Abstract

The aim of the present work is to obtain expressions for both the wavefront train and the caustic associated with the light rays reflected by an arbitrary smooth surface after being emitted by a point light source located at an arbitrary position in free space. To this end, we obtain an expression for the k-function associated with the general integral of Stavroudis to the eikonal equation that describes the evolution of the reflected light rays. The caustic is computed by using the definitions of the critical and caustic sets of the map that describes the evolution of an arbitrary wavefront associated with the general integral. It is shown that the expression for the caustic is the same as that—reported in the literature—obtained by using an exact ray tracing. The general results are applied to a parabolic mirror. For this case, we find that when the point light source is off the optical axis, the caustic locally has singularities of the hyperbolic umbilic type while the reflected wavefront at the caustic region locally has singularities of the cusp ridge and swallowtail types.

© 2009 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, 1999), Chap. 3.
  2. O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972).
  3. O. N. Stavroudis and R. C. Fronczek, “Caustic surfaces and the structure of the geometrical image,” J. Opt. Soc. Am. 66, 795-800 (1976).
    [CrossRef]
  4. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics(Wiley-VCH Verlag, 2006).
    [CrossRef]
  5. O. N. Stavroudis, “The k function in geometrical optics and its relationship to the archetypal wave front and the caustic surface,” J. Opt. Soc. Am. A 12, 1010-1016 (1995).
    [CrossRef]
  6. D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370-2382 (2008).
    [CrossRef]
  7. G. Silva-Ortigoza, “An analytical expression for the singularities developed by an aberrated wavefront,” Rev. Mex. Fís. 46, 507-511 (2000).
  8. 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]
  9. G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358-365 (2002).
    [CrossRef]
  10. J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, “Computation of the disk of least confusion for conic mirrors,” Appl. Opt. 43, 6080-6089 (2004).
    [CrossRef] [PubMed]
  11. E. Román-Hernández and G. Silva-Ortigoza, “Exact computation of image disruption under reflection on a smooth surface and Ronchigrams,” Appl. Opt. 47, 5500-5518 (2008).
    [CrossRef] [PubMed]
  12. L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed.(Butterworth Heinemann, 2000).
  13. J. W. Bruce and P. J. Giblin, Curves and Singularities, 2nd ed.(Cambridge Univ. Press, 1992).
  14. V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1986).
    [CrossRef]
  15. V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps, Vol. I (Birkhäuser, 1995).
  16. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1980).
  17. 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]
  18. D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588-2596 (1976).
    [CrossRef] [PubMed]
  19. 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]
  20. P. S. Theocaris and J. G. Michopoulos, “Generalization of the theory of far-field caustics by the catastrophe theory,” Appl. Opt. 21, 1080-1091 (1982).
    [CrossRef] [PubMed]
  21. D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897-909 (1981).
    [CrossRef] [PubMed]
  22. P. S. Theocaris, “Surface topography by caustics,” Appl. Opt. 15, 1629-1638 (1976).
    [CrossRef] [PubMed]
  23. P. S. Theocaris, “Properties of caustics from conic reflectors. 1: Meridional rays,” Appl. Opt. 16, 1705-1716 (1977).
    [CrossRef] [PubMed]

2008 (2)

2004 (1)

2002 (1)

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358-365 (2002).
[CrossRef]

2001 (1)

2000 (1)

G. Silva-Ortigoza, “An analytical expression for the singularities developed by an aberrated wavefront,” Rev. Mex. Fís. 46, 507-511 (2000).

1995 (1)

1982 (1)

1981 (1)

1977 (1)

1976 (3)

1973 (2)

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]

Arnold, V. I.

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

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

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

Born, M.

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

Bruce, J. W.

J. W. Bruce and P. J. Giblin, Curves and Singularities, 2nd ed.(Cambridge Univ. Press, 1992).

Burkhard, D. G.

Carvente-Muñoz, O.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358-365 (2002).
[CrossRef]

Castro-Ramos, J.

Cordero-Dávila, A.

de Ita Prieto, O.

Fronczek, R. C.

Giblin, P. J.

J. W. Bruce and P. J. Giblin, Curves and Singularities, 2nd ed.(Cambridge Univ. Press, 1992).

Gusein-Zade, S. M.

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

Hoffnagle, J. A.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed.(Butterworth Heinemann, 2000).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed.(Butterworth Heinemann, 2000).

Marciano-Melchor, M.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358-365 (2002).
[CrossRef]

Michopoulos, J. G.

Román-Hernández, E.

Shealy, D. L.

Silva-Ortigoza, G.

Silva-Ortigoza, Ramón

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358-365 (2002).
[CrossRef]

Stavroudis, O. N.

Theocaris, P. S.

Varchenko, A. N.

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

Wolf, E.

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

Appl. Opt. (9)

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, 6080-6089 (2004).
[CrossRef] [PubMed]

E. Román-Hernández and G. Silva-Ortigoza, “Exact computation of image disruption under reflection on a smooth surface and Ronchigrams,” Appl. Opt. 47, 5500-5518 (2008).
[CrossRef] [PubMed]

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 and J. G. Michopoulos, “Generalization of the theory of far-field caustics by the catastrophe theory,” Appl. Opt. 21, 1080-1091 (1982).
[CrossRef] [PubMed]

D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897-909 (1981).
[CrossRef] [PubMed]

P. S. Theocaris, “Surface topography by caustics,” Appl. Opt. 15, 1629-1638 (1976).
[CrossRef] [PubMed]

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

J. Opt. A, Pure Appl. Opt. (1)

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A, Pure Appl. Opt. 4, 358-365 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

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]

Rev. Mex. Fís. (1)

G. Silva-Ortigoza, “An analytical expression for the singularities developed by an aberrated wavefront,” Rev. Mex. Fís. 46, 507-511 (2000).

Other (8)

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics(Wiley-VCH Verlag, 2006).
[CrossRef]

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

O. N. Stavroudis, The Optics of Rays, Wavefronts and Caustics (Academic, 1972).

L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed.(Butterworth Heinemann, 2000).

J. W. Bruce and P. J. Giblin, Curves and Singularities, 2nd ed.(Cambridge Univ. Press, 1992).

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

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

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

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Figures (7)

Fig. 1
Fig. 1

Schematic drawing of the optical system and the vectors used to compute the optical path length from some point in the object space through the optical system to the point in image space whose coordinates are X, Y, and Z. In this diagram s = ( s 1 , s 2 , s 3 ) denotes the position of the point light source, I ̂ the direction of an emitted light ray, r = ( x , y , f ( x , y ) ) the point on the smooth surface where the emitted light ray is reflected in the direction R ̂ , and N ̂ the normal vector to the smooth surface at the point of reflection.

Fig. 2
Fig. 2

(a) Critical set C = C ± . (b) Intersection of the caustic with the plane Y = 0 . (c) Intersection of the caustic and some singular wavefronts with the plane Y = 0 . In these plots we take c = 1 ( 2415 mm ) and s = ( 0 , 0 , 2400 mm ) .

Fig. 3
Fig. 3

Intersection of some reflected wavefronts with the plane Y = 0 before the caustic ( C = 4400 mm ) , at the caustic ( C = 4900 mm , 5000 mm , 5200 , 5500 mm ) and after the caustic ( C = 6600 mm ) , for c = 1 ( 2415 mm ) and s = ( 0 , 0 , 2400 mm ) . Before and after the caustic the wavefronts are smooth.

Fig. 4
Fig. 4

Two branches of the caustic: (a) corresponds to X = X , (b) corresponds to X = X + for c = 1 ( 2415 mm ) and s = ( 200 mm , 200 mm , 2400 mm ) .

Fig. 5
Fig. 5

Caustic for c = 1 ( 2415 mm ) and s = ( 200 mm , 200 mm , 2400 mm ) . That is, the superposition of Figs. 4a, 4b.

Fig. 6
Fig. 6

Intersection of the caustic shown in Fig. 5, with some planes Z = constant for c = 1 ( 2415 mm ) and s = ( 200 mm , 200 mm , 2400 mm ) . From these plots it is evident that for this case the caustic locally has singularities of the hyperbolic umbilic type.

Fig. 7
Fig. 7

Some reflected wavefronts: before the caustic ( C = 4400 mm ) , at the caustic ( C = 4900 mm , 5000 mm , 5200 , 5500 mm ) and after the caustic ( C = 6600 mm ) , for c = 1 ( 2415 mm ) and s = ( 500 mm , 500 mm , 2400 mm ) . Before and after the caustic the wavefronts are smooth. In order to show the differences between the plots with the point light source on and off the optical axis we have taken for C exactly the same values as those used to obtain Fig. 3.

Equations (84)

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( Φ X ) 2 + ( Φ Y ) 2 + ( Φ Z ) 2 = n 2 .
Φ = u X + v Y + w Z + A ,
w 2 n 2 u 2 v 2 ,
u = n cos φ sin θ , v = n sin φ sin θ .
Φ = n X R ̂ ( θ , φ ) ,
X ( X , Y , Z ) ,
R ̂ ( cos φ sin θ , sin φ sin θ , cos θ ) .
n X R ̂ ( θ , φ ) = C ,
Φ = n X R ̂ ( θ , φ ) = C ,
Φ = n X R ̂ ( θ , φ ) = C ,
Φ θ = n X L ( θ , φ ) = 0 ,
Φ φ = n X M ( θ , φ ) = 0 ,
L R ̂ θ = ( cos φ cos θ , sin φ cos θ , sin θ ) ,
M R ̂ φ = ( sin φ sin θ , cos φ sin θ , 0 ) .
R ̂ R ̂ = 1 , R ̂ L = 0 , R ̂ M = 0 ,
L L = 1 , L M = 0 , M M = sin 2 θ .
X = a ̃ R ̂ ( θ , φ ) + b ̃ L ( θ , φ ) + c ̃ M ( θ , φ ) ,
X = ( C n ) R ̂ ( θ , φ ) .
θ ( X , Y , Z ) = arccos ( Z X 2 + Y 2 + Z 2 ) ,
φ ( X , Y ) = arctan ( Y X ) .
Φ ̃ ( X , Y , Z ) Φ ( X , Y , Z , θ ( X , Y , Z ) , φ ( X , Y ) ) = n X 2 + Y 2 + Z 2 .
Φ * = n X R ̂ ( θ , φ ) + k ( θ , φ ) ,
n X R ̂ ( θ , φ ) + k ( θ , φ ) = C ,
Φ * = n X R ̂ + k = C ,
Φ θ * = n X L + k θ = 0 ,
Φ φ * = n X M + k φ = 0 ,
X ( θ , φ , C ) = ( 1 n ) [ ( C k ) R ̂ k θ L k φ csc 2 θ M ] .
θ = θ ( X , Y , Z ) ,
φ = φ ( X , Y , Z ) .
Φ ̃ ( X , Y , Z ) Φ * ( X , θ ( X , Y , Z ) , φ ( X , Y , Z ) ) .
X ( θ , φ , τ ) = ( 1 n ) [ ( C 0 τ k ) R ̂ k θ L k φ csc 2 θ M ] ,
J ( θ , φ , τ ) = det ( ( X , Y , Z ) ( θ , φ , τ ) ) = ( X θ ) ( X φ × X τ ) = 0 .
τ = τ ± = C 0 k ( 1 2 ) ( k θ θ + k φ φ csc 2 θ + k θ cot θ ) ± ( Δ 2 ) ,
Δ 2 = ( k θ θ k φ φ csc 2 θ k θ cot θ ) 2 + 4 csc 2 θ ( k φ cot θ k θ φ ) 2 .
X = X c ± ( θ , φ ) = 1 n [ ( 1 2 ) ( k θ θ + k φ φ csc 2 θ + k θ cot θ Δ ) R ̂ k θ L k φ csc 2 θ M ] ,
X = 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 ) .
X = r + l ( h α ) ,
h = ( h 1 , h 2 , h 3 ) ,
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 ,
Φ = ( r s ) I ̂ + ( X r ) R ̂ .
Φ = X R ̂ + k ,
k ( s , x , y ) | r s | ( r R ̂ ) .
R ̂ ( s , x , y ) = h α ,
X R ̂ + k = C ,
X R ̂ + k = C ,
X R ̂ x + k x = 0 ,
X R ̂ y + k y = 0 ,
X ( s , x , y ) = r ( x , y ) + [ C | r ( x , y ) s | ] R ̂ ( s , x , y ) .
X ( s , x , y ) = r ( x , y ) + [ C | r ( x , y ) s | ] ( h α ) .
l = C | r ( x , y ) s | = C ( x s 1 ) 2 + ( y s 2 ) 2 + ( f ( x , y ) s 3 ) 2 .
x = x ( X , Y , Z ) , y = y ( X , Y , Z ) .
Φ ̃ ( X , Y , Z ) X R ̂ ( x ( X , Y , Z ) , y ( X , Y , Z ) ) + k ( x ( X , Y , Z ) , y ( X , Y , Z ) ) .
J ( x , y , C ) = det ( ( X , Y , Z ) ( x , y , C ) ) = ( X x ) ( X y × X C ) = 0 .
H 2 ( C | r ( x , y ) s | α ) 2 + H 1 ( C | r ( x , y ) s | α ) + H 0 = 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 ) ] .
C = C ± ( x , y ) | r ( x , y ) s | + α ( H 1 ± H 1 2 4 H 2 H 0 2 H 2 ) .
X = X c ± = r + ( H 1 ± H 1 2 4 H 2 H 0 2 H 2 ) h .
z = f ( x , y ) = c 2 ( x 2 + y 2 ) ,
h 1 = x ( 1 2 c s 3 + 2 c 2 s 2 y ) s 1 [ 1 c 2 ( x 2 y 2 ) ] ,
h 2 = y ( 1 2 c s 3 + 2 c 2 s 1 x ) s 2 [ 1 + c 2 ( x 2 y 2 ) ] ,
h 3 = 2 c [ x ( x s 1 ) + y ( y s 2 ) ] 1 2 [ c ( x 2 + y 2 ) 2 s 3 ] [ 1 c 2 ( x 2 + y 2 ) ] ,
α = [ 1 + c 2 ( x 2 + y 2 ) ] ( x s 1 ) 2 + ( y s 2 ) 2 + [ c 2 ( x 2 + y 2 ) s 3 ] 2 .
k ( s , x , y ) = 1 2 [ 1 + c 2 ( x 2 + y 2 ) ] Ω { 2 ( s 2 2 + s 3 2 ) 2 s 2 y + c s 3 ( x 2 + y 2 ) [ 1 c 2 ( x 2 + y 2 ) ] + c 2 ( x 2 + y 2 ) [ 2 s 2 2 + 2 s 3 2 + x 2 4 s 2 y + y 2 ] 2 s 1 ( x + 2 c 2 x 3 + 2 c 2 x y 2 ) + 2 s 1 2 [ 1 + c 2 ( x 2 + y 2 ) ] } ,
Ω = | r ( x , y ) s | = ( x s 1 ) 2 + ( y s 2 ) 2 + [ c 2 ( x 2 + y 2 ) s 3 ] 2 .
X = x + ( C Ω ) { x ( 1 2 c s 3 + 2 c 2 s 2 y ) s 1 [ 1 c 2 ( x 2 y 2 ) ] } [ 1 + c 2 ( x 2 + y 2 ) ] Ω ,
Y = y + ( C Ω ) { y ( 1 2 c s 3 + 2 c 2 s 1 x ) s 2 [ 1 + c 2 ( x 2 y 2 ) ] } [ 1 + c 2 ( x 2 + y 2 ) ] Ω ,
Z = c 2 ( x 2 + y 2 ) + ( C Ω ) { 2 c [ x ( x s 1 ) + y ( y s 2 ) ] 1 2 [ c ( x 2 + y 2 ) 2 s 3 ] [ 1 c 2 ( x 2 + y 2 ) ] } [ 1 + c 2 ( x 2 + y 2 ) ] Ω .
C = C ± ( x , y ) = Ω ( 1 + [ 1 + c 2 ( x 2 + y 2 ) ] [ H 1 ± Σ ] 2 H 2 ) ,
H 2 = 1 2 [ 1 + c 2 ( x 2 + y 2 ) ] { 2 s 3 + c { 8 c 2 s 1 3 x + 8 c 2 s 2 3 y + 3 ( x 2 + y 2 ) + 4 s 3 ( 1 + c s 3 ) [ 2 s 3 + 3 c ( x 2 + y 2 ) ] 2 s 2 y ( 1 + 2 c s 3 ) [ 3 2 c s 3 + 4 c 2 ( x 2 + y 2 ) ] + 4 s 2 2 [ 1 2 c s 3 + c 2 ( x 2 3 y 2 ) + c 4 ( x 2 + y 2 ) 2 ] + 4 s 1 2 [ 1 2 c s 3 + c 4 ( x 2 + y 2 ) 2 + c 2 ( 3 x 2 + 2 s 2 y + y 2 ) ] + 2 s 1 x [ 3 + 4 c ( s 3 + c ( s 2 2 + x 2 4 s 2 y + y 2 + s 3 [ s 3 2 c ( x 2 + y 2 ) ] ) ) ] } } ,
H 1 = [ 1 + c 2 ( x 2 + y 2 ) ] { 2 s 3 + 2 c 3 ( s 1 2 + s 2 2 ) ( x 2 + y 2 ) + c [ 2 s 1 2 + 2 s 2 2 + 4 s 3 2 2 s 1 x + x 2 2 s 2 y + y 2 ] 2 c 2 s 3 [ 2 s 1 x + x 2 + 2 s 2 y + y 2 ] } ,
Σ 2 = 4 c 2 [ ( s 1 x ) 2 + ( s 2 y ) 2 ] [ 1 + c 2 ( x 2 + y 2 ) ] 2 { ( 1 2 c s 3 ) 2 ( x 2 + y 2 ) 2 s 2 y ( 1 + 2 c s 3 ) [ 1 + c 2 ( x 2 + y 2 ) ] + s 1 2 [ 1 2 c 2 ( x 2 y 2 ) + c 4 ( x 2 + y 2 ) 2 ] + s 2 2 [ 1 + 2 c 2 ( x 2 y 2 ) + c 4 ( x 2 + y 2 ) 2 ] + 2 s 1 x [ 1 + 2 c s 3 2 c 3 s 3 ( x 2 + y 2 ) + c 2 ( x 2 4 s 2 y + y 2 ) ] } .
X c ± = x + ( H 1 ± 2 H 2 ) { x ( 1 2 c s 3 + 2 c 2 s 2 y ) s 1 [ 1 c 2 ( x 2 y 2 ) ] } ,
Y c ± = y + ( H 1 ± 2 H 2 ) { y ( 1 2 c s 3 + 2 c 2 s 1 x ) s 2 [ 1 + c 2 ( x 2 y 2 ) ] } ,
Z c ± = c 2 ( x 2 + y 2 ) + ( H 1 ± 2 H 2 ) { 2 c [ x ( x s 1 ) + y ( y s 2 ) ] 1 2 [ c ( x 2 + y 2 ) 2 s 3 ] [ 1 c 2 ( x 2 + y 2 ) ] } .
X c ± = ( 0 , 0 , 1 2 c ) ,

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