Abstract

The second-order derivative matrix of a scalar function with respect to a variable vector is called a Hessian matrix, which is a square matrix. Our research group previously presented a method for determination of the first-order derivatives (i.e., the Jacobian matrix) of a skew ray with respect to the variable vector of an optical system. This paper extends our previous methodology to determine the second-order derivatives (i.e., the Hessian matrix) of a skew ray with respect to the variable vector of its source ray when this ray is reflected/refracted by spherical boundary surfaces. The traditional finite-difference methods using ray-tracing data to compute the Hessian matrix suffer from various cumulative rounding and truncation errors. The proposed method uses differential geometry, giving it an inherently greater accuracy. The proposed Hessian matrix methodology has potential use in optimization methods where the merit function is defined as ray aberrations. It also can be used to investigate the shape of the wavefront for a ray traveling through an optical system.

© 2011 Optical Society of America

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References

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  16. M. Laikin, Lens Design (Marcel Dekker1995), pp. 71–72.
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2011 (2)

2008 (1)

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628 (2008).
[CrossRef]

2007 (1)

1999 (1)

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

1997 (2)

1988 (2)

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–18(1988).

W. Oertmann, “Differential ray tracing formulae: applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).

1985 (1)

1982 (1)

1980 (1)

W. Mandler, “Uber die Berechnung einfacher GauB-Objektive,” Optik 55, 219–240 (1980).

1976 (1)

1968 (1)

1957 (1)

Andersen, T. B.

Feder, D. P.

Forbes, G. W.

Kross, J.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–18(1988).

Laikin, M.

M. Laikin, Lens Design (Marcel Dekker1995), pp. 71–72.

Leveque, R. J.

R. J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems (SIAM, 2007), pp. 3–4.
[CrossRef]

Lin, P. D.

Mandler, W.

W. Mandler, “Uber die Berechnung einfacher GauB-Objektive,” Optik 55, 219–240 (1980).

Oertmann, W.

W. Oertmann, “Differential ray tracing formulae: applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).

Shi, R.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

Smith, G. D.

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed. (Oxford University Press, 1985).

Stavroudis, O.

Stone, B. D.

Tsai, C. Y.

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628 (2008).
[CrossRef]

P. D. Lin and C. Y. Tsai, “First order gradients of skew rays of axis-symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776–784 (2007).
[CrossRef]

Wu, W.

Appl. Opt. (2)

Appl. Phys. B (1)

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628 (2008).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Optik (1)

W. Mandler, “Uber die Berechnung einfacher GauB-Objektive,” Optik 55, 219–240 (1980).

Proc. SPIE (3)

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–18(1988).

W. Oertmann, “Differential ray tracing formulae: applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).

Other (3)

R. J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems (SIAM, 2007), pp. 3–4.
[CrossRef]

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed. (Oxford University Press, 1985).

M. Laikin, Lens Design (Marcel Dekker1995), pp. 71–72.

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Figures (3)

Fig. 1
Fig. 1

Skew-ray tracing at a spherical boundary surface.

Fig. 2
Fig. 2

Petzval lens system with n = 11 boundary surfaces [16].

Fig. 3
Fig. 3

Representation of the unit directional vector ¯ 0 originating from source point P ¯ 0 .

Tables (3)

Tables Icon

Table 1 Variables of the Petzval Lens System [16] (units: mm)

Tables Icon

Table 2 Results of 2 R ¯ 11 / 2 X ¯ e 0 a

Tables Icon

Table 3 P 11 z 2 / β 0 β 0 from FD#1 and FD#2 Methods for Different Δ β 0 a

Equations (105)

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r ¯ i i = [ R i C β i C α i R i C β i S α i R i S β i 1 ] T ( π / 2 β i π / 2 and 0 α i < 2 π ) .
n ¯ i i = s i ( r ¯ i i β i × r ¯ i i α i ) | r ¯ i i β i × r ¯ i i α i | = s i [ C β i C α i C β i S α i S β i 0 ] T ,
A i 0 = [ I i x J i x K i x t i x I i y J i y K i y t i y I i z J i z K i z t i z 0 0 0 1 ] .
n ¯ i = [ n i x n i y n i z 0 ] = A i 0 n ¯ i i = s i [ I i x C β i C α i + J i x C β i S α i + K i x S β i I i y C β i C α i + J i y C β i S α i + K i y S β i I i z C β i C α i + J i z C β i S α i + K i z S β i 0 ] .
P ¯ i = [ P i x P i y P i z 1 ] T = [ P i 1 x + i 1 x λ i P i 1 y + i 1 y λ i P i 1 z + i 1 z λ i 1 ] T ,
λ i = D i ± D i 2 E i ,
D i = t i x i 1 x t i y i 1 y t i z i 1 z + P i 1 x i 1 x + P i 1 y i 1 y + P i 1 z i 1 z ,
E i = P i 1 x 2 + P i 1 y 2 + P i 1 z 2 R i 2 + t i x 2 + t i y 2 + t i z 2 2 ( t i x P i 1 x + t i y P i 1 y + t i z P i 1 z ) .
α i = atan 2 ( ρ i , σ i ) ,
β i = atan 2 ( τ i , ρ i 2 + σ i 2 ) ,
σ i = I i x ( P i 1 x + i 1 x λ i ) + I i y ( P i 1 y + i 1 y λ i ) + I i z ( P i 1 z + i 1 z λ i ) ( I i x t i x + I i y t i y + I i z t i z ) ,
ρ i = J i x ( P i 1 x + i 1 x λ i ) + J i y ( P i 1 y + i 1 y λ i ) + J i z ( P i 1 z + i 1 z λ i ) ( J i x t i x + J i y t i y + J i z t i z ) ,
τ i = K i x ( P i 1 x + i 1 x λ i ) + K i y ( P i 1 y + i 1 y λ i ) + K i z ( P i 1 z + i 1 z λ i ) ( K i x t i x + K i y t i y + K i z t i z ) ,
C θ i = ¯ i 1 n ¯ i = s i [ i 1 x ( I i x C β i C α i + J i x C β i S α i + K i x S β i ) + i 1 y ( I i y C β i C α i + J i y C β i S α i + K i y S β i ) + i 1 z ( I i z C β i C α i + J i z C β i S α i + K i z S β i ) ] .
S θ ̲ i = ( ξ i 1 / ξ i ) S θ i = N i S θ i ,
¯ i = [ i x i y i z 0 ] = [ i 1 x + 2 n i x C θ i i 1 y + 2 n i y C θ i i 1 z + 2 n i z C θ i 0 ] ,
¯ i = [ i x i y i z 0 ] = [ n i x 1 N i 2 + ( N i C θ i ) 2 + N i ( i 1 x + n i x C θ i ) n i y 1 N i 2 + ( N i C θ i ) 2 + N i ( i 1 y + n i y C θ i ) n i z 1 N i 2 + ( N i C θ i ) 2 + N i ( i 1 z + n i z C θ i ) 0 ] ,
R ¯ 0 = [ P ¯ 0 ¯ 0 ] T = [ P 0 x P 0 y P 0 z 0 x 0 y 0 z ] T = [ P 0 x P 0 y P 0 z C β 0 C α 0 C β 0 S α 0 S β 0 ] T ,
X ¯ e 0 = [ P 0 x P 0 y P 0 z α 0 β 0 ] T = [ x 1 x 2 x 3 x 4 x 5 ] T .
R ¯ i X ¯ e 0 = R ¯ i R ¯ i 1 R ¯ i 1 X ¯ e 0 = [ P ¯ i R ¯ i 1 ¯ i R ¯ i 1 ] T R ¯ i 1 X ¯ e 0 .
P ¯ i R ¯ i 1 = [ 1 0 0 λ i 0 0 0 1 0 0 λ i 0 0 0 1 0 0 λ i ] + [ i 1 x i 1 y i 1 z ] [ λ i , u ] ( u = 1 6 ) ,
λ i , u = D i , u ± 2 D i D i , u E i , u 2 D i 2 E i ,
¯ i R ¯ i 1 = [ n i x N i 2 C θ i / 1 N i 2 + ( N i C θ i ) 2 + N i n i x n i y N i 2 C θ i / 1 N i 2 + ( N i C θ i ) 2 + N i n i y n i z N i 2 C θ i / 1 N i 2 + ( N i C θ i ) 2 + N i n i z ] [ C θ i , u ] + [ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ] + ( 1 N i 2 + ( N i C θ i ) 2 + N i C θ i ) [ n i x , u n i y , u n i z , u ] ( u = 1 6 ) ,
¯ i R ¯ i 1 = 2 C θ i [ n i x , u n i y , u n i z , u ] + [ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ] + [ 2 n i x 2 n i y 2 n i z ] [ C θ i , u ] .
R ¯ i R ¯ i 1 = [ P ¯ i P ¯ i 1 P ¯ i ¯ i 1 ¯ i P ¯ i 1 ¯ i ¯ i 1 ] .
f = f ( R ¯ i 1 ( X ¯ e 0 ) ) = f ( P i 1 x ( X ¯ e 0 ) , P i 1 y ( X ¯ e 0 ) , P i 1 z ( X ¯ e 0 ) , i 1 x ( X ¯ e 0 ) , i 1 y ( X ¯ e 0 ) , i 1 z ( X ¯ e 0 ) ) .
f 2 X ¯ e 0 2 = [ f 2 P 0 x P 0 x f 2 P 0 x P 0 y f 2 P 0 x P 0 z f 2 P 0 x α 0 f 2 P 0 x β 0 f 2 P 0 y P 0 y f 2 P 0 y P 0 z f 2 P 0 y α 0 f 2 P 0 y β 0 f 2 P 0 z P 0 z f 2 P 0 z α 0 f 2 P 0 z β 0 symm. f 2 α 0 α 0 f 2 α 0 β 0 f 2 β 0 β 0 ] = [ f u v ] ,
f u v = f 2 x u x v = ( R ¯ i 1 x u ) T f 2 R ¯ i 1 2 R ¯ i 1 x v + f R ¯ i 1 R ¯ i 1 2 x u x v .
P i x 2 R ¯ i 1 2 = [ 0 0 0 λ i , 1 0 0 0 0 0 λ i , 2 0 0 0 0 0 λ i , 3 0 0 λ i , 1 λ i , 2 λ i , 3 2 λ i , 4 λ i , 5 λ i , 6 0 0 0 λ i , 5 0 0 0 0 0 λ i , 6 0 0 ] + i 1 x [ λ i , 11 λ i , 21 λ i , 31 λ i , 41 λ i , 51 λ i , 61 λ i , 22 λ i , 32 λ i , 42 λ i , 52 λ i , 62 λ i , 33 λ i , 43 λ i , 53 λ i , 63 λ i , 44 λ i , 54 λ i , 64 symm. λ i , 55 λ i , 65 λ i , 66 ] ,
P i y 2 R ¯ i 1 2 = [ 0 0 0 0 λ i , 1 0 0 0 0 0 λ i , 2 0 0 0 0 0 λ i , 3 0 0 0 0 0 λ i , 4 0 λ i , 1 λ i , 2 λ i , 3 λ i , 4 2 λ i , 5 λ i , 6 0 0 0 0 λ i , 6 0 ] + i 1 y [ λ i , 11 λ i , 21 λ i , 31 λ i , 41 λ i , 51 λ i , 61 λ i , 22 λ i , 32 λ i , 42 λ i , 52 λ i , 62 λ i , 33 λ i , 43 λ i , 53 λ i , 63 λ i , 44 λ i , 54 λ i , 64 symm. λ i , 55 λ i , 65 λ i , 66 ] ,
P i z 2 R ¯ i 1 2 = [ 0 0 0 0 0 λ i , 1 0 0 0 0 0 λ i , 2 0 0 0 0 0 λ i , 3 0 0 0 0 0 λ i , 4 0 0 0 0 0 λ i , 5 λ i , 1 λ i , 2 λ i , 3 λ i , 4 λ i , 5 2 λ i , 6 ] + i 1 z [ λ i , 11 λ i , 21 λ i , 31 λ i , 41 λ i , 51 λ i , 61 λ i , 22 λ i , 32 λ i , 42 λ i , 52 λ i , 62 λ i , 33 λ i , 43 λ i , 53 λ i , 63 λ i , 44 λ i , 54 λ i , 64 symm. λ i , 55 λ i , 65 λ i , 66 ] ,
i x 2 R ¯ i 1 2 = ( n i x N i 2 C θ i 1 N i 2 + ( N i C θ i ) 2 + N i n i x ) [ C θ i , u v ] + ( 1 N i 2 + ( N i C θ i ) 2 + N i C θ i ) [ n i x , u v ] + n i x N i 2 ( N i 2 1 ) ( 1 N i 2 + N i 2 C θ i 2 ) 1 N i 2 + ( N i C θ i ) 2 [ C θ i , u ] T [ C θ i , v ] + ( N i 2 C θ i 1 N i 2 + ( N i C θ i ) 2 + N i ) ( [ C θ i , u ] T [ n i x , v ] + [ n i x , u ] T [ C θ i , v ] ) ,
i y 2 R ¯ i 1 2 = ( n i y N i 2 C θ i 1 N i 2 + ( N i C θ i ) 2 + N i n i y ) [ C θ i , u v ] + ( 1 N i 2 + ( N i C θ i ) 2 + N i C θ i ) [ n i y , u v ] + n i y N i 2 ( N i 2 1 ) ( 1 N i 2 + ( N i C θ i ) 2 ) 1 N i 2 + ( N i C θ i ) 2 [ C θ i , u ] T [ C θ i , v ] + ( N i 2 C θ i 1 N i 2 + ( N i C θ i ) 2 + N i ) ( [ C θ i , u ] T [ n i y , v ] + [ n i y , u ] T [ C θ i , v ] ) ,
i z 2 R ¯ i 1 2 = ( n i z N i 2 C θ i 1 N i 2 + ( N i C θ i ) 2 + N i n i z ) [ C θ i , u v ] + ( 1 N i 2 + ( N i C θ i ) 2 + N i C θ i ) [ n i z , u v ] + n i z N i 2 ( N i 2 1 ) ( 1 N i 2 + ( N i C θ i ) 2 ) 1 N i 2 + ( N i C θ i ) 2 [ C θ i , u ] T [ C θ i , v ] + ( N i 2 C θ i 1 N i 2 + ( N i C θ i ) 2 + N i ) ( [ C θ i , u ] T [ n i z , v ] + [ n i z , u ] T [ C θ i , v ] ) ,
i x 2 R ¯ i 1 2 = 2 C θ i [ n i x , u v ] + 2 [ C θ i , v ] T [ n i x , u ] + 2 [ n i x , v ] T [ C θ i , u ] + 2 n i x [ C θ i , u v ] ,
i y 2 R ¯ i 1 2 = 2 C θ i [ n i y , u v ] + 2 [ C θ i , v ] T [ n i y , u ] + 2 [ n i y , v ] T [ C θ i , u ] + 2 n i y [ C θ i , u v ] ,
i z 2 R ¯ i 1 2 = 2 C θ i [ n i z , u v ] + 2 [ C θ i , v ] T [ n i z , u ] + 2 [ n i z , v ] T [ C θ i , u ] + 2 n i z [ C θ i , u v ] .
X ¯ e 0 = [ P 0 x P 0 y P 0 z α 0 β 0 ] T = [ 0.00000 507.00000 170.00000 89.69974 ° 17.94211 ° ] T ,
R ¯ 0 = [ 0.00000 507.00000 170.00000 0.00498 0.95135 0.30806 ] T .
P 11 z 2 x u x v = P 11 z ( x v + Δ x v ) x u P 11 z ( x v ) x u Δ x v .
P 11 z 2 x u x v = P 11 z ( x u + Δ x u , x v + Δ x v ) P 11 z ( x u , x v + Δ x v ) Δ x u P 11 z ( x u + Δ x u , x v ) P 11 z ( x u , x v ) Δ x u Δ x v ,
λ i , u = D i , u ± 2 D i D i , u E i , u 2 D i 2 E i ,
D i , 1 = i 1 x , D i , 2 = i 1 y , D i , 3 = i 1 z , D i , 4 = P i 1 x t i x , D i , 5 = P i 1 y t i y , D i , 6 = P i 1 z t i z ,
E i , 1 = 2 ( P i 1 x t i x ) , E i , 2 = 2 ( P i 1 y t i y ) , E i , 3 = 2 ( P i 1 z t i z ) , E i , 4 = E i , 5 = E i , 6 = 0 .
C θ i , 1 = ( i 1 x n i x , 1 + i 1 y n i y , 1 + i 1 z n i z , 1 ) ,
C θ i , 2 = ( i 1 x n i x , 2 + i 1 y n i y , 2 + i 1 z n i z , 2 ) ,
C θ i , 3 = ( i 1 x n i x , 3 + i 1 y n i y , 3 + i 1 z n i z , 3 ) ,
C θ i , 4 = ( i 1 x n i x , 4 + i 1 y n i y , 4 + i 1 z n i z , 4 ) n i x ,
C θ i , 5 = ( i 1 x n i x , 5 + i 1 y n i y , 5 + i 1 z n i z , 5 ) n i y ,
C θ i , 6 = ( i 1 x n i x , 6 + i 1 y n i y , 6 + i 1 z n i z , 6 ) n i z ,
n i x , u = s i ( I i x S β i C α i J i x S β i S α i + K i x C β i ) β i , u + s i ( I i x C β i S α i + J i x C β i C α i ) α i , u ,
n i y , u = s i ( I i y S β i C α i J i y S β i S α i + K i y C β i ) β i , u + s i ( I i y C β i S α i + J i y C β i C α i ) α i , u ,
n i z , u = s i ( I i z S β i C α i J i z S β i S α i + K i z C β i ) β i , u + s i ( I i z C β i S α i + J i z C β i C α i ) α i , u ,
β i , u = ( σ i 2 + ρ i 2 ) τ i , u τ i ( σ i σ i , u + ρ i ρ i , u ) ( σ i 2 + ρ i 2 + τ i 2 ) ( σ i 2 + ρ i 2 ) ,
α i , u = σ i ρ i , u ρ i σ i , u σ i 2 + ρ i 2 ,
σ i , 1 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 1 + I i x ,
σ i , 2 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 2 + I i y ,
σ i , 3 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 3 + I i z ,
σ i , 4 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 4 + I i x λ i ,
σ i , 5 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 5 + I i y λ i ,
σ i , 6 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 6 + I i z λ i ,
ρ i , 1 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 1 + J i x ,
ρ i , 2 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 2 + J i y ,
ρ i , 3 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 3 + J i z ,
ρ i , 4 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 4 + J i x λ i ,
ρ i , 5 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 5 + J i y λ i ,
ρ i , 6 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 6 + J i z λ i
τ i , 1 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 1 + K i x ,
τ i , 2 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 2 + K i y ,
τ i , 3 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 3 + K i z ,
τ i , 4 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 4 + K i x λ i ,
τ i , 5 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 5 + K i y λ i ,
τ i , 6 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 6 + K i z λ i .
λ i , u v = D i , u v ± ( 2 D i . u D i , v + 2 D i D i , u v E i , u v ) 2 D i 2 E i ± ( 2 D i D i , u + E i , u ) ( 2 D i D i , v E i , v ) 4 ( D i 2 E i ) D i 2 E i ,
D i , 11 = D i , 12 = D i , 21 = D i , 13 = D i , 31 = D i , 15 = D i , 51 = D i , 16 = D i , 61 = D i , 22 = D i , 23 = D i , 32 = D i , 24 = D i , 42 = D i , 26 = D i , 62 = D i , 33 = D i , 34 = D i , 43 = D i , 35 = D i , 53 = D i , 44 = D i , 45 = D i , 54 = D i , 46 = D i , 64 = D i , 55 = D i , 56 = D i , 65 = D i , 66 = 0 , D i , 14 = D i , 25 = D i , 36 = D i , 41 = D i , 52 = D i , 63 = 1 ,
E i , 12 = E i , 21 = E i , 13 = E i , 31 = E i , 14 = E i , 41 = E i , 15 = E i , 51 = E i , 16 = E i , 61 = E i , 23 = E i , 32 = E i , 24 = E i , 42 = E i , 25 = E i , 52 = E i , 26 = E i , 62 = E i , 34 = E i , 43 = E i , 35 = E i , 53 = E i , 36 = E i , 63 = E i , 44 = E i , 45 = E i , 54 = E i , 46 = E i , 64 = E i , 55 = E i , 56 = E i , 65 = E i , 66 = 0 , E i , 11 = E i , 22 = E i , 33 = 2 .
C θ i , 11 = ( i 1 x n i x , 11 + i 1 y n i y , 11 + i 1 z n i z , 11 ) , C θ i , 12 = ( i 1 x n i x , 12 + i 1 y n i y , 12 + i 1 z n i z , 12 ) = C θ i , 21 , C θ i , 13 = ( i 1 x n i x , 13 + i 1 y n i y , 13 + i 1 z n i z , 13 ) = C θ i , 31 , C θ i , 14 = ( i 1 x n i x , 14 + i 1 y n i y , 14 + i 1 z n i z , 14 ) n i x , 1 = C θ i , 41 , C θ i , 15 = ( i 1 x n i x , 15 + i 1 y n i y , 15 + i 1 z n i z , 15 ) n i y , 1 = C θ i , 51 , C θ i , 16 = ( i 1 x n i x , 16 + i 1 y n i y , 16 + i 1 z n i z , 16 ) n i z , 1 = C θ i , 61 .
C θ i , 22 = ( i 1 x n i x , 22 + i 1 y n i y , 22 + i 1 z n i z , 22 ) , C θ i , 23 = ( i 1 x n i x , 23 + i 1 y n i y , 23 + i 1 z n i z , 23 ) = C θ i , 32 , C θ i , 24 = ( i 1 x n i x , 24 + i 1 y n i y , 24 + i 1 z n i z , 24 ) n i x , 2 = C θ i , 42 , C θ i , 25 = ( i 1 x n i x , 25 + i 1 y n i y , 25 + i 1 z n i z , 25 ) n i y , 2 = C θ i , 52 , C θ i , 26 = ( i 1 x n i x , 26 + i 1 y n i y , 26 + i 1 z n i z , 26 ) n i z , 2 = C θ i , 62 .
C θ i , 33 = ( i 1 x n i x , 33 + i 1 y n i y , 33 + i 1 z n i z , 33 ) , C θ i , 34 = ( i 1 x n i x , 34 + i 1 y n i y , 34 + i 1 z n i z , 34 ) n i x , 3 = C θ i , 43 , C θ i , 35 = ( i 1 x n i x , 35 + i 1 y n i y , 35 + i 1 z n i z , 35 ) n i y , 3 = C θ i , 53 , C θ i , 36 = ( i 1 x n i x , 36 + i 1 y n i y , 36 + i 1 z n i z , 36 ) n i z , 3 = C θ i , 63 .
C θ i , 44 = ( i 1 x n i x , 44 + i 1 y n i y , 44 + i 1 z n i z , 44 ) n i x , 4 n i x , 4 , C θ i , 45 = ( i 1 x n i x , 45 + i 1 y n i y , 45 + i 1 z n i z , 45 ) n i x , 5 n i y , 4 = C θ i , 53 , C θ i , 46 = ( i 1 x n i x , 46 + i 1 y n i y , 46 + i 1 z n i z , 46 ) n i x , 6 n i z , 4 = C θ i , 64 .
C θ i , 55 = ( i 1 x n i x , 55 + i 1 y n i y , 55 + i 1 z n i z , 55 ) n i y , 5 n i y , 5 , C θ i , 56 = ( i 1 x n i x , 56 + i 1 y n i y , 56 + i 1 z n i z , 56 ) n i y , 6 n i z , 5 = C θ i , 65 .
C θ i , 66 = ( i 1 x n i x , 66 + i 1 y n i y , 66 + i 1 z n i z , 66 ) n i z , 6 n i z , 6 .
n i x , u v = s i ( I i x C β i C α i I i y C β i S α i I i z S β i ) β i , u β i , v + s i ( I i x S β i S α i I i y S β i C α i ) β i , u α i , v + s i ( I i x S β i S α i I i y S β i C α i ) α i , u β i , v + s i ( I i x C β i C α i I i y C β i S α i ) α i , u α i , v + s i ( I i x S β i C α i I i y S β i S α i + I i z C β i ) β i , u v + s i ( I i x C β i S α i + I i y C β i C α i ) α i , u v ,
n i y , u v = s i ( J i x C β i C α i J i y C β i S α i J i z S β i ) β i , u β i , v + s i ( J i x S β i S α i J i y S β i C α i ) β i , u α i , v + s i ( J i x S β i S α i J i y S β i C α i ) α i , u β i , v + s i ( J i x C β i C α i J i y C β i S α i ) α i , u α i , v + s i ( J i x S β i C α i J i y S β i S α i + J i z C β i ) β i , u v + s i ( J i x C β i S α i + J i y C β i C α i ) α i , u v ,
n i z , u v = s i ( K i x C β i C α i K i y C β i S α i K i z S β i ) β i , u β i , v + s i ( K i x S β i S α i K i y S β i C α i ) β i , u α i , v + s i ( K i x S β i S α i K i y S β i C α i ) α i , u β i , v + s i ( K i x C β i C α i K i y C β i S α i ) α i , u α i , v + s i ( K i x S β i C α i K i y S β i S α i + K i z C β i ) β i , u v + s i ( K i x C β i S α i + K i y C β i C α i ) α i , u v .
β i , u v = ( σ i 2 + ρ i 2 ) τ i , u v + 2 ( σ i σ i , v + ρ i ρ i , v ) τ i , u τ i ( σ i , v σ i , u + σ i σ i , u v + ρ i , v ρ i , u + ρ i ρ i , u v ) ( σ i σ i , u + ρ i ρ i , u ) τ i , v ( σ i 2 + ρ i 2 + τ i 2 ) ( σ i 2 + ρ i 2 ) + [ ( σ i 2 + ρ i 2 ) τ i , u τ i ( σ i σ i , u + ρ i ρ i , u ) ] [ 2 ( σ i σ i , v + ρ i ρ i , v + τ i τ i , v ) ( σ i 2 + ρ i 2 + τ i 2 ) 2 ( σ i 2 + ρ i 2 ) ( σ i σ i , v + ρ i ρ i , v ) ( σ i 2 + ρ i 2 + τ i 2 ) ( σ i 2 + ρ i 2 ) ( σ i 2 + ρ i 2 ) ] ,
α i , u v = σ i ρ i , u v + σ i , v ρ i , u ρ i , v σ i , u ρ i σ i , u v σ i 2 + ρ i 2 2 ( σ i ρ i , u ρ i σ i , u ) ( σ i σ i , v + ρ i ρ i , v ) ( σ i 2 + ρ i 2 ) 2 ,
σ i , 11 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 11 , σ i , 12 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 12 = σ i , 21 , σ i , 13 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 13 = σ i , 31 , σ i , 14 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 14 + I i x λ i , 1 = σ i , 41 , σ i , 15 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 15 + I i y λ i , 1 = σ i , 52 , σ i , 16 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 16 + I i z λ i , 1 = σ i , 62 .
σ i , 22 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 22 , σ i , 23 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 23 = σ i , 32 , σ i , 24 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 24 + I i x λ i , 2 = σ i , 42 , σ i , 25 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 25 + I i y λ i , 2 = σ i , 52 , σ i , 26 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 26 + I i z λ i , 2 = σ i , 62 .
σ i , 33 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 33 , σ i , 34 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 34 + I i x λ i , 3 = σ i , 43 , σ i , 35 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 35 + I i y λ i , 3 = σ i , 53 , σ i , 36 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 36 + I i z λ i , 3 = σ i , 63 .
σ i , 44 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 44 + I i x λ i , 4 + I i x λ i , 4 , σ i , 45 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 45 + I i x λ i , 5 + I i y λ i , 4 = σ i , 54 , σ i , 46 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 46 + I i x λ i , 6 + I i z λ i , 4 = σ i , 64 .
σ i , 55 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 55 + I i y λ i , 5 + I i y λ i , 5 , σ i , 56 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 56 + I i y λ i , 6 + I i z λ i , 5 = σ i , 65 .
σ i , 66 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 66 + I i z λ i , 6 + I i z λ i , 6 .
ρ i , 11 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 11 , ρ i , 12 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 12 = ρ i , 21 , ρ i , 13 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 13 = ρ i , 31 , ρ i , 14 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 14 + J i x λ i , 1 = ρ i , 41 , ρ i , 15 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 15 + J i y λ i , 1 = ρ i , 51 , ρ i , 16 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 16 + J i z λ i , 1 = ρ i , 61 .
ρ i , 22 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 22 , ρ i , 23 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 23 = ρ i , 32 , ρ i , 24 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 24 + J i x λ i , 2 = ρ i , 42 , ρ i , 25 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 25 + J i y λ i , 2 = ρ i , 52 , ρ i , 26 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 26 + J i z λ i , 2 = ρ i , 62 .
ρ i , 33 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 33 , ρ i , 34 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 34 + J i x λ i , 3 = ρ i , 43 , ρ i , 35 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 35 + J i y λ i , 3 = ρ i , 53 , ρ i , 36 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 36 + J i z λ i , 3 = ρ i , 63 .
ρ i , 44 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 44 + J i x λ i , 4 + J i x λ i , 4 , ρ i , 45 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 45 + J i x λ i , 5 + J i y λ i , 4 = ρ i , 54 , ρ i , 46 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 46 + J i x λ i , 6 + J i z λ i , 4 = ρ i , 64 .
ρ i , 55 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 55 + J i y λ i , 5 + J i y λ i , 5 , ρ i , 56 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 56 + J i y λ i , 6 + J i z λ i , 5 = ρ i , 65 .
ρ i , 66 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 66 + J i z λ i , 6 + J i z λ i , 6 .
τ i , 11 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 11 , τ i , 12 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 12 = τ i , 21 , τ i , 13 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 13 = τ i , 31 , τ i , 14 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 14 + K i x λ i , 1 = τ i , 41 , τ i , 15 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 15 + K i y λ i , 1 = τ i , 51 , τ i , 16 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 16 + K i z λ i , 1 = τ i , 61 .
τ i , 22 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 22 , τ i , 23 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 23 = τ i , 32 , τ i , 24 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 24 + K i x λ i , 2 = τ i , 42 , τ i , 25 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 25 + K i y λ i , 2 = τ i , 52 , τ i , 26 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 26 + K i z λ i , 2 = τ i , 62 .
τ i , 33 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 33 , τ i , 34 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 34 + K i x λ i , 3 = τ i , 43 , τ i , 35 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 35 + K i y λ i , 3 = τ i , 53 , τ i , 36 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 36 + K i z λ i , 3 = τ i , 63 .
τ i , 44 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 44 + K i x λ i , 4 + K i x λ i , 4 , τ i , 45 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 45 + K i x λ i , 5 + K i y λ i , 4 = τ i , 54 , τ i , 46 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 46 + K i x λ i , 6 + K i z λ i , 4 = τ i , 46 .
τ i , 55 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 55 + K i y λ i , 5 + K i y λ i , 5 , τ i , 56 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 56 + K i y λ i , 6 + K i z λ i , 5 = τ i , 65 .
τ i , 66 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 66 + K i z λ i , 6 + K i z λ i , 6 .

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