrossRef]

Campos-García, M.

Carmona-Paredes, L.

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).

Cornejo-Rodríguez, A.

Diaz-Uribe, R.

Díaz-Uribe, R.

V. I. Moreno-Oliva, M. Campos-Garcia, F. Granados-Agustin, M. J. Arjona-Pérez, R. Díaz-Uribe, and M. Avendaño-Alejo, “Optical testing of a parabolic trough solar collector by a null screen with stitching,” Proc. SPIE 7390, 739012(2009).
[CrossRef]

M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. 48, 1008–1015 (2009).
[CrossRef] [PubMed]

M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing fast aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47, 849–859 (2008).
[CrossRef] [PubMed]

V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point-shifting in the optical testing of fast aspheric concave surfaces by a cylindrical screen,” Appl. Opt. 47, 644–651 (2008).
[CrossRef] [PubMed]

V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Improving the quantitative testing of fast aspherics with two-dimensional point-shifting by only rotating a cylindrical null screen,” J. Opt. A 10, 104029–104035(2008).
[CrossRef]

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).

M. F. González-Cardel and R. Díaz-Uribe, “Profile and deformation coefficients measurement of fast optical surfaces,” Opt. Express 14, 9917–9930 (2006).
[CrossRef] [PubMed]

M. Campos-García, R. Díaz-Uribe, and F. S. Granados-Agustin, “Testing fast aspheric surfaces with a linear array of sources,” Appl. Opt. 43, 6255–6264 (2004).
[CrossRef] [PubMed]

R. Díaz-Uribe and M. Campos-García, “Null-screen testing of fast convex aspheric surfaces,” Appl. Opt. 39, 2670–2677(2000).
[CrossRef]

R. Díaz-Uribe, “Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. 39, 2790–2804(2000).
[CrossRef]

R. Díaz-Uribe and A. Cornejo-Rodríguez, “Conic constant and paraxial radius of curvature measurement for conic surfaces,” Appl. Opt. 25, 3731–3734 (1986).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1990).

González-Cardel, M. F.

Granados-Agustin, F.

V. I. Moreno-Oliva, M. Campos-Garcia, F. Granados-Agustin, M. J. Arjona-Pérez, R. Díaz-Uribe, and M. Avendaño-Alejo, “Optical testing of a parabolic trough solar collector by a null screen with stitching,” Proc. SPIE 7390, 739012(2009).
[CrossRef]

Granados-Agustin, F. S.

Moreno-Oliva, V. I.

V. I. Moreno-Oliva, M. Campos-Garcia, F. Granados-Agustin, M. J. Arjona-Pérez, R. Díaz-Uribe, and M. Avendaño-Alejo, “Optical testing of a parabolic trough solar collector by a null screen with stitching,” Proc. SPIE 7390, 739012(2009).
[CrossRef]

M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. 48, 1008–1015 (2009).
[CrossRef] [PubMed]

V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point-shifting in the optical testing of fast aspheric concave surfaces by a cylindrical screen,” Appl. Opt. 47, 644–651 (2008).
[CrossRef] [PubMed]

V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Improving the quantitative testing of fast aspherics with two-dimensional point-shifting by only rotating a cylindrical null screen,” J. Opt. A 10, 104029–104035(2008).
[CrossRef]

Parks, R. E.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412(2010).
[CrossRef] [PubMed]

L. Wang, P. Su, R. E. Parks, J. M. Sasian, and J. H. Burge, “Low-cost, flexible, high dynamic range test for free-form illumination optics,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuE3.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “SCOTS: A quantitative slope measuring method for optical shop use,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuB3.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1990).

Rasban, W.

W. Rasban, National Institutes of Health, USA. Image J V. 1.312u, Image Processing and Analysis in Javahttp://rsb.info.nih.gov/ij/.

Robinson, D. K.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161–166.

Sasian, J. M.

L. Wang, P. Su, R. E. Parks, J. M. Sasian, and J. H. Burge, “Low-cost, flexible, high dynamic range test for free-form illumination optics,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuE3.

Su, P.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412(2010).
[CrossRef] [PubMed]

L. Wang, P. Su, R. E. Parks, J. M. Sasian, and J. H. Burge, “Low-cost, flexible, high dynamic range test for free-form illumination optics,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuE3.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “SCOTS: A quantitative slope measuring method for optical shop use,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuB3.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1990).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1990).

Wang, L.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412(2010).
[CrossRef] [PubMed]

L. Wang, P. Su, R. E. Parks, J. M. Sasian, and J. H. Burge, “Low-cost, flexible, high dynamic range test for free-form illumination optics,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuE3.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “SCOTS: A quantitative slope measuring method for optical shop use,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuB3.

Appl. Opt.

R. Díaz-Uribe and A. Cornejo-Rodríguez, “Conic constant and paraxial radius of curvature measurement for conic surfaces,” Appl. Opt. 25, 3731–3734 (1986).
[CrossRef] [PubMed]

R. Díaz-Uribe and M. Campos-García, “Null-screen testing of fast convex aspheric surfaces,” Appl. Opt. 39, 2670–2677(2000).
[CrossRef]

R. Díaz-Uribe, “Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. 39, 2790–2804(2000).
[CrossRef]

M. Campos-García, R. Díaz-Uribe, and F. S. Granados-Agustin, “Testing fast aspheric surfaces with a linear array of sources,” Appl. Opt. 43, 6255–6264 (2004).
[CrossRef] [PubMed]

M. Avendano-Alejo and R. Dıaz-Uribe, “Testing a fast off-axis parabolic mirror using tilted null screens,” Appl. Opt. 45, 2607–2614 (2006).
[CrossRef] [PubMed]

V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point-shifting in the optical testing of fast aspheric concave surfaces by a cylindrical screen,” Appl. Opt. 47, 644–651 (2008).
[CrossRef] [PubMed]

M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing fast aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47, 849–859 (2008).
[CrossRef] [PubMed]

M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. 48, 1008–1015 (2009).
[CrossRef] [PubMed]

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412(2010).
[CrossRef] [PubMed]

J. Opt. A

V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Improving the quantitative testing of fast aspherics with two-dimensional point-shifting by only rotating a cylindrical null screen,” J. Opt. A 10, 104029–104035(2008).
[CrossRef]

Opt. Express

Proc. SPIE

V. I. Moreno-Oliva, M. Campos-Garcia, F. Granados-Agustin, M. J. Arjona-Pérez, R. Díaz-Uribe, and M. Avendaño-Alejo, “Optical testing of a parabolic trough solar collector by a null screen with stitching,” Proc. SPIE 7390, 739012(2009).
[CrossRef]

Rev. Mex. Fís.

L. Carmona-Paredes and R. Díaz-Uribe, “Geometric analysis of the null screens used for testing convex optical surfaces,” Rev. Mex. Fís. 53, 421–430 (2007).

Other

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161–166.

W. Rasban, National Institutes of Health, USA. Image J V. 1.312u, Image Processing and Analysis in Javahttp://rsb.info.nih.gov/ij/.

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “SCOTS: A quantitative slope measuring method for optical shop use,” in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuB3.

L. Wang, P. Su, R. E. Parks, J. M. Sasian, and J. H. Burge, “Low-cost, flexible, high dynamic range test for free-form illumination optics,” in International Optical Design Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper ITuE3.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University, 1990).

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Figures (13)

Fig. 1
Fig. 1

Proposed schematic arrangement of the LCD monitors.

Fig. 2
Fig. 2

Layout of the test configuration.

Fig. 3
Fig. 3

Schematic representation of the PSM (image plane).

Fig. 4
Fig. 4

LCD monitor displaying the flat null screen and test surface.

Fig. 5
Fig. 5

Rms differences in the sagitta obtained in a simulation when a random displacement is added to the coordinates of the centroids of the spots at the CCD and to the positions of the spots of the null screen.

Fig. 6
Fig. 6

Alignment of the test surface.

Fig. 7
Fig. 7

Sequence of flat null screens displayed on LCD 1 and LCD 1 to increase the number of evaluation points.

Fig. 8
Fig. 8

First screen image of the sequence for: (a) LCD 1 and (b) LCD 1 .

Fig. 9
Fig. 9

Sequence of flat null screens displayed on LCD 2 and LCD 2 .

Fig. 10
Fig. 10

First screen image of the sequence for: (a) LCD 2, and (b) LCD 2 .

Fig. 11
Fig. 11

Coordinates of the positions of all the centroid spots; integration paths for all the points are shown.

Fig. 12
Fig. 12

Evaluated surface.

Fig. 13
Fig. 13

Difference in the sagitta between the measured surface and the best-fitting aspheric.

Tables (3)

Tables Icon

Table 1 Setup Parameters Used for the Design of the LCD Null Screen

Tables Icon

Table 2 Parameters Resulting from Least Squares Fitting of Sagitta Data: Methods

Tables Icon

Table 3 Parameters Resulting from Least Squares Fitting of Sagitta Data: Deformation Coefficients

Equations (14)

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

x 2 x 1 x 1 = y 2 y 1 y 1 = z 2 + a + b a .
R = I 2 ( I · N ) N = ( R x , R y , R z ) ,
I = ( x 1 , y 1 , a ) ,
N = f ( x , y , z ) | f ( x , y , z ) | | P 2 ,
f ( x , y , z ) = Q z 2 2 r z + x 2 + y 2
R x = x 1 [ x 2 2 + y 2 2 + ( Q z 2 r ) 2 ] 2 x 2 [ x 1 x 2 y 1 y 2 + a ( Q z 2 r ) ] x 2 2 + y 2 2 + ( Q z 2 r ) 2 , R y = y 1 [ x 2 2 + y 2 2 + ( Q z 2 r ) 2 ] 2 y 2 [ x 1 x 2 y 1 y 2 + a ( Q z 2 r ) ] x 2 2 + y 2 2 + ( Q z 2 r ) 2 , R z = a [ x 2 2 + y 2 2 + ( Q z 2 r ) 2 ] 2 ( Q z 2 r ) [ x 1 x 2 y 1 y 2 + a ( Q z 2 r ) ] x 2 2 + y 2 2 + ( Q z 2 r ) 2 .
x 3 = ± ρ , y 3 = R y R x ( x 3 x 2 ) + y 2 , z 3 = R z R x ( x 3 x 2 ) + z 2 ;
y 3 = ± ρ , x 3 = R x R y ( y 3 y 2 ) + x 2 , z 3 = R z R y ( y 3 y 2 ) + z 2 .
z z 0 = p 0 p ( n x n z d x + n y n z d y ) ,
N a = R I | R I | ,
z N = i = 1 N 1 { ( n x i n z i + n x i + 1 n z i + 1 ) ( x i + 1 x i ) 2 + ( n y i n z i + n y i + 1 n z i + 1 ) ( y i + 1 y i ) 2 } + z o ,
δ x = η 2 ( 2 ln r 1 ) 1 / 2 cos ( 2 π r 2 ) , δ y = η 2 ( 2 ln r 1 ) 1 / 2 sin ( 2 π r 2 ) , δ z = η 2 ( 2 ln r 3 ) 1 / 2 cos ( 2 π r 2 ) ,
Δ z { ( n x a n z a n x n z ) d x + ( n y a n z a n y n z ) d y } ,
z = s 2 2 r + A x + B y + D 1 s 4 + D 2 s 6 + D 3 s 8 + D 4 s 10 + z 0 ,

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