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parameters as well as ellipsoidal form parameters. In contrast, J
23
includes the relevant 2
×
3
submatrix
∂
(
l, b
)
/∂
(
x, y, z
) at the Taylor point of the general 3
3 matrix
∂
(
l,b,h
)
/∂
(
x, y, z
).
Indeed, the complicated derivative matrix
∂
(
l, b, h
)
/∂
(
x, y, z
) is computed from its simple
inverse
∂
(
x, y, z
)
/∂
(
l,b,h
). Equation (
21.28
) is the closed form representation of the Jacobi matrix
A given as a function of
×
{
L, B, H
}
of the global curvilinear coordinate system!
Box 21.3 (Curvilinear coordinate conformal transformation (datum transformation) extended
by ellipsoid parameters, ellipsoid-of-revolution
E
A
1
,A
2
versus
E
a
1
,a
2
, ellipsoidal coordinates
{
l,b,h
}
of local type versus ellipsoidal coordinates
{
L, B, H
}
of global type).
+
semi-major axes;
E
:=
(
A
1
−
Λ
2
)
/A
1
,
e
=
(
α
1
−
A
1
,a
1
∈
R
a
2
)
/a
1
relative
eccentricities.
Direct transformation
{
l,b,h
} →{
x, y, z
}
:
x
=
+
h
(
l,b
)
cos
b
cos
l
=[
n
(
b
)+
h
(
l,b
)] cos
b
cos
l,
a
1
1
e
2
sin
2
b
−
y
=
+
h
(
l,b
)
cos
b
sin
l
=[
n
(
b
)+
h
(
l,b
)] cos
b
sin
l,
a
1
1
e
2
sin
2
b
−
(21.13)
z
=
a
1
(1
−
e
2
)
+
h
(
l,b
)
sin
b
=[(1
− e
2
)
n
(
b
)+
h
(
l,b
)] sin
b,
1
e
2
sin
2
b
−
e
2
)
a
1
a
1
(1
−
1
(1
n
(
b
):=
, m
(
b
):=
e
2
sin
2
b
)
3
.
(21.14)
e
2
sin
2
b
−
−
Direct transformation
{L, B, H} →{X,Y,Z}
:
X
=
+
H
(
L, B
)
cos
B
sin
L
A
1
1
− E
2
sin
2
B
=[
N
(
B
)+
H
(
L, B
)] cos
B
sin
L,
Y
=
+
H
(
L, B
)
cos
B
sin
L
A
1
1
E
2
sin
2
B
−
=[
N
(
B
)+
H
(
L, B
)] cos
B
sin
L, l
(21.15)
Z
=
A
1
(1
+
H
(
L, B
)
sin
B
E
2
)
−
1
E
2
sin
2
B
−
E
2
)
N
(
B
)+
H
(
L, B
)] sin
B,
=[(1
−
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