21-1 Analysis of a datum problem 459
Box 21.4 (Curvilinear conformal coordinate transformation (datum transformation) extended by ellipsoid
parameters close to identity, Jacobi matrix ).
y =
"
l − L
b − B
#
= A
2
6
6
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
7
7
5
+ O 2lb = Ax + O 2x .
(21.23)
Jacobi matrix:
A :=
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 } .
(21.24)
Chain Jacobi matrix:
A := 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
,
(21.25)
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
,
(21.26)
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 , (21.27)
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
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
5 ∈ R
3×9 .
(21.28)