21-2 Synthesis of a datum problem 465
Box 21.7 (Inverse curvilinear conformal coordinate transformation (datum transformation) extended by
ellipsoid parameters close to identity, Jacobi matrix ).
"
L
B
#
=
"
l
b
#
+ B
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
t x
t y
t z
α
β
γ
s
δa
δe
2
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
+ O 2LB =
"
l
b
#
+ Bx + O 2LB .
(21.37)
Jacobi matrix:
B :=
j
∂(L, B)
∂(t x , t y , t z , α, β, γ, s, a 1 , e 2 )
ff
taylor
∈ R
2×9 ,
taylor := {t x = t y = t z = α = β = γ = s = 0, a 1 = A 1 , e
2 = E
2 } .
(21.38)
Chain Jacobi matrix:
B := J
23 J
1 =
j
∂(L, B)
∂(X, Y, Z)
ff
taylor
j
∂(X, Y, Z)
∂(t x , t y , t z , α, β, γ, s, a 1 , e 2 )
ff
taylor
,
(21.39)
j
∂(X, Y, Z)
∂(L, B, H)
ff −1
=
j
∂(L, B, H)
∂(X, Y, Z)
ff
,
J
2 :=
j
∂(X, Y, Z)
∂(L, B, H)
ff
=
2
6
4
∂X/∂L ∂X/∂B ∂X/∂H
∂Y /∂L ∂Y /∂B ∂Y /∂H
∂Z/∂L ∂Z/∂B ∂Z/∂H
3
7
5 , J
2 :=
`
J
2 ´ −1 =
j
∂(L, B, H)
∂(X, Y, Z)
ff
,
(21.40)
J
2 =
2
6
4
−(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
3
7
5,
J
2 =
2
6
6
4
−
sin L
(N +H) cos B
cos L
(N +H) cos B
0
−
cos L sin B
M+H
−
sin L sin B
M+H
cos B
M+H
cos B cos L cos B sin L sin B
3
7
7
5 ,
(21.41)
J
23 =
j
∂(L, B)
∂(X, Y, Z)
ff
taylor
=
" −
sin l
(n+h) cos b
+
cos l
(n+h) cos b
0
−
cos l sin b
m+h
−
sin l sin b
m+h
cos b
m+h
#
∈ R
2×3 ,
J
1 =
j
∂(X, Y, Z)
∂(t x , t y , t z , α, β, γ, s, a 1 , e 2 )
ff
taylor
=
=
2
6
6
6
4
−1 0 0 | 0 z −y | −x |
n cos b cos l
a 1
m cos b sin
2 b cos l
2(1−e 2 )
|
|
|
|
|
|
0 −1 0 | −z 0 x | −y |
n cos b sin l
a 1
m cos b sin
2 b sin l
2(1−e 2 )
|
|
|
|
|
|
0 0 −1 | y −x 0 | −z |
n sin b(1−e
2 )
a 1
m sin
3 b−2n sin b
2
3
7
7
7
5
∈ R
3×9 .
(21.42)