Geography Reference
In-Depth Information
0
([01]
0
+
∂
[
μν
]
∂B
0
y
2
+ [03]
0
y
3
+ [04]
0
y
4
+
[
μν
]
|
F
=[
μν
]
|
|
0
y
+ [02]
|
|
|
···
)+
∂B
2
0
∂
2
[
μν
]
+
1
2!
|
0
y
2
+ [03]
|
0
y
3
+ [04]
|
0
y
4
+
)
2
([01]
|
0
y
+ [02]
···
+
···
+
(20.120)
∂B
n
0
∂
n
[
μν
]
1
n
!
|
0
y
2
+ [03]
|
0
y
3
+ [04]
|
0
y
4
+
...
)
n
,
+
([01]
|
0
y
+ [02]
0
[
μν
]
α
|
F
=[
μν
]
α
|
0
+
∂
[
μν
]
α
([01]
α
|
0
y
+ [02]
α
|
0
y
2
+ [03]
α
|
0
y
3
∂B
+[04]
α
|
0
y
4
+
···
)+
0
([01]
α
∂
2
[
μν
]
α
∂B
2
+
1
2!
0
y
2
+ [03]
α
0
y
3
|
0
y
+ [02]
α
|
|
+[04]
α
|
0
y
4
+
···
)
2
+
···
+
(20.121)
0
([01]
α
∂
n
[
μν
]
α
∂B
n
+
1
n
!
0
y
2
+ [03]
α
0
y
3
|
0
y
+ [02]
α
|
|
0
y
4
+
)
n
.
+[04]
α
|
···
The final step consists of substituting (
20.120
)and(
20.121
)into(
20.115
), and summing
up (
20.115
)and(
20.114
). The finally resulting representation for the ellipsoidal coordinates
{L, B}
of point
P
, given the Soldner coordinates
{x, y}
with respect to the point
P
0
(
L
0
,B
0
),
is of the form (
20.73
)-(
20.75
) with Soldner coecients [
μν
]
L
and [
μν
]
B
:
L
=
L
P
=
L
0
+
∞
∞
[
μν
]
L
x
μ
y
ν
,
μ
=0
ν
=0
B
=
B
P
=
B
0
+
∞
∞
[
μν
]
B
x
μ
y
ν
,
(20.122)
μ
=0
ν
=0
∞
∞
[
μν
]
α
x
μ
y
ν
.
γ
=
α
PF
=
μ
=0
ν
=0
As an alternative, the numerical computation of
is done using the two-step-method
combined with a numerical integration of (
20.69
)and(
20.71
), for example, by classical Runge-
Kutta techniques. First integrate (
20.69
)and(
20.71
) numerically with initial values (
20.123
)
in order to obtain (
20.124
), then integrate (
20.69
)and(
20.71
) a second time with initial values
(
20.125
) with the result for (
20.126
):
{
L, B, γ
}
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