436 20 Geodetic mapping
Continuation of Box.
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 coefficients [µν]
S
L and [µν]
S
B :
L = L P = L 0 +
∞
X
µ=0
∞
X
ν=0
[µν]
S
L x
µ y
ν ,
B = B P = B 0 +
∞
X
µ=0
∞
X
ν=0
[µν]
S
B x
µ y
ν ,
γ = α PF =
∞
X
µ=0
∞
X
ν=0
[µν]
S
α x
µ y
ν .
(20.122)
As an alternative, the numerical computation of {L, B, γ} 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 0 = L P 0 , B 0 = B P 0 , s = x c = y, α 0 = α 0F = π/2} ,
(20.123)
{L F , B F , α F 0 = α 0F } ,
(20.124)
{L 0 = L F , B 0 = B F , s = y c = x, α 0 = α F P = 0} ,
(20.125)
{L = L P , B = B P , γ = α PF } .
(20.126)
A numerical example is given in Table 20.2.
Table 20.2. First problem of Soldner coordinates: input {L 0 , B 0 , x, y} versus output {L, B, γ = α PF }.
Reference ellipsoid, Bessel:
A 1 = 6 377 397.155 m, E
2 = 0.006 674 372 20.
Soldner origin, T¨ ubingen observatory:
B 0 = 48
◦ 31
15”.723 4 N, L 0 = 9
◦ 3
7”.144 5 E.
Soldner coordinates:
y = 29 682.228 m, x = 2 469.517 m.
Ellipsoidal coordinates, footpoint P F :
B F = 48
◦ 47
16”.741 0, L F = 9
◦ 3
7”.144 5.
Ellipsoidal coordinates, point P :
B = B P = 48
◦ 47
16”.723 4, L = L P = 9
◦ 5
8”.145 0.
Continuation of Box.
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 coefficients [µν]
S
L and [µν]
S
B :
L = L P = L 0 +
∞
X
µ=0
∞
X
ν=0
[µν]
S
L x
µ y
ν ,
B = B P = B 0 +
∞
X
µ=0
∞
X
ν=0
[µν]
S
B x
µ y
ν ,
γ = α PF =
∞
X
µ=0
∞
X
ν=0
[µν]
S
α x
µ y
ν .
(20.122)
As an alternative, the numerical computation of {L, B, γ} 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 0 = L P 0 , B 0 = B P 0 , s = x c = y, α 0 = α 0F = π/2} ,
(20.123)
{L F , B F , α F 0 = α 0F } ,
(20.124)
{L 0 = L F , B 0 = B F , s = y c = x, α 0 = α F P = 0} ,
(20.125)
{L = L P , B = B P , γ = α PF } .
(20.126)
A numerical example is given in Table 20.2.
Table 20.2. First problem of Soldner coordinates: input {L 0 , B 0 , x, y} versus output {L, B, γ = α PF }.
Reference ellipsoid, Bessel:
A 1 = 6 377 397.155 m, E
2 = 0.006 674 372 20.
Soldner origin, T¨ ubingen observatory:
B 0 = 48
◦ 31
15”.723 4 N, L 0 = 9
◦ 3
7”.144 5 E.
Soldner coordinates:
y = 29 682.228 m, x = 2 469.517 m.
Ellipsoidal coordinates, footpoint P F :
B F = 48
◦ 47
16”.741 0, L F = 9
◦ 3
7”.144 5.
Ellipsoidal coordinates, point P :
B = B P = 48
◦ 47
16”.723 4, L = L P = 9
◦ 5
8”.145 0.