Datum Problems - Map Projections: Cartographic Information Systems - page 522

Jacobi matrix:

B:=

∂ ( L, B )
∂ ( t x ,t y ,t z ,α,β,γ,s,a 1 ,e 2 )

2 × 9 ,

taylor ∈ R

(21.38)

t x = t y = t z = α = β = γ = s =0 ,a 1 = A 1 ,e 2 = E 2 taylor := { } .

Chain Jacobi matrix:

B:=J 23 J 1 = ∂ ( L, B )

∂ ( X,Y,Z )
∂ ( t x ,t y ,t z ,α,β,γ,s,a 1 ,e 2 )

,

(21.39)

∂ ( X,Y,Z )

taylor taylor

∂ ( X,Y,Z )
∂ ( L, B, H )

− 1

= ∂ ( L, B, H )
∂ ( X,Y,Z )

,

⎡ ⎤

J 2 := ∂ ( X,Y,Z )
∂ ( L, B, H )

=

∂X/∂L ∂X/∂B ∂X/∂H
∂Y/∂L ∂Y/∂B ∂Y/∂H
∂Z/∂L ∂Z/∂B ∂Z/∂H

⎣ ⎦ ,

J 2 := (J 2 ) − 1 = ∂ ( L, B, H )

,

(21.40)

∂ ( X,Y,Z )

J 2 =

⎡ ⎤

− ( N + H )cos B sin L − ( M + H )sin B cos L cos B cos L
( N + H )cos B cos L

⎣ ⎦ ,

− ( M + H )sin B sin L cos B sin L

0 ( M + H )cos B sin B

(21.41)

J 2 =

⎡ ⎤

sin L
( N + H )cos B

cos L
( N + H )cos B

− 0

⎣ ⎦ ,

cos B
M + H
cos B cos L cos B sin L sin B

cos L sin B
M + H

sin L sin B
M + H

− −

J 23 = ∂ ( L, B ) = −

sin l cos l

( n + h )cos b + ( n + h )cos b 0 2 × 3 ,

∈ R

cos l sin b
m + h

sin l sin b
m + h

cos b
m + h

∂ ( X,Y,Z ) − −

taylor

J 1 =

∂ ( X,Y,Z )
∂ ( t x ,t y ,t z ,α,βγ,s,a 1 ,e 2 )

= (21.42)

taylor

⎡ ⎤

m cos b sin 2 b cos l
2(1 −e 2 )

n cos b cos l
a 1

− 10 0
0 − 10
00 − 1

0 z −y
−z 0 x
y −x 0

−x
−y
−z

⎣ ⎦ ∈ R m cos b sin 2 b sin l

2(1 −e 2 )

n cos b sin l
a 1

3 × 9 .

=

n sin b (1 − e 2 )
a 1

m sin 3 b − 2 n sin b
2