446 20 Geodetic mapping
The total deformation energy (total distortion energy, total distance distortion) over the symmetric
strip [−l E , +l E ] × [B S = B 0 + b S , B N = B 0 + b N ], L 0 = (L W + L E )/2, and B 0 = (B S + B N )/2 are given
as follows.
Conformal coordinates of type Gauss–Krueger:
I AGK =
1
20
l
4
E
(1 − E 2 ) 2 ×
×
1 −
2
3
sin
3 B N − sin
3 B S
sin B N − sin B S
+
1
5
sin
5 B N − sin
5 B S
sin B N − sin B S
×
×
1 −
2
3
E
2 sin
3 B N − sin
3 B S
sin B N − sin B S
+ O 2 (E
4 )
+
+O GK (l
6
E ) ,
(20.147)
I AGK (B N = −B S ) =
1
20
l
4
E
(1 − E 2 ) 2 ×
×
1 −
2
3
sin
2 B N (1 + E
2 ) + sin
4 B N
4
9
E
2 +
1
5
−
2
15
E
2 sin
6 B N + O 2 (E
4 )
+
+O GK (l
6
E ) .
(20.148)
Parallel coordinates of type Soldner:
I AS =
1
2
I AGK + O S (l
6
E ) .
(20.149)
Normal coordinates of type Riemann:
I AR =
1
2 cos B 0
A
5
l
4
E +
B
9
b
3
N − b
3
S
b N − b S
l
2
E +
C
5
b
5
N − b
5
S
b N − b S
+
+
D
10
(b N + b S )l
4
E +
P
12
b
4
N − b
4
S
b N − b S
l
2
E +
Q
6
b
6
N − b
6
S
b N − b S
×
×
1 − 2E
2 sin
2 B 0 + sin B 0 (1 − 2E
2 sin
2 B 0 − 4E
2 cos
2 B 0 )(b N + b S )/2 cos B 0
+
+O R (6) ,
(20.150)
I AR (b N = −b S ) =
cos
4 B 0
360
1 + 2E
2 cos
2 B 0 + O(E
4 )
l
4
E +
+
cos
2 B 0
324
1 + O(E
4 )
b
2
N l
2
E +
1 − 2E
2 cos
2 B 0 + O(E
4 )
360
b
4
N +
+O R (6) .
(20.151)
(A, B, C, D, P, Q are given by (20.173)).
End of Corollary.