Abstract

Given an optical system with a plane of symmetry and an arbitrarily chosen ray contained in this plane, it is shown that it is possible to determine exact analytic formulas for a set of aberration coefficients characterizing the image-forming properties of bundles of rays in the neighborhood (but not necessarily centered on) the given ray. The foundations of the theory are laid, all formulas needed to compute the coefficients of orders one, two, and three are derived, and those simplifications which arise when a surface is a sphere are considered in detail.

© 1972 Optical Society of America

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References

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  1. P. J. Sands, thesis, Australian National University, 1967.
  2. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  3. See, for instance, P. J. Sands, J. Opt. Soc. Am. 62, 369 (1972).
    [CrossRef]
  4. C. G. Wynne, Proc. Phys. Soc. (London) B67, 529 (1954).
  5. M. Herzberger, Modern GeometricalOptics (Wiley–Interscience, New York, 1954), p. 385.
  6. P. J. Sands, Appl. Opt. 9, 828 (1970).
    [CrossRef] [PubMed]
  7. P. J. Sands, J. Opt. Soc. Am. 58, 1365 (1968).
    [CrossRef]

1972 (1)

1970 (1)

1968 (1)

1954 (1)

C. G. Wynne, Proc. Phys. Soc. (London) B67, 529 (1954).

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Herzberger, M.

M. Herzberger, Modern GeometricalOptics (Wiley–Interscience, New York, 1954), p. 385.

Sands, P. J.

Wynne, C. G.

C. G. Wynne, Proc. Phys. Soc. (London) B67, 529 (1954).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Proc. Phys. Soc. (London) (1)

C. G. Wynne, Proc. Phys. Soc. (London) B67, 529 (1954).

Other (3)

M. Herzberger, Modern GeometricalOptics (Wiley–Interscience, New York, 1954), p. 385.

P. J. Sands, thesis, Australian National University, 1967.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

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Tables (12)

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Table I Identifications of coefficients.

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Table II The x ̂ coefficients for an arbitrary plane symmetric surface.

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Table III The î coefficients.

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Table IV The σ ̂ and τ ̂ coefficients.

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Table V The r ̂ coefficients.

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Table VI The p ̂ coefficients.

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Table VII The second- and third-order ê coefficients.

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Table VIII The second-order coefficients ωαβ.

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Table IX Third-order coefficients ωαβ.

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Table X Third-order iteration equations. The subscript A has been omitted from all tα, t α, and s α.

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Table XI The x ̂ coefficients for spherical surfaces.

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Table XII The σ ̂ and τ ̂ coefficients for spherical surfaces.

Equations (312)

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V ̂ j = ( V ̂ j , W ̂ j ) : = ( β j / α j , γ j / α j ) .
y j = Y j + x j ( V j + V ̂ 0 j ) ,
Y j + 1 = Y j + d j V j , V j + 1 = V j ,
Q j = Q 0 j + n = 1 q j ( n ) ( S , T ) , q j ( n ) = μ = 0 n ν = 0 μ τ = 0 ν q ( n ) μ ν τ j S y n μ T y μ ν S z ν τ T z τ , Q j = Q 0 j + n = 1 q ̂ j ( n ) ( Y , V ) , q ̂ j ( n ) = μ = 0 n ν = 0 μ τ = 0 ν q ̂ ( n ) μ ν τ j Y j n μ V j μ ν Z j ν τ W j τ ,
n even : N y n = 1 12 ( n + 2 ) ( n 2 + 4 n + 6 ) , N z n = 1 12 n ( n + 2 ) ( n + 4 ) ; n odd : N y n = N z n = 1 12 ( n + 1 ) ( n + 2 ) ( n + 3 ) .
α : = 1 + τ + 1 2 ν ( ν + 1 ) + 1 6 μ ( μ + 1 ) ( μ + 2 ) .
q a : = q ( 1 ) 000 , q b : = q ( 1 ) 100 , q c : = q ( 1 ) 110 , q d : = q ( 1 ) 111 .
Δ ( N e ) = N α R ¯ n , R ¯ : = ( Δ N cos I ) / N α | n | .
Δ V ̂ = R ¯ 0 R Î , Î : = n x V ̂ n , n : = ( n y , n z ) , R : = R ¯ / R ¯ 0 ;
Δ Y = x Δ V ̂ = R ¯ 0 P Î , P : = x R ;
Δ ( N α Î ) = 0 .
τ T 2 + 2 σ R + μ = 0 ,
μ = ( k 2 1 ) ( 1 + V ̂ · V ̂ ) , σ = R ¯ 0 [ k 2 Î · V ̂ · ( 1 + V ̂ · V ̂ ) n x ] , τ = R ¯ 0 2 [ k 2 Î · Î ( 1 + V ̂ · V ̂ ) n x 2 ] .
cos I 0 = ( n 0 x α 0 + n 0 · β 0 ) / | n 0 | , Δ ( N e 0 ) = N α 0 R ¯ 0 n 0 ; Δ V ̂ 0 = R ¯ 0 Î 0 , Î 0 = n 0 x V ̂ 0 n 0 , Δ ( N α 0 Î 0 ) = 0 .
n = ( F / x , F / y , F / z ) .
F = x + n 0 y 1 2 ( c y y 2 + c z z 2 ) θ 30 y 3 θ 32 y z 2 θ 40 y 4 θ 42 y 2 z 2 θ 44 z 4 = 0 ,
x ̂ a = n 0 y q , x ̂ b = 0 , q : = ( 1 + n 0 y V ̂ 0 ) 1 .
Δ ( q Y ) = O ( 2 ) , Δ Z = O ( 2 ) .
Î y = Î 0 y + I y + O ( 2 ) , I y = î y ( 1 ) = q c y Y + V , Î z = I z + O ( 2 ) I z = î z ( 1 ) = c z Z + W .
α = ( 1 + V ̂ · V ̂ ) 1 2 = α 0 ( 1 α 0 β 0 V ) + O ( 2 )
Δ [ N α 0 ( 1 α 0 β 0 Î 0 y ) V ] + q c y ( Δ N α 0 ) Y = O ( 2 ) ; Δ [ N α 0 ( c z Z + W ) ] = O ( 2 ) .
Δ ( N α 0 3 V / q ) = ( N α 0 q c y R ¯ 0 ) Y + O ( 2 ) , Δ ( N α 0 W ) = ( N α 0 q c z R ¯ 0 ) W + O ( 2 ) .
Y = ( q / q ) Y + O ( 2 ) , Z = Z + O ( 2 ) , V = ( N α 0 3 q / N α 0 3 q ) V ( c y R ¯ 0 q q / α 0 2 ) Y + O ( 2 ) , W = ( N α 0 / N α 0 ) W c z R ¯ 0 Z + O ( 2 ) ,
Y j = y a j S y + y b j T y + O ( 2 ) , Z j = z c j S z + z d j T z + O ( 2 ) , V j = υ a j S y + υ b j T y + O ( 2 ) , W j = w c j S z + w d j T z + O ( 2 ) .
Λ y : = N y ( y V υ Y ) , Λ z : = N z ( z W w Z ) , N y : = N α 0 3 / λ y , λ y : = N 1 α 01 3 ( y a 1 υ b 1 y b 1 υ a 1 ) , N z : = N α 0 / λ z , λ z : = N 1 α 01 ( z c 1 w d 1 z d 1 w c 1 ) ,
Λ y j = N y j ( y a j υ b j y b j υ a j ) Λ y 1 + O ( 2 ) , Λ z j = N z j ( z c j w d j z d j w c j ) Λ z 1 + O ( 2 ) , Λ y 1 = s y T y t y S y , Λ z 1 = s z T z t z S z .
N j α 0 j 3 ( y a j υ b j y b j υ a j ) = λ y , N j α 0 j ( z c j w d j z d j w c j ) = λ y .
Λ y = Λ a s y + Λ b t y , Λ z = Λ c s z + Λ d t z ;
Λ a = N y ( y a V υ a Y ) , Λ b = N y ( y b V υ b Y ) ; Λ c = N z ( z c W w c Z ) , Λ d = N z ( z d W w d Z ) .
G A j : = Λ A j Λ A 1 = i = 1 j 1 g A i , g A i : = Δ Λ A i ,
N y ( y a V υ a Y ) = G a + T y ,
Y = y a ( S y G b ) + y b ( T y + G a ) , V = υ a ( S y G b ) + υ b ( T y + G a ) , Z = z c ( S z G d ) + z d ( T z + G c ) , W = w c ( S z G d ) + w d ( T z + G c ) .
H = Y k + l k V k ,
H y = h a S y + h b T y + ( G a k h b G b k h a ) , H z = h c S z + h d T z + ( G c k h d G d k h c ) ,
G A k : = i = 1 k g A i .
H y = h b ( T y + G a k ) , H z = h d ( T z + G c k ) ;
Q : = n = 2 q ̂ ( n ) ,
E = ( E y , E z ) : = ( R Î ) = R Î r ̂ ( 1 ) Î 0 I Î 0 , F = ( F y , F z ) : = ( P Î ) = P Î x ̂ ( 1 ) Î 0 .
g y = N y R ¯ 0 ( y E y + υ F y ) + { V Δ ( N y y ) Y Δ ( N y υ ) N y R ¯ 0 [ y ( r ̂ ( 1 ) I 0 y + I y ) + υ x ̂ ( 1 ) I 0 y ] } ,
g y = N y R ¯ 0 ( E y y + F y υ ) , g z = N z R ¯ 0 ( E z z + F z w ) .
( 1 + n 0 y V ̂ 0 ) x ̂ ( 2 ) = n 0 y V x ̂ ( 1 ) + 1 2 ( c y y ̂ ( 1 ) 2 + c z Z 2 ) .
μ = μ 0 ( 1 + 2 α 0 β 0 V + α 0 2 V 2 + α 0 2 W 2 ) , σ = ( ρ 1 / μ 0 ) μ n x + ρ 2 V ̂ · Î , τ = ( R ¯ 0 ρ 1 / μ 0 ) μ n x 2 R ¯ 0 ρ 2 Î · Î ,
μ 0 : = ( 1 k 2 ) ρ 1 / R ¯ 0 , ρ 1 : = q / 2 k α 0 α 0 , ρ 2 : = k α 0 q / 2 α 0 , ρ 3 : = 2 R ¯ 0 ρ 2 ,
μ ̂ b = 2 μ 0 α 0 β 0 , ŝ μ 5 = ŝ μ 10 = μ 0 α 0 2 ,
m = 0 n ( l = 0 n τ ̂ ( n m ) r ̂ ( m l ) r ̂ ( l ) + 2 σ ̂ ( n m ) q ̂ ( m ) ) + μ ̂ ( n ) 0 .
r ̂ ( n ) = ϕ ̂ ( n ) + m = 1 n 1 ( r ̂ ( n m ) ψ ̂ ( m ) + τ ̂ ( n m ) l = 1 m = 1 r ̂ ( m l ) r ̂ l ) ,
ψ ̂ ( n ) : = τ 0 r ̂ ( n ) + 2 σ ̂ ( n ) + 2 R 0 τ ̂ ( n ) , ϕ ̂ ( n ) : = R 0 2 τ ̂ ( n ) + 2 R 0 σ ̂ ( n ) + μ ̂ ( n ) ;
p ̂ ( n ) = m = 1 n r ̂ ( n m ) x ̂ ( m )
k ̂ α A = N y R ¯ 0 ( y A k ̂ e α + υ A k ̂ j α ) ( A = a , b ; ν even ) , = N z R ¯ 0 ( z A k ̂ e α + w A k ̂ f α ) ( A = c , d ; ν odd ) .
S j * : = ( S y G b j , S z G d j ) , T j * : = ( T y + G a j , T z + G c j ) ,
n = 2 μ = 0 n ν = 0 μ τ = 0 ν [ q ( n ) μ ν τ S y * n μ T y * μ ν S z * ν τ T z * τ q ̂ ( n ) μ ν τ ( y a S y * + y b T y * ) n μ ( υ a S y * + υ b T y * ) μ ν × ( z c S z * + z d T z * ) ν τ ( w c S z * + w d T z * ) τ ] 0 ,
k q α = β ω α β k ̂ q β ,
t 8 α = y a z c 2 t ̂ 8 a + y a z c w c t ̂ 9 a + y a w c 2 t ̂ 10 a + υ a z c 2 t ̂ 14 a + υ a z c w c t ̂ 15 a + υ a w c 2 t ̂ 16 a .
n = 2 μ = 0 n ν = 0 μ τ = 0 ν [ g ( n ) μ ν τ A j S y n μ T y μ ν S z ν τ T z τ g ( n ) μ ν τ A j ( S y G b j ) n μ ( T y + G a j ) μ ν × ( S z G d j ) ν τ ( T z + G c j ) τ ] 0 ;
S α A j = i = 1 j 1 s α A i , T α A j = i = 1 j 1 t α A i ;
G a = S 1 a S y 2 + S 2 a S y T y + S 5 a T y 2 + S 8 a S z 2 + S 9 a S z T z + S 10 a T z 2 + O ( 3 ) , G c = S 3 c S y S z + S 4 c S y T z + S 6 c T y S z + S 7 c T y T z + O ( 3 ) .
S α A = i = 1 k s α A i , T α A = i = 1 k t α A i .
( n 0 x c x ) 2 + ( n 0 y c y ) 2 + c 2 z 2 = 1
Î = Î 0 + I , Î 0 = n 0 x V ̂ 0 n 0 , I : = c Y + n 0 x V ,
î a = î c = c , î b = î d = n 0 x , î ( n ) μ ν τ = 0 ( n > 1 ) .
q : = 1 / ( n 0 x + n 0 y V ̂ 0 ) = α 0 / cos I 0 .
ρ 3 : = 2 ρ 2 R ¯ 0 .
E = ( R 1 ) I + R Î 0 , F = P I + P Î 0 .
k r α = β ω α β k ̂ r α ,
R = R 0 + r ̂ ( 1 ) + R = R 0 + r ̂ a Y + r ̂ b V + R ,
r A = r ̂ a y A + r ̂ b υ A , A = a , b , r ( n ) μ ν τ = r ( n ) μ ν τ + G ( n ) μ ν τ a r b G ( n ) μ ν τ b r a ;
i A = c y A + n 0 x υ A ( A = a or b ) , i A = c z A + n 0 x w A ( A = c , d ) , i ( n ) μ ν τ = G ( n ) μ ν τ i b G ( n ) μ ν τ b i a ( ν even ) = G ( n ) μ ν τ c i d G ( n ) μ ν τ d i c ( ν odd ) .
q ̂ 000 ( 1 )
q ̂ a
q ̂ 110 ( 1 )
q ̂ c
q ̂ 100 ( 1 )
q ̂ b
q ̂ 111 ( 1 )
q ̂ d
q ̂ 000 ( 2 )
q ̂ 110 ( 2 )
q ̂ 100 ( 2 )
q ̂ 111 ( 2 )
q ̂ 200 ( 2 )
q ̂ 210 ( 2 )
q ̂ 220 ( 2 )
q ̂ 211 ( 2 )
q ̂ 221 ( 2 )
q ̂ 222 ( 2 )
q ̂ 000 ( 3 )
t ̂ q 1
q ̂ 110 ( 3 )
t ̂ q 3
q ̂ 100 ( 3 )
t ̂ q 2
q ̂ 111 ( 3 )
t ̂ q 4
q ̂ 200 ( 3 )
t ̂ q 5
q ̂ 210 ( 3 )
t ̂ q 6
q ̂ 220 ( 3 )
t ̂ q 8
q ̂ 211 ( 3 )
t ̂ q 7
q ̂ 221 ( 3 )
t ̂ q 9
q ̂ 310 ( 3 )
t ̂ q 12
q ̂ 222 ( 3 )
t ̂ q 10
q ̂ 311 ( 3 )
t ̂ q 13
q ̂ 300 ( 3 )
t ̂ q 11
q ̂ 330 ( 3 )
t ̂ q 17
q ̂ 320 ( 3 )
t ̂ q 14
q ̂ 331 ( 3 )
t ̂ q 18
q ̂ 321 ( 3 )
t ̂ q 15
q ̂ 322 ( 3 )
t ̂ q 19
q ̂ 322 ( 3 )
t ̂ q 16
q ̂ 333 ( 3 )
t ̂ q 20
x ̂ a = n 0 y q
x ̂ b = 0
ŝ x 1 = 1 2 q 3 c y
S ̂ x 2 = x ̂ a 2
ŝ x 8 = 1 2 q c z
t ̂ x 1 = q 2 ( c y V ̂ 0 ŝ x 1 + q 2 θ 30 )
t ̂ x 2 = 3 x ̂ a ŝ x 1
t ̂ x 5 = x ̂ a ŝ x 2
t ̂ x 8 = q 2 ( c y V ̂ 0 ŝ x 8 + θ 32 )
t ̂ x 9 = 2 x ̂ a ŝ x 8
t ̂ x 10 = 0
t ̂ x 11 = 0
t ̂ x 14 = x ̂ a ŝ x 8
t ̂ x 15 = 0
t ̂ x 16 = 0
ŝ i 1 = 3 q 2 θ 30 + c y V ̂ 0 ŝ x 1
ŝ i 2 = c y q x ̂ a
ŝ i 4 = c 2 x ̂ a
ŝ i 8 = θ 32 + c y V ̂ 0 ŝ x 8
t ̂ i 1 = c y V ̂ 0 t ̂ x 1 + 6 θ 30 V ̂ 0 q ŝ x 1 + 4 θ 40 q 3
t ̂ i 3 = 2 V ̂ 0 θ 32 ŝ x 1 + 2 θ 42 q 2
t ̂ i 2 = c y V ̂ 0 t ̂ x 2 + c y ŝ x 1 + 6 θ 30 q 2 x ̂ a
t ̂ i 4 = c z ŝ x 1 + 2 θ 32 q x ̂ a
t ̂ i 5 = c y q ŝ x 2
t ̂ i 6 = 2 θ 32 q x ̂ a
t ̂ i 8 = c y V ̂ 0 t ̂ x 8 + 6 θ 30 V ̂ 0 q ŝ x 8 + 2 θ 42 q
t ̂ i 7 = c z ŝ x 2
t ̂ i 9 = c y V ̂ 0 t ̂ x 9 + 2 θ 32 x ̂ a
t ̂ i 12 = 0
t ̂ i 10 = 0
t ̂ i 13 = 0
t ̂ i 11 = 0
t ̂ i 17 = 2 V ̂ 0 θ 32 ŝ x 8 + 4 θ 44
t ̂ i 14 = c y q ŝ x 8
t ̂ i 18 = c z ŝ x 8
t ̂ i 15 = 0
t ̂ i 19 = 0
t ̂ i 16 = 0
t ̂ i 20 = 0
σ ̂ a = q ρ 2 V ̂ 0 c y
τ ̂ a = q ρ 3 Î 0 y c y
σ ̂ b = 2 ρ 1 α 0 β 0 + ρ 2 ( V ̂ 0 + Î 0 y )
τ ̂ b = R ¯ 0 ( ρ 2 n 0 y σ ̂ b )
ŝ σ 1 = ρ 2 V ̂ 0 ŝ i 1
ŝ τ 1 = ρ 3 ( Î 0 y ŝ i 1 + 1 2 q 2 c y 2 )
ŝ σ 2 = ρ 2 ( V ̂ 0 ŝ i 2 + q c y )
ŝ τ 2 = ρ 3 ( Î 0 y ŝ i 2 + q c y )
ŝ τ 5 = R ¯ 0 ŝ σ 5
ŝ σ 8 = ρ 2 V ̂ 0 ŝ i 8
ŝ τ 8 = ρ 3 ( Î 0 y ŝ i 8 + 1 2 c z 2 )
t ̂ σ 1 = ρ 2 V ̂ 0 t ̂ i 1
t ̂ τ 1 = ρ 3 ( Î 0 y t ̂ i 1 + q c y ŝ i 1 )
t ̂ σ 2 = ρ 2 ( V ̂ 0 t ̂ i 2 + ŝ i 1 )
t ̂ τ 2 = ρ 3 ( Î 0 y t ̂ i 2 + q c y ŝ i 2 + ŝ i 1 )
t ̂ σ 5 = ρ 2 q ŝ i 2
t ̂ τ 5 = ρ 3 ( Î 0 y t ̂ i 5 + ŝ i 2 )
t ̂ σ 8 = ρ 2 V ̂ 0 t ̂ i 8
t ̂ τ 8 = ρ 3 ( Î 0 y t ̂ i 8 + q c y ŝ i 8 + c z ŝ i 3 )
t ̂ σ 9 = ρ 2 ( V ̂ 0 t ̂ i 9 + ŝ i 3 )
t ̂ τ 9 = ρ 3 ( Î 0 y t ̂ i 9 + c z ŝ i 4 + ŝ i 3 )
t ̂ σ 10 = ρ 2 ŝ i 4
t ̂ τ 10 = ρ 3 ŝ i 4
t ̂ σ 11 = 0
t ̂ τ 11 = 0
t ̂ σ 14 = ρ 2 ( V ̂ 0 t ̂ i 14 + ŝ i 8 )
t ̂ τ 14 = ρ 3 ( Î 0 y t ̂ i 14 + ŝ i 8 )
t ̂ σ 15 = 0
t ̂ τ 15 = 0
t ̂ σ 16 = 0
t ̂ τ 16 = 0
r ̂ a = ϕ ̂ a
r ̂ b = ϕ ̂ b
ŝ r 1 = ŝ ϕ 1 + r ̂ a ψ ̂ a
ŝ r 2 = ŝ ϕ 2 + r ̂ a ψ ̂ b + r ̂ b ψ ̂ a
ŝ r 5 = ŝ ϕ 5 + r ̂ b ψ ̂ b
t ̂ r 1 = t ̂ ϕ 1 + r ̂ a ( ŝ ψ 1 + τ ̂ a r ̂ a ) + ŝ τ 1 ψ ̂ a
t ̂ r 2 = t ̂ ϕ 3 + r ̂ a ( ŝ ψ 2 + τ ̂ a r ̂ b + τ ̂ b r ̂ a ) + r ̂ b ( ŝ ψ 1 + τ ̂ a r ̂ a ) + ψ ̂ a ŝ r 2 + ψ ̂ a ŝ r 1
t ̂ r 5 = t ̂ ϕ 5 + r ̂ a ( ŝ ψ 5 + τ ̂ a r ̂ b ) + r ̂ b ( ŝ ψ 2 + τ ̂ a r ̂ b + τ ̂ b r ̂ a ) + ψ ̂ a ŝ r 5 + ψ ̂ b ŝ r 2
t ̂ r 8 = t ̂ ϕ 8 + r ̂ a ŝ ψ 8 + ψ ̂ a ŝ r 8
t ̂ r 9 = t ̂ ϕ 9 + r ̂ a ŝ ψ 9 + ψ ̂ a ŝ r 9
t ̂ r 10 = t ̂ ϕ 10 + r ̂ a ŝ ψ 10 + ψ ̂ a ŝ r 10
t ̂ r 11 = t ̂ ϕ 11 + r ̂ b ( ŝ ψ 5 + τ ̂ b r ̂ b ) + ψ ̂ b ŝ r 5
t ̂ r 14 = t ̂ ϕ 14 + r ̂ b ŝ ϕ 8 + ψ ̂ b ŝ r 8
t ̂ r 15 = t ̂ ϕ 15 + r ̂ b ŝ ψ 9 + ψ ̂ b ŝ r 9
t ̂ r 16 = t ̂ ϕ 16 + r ̂ b ŝ ψ 10 + ψ ̂ b ŝ r 10
p ̂ a = x ̂ a
p ̂ b = 0
ŝ p 1 = ŝ x 1 + x ̂ a r ̂ a
ŝ p 2 = ŝ x 2 + x ̂ a r ̂ b
t ̂ p 1 = t ̂ x 1 + r ̂ a ŝ z 1 + x ̂ a ŝ r 1
t ̂ p 2 = t ̂ x 2 + r ̂ a ŝ x 2 + r ̂ b ŝ x 1 + x ̂ a ŝ r 2
t ̂ p 5 = t ̂ x 5 + r ̂ a ŝ x 2 + x ̂ a ŝ r 5
t ̂ p 8 = t ̂ x 8 + r ̂ a ŝ x 8 + x ̂ a ŝ r 8
t ̂ p 9 = t ̂ x 9 + x ̂ a ŝ r 9
t ̂ p 10 = x ̂ a ŝ r 10
t ̂ p 11 = 0
t ̂ p 14 = t ̂ x 14 + r ̂ b ŝ x 8
t ̂ p 15 = 0
t ̂ p 16 = 0
ŝ e 1 = R 0 ŝ i 1 + Î 0 y ŝ r 1 + î a r ̂ a
ŝ e 3 = R 0 ŝ i 3 + î c r ̂ a
ŝ e 2 = R 0 ŝ i 2 + Î 0 y ŝ r 2 + î a r ̂ b + î b r ̂ a
ŝ e 4 = R 0 ŝ i 4 + î d r ̂ a
ŝ e 5 = R 0 ŝ i 5 + Î 0 y ŝ r 5 + î b r ̂ b
ŝ e 6 = R 0 ŝ i 6 + î c r ̂ b
ŝ e 7 = R 0 ŝ i 7 + î d r ̂ b
t ̂ e 1 = R 0 t ̂ i 1 + Î 0 y t ̂ r 1 + î a ŝ r 1 + ŝ a ŝ i 1
t ̂ e 2 = R 0 î i 2 + Î 0 y t ̂ r 2 + î a ŝ r 2 + î b ŝ r 1 + r ̂ a ŝ i 2 + r ̂ b ŝ i 1
t ̂ e 5 = R 0 t ̂ i 5 + Î 0 y t ̂ r 5 + î a ŝ r 5 + î a ŝ r 2 + î b ŝ r 2 + r ̂ a ŝ i 5 + r ̂ b ŝ i 2
t ̂ e α = R 0 t ̂ i α + Î 0 y t ̂ r α + î a ŝ r α + r ̂ a ŝ i α ( α = 8 , 9 , 10 )
t ̂ e 11 = R 0 t ̂ i 11 + Î 0 y t ̂ r 11 + î b ŝ r 5 + r ̂ b ŝ i 5
t ̂ e 14 = R 0 t ̂ i 14 + Î 0 y t ̂ r 14 + î b ŝ r 8 + r ̂ b ŝ i 8
t ̂ e 15 = R 0 t ̂ i 15 + Î 0 y t ̂ r 15 + î b ŝ r 9 + r ̂ b ŝ i 9
t ̂ e 16 = R 0 t ̂ i 16 + Î 0 y t ̂ r 16 + î b ŝ r 10 + r ̂ b ŝ i 10
t ̂ e 3 = R 0 t ̂ i 3 + î c ŝ r 1 + r ̂ a ŝ i 3
t ̂ e 4 = R 0 t ̂ i 4 + î d ŝ r 1 + r ̂ a ŝ i 4
t ̂ e 6 = R 0 t ̂ i 6 + î c ŝ r 2 + r ̂ a ŝ i 6 + r ̂ b ŝ i 3
t ̂ e 7 = R 0 t ̂ i 7 + î d ŝ r 2 + r ̂ a ŝ i 7 + r ̂ b ŝ i 4
t ̂ e 12 = R 0 t ̂ i 12 + î c ŝ r 5 + r ̂ b ŝ i 6
t ̂ e 13 = R 0 t ̂ i 13 + î d ŝ r 5 + r ̂ b ŝ i 7
t ̂ e 17 = R 0 t ̂ i 17 + î c ŝ r 8
t ̂ e 18 = R 0 t ̂ i 18 + î c ŝ r 9 + î d ŝ r 8
t ̂ e 19 = R 0 t ̂ i 19 + î c ŝ r 10 + î d ŝ r 9
t ̂ e 20 = R 0 t ̂ i 20 + î d ŝ r 10
t 1 = t 1 2 s 1 S 1 b + s 2 S 1 a
t 2 = t 2 2 s 1 S 2 b + s 2 ( S 2 a S 1 b ) + 2 s 5 S 1 a
t 5 = t 5 2 s 1 S 5 b + s 2 ( S 5 a S 2 b ) + 2 s 5 S 2 a
t 8 = t 8 2 s 1 S 8 b + s 2 S 8 a 2 s 8 S 3 d + s 9 S 3 c
t 9 = t 9 2 s 1 S 9 b + s 2 S 9 a 2 s 8 S 4 d + s 9 ( S 4 c S 3 d ) + 2 s 10 s 3 c
t 10 = t 10 2 s 1 S 10 b + s 2 S 10 a s 9 S 4 d + 2 s 10 S 4 c
t 11 = t 11 s 2 S 5 b + 2 s 5 S 5 a
t 14 = t 14 s 2 S 8 b + 2 s 5 S 8 a 2 s 8 S 6 d + s 9 S 6 c
t 15 = t 15 s 2 S 9 b + 2 s 5 S 9 a 2 s 8 S 7 d + s 9 ( S 7 c S 6 d ) + 2 s 10 S 6 c
t 16 = t 16 s 2 S 10 b + 2 s 5 S 10 a s 9 S 7 d + 2 s 10 S 7 c
t 3 = t 3 s 3 ( S 1 b + S 3 d ) + s 4 S 3 c + s 6 S 1 a
t 4 = t 4 s 3 S 4 d + s 4 ( S 4 c S 1 b ) + s 7 S 1 a
t 6 = t 6 s 3 ( S 2 b + S 6 d ) + s 4 S 6 c + s 6 ( S 2 a S 3 d ) + s 7 S 3 c
t 7 = t 7 s 3 S 7 d + s 4 ( S 7 c S 2 b ) s 6 S 4 d + s 7 ( S 4 c + S 2 a )
t 12 = t 12 s 3 S 5 b + s 6 ( S 5 a S 6 b ) + s 7 S 6 c
t 13 = t 13 s 4 S 5 b s 6 S 7 d + s 7 ( S 7 c + S 5 a )
t 17 = t 17 s 3 S 8 b + s 6 S 8 a
t 18 = t 18 s 3 S 9 d s 4 S 8 b + s 6 S 9 a + s 7 S 8 a
t 19 = t 19 s 3 S 10 b s 4 S 9 b + s 6 S 10 a + s 7 S 9 a
t 20 = t 20 s 4 S 10 b + s 7 S 10 a
x ̂ a = n 0 y q
x ̂ b = 0
ŝ x 1 = 1 2 c q 3
ŝ x 2 = x ̂ a 2
ŝ x 8 = 1 2 c q
t ̂ x 1 = c Î 0 y q 2 ŝ x 1
t ̂ x 2 = x ̂ a ŝ x 1
t ̂ x 5 = x ̂ a ŝ x 2
t ̂ x 8 = c Î 0 y q 2 ŝ x 8
t ̂ x 9 = 2 x ̂ a ŝ x 8
t ̂ x 10 = 0
t ̂ x 11 = 0
t ̂ x 14 = x ̂ a ŝ x 8
t ̂ x 15 = 0
t ̂ x 16 = 0
σ ̂ a = c ( ρ 2 V ̂ 0 + ρ 1 x ̂ a )
τ ̂ a = ρ 3 σ ̂ a + 2 c R ¯ 0 ρ 2 n 0 y
σ ̂ b = ρ 2 ( Î 0 y + n 0 x V ̂ 0 ) 2 ρ 1 α 0 β 0 n 0 x
τ ̂ b = 1 2 ρ 3 ( σ ̂ b ρ 2 n 0 y )
ŝ τ 1 = ρ 3 ŝ σ 1 + c 2 R ¯ 0 ( ρ 1 ŝ x 2 ρ 3 )
ŝ σ 2 = c ρ 2 + c ρ 1 ( 2 α 0 β 0 x ̂ a + ŝ x 2 )
ŝ τ 5 = 1 2 ρ 3 ŝ σ 5
ŝ τ 8 = ρ 3 ŝ σ 8 R ¯ 0 c 2 ρ 2
t ̂ σ 1 = c ρ 1 t ̂ x 1
t ̂ τ 1 = ρ 3 t ̂ σ 1 + 2 c 2 R ¯ 0 ρ 1 ŝ x 1 x ̂ a
t ̂ σ 2 = c ρ 1 ( t ̂ x 2 + 2 α 0 β 0 ŝ x 1 )
t ̂ τ 2 = ρ 3 t ̂ σ 2 + 2 c 2 R ¯ 0 ρ 1 ŝ x 2 ( x ̂ a + α 0 β 0 )
t ̂ σ 5 = c ρ 1 ( t ̂ x 5 + 2 α 0 β 0 ŝ x 2 + α 0 2 x ̂ a )
t ̂ τ 5 = ρ 3 t ̂ σ 5
t ̂ σ 8 = c ρ 1 t ̂ x 8
t ̂ τ 8 = ρ 3 t ̂ σ 8 + c R ¯ 0 t ̂ σ 8
t ̂ σ 9 = c ρ 1 t ̂ x 9
t ̂ τ 9 = ρ 3 t ̂ σ 9
t ̂ σ 10 = c ρ 1 α 0 2 x ̂ a
t ̂ τ 10 = ρ 3 t ̂ σ 10
t ̂ σ 11 = 0
t ̂ τ 11 = 0
t ̂ σ 14 = c ρ 1 ( t ̂ x 14 + 2 α 0 β 0 ŝ x 8 )
t ̂ τ 14 = ρ 3 t ̂ σ 14
t ̂ σ 15 = 0
t ̂ τ 15 = 0
t ̂ σ 16 = 0
t ̂ τ 16 = 0