Abstract

We use geometrical optics and the caustic-touching theorem to study, in an exact way, the change in the topology of the image of an object obtained by reflections on an arbitrary smooth surface. Since the procedure that we use to compute the image is exactly the same as that used to simulate the ideal patterns, referred to as Ronchigrams, in the Ronchi test used to test mirrors, we remark that the closed loop fringes commonly observed in the Ronchigrams when the grating, referred to as a Ronchi ruling, is located at the caustic place are due to a disruption of fringes, or, more correctly, as disruption of shadows corresponding to the ruling bands. To illustrate our results, we assume that the reflecting surface is a spherical mirror and we consider two kinds of objects: circles and line segments.

© 2008 Optical Society of America

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References

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  1. M. V. Berry, “Disruption of images: the caustic-touching theorem,” J. Opt. Soc. Am. A 4, 561-569 (1987).
    [CrossRef]
  2. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).
  3. R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, 1980).
  4. A. B. Fraser and W. H. Mach, “Mirages,” Sci. Am. 234, 102-111(1976).
    [CrossRef]
  5. W. Tape, “The topology of mirages,” Sci. Am. 252, 120-129(1985).
    [CrossRef]
  6. R. Narayan, R. Blandford, and R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112-115 (1984).
    [CrossRef]
  7. C. Hogan and R. Narayan, “Gravitational lensing by cosmic strings,” Mon. Not. R. Astron. Soc. 211, 575-591 (1984).
  8. A. O. Petters, H. Levine, and J. Wambsganss, Singularity Theory and Gravitational Lensing (Birkhäuser, 2001).
    [CrossRef]
  9. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437-451 (1964).
    [CrossRef]
  10. A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1978), Chap. 9.
  11. D. Malacara, “Geometrical Ronchi test of aspherical mirrors,” Appl. Opt. 4, 1371-1374 (1965).
    [CrossRef]
  12. A. A. Sherwood, “Quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron. Assoc. 68, 180-191 (1958).
  13. A. Cordero-Dávila, A. Cornejo-Rodriguez, and O. Cardona-Nuñez, “Ronchi and Hartmann tests with the same mathematical theory,” Appl. Opt. 31, 2370-2376 (1992).
    [CrossRef] [PubMed]
  14. A. Cordero-Dávila, J. Díaz-Anzures, and V. Cabrera-Peláez, “Algorithm for the simulation of Ronchigrams of arbitrary optical systems and Ronchi grids in generalized coordinates,” Appl. Opt. 41, 3866-3873 (2002).
    [CrossRef] [PubMed]
  15. G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021-1028(2001).
    [CrossRef]
  16. J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, “Computation of the disk of least confusion for conic mirrors,” Appl. Opt. 43, 6080-6088 (2004).
    [CrossRef] [PubMed]
  17. V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1986).
  18. V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, 1995), Vol I.
  19. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1980).
  20. D. L. Shealy and D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the reflecting surface,” Opt. Acta 20, 287-301(1973).
    [CrossRef]
  21. D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588-2596(1976).
    [CrossRef] [PubMed]
  22. D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic, paraboloid, and elliptic cone,” Appl. Opt. 12, 2955-2959(1973).
    [CrossRef] [PubMed]
  23. P. S. Theocaris, “Properties of caustics from conic reflectors. 1. Meridional rays,” Appl. Opt. 16, 1705-1716 (1977).
    [CrossRef] [PubMed]
  24. A. Cordero-Dávila and J. Castro-Ramos, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror,” Appl. Opt. 37, 6774-6778 (1998).
    [CrossRef]
  25. R. W. Hosken, “Circle of least confusion of a spherical reflector,” Appl. Opt. 46, 3107-3117 (2007).
    [CrossRef] [PubMed]
  26. M. V. R. K. Murty and A. H. Shoemaker, “Theory of concentric circular grid,” Appl. Opt. 5, 323-326 (1966).
    [CrossRef] [PubMed]

2007

2004

2002

2001

1998

1995

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, 1995), Vol I.

1992

1987

1986

V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1986).

1985

W. Tape, “The topology of mirages,” Sci. Am. 252, 120-129(1985).
[CrossRef]

1984

R. Narayan, R. Blandford, and R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112-115 (1984).
[CrossRef]

C. Hogan and R. Narayan, “Gravitational lensing by cosmic strings,” Mon. Not. R. Astron. Soc. 211, 575-591 (1984).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

1980

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, 1980).

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1980).

1978

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1978), Chap. 9.

1977

1976

1973

D. L. Shealy and D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the reflecting surface,” Opt. Acta 20, 287-301(1973).
[CrossRef]

D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic, paraboloid, and elliptic cone,” Appl. Opt. 12, 2955-2959(1973).
[CrossRef] [PubMed]

1966

1965

1964

1958

A. A. Sherwood, “Quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron. Assoc. 68, 180-191 (1958).

Arnold, V. I.

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, 1995), Vol I.

V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1986).

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1980).

Berry, M. V.

Blandford, R.

R. Narayan, R. Blandford, and R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112-115 (1984).
[CrossRef]

Burkhard, D. G.

D. L. Shealy and D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the reflecting surface,” Opt. Acta 20, 287-301(1973).
[CrossRef]

D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic, paraboloid, and elliptic cone,” Appl. Opt. 12, 2955-2959(1973).
[CrossRef] [PubMed]

Cabrera-Peláez, V.

Cardona-Nuñez, O.

Castro-Ramos, J.

Cordero-Dávila, A.

Cornejo-Rodriguez, A.

de Ita Prieto, O.

Díaz-Anzures, J.

Fraser, A. B.

A. B. Fraser and W. H. Mach, “Mirages,” Sci. Am. 234, 102-111(1976).
[CrossRef]

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, 1980).

Gusein-Zade, S. M.

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, 1995), Vol I.

Hogan, C.

C. Hogan and R. Narayan, “Gravitational lensing by cosmic strings,” Mon. Not. R. Astron. Soc. 211, 575-591 (1984).

Hosken, R. W.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

Levine, H.

A. O. Petters, H. Levine, and J. Wambsganss, Singularity Theory and Gravitational Lensing (Birkhäuser, 2001).
[CrossRef]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

Mach, W. H.

A. B. Fraser and W. H. Mach, “Mirages,” Sci. Am. 234, 102-111(1976).
[CrossRef]

Malacara, D.

Murty, M. V. R. K.

Narayan, R.

C. Hogan and R. Narayan, “Gravitational lensing by cosmic strings,” Mon. Not. R. Astron. Soc. 211, 575-591 (1984).

R. Narayan, R. Blandford, and R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112-115 (1984).
[CrossRef]

Nityananda, R.

R. Narayan, R. Blandford, and R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112-115 (1984).
[CrossRef]

Petters, A. O.

A. O. Petters, H. Levine, and J. Wambsganss, Singularity Theory and Gravitational Lensing (Birkhäuser, 2001).
[CrossRef]

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

Ronchi, V.

Shealy, D. L.

Sherwood, A. A.

A. A. Sherwood, “Quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron. Assoc. 68, 180-191 (1958).

Shoemaker, A. H.

Silva-Ortigoza, G.

Tape, W.

W. Tape, “The topology of mirages,” Sci. Am. 252, 120-129(1985).
[CrossRef]

Theocaris, P. S.

Varchenko, A. N.

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, 1995), Vol I.

Wambsganss, J.

A. O. Petters, H. Levine, and J. Wambsganss, Singularity Theory and Gravitational Lensing (Birkhäuser, 2001).
[CrossRef]

Appl. Opt.

V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437-451 (1964).
[CrossRef]

A. Cordero-Dávila, A. Cornejo-Rodriguez, and O. Cardona-Nuñez, “Ronchi and Hartmann tests with the same mathematical theory,” Appl. Opt. 31, 2370-2376 (1992).
[CrossRef] [PubMed]

A. Cordero-Dávila, J. Díaz-Anzures, and V. Cabrera-Peláez, “Algorithm for the simulation of Ronchigrams of arbitrary optical systems and Ronchi grids in generalized coordinates,” Appl. Opt. 41, 3866-3873 (2002).
[CrossRef] [PubMed]

G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021-1028(2001).
[CrossRef]

J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, “Computation of the disk of least confusion for conic mirrors,” Appl. Opt. 43, 6080-6088 (2004).
[CrossRef] [PubMed]

D. Malacara, “Geometrical Ronchi test of aspherical mirrors,” Appl. Opt. 4, 1371-1374 (1965).
[CrossRef]

D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588-2596(1976).
[CrossRef] [PubMed]

D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic, paraboloid, and elliptic cone,” Appl. Opt. 12, 2955-2959(1973).
[CrossRef] [PubMed]

P. S. Theocaris, “Properties of caustics from conic reflectors. 1. Meridional rays,” Appl. Opt. 16, 1705-1716 (1977).
[CrossRef] [PubMed]

A. Cordero-Dávila and J. Castro-Ramos, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror,” Appl. Opt. 37, 6774-6778 (1998).
[CrossRef]

R. W. Hosken, “Circle of least confusion of a spherical reflector,” Appl. Opt. 46, 3107-3117 (2007).
[CrossRef] [PubMed]

M. V. R. K. Murty and A. H. Shoemaker, “Theory of concentric circular grid,” Appl. Opt. 5, 323-326 (1966).
[CrossRef] [PubMed]

J. Br. Astron. Assoc.

A. A. Sherwood, “Quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron. Assoc. 68, 180-191 (1958).

J. Opt. Soc. Am. A

Mon. Not. R. Astron. Soc.

C. Hogan and R. Narayan, “Gravitational lensing by cosmic strings,” Mon. Not. R. Astron. Soc. 211, 575-591 (1984).

Nature

R. Narayan, R. Blandford, and R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112-115 (1984).
[CrossRef]

Opt. Acta

D. L. Shealy and D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the reflecting surface,” Opt. Acta 20, 287-301(1973).
[CrossRef]

Sci. Am.

A. B. Fraser and W. H. Mach, “Mirages,” Sci. Am. 234, 102-111(1976).
[CrossRef]

W. Tape, “The topology of mirages,” Sci. Am. 252, 120-129(1985).
[CrossRef]

Other

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, 1980).

A. O. Petters, H. Levine, and J. Wambsganss, Singularity Theory and Gravitational Lensing (Birkhäuser, 2001).
[CrossRef]

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1978), Chap. 9.

V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1986).

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, 1995), Vol I.

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1980).

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Figures (23)

Fig. 1
Fig. 1

Schematic of the object curve, which we assume is lying on a plane perpendicular to the z axis, the two-dimensional reflecting smooth surface is locally given by z = f ( x , y ) , and the position of the observing eye is given by S = ( s 1 , s 2 , s 3 ) .

Fig. 2
Fig. 2

New geometric arrangement to compute the image of a one-dimensional object under reflection on the arbitrary smooth surface z = f ( x , y ) . The observing eye has been replaced by an imaginary point light source. In this diagram, we show an imaginary emitted light ray and the corresponding reflected light ray, which we assume arrives at a point of the one-dimensional object. Therefore, the point of the surface where the imaginary light ray is reflected belongs to the image of the one-dimensional object.

Fig. 3
Fig. 3

Schematic of the Ronchi test arrangement. In this diagram, we have the surface under test locally given by z = f ( x , y ) , a real point light source located on the optical system, and a Ronchi ruling. The pattern observed through the grating on the surface of the mirror is referred to as the real Ronchigram.

Fig. 4
Fig. 4

Schematic of the optical system and the vectors used to compute the image of a one-dimensional object obtained by reflection on an arbitrary smooth surface locally given by z = f ( x , y ) . In this diagram, S = ( s 1 , s 2 , s 3 ) denotes the position of the point light source, I ^ is the direction of an emitted light ray, r = ( x , y , f ( x , y ) ) is the point on the smooth surface where the emitted light ray is reflected in the direction R ^ , and N ^ is the normal vector to the smooth surface at the point of reflection.

Fig. 5
Fig. 5

Schematic of the two sets of coordinate systems used to compute the image of a one-dimensional object obtained by reflection on an arbitrary smooth surface locally given by z = f ( x , y ) . We also have included an emitted light ray such that its associated reflected light ray connects a point of the smooth surface with a point of the one-dimensional object. The point on the smooth surface where the emitted light ray is reflected belongs to the image of the one-dimensional object.

Fig. 6
Fig. 6

Intersection of the caustic given by Eqs. (40, 41) with the plane y = 0 , for the special case r = 2415 mm , D = 1470 mm = 2 ρ max , z = z 0 = 11 , 000 mm , and the point light source is located at (0, 0, 1350 mm ).

Fig. 7
Fig. 7

Intersection of the two branches of the caustic with the plane z = 11 , 000 .

Fig. 8
Fig. 8

(a) Object space and (b) image space for k = 7 .

Fig. 9
Fig. 9

(a) Object space and (b) image space for k = R c .

Fig. 10
Fig. 10

(a) Object space and (b) image space for k = 1.5 .

Fig. 11
Fig. 11

(a) Object space and (b) image space for k = 0 .

Fig. 12
Fig. 12

Set of circles with their centers on T x .

Fig. 13
Fig. 13

Image of the set of circles with their centers on T x .

Fig. 14
Fig. 14

Set of concentric circles with their centers on T x .

Fig. 15
Fig. 15

Image of the set of concentric circles with their centers on T x .

Fig. 16
Fig. 16

(a) Object space: set of line segments parallel to the T y axis with T y [ 6 , 6 ] and the caustic, which is a circle of radius R c = 4.1643 mm and its center. (b) Image space: the corresponding images. In the Ronchi test, the set of lines (a) is the grating or Ronchi ruling and its image (b) is referred to as the associated Ronchigram.

Fig. 17
Fig. 17

(a) Object space and (b) image space for k = 6 .

Fig. 18
Fig. 18

(a) Object space and (b) image space for k = 4 .

Fig. 19
Fig. 19

(a) Object space and (b) image space for k = 2 .

Fig. 20
Fig. 20

(a)–(d) Object space and (e)–(h) image space for k = 0 , 2, 4, and 6, respectively.

Fig. 21
Fig. 21

Pattern obtained by Ronchi.

Fig. 22
Fig. 22

Simulation of the Ronchi ruling and the caustic curve.

Fig. 23
Fig. 23

Image of the Ronchi rulings or Ronchigrams.

Equations (48)

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T = r + l R ^ ,
R ^ = I ^ 2 ( I ^ · N ^ ) N ^ ,
I ^ = I | I | = ( x s 1 , y s 2 , f s 3 ) ( x s 1 ) 2 + ( y s 2 ) 2 + ( f s 3 ) 2 .
N = ( f x , f y , 1 ) ,
N ^ = ( f x , f y , 1 ) ( 1 + f x 2 + f y 2 ) .
T 1 = x + l h 1 α ,
T 2 = y + l h 2 α ,
T 3 = f ( x , y ) + l h 3 α ,
h 1 = ( x s 1 ) ( 1 f x 2 + f y 2 ) 2 f x [ f y ( y s 2 ) + s 3 f ] , h 2 = ( y s 2 ) ( 1 + f x 2 f y 2 ) 2 f y [ f x ( x s 1 ) + s 3 f ] , h 3 = ( f s 3 ) ( 1 + f x 2 + f y 2 ) + 2 [ f x ( x s 1 ) + f y ( y s 2 ) ] , α = ( 1 + f x 2 + f y 2 ) ( s 1 x ) 2 + ( s 2 y ) 2 + ( s 3 f ) 2 ,
l = α ( T 3 f ( x , y ) h 3 ) = α ( z 0 f ( x , y ) h 3 ) ,
T 1 ( x , y , z 0 ) = x + [ z 0 f ( x , y ) ] ( h 1 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , y , s 1 , s 2 , s 3 ) ) , T 2 ( x , y , z 0 ) = y + [ z 0 f ( x , y ) ] ( h 2 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , y , s 1 , s 2 , s 3 ) ) , T 3 ( x , y , z 0 ) = z 0 ,
T 1 ( x , y , z 0 ) = x + [ z 0 f ( x , y ) ] ( h 1 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , y , s 1 , s 2 , s 3 ) ) , T 2 ( x , y , z 0 ) = y + [ z 0 f ( x , y ) ] ( h 2 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , y , s 1 , s 2 , s 3 ) ) .
T x = Γ ( σ ) , T y = Σ ( σ ) ,
T x = Λ ( T y ) .
T x ( σ ) = x + [ z ˜ 0 f ( x , y ) ] ( h 1 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , y , s 1 , s 2 , s 3 ) ) , T y ( σ ) = y + [ z ˜ 0 f ( x , y ) ] ( h 2 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , y , s 1 , s 2 , s 3 ) ) .
y i = h i ( x j ) , where     i , j = 1 , , n .
J det ( y i x j ) = 0 .
F ( x 1 , , x n ) = 0 ,
x n = g ( x 1 , , x n 1 ) ,
y 1 = h 1 ( x 1 , , x n 1 , g ( x 1 , , x n 1 ) ) , y 2 = h 2 ( x 1 , , x n 1 , g ( x 1 , , x n 1 ) ) , y n = h n ( x 1 , , x n 1 , g ( x 1 , , x n 1 ) ) .
J ( x , y , z 0 ) = det ( ( T 1 , T 2 , T 3 ) ( x , y , z 0 ) ) = ( T 1 x ) ( T 2 y ) ( T 1 y ) ( T 2 x ) = 0 .
J ( x , y , z 0 ) = H 2 ( x , y ) ( z 0 f h 3 ) 2 + H 1 ( x , y ) ( z 0 f h 3 ) + H 0 ( x , y ) = 0 ,
H 2 ( x , y ) = h · [ ( h x ) × ( h y ) ] , H 1 ( x , y ) = h · [ ( r x ) × ( h y ) + ( h x ) × ( r y ) ] , H 0 ( x , y ) = h · [ ( r x ) × ( r y ) ] ,
r = ( x , y , f ( x , y ) ) , h = ( h 1 , h 2 , h 3 ) .
z 0 = z 0 ± ( x , y ) f + h 3 ( H 1 ± H 1 2 4 H 2 H 0 2 H 2 ) .
T c ± = r + ( H 1 ± H 1 2 4 H 2 H 0 2 H 2 ) h .
T 1 ( x , y ) = x + [ z ˜ 0 f ( x , y ) ] ( h 1 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , y , s 1 , s 2 , s 3 ) ) , T 2 ( x , y ) = y + [ z ˜ 0 f ( x , y ) ] ( h 2 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , y , s 1 , s 2 , s 3 ) ) ,
J ˜ ( x , y ) = det ( ( T 1 , T 2 ) ( x , y ) ) = ( T 1 x ) ( T 2 y ) ( T 1 y ) ( T 2 x ) = 0 ,
J ˜ ( x , y ) = H 2 ( x , y ) ( z ˜ 0 f h 3 ) 2 + H 1 ( x , y ) ( z ˜ 0 f h 3 ) + H 0 ( x , y ) = 0 .
z c min ± z ˜ 0 z c max ± ,
y = Ψ ( x , z ˜ 0 ) .
T 1 c ( x ) = x + [ z ˜ 0 f ( x , Ψ ) ] ( h 1 ( x , Ψ , s 1 , s 2 , s 3 ) h 3 ( x , Ψ , s 1 , s 2 , s 3 ) ) , T 2 c ( x ) = Ψ + [ z ˜ 0 f ( x , Ψ ) ] ( h 2 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , Ψ , s 1 , s 2 , s 3 ) ) .
T x ( σ ) = x + [ z ˜ 0 f ( x , y ) ] ( h 1 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , y , s 1 , s 2 , s 3 ) ) , T y ( σ ) = y + [ z ˜ 0 f ( x , y ) ] ( h 2 ( x , y , s 1 , s 2 , s 3 ) h 3 ( x , y , s 1 , s 2 , s 3 ) ) ,
T c ± = r + ( H 1 ± H 1 2 4 H 2 H 0 2 H 2 ) h ,
z = f ( x , y ) = r r 2 x 2 y 2 ,
z = f ( ρ ) = r r 2 ρ 2 ,
T 1 ( x , y , z 0 ) = x [ 1 + ( z 0 r + r 2 ρ 2 ) G ( ρ ) ] , T 2 ( x , y , z 0 ) = y [ 1 + ( z 0 r + r 2 ρ 2 ) G ( ρ ) ] , T 2 ( x , y , z 0 ) = z 0 ,
G ( ρ ) = 2 ( r s ) ( r 2 ρ 2 ) r 2 r 2 ρ 2 r 2 ρ 2 [ 2 ρ 2 ( r s ) + r 2 ( s r + r 2 ρ 2 ) ] .
z 0 = r [ 2 r 3 + 2 s ρ 2 2 r 2 ( s + r 2 ρ 2 ) r ( 2 ρ 2 s r 2 ρ 2 ) ] 2 r 3 2 r ρ 2 + 2 s ρ 2 r 2 ( 2 s + r 2 ρ 2 ) , z 0 + = r 3 ( 4 r 2 5 r s + 2 s 2 ) ( r s ) [ r 2 ( 4 r s ) + 2 ( r s ) ρ 2 ] r 2 ρ 2 r 2 [ 3 r 2 4 r s + 2 s 2 + 3 ( s r ) r 2 ρ 2 ] ,
T 1 c ( x , y ) = 0 , T 2 c ( x , y ) = 0 , T 3 c ( x , y ) = z 0 ,
T 1 c + ( x , y ) = 2 x ( r s ) 2 ρ 2 r 2 [ 3 r 2 4 r s + 2 s 2 + 3 ( s r ) r 2 ρ 2 ] , T 1 c + ( x , y ) = 2 y ( r s ) 2 ρ 2 r 2 [ 3 r 2 4 r s + 2 s 2 + 3 ( s r ) r 2 ρ 2 ] , T 3 c + ( x , y ) = z 0 + .
T 3 c ( ρ max ) T 3 c ( ρ ) r s 2 s r , T 3 c + ( ρ max ) T 3 c + ( ρ ) r s 2 s r .
T 1 ( x , y ) = x [ 1 + ( z 0 r + r 2 ρ 2 ) G ( ρ ) ] , T 2 ( x , y ) = y [ 1 + ( z 0 r + r 2 ρ 2 ) G ( ρ ) ] .
T x ( σ ) = x [ 1 + ( z 0 r + r 2 ρ 2 ) G ( ρ ) ] , T y ( σ ) = y [ 1 + ( z 0 r + r 2 ρ 2 ) G ( ρ ) ] .
( T x k ) 2 + T y 2 = 1 ,
{ x [ 1 + ( 11000 r + r 2 ρ 2 ) G ( ρ ) ] k } 2 + y 2 [ 1 + ( 11000 r + r 2 ρ 2 ) G ( ρ ) ] 2 = 1 .
T x = k ,
x [ 1 + ( 11000 r + r 2 ρ 2 ) G ( ρ ) ] = k .

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