Abstract

Closed-form analytical solutions are obtained for ray tracing in several types of optically anisotropic inhomogeneous media whose optical properties are characterized by a matrix form of the inhomogeneous dielectric tensor in principal coordinates. The first solution is for anisotropic axial media, the second solution is for meridional rays in epsilon-negative metamaterial, and the third solution is an approximate one for rectangular lenses fabricated by molding procedures. The validation of numerical ray-tracing procedures for optically anisotropic inhomogeneous media was widely ignored since the solution was not available, and thus the present solutions are also useful for the validation. Furthermore, as examples of validation, ray trajectories are calculated by the closed-form solutions, and their results are compared with those obtained by a numerical solution of the geodesic equation which can be interpreted as a generalized ray equation.

© 2013 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics III, 7th ed. (Cambridge University, 1999), Chap. 15.5.
  2. H. Suhara, “Interferometric measurement of the refractive-index distribution in plastic lenses by use of computed tomography,” Appl. Opt. 41, 5317–5325 (2002).
    [CrossRef]
  3. W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Refractive index and dispersion variation in precision optical glass molding by computed tomography,” Appl. Opt. 48, 3588–3595 (2009).
    [CrossRef]
  4. A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. Lond. A 223, 216–225 (1954).
    [CrossRef]
  5. S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
    [CrossRef]
  6. P. J. Sands, “Inhomogeneous lenses, IV. Aberrations of lenses with axial index distributions,” J. Opt. Soc. Am. A 61, 1086–1091 (1971).
    [CrossRef]
  7. E. W. Marchand, Gradient Index Optics (Academic, 1978).
  8. M. Born and E. Wolf, Principles of Optics I, 7th ed. (Cambridge University, 1999), Chap. 3.
  9. T. Sakamoto, “Analytic solutions of the eikonal equation for a GRIN-rod lens 1. Meridional rays,” J. Mod. Opt. 40, 503–516 (1993).
    [CrossRef]
  10. T. Sakamoto, “Analytic solutions of the eikonal equation for a GRIN-rod lens 2. Skew rays,” J. Mod. Opt. 42, 1575–1592 (1995).
    [CrossRef]
  11. D. T. Moore, “Ray tracing in gradient-index media,” J. Opt. Soc. Am. 65, 451–455 (1975).
    [CrossRef]
  12. A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21, 984–987 (1982).
    [CrossRef]
  13. A. Sharma, “Computing optical path length in gradient-index media: a fast and accurate method,” Appl. Opt. 24, 4367–4370 (1985).
    [CrossRef]
  14. A. Sharma and A. K. Ghatak, “Ray tracing in gradient-index lenses: computation of ray-surface intersection,” Appl. Opt. 25, 3409–3412 (1986).
    [CrossRef]
  15. B. D. Stone and G. W. Forbes, “Optimal interpolants for Runge-Kutta ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A 7, 248–254 (1990).
    [CrossRef]
  16. J. Puchalski, “Numerical determination of ray tracing: a new method,” Appl. Opt. 31, 6789–6799 (1992).
    [CrossRef]
  17. B. Richerzhagen, “Finite element ray tracing: a new method for ray tracing in gradient-index media,” Appl. Opt. 35, 6186–6189 (1996).
    [CrossRef]
  18. S. Deng, X. Li, Z. Cen, and S. Jian, “Simulation of the inhomogeneous medium with a self-adapting grid,” Appl. Opt. 46, 3102–3106 (2007).
    [CrossRef]
  19. Y. Nishidate, T. Nagata, S. Morita, and Y. Yamagata, “Ray-tracing method for isotropic inhomogeneous refractive-index media from arbitrary discrete input,” Appl. Opt. 50, 5192–5199 (2011).
    [CrossRef]
  20. M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
    [CrossRef]
  21. M. C. Simon, “Refraction in biaxial crystals: a formula for the indices,” J. Opt. Soc. Am. A 4, 2201–2204 (1987).
    [CrossRef]
  22. T. A. Maldonado and T. K. Gaylord, “Light propagation characteristics for arbitrary wavevector directions in biaxial media by a coordinate-free approach,” Appl. Opt. 30, 2465–2480 (1991).
    [CrossRef]
  23. J. Lekner, “Reflection and Refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
    [CrossRef]
  24. S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993).
    [CrossRef]
  25. A. L. Rivera, S. M. Chumakov, and K. B. Wolf, “Hamiltonian foundation of geometrical anisotropic optics,” J. Opt. Soc. Am. A 121380–1389 (1995).
    [CrossRef]
  26. M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25, 1260–1272 (2008).
    [CrossRef]
  27. M. Sluijter, D. K. G. de Boer, and H. P. Urbach, “Ray-optics analysis of inhomogeneous biaxially anisotropic media,” J. Opt. Soc. Am. A 26, 317–329 (2009).
    [CrossRef]
  28. H. Guo and X. Deng, “Differential geometrical methods in the study of optical transmission (scalar theory). I. Static transmission case,” J. Opt. Soc. Am. A 12, 600–606 (1995).
    [CrossRef]
  29. W. Shen, J. Zhang, S. Wang, and S. Zhu, “Fermat’s principle, the general eikonal equation, and space geometry in a static anisotropic medium,” J. Opt. Soc. Am. A 14, 2850–2854 (1997).
    [CrossRef]
  30. M. A. Slawinski, Seismic Waves and Rays in Elastic Media (Pergamon, 2003).
  31. Y. Rogister and M. A. Slawinski, “Analytic solution of ray-tracing equations for a linearly inhomogeneous and elliptically anisotropic velocity model,” Geophysics 70, D37–D41 (2005).
    [CrossRef]
  32. J. Machac, P. Protiva, and J. Zehentner, “Isotropic epsilon-negative particles,” in Proceedings of IEEE/MTT-S International Microwave Symposium (IEEE/MTT-S, 2007), pp. 1831–1834.
  33. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
    [CrossRef]
  34. J-H. Park, S. Jung, H. Choi, Y. Kim, and B. Lee, “Depth extraction by use of a rectangular lens array and one-dimensional elemental image modification,” Appl. Opt. 43, 4882–4895 (2004).
    [CrossRef]
  35. E. Fehlberg, “Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems,” NASA Tech. Rep. 315, (NASA, 1969).

2011 (1)

2009 (2)

2008 (1)

2007 (1)

2005 (1)

Y. Rogister and M. A. Slawinski, “Analytic solution of ray-tracing equations for a linearly inhomogeneous and elliptically anisotropic velocity model,” Geophysics 70, D37–D41 (2005).
[CrossRef]

2004 (1)

2003 (1)

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

2002 (1)

1997 (1)

1996 (1)

1995 (3)

1993 (2)

S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993).
[CrossRef]

T. Sakamoto, “Analytic solutions of the eikonal equation for a GRIN-rod lens 1. Meridional rays,” J. Mod. Opt. 40, 503–516 (1993).
[CrossRef]

1992 (1)

1991 (2)

1990 (1)

1987 (1)

1986 (1)

1985 (1)

1983 (1)

1982 (1)

1975 (1)

1971 (1)

P. J. Sands, “Inhomogeneous lenses, IV. Aberrations of lenses with axial index distributions,” J. Opt. Soc. Am. A 61, 1086–1091 (1971).
[CrossRef]

1958 (1)

S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
[CrossRef]

1954 (1)

A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. Lond. A 223, 216–225 (1954).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics III, 7th ed. (Cambridge University, 1999), Chap. 15.5.

M. Born and E. Wolf, Principles of Optics I, 7th ed. (Cambridge University, 1999), Chap. 3.

Braat, J. J. M.

Cen, Z.

Chen, Y.

Chipman, R. A.

Choi, H.

Chumakov, S. M.

de Boer, D. K. G.

Deng, S.

Deng, X.

Fehlberg, E.

E. Fehlberg, “Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems,” NASA Tech. Rep. 315, (NASA, 1969).

Fletcher, A.

A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. Lond. A 223, 216–225 (1954).
[CrossRef]

Forbes, G. W.

Gaylord, T. K.

Ghatak, A. K.

Guo, H.

Hillman, L. W.

Jian, S.

Jung, S.

Kim, Y.

Kumar, D. V.

Lee, B.

Lekner, J.

J. Lekner, “Reflection and Refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

Li, X.

Machac, J.

J. Machac, P. Protiva, and J. Zehentner, “Isotropic epsilon-negative particles,” in Proceedings of IEEE/MTT-S International Microwave Symposium (IEEE/MTT-S, 2007), pp. 1831–1834.

Maldonado, T. A.

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, 1978).

McClain, S. C.

Moore, D. T.

Morgan, S. P.

S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
[CrossRef]

Morita, S.

Murphy, T.

A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. Lond. A 223, 216–225 (1954).
[CrossRef]

Nagata, T.

Nishidate, Y.

Park, J-H.

Protiva, P.

J. Machac, P. Protiva, and J. Zehentner, “Isotropic epsilon-negative particles,” in Proceedings of IEEE/MTT-S International Microwave Symposium (IEEE/MTT-S, 2007), pp. 1831–1834.

Puchalski, J.

Richerzhagen, B.

Rivera, A. L.

Rogister, Y.

Y. Rogister and M. A. Slawinski, “Analytic solution of ray-tracing equations for a linearly inhomogeneous and elliptically anisotropic velocity model,” Geophysics 70, D37–D41 (2005).
[CrossRef]

Sakamoto, T.

T. Sakamoto, “Analytic solutions of the eikonal equation for a GRIN-rod lens 2. Skew rays,” J. Mod. Opt. 42, 1575–1592 (1995).
[CrossRef]

T. Sakamoto, “Analytic solutions of the eikonal equation for a GRIN-rod lens 1. Meridional rays,” J. Mod. Opt. 40, 503–516 (1993).
[CrossRef]

Sands, P. J.

P. J. Sands, “Inhomogeneous lenses, IV. Aberrations of lenses with axial index distributions,” J. Opt. Soc. Am. A 61, 1086–1091 (1971).
[CrossRef]

Schurig, D.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Sharma, A.

Shen, L.

Shen, W.

Simon, M. C.

Slawinski, M. A.

Y. Rogister and M. A. Slawinski, “Analytic solution of ray-tracing equations for a linearly inhomogeneous and elliptically anisotropic velocity model,” Geophysics 70, D37–D41 (2005).
[CrossRef]

M. A. Slawinski, Seismic Waves and Rays in Elastic Media (Pergamon, 2003).

Sluijter, M.

Smith, D. R.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Stone, B. D.

Suhara, H.

Urbach, H. P.

Wang, S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics III, 7th ed. (Cambridge University, 1999), Chap. 15.5.

M. Born and E. Wolf, Principles of Optics I, 7th ed. (Cambridge University, 1999), Chap. 3.

Wolf, K. B.

Yamagata, Y.

Yi, A. Y.

Young, A.

A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. Lond. A 223, 216–225 (1954).
[CrossRef]

Zehentner, J.

J. Machac, P. Protiva, and J. Zehentner, “Isotropic epsilon-negative particles,” in Proceedings of IEEE/MTT-S International Microwave Symposium (IEEE/MTT-S, 2007), pp. 1831–1834.

Zhang, J.

Zhao, W.

Zhu, S.

Appl. Opt. (12)

H. Suhara, “Interferometric measurement of the refractive-index distribution in plastic lenses by use of computed tomography,” Appl. Opt. 41, 5317–5325 (2002).
[CrossRef]

W. Zhao, Y. Chen, L. Shen, and A. Y. Yi, “Refractive index and dispersion variation in precision optical glass molding by computed tomography,” Appl. Opt. 48, 3588–3595 (2009).
[CrossRef]

A. Sharma, D. V. Kumar, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. 21, 984–987 (1982).
[CrossRef]

A. Sharma, “Computing optical path length in gradient-index media: a fast and accurate method,” Appl. Opt. 24, 4367–4370 (1985).
[CrossRef]

A. Sharma and A. K. Ghatak, “Ray tracing in gradient-index lenses: computation of ray-surface intersection,” Appl. Opt. 25, 3409–3412 (1986).
[CrossRef]

J. Puchalski, “Numerical determination of ray tracing: a new method,” Appl. Opt. 31, 6789–6799 (1992).
[CrossRef]

B. Richerzhagen, “Finite element ray tracing: a new method for ray tracing in gradient-index media,” Appl. Opt. 35, 6186–6189 (1996).
[CrossRef]

S. Deng, X. Li, Z. Cen, and S. Jian, “Simulation of the inhomogeneous medium with a self-adapting grid,” Appl. Opt. 46, 3102–3106 (2007).
[CrossRef]

Y. Nishidate, T. Nagata, S. Morita, and Y. Yamagata, “Ray-tracing method for isotropic inhomogeneous refractive-index media from arbitrary discrete input,” Appl. Opt. 50, 5192–5199 (2011).
[CrossRef]

M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
[CrossRef]

T. A. Maldonado and T. K. Gaylord, “Light propagation characteristics for arbitrary wavevector directions in biaxial media by a coordinate-free approach,” Appl. Opt. 30, 2465–2480 (1991).
[CrossRef]

J-H. Park, S. Jung, H. Choi, Y. Kim, and B. Lee, “Depth extraction by use of a rectangular lens array and one-dimensional elemental image modification,” Appl. Opt. 43, 4882–4895 (2004).
[CrossRef]

Geophysics (1)

Y. Rogister and M. A. Slawinski, “Analytic solution of ray-tracing equations for a linearly inhomogeneous and elliptically anisotropic velocity model,” Geophysics 70, D37–D41 (2005).
[CrossRef]

J. Appl. Phys. (1)

S. P. Morgan, “General solution of the Luneberg lens problem,” J. Appl. Phys. 29, 1358–1368 (1958).
[CrossRef]

J. Mod. Opt. (2)

T. Sakamoto, “Analytic solutions of the eikonal equation for a GRIN-rod lens 1. Meridional rays,” J. Mod. Opt. 40, 503–516 (1993).
[CrossRef]

T. Sakamoto, “Analytic solutions of the eikonal equation for a GRIN-rod lens 2. Skew rays,” J. Mod. Opt. 42, 1575–1592 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

P. J. Sands, “Inhomogeneous lenses, IV. Aberrations of lenses with axial index distributions,” J. Opt. Soc. Am. A 61, 1086–1091 (1971).
[CrossRef]

M. C. Simon, “Refraction in biaxial crystals: a formula for the indices,” J. Opt. Soc. Am. A 4, 2201–2204 (1987).
[CrossRef]

B. D. Stone and G. W. Forbes, “Optimal interpolants for Runge-Kutta ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A 7, 248–254 (1990).
[CrossRef]

S. C. McClain, L. W. Hillman, and R. A. Chipman, “Polarization ray tracing in anisotropic optically active media. I. Algorithms,” J. Opt. Soc. Am. A 10, 2371–2382 (1993).
[CrossRef]

A. L. Rivera, S. M. Chumakov, and K. B. Wolf, “Hamiltonian foundation of geometrical anisotropic optics,” J. Opt. Soc. Am. A 121380–1389 (1995).
[CrossRef]

M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25, 1260–1272 (2008).
[CrossRef]

M. Sluijter, D. K. G. de Boer, and H. P. Urbach, “Ray-optics analysis of inhomogeneous biaxially anisotropic media,” J. Opt. Soc. Am. A 26, 317–329 (2009).
[CrossRef]

H. Guo and X. Deng, “Differential geometrical methods in the study of optical transmission (scalar theory). I. Static transmission case,” J. Opt. Soc. Am. A 12, 600–606 (1995).
[CrossRef]

W. Shen, J. Zhang, S. Wang, and S. Zhu, “Fermat’s principle, the general eikonal equation, and space geometry in a static anisotropic medium,” J. Opt. Soc. Am. A 14, 2850–2854 (1997).
[CrossRef]

J. Phys. Condens. Matter (1)

J. Lekner, “Reflection and Refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

Phys. Rev. Lett. (1)

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Proc. R. Soc. Lond. A (1)

A. Fletcher, T. Murphy, and A. Young, “Solutions of two optical problems,” Proc. R. Soc. Lond. A 223, 216–225 (1954).
[CrossRef]

Other (6)

E. W. Marchand, Gradient Index Optics (Academic, 1978).

M. Born and E. Wolf, Principles of Optics I, 7th ed. (Cambridge University, 1999), Chap. 3.

M. Born and E. Wolf, Principles of Optics III, 7th ed. (Cambridge University, 1999), Chap. 15.5.

E. Fehlberg, “Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems,” NASA Tech. Rep. 315, (NASA, 1969).

J. Machac, P. Protiva, and J. Zehentner, “Isotropic epsilon-negative particles,” in Proceedings of IEEE/MTT-S International Microwave Symposium (IEEE/MTT-S, 2007), pp. 1831–1834.

M. A. Slawinski, Seismic Waves and Rays in Elastic Media (Pergamon, 2003).

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Figures (4)

Fig. 1.
Fig. 1.

Comparisons of the numerical solution (Numerical) with the closed-form solution (39) for different values α = 0.125 , 0.25, and 0.5. The result of closed-form solution is also shown for α = 0 as a reference.

Fig. 2.
Fig. 2.

Comparisons of the numerical solution (Numerical) with the closed-form solution (40) for different parameter c values.

Fig. 3.
Fig. 3.

Comparison of the results of closed-form solutions (41) and (43) with the numerical solution for different ray directions u 0 .

Fig. 4.
Fig. 4.

Comparison of the results of closed-form solutions (41) and (44) with the numerical solution for different ray directions u 0 .

Tables (1)

Tables Icon

Table 1. Coefficients a i j , b i , b ^ i , and c i for the RKF45

Equations (44)

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

D = ε 0 ε E ,
B = μ 0 μ H ,
[ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] .
[ ε 1 0 0 0 ε 2 0 0 0 ε 3 ] ,
[ g 1 ( x ) 0 0 0 g 2 ( x ) 0 0 0 g 3 ( x ) ] .
δ d s = 0 ,
δ [ g i j ( x ) d x i d x j ] 1 / 2 = 0 .
δ [ g 1 ( x ) d x 2 + g 2 ( x ) d y 2 + g 3 ( x ) d z 2 ] 1 / 2 = 0 .
δ [ g 1 ( x ) x ˙ 2 + g 2 ( x ) y ˙ 2 + g 3 ( x ) ] 1 / 2 d z = 0 ,
d d z ( F x ˙ ) = F x ,
d d z ( F y ˙ ) = F y .
g 1 g 3 x ¨ + ( g ˙ 1 g 3 g 1 g ˙ 3 ) x ˙ + 1 2 ( g 1 x ˙ L z g 3 L x ) = 0 ,
g 2 g 3 y ¨ + ( g ˙ 2 g 3 g 2 g ˙ 3 ) y ˙ + 1 2 ( g 2 y ˙ L z g 3 L y ) = 0 ,
L p = g 1 p x ˙ 2 + g 2 p y ˙ 2 + g 3 p
g 3 x ¨ 1 2 g ˙ 3 x ˙ = 0 ,
g 3 y ¨ 1 2 g ˙ 3 y ˙ = 0 .
x = x 0 + u 0 g 3 ( z 0 ) z 0 z g 3 ( ζ ) d ζ ,
y = y 0 + v 0 g 3 ( z 0 ) z 0 z g 3 ( ζ ) d ζ ,
g 1 g 3 x ¨ + ( g ˙ 1 g 3 g 1 g ˙ 3 ) x ˙ = 0 ,
g 2 g 3 y ¨ + ( g ˙ 2 g 3 g 2 g ˙ 3 ) y ˙ = 0
x = x 0 + u 0 g 1 ( z 0 ) g 3 ( z 0 ) z 0 z g 3 ( ζ ) g 1 ( ζ ) d ζ ,
y = y 0 + v 0 g 2 ( z 0 ) g 3 ( z 0 ) z 0 z g 3 ( ζ ) g 2 ( ζ ) d ζ .
g 1 g 3 x ¨ g 1 g ˙ 3 x ˙ + 1 2 g 1 ( g 1 x x ˙ 3 + g 2 y y ˙ 3 + g 3 z ) x ˙ 1 2 g 3 g 1 x x ˙ 2 = 0 ,
g 2 g 3 x ¨ g 2 g ˙ 3 x ˙ + 1 2 g 2 ( g 1 x x ˙ 3 + g 2 y y ˙ 3 + g 3 z ) y ˙ 1 2 g 3 g 2 y y ˙ 2 = 0 .
g 1 g 3 x ¨ 1 2 g 1 g ˙ 3 x ˙ 1 2 g 3 g 1 x x ˙ 2 = 0 ,
g 2 g 3 x ¨ 1 2 g 2 g ˙ 3 x ˙ 1 2 g 3 g 2 y y ˙ 2 = 0 .
x 0 x 1 g 1 ( ξ ) d ξ = u 0 g 1 ( x 0 ) g 3 ( z 0 ) z 0 z g 3 ( ζ ) d ζ ,
y 0 y 1 g 2 ( η ) d η = v 0 g 2 ( y 0 ) g 3 ( z 0 ) z 0 z g 3 ( ζ ) d ζ .
d 2 x i d s 2 + Γ j k i d x j d s d x k d s = 0 ,
Γ j k i = 1 2 ε ^ i l ( ε l j x k + ε l k x j ε j k x l ) ,
d x i d s = t i ,
d t i d s = Γ j k i t j t k ,
d x d s = f ( s , x ) .
x ( s + Δ s ) = x ( s ) + Δ s i = 1 6 b i k i + O ( Δ s 5 ) ,
x ^ ( s + Δ s ) = x ^ ( s ) + Δ s i = 1 6 b ^ i k i + O ( Δ s 6 ) ,
k i = f ( s + c i Δ s , x ( s ) + Δ s j = 1 i 1 a i j k j ) ,
δ = max i | x ^ i ( s + Δ s ) x i ( s + Δ s ) | / ε i ,
Δ s n + 1 = S Δ s n δ 1 5 ,
x = x 0 + u 0 g 3 ( z 0 ) [ z z 0 α ( cos z cos z 0 ) ] .
x = x 0 + b u 0 g 1 ( z 0 ) a g 3 ( z 0 ) log z + z 2 + c z 0 + z 0 2 + c .
x = b 1 2 a 1 + ( x 0 + b 1 2 a 1 ) cos β a 1 ( c 1 + b 1 x 0 + a 1 x 0 2 ) a 1 sin β ,
β = u 0 a 1 g 1 ( x 0 ) g 3 ( z 0 ) z 0 z g 3 ( ζ ) d ζ .
β = u 0 a 1 g 1 ( x 0 ) g 3 ( z 0 ) [ γ 1 ( z z 0 ) α 1 ( cos z cos z 0 ) ] .
β = u 0 a 1 g 1 ( x 0 ) g 3 ( z 0 ) ( z z 0 ) [ γ 2 α 2 ( z + z 0 ) ] .

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