20-5 Deformation analysis: Riemann, Soldner, Gauss–Krueger coordinates 445
For a local representation of the ellipsoidal surface, namely the reference figure of the Earth,
conformal Gauss-Krueger and parallel Soldner coordinates are the most popular. Therefore, they are
compared with (Riemann) polar/normal coordinates. As a measure of the total deformation energy
(total distortion energy), according to G. B. Airy (1861), we introduce (20.142) once we map the area
over the symmetric strip (20.141).
[l W = −l E , +l E = −l W ] × [B S = B 0 + b S , B N = B 0 + b N ] .
(20.141)
The indices W, E, S, and N refer to “West”, “East”, “South”, and “ North”. The area of the symmetric
strip is provided by (20.143). Results characterizing the total deformation energy are collected in
Corollary 20.5.
I A :=
:=
1
2S E 2
A 1 ,A 2
l E
l W
dl
B N
B S
dB
cos B{(Λ 1 − 1)
2 + (Λ 2 − 1)
2
}
(1 − E 2 sin
2 B) 2
,
(20.142)
S E 2
A 1 ,A 2
=
dS E 2
A 1 ,A 2
:=
:= A
2
1 (1 − E
2 )
l E
−l E
dl
B N
B S
dB
cos B
(1 − E 2 sin
2 B) 2 =
= 2A
2
1 (1 − E
2 )l E [sin B N +
2
3
E
2 sin
3 B N − (sin B S +
2
3
E
2 sin
3 B S ) + O(E
4 )] .
(20.143)
Corollary 20.5 (E
2
A 1 ,A 2
, the total deformation energy, the Airy measure, Gauss–Krueger coordinates,
Soldner coordinates, and Riemann coordinates).
The principal distortions read as follows (l := L − L 0 , b := B − B 0 ).
Conformal coordinates of type Gauss–Krueger:
Λ 1 = Λ 2 = 1 +
cos
2 B
2
1 +
E
2
1 − E 2
l
2 + O GK (l
4 ) .
(20.144)
Parallel coordinates of type Soldner:
Λ 1 = 1 +
cos
2 B
2
1 +
E
2
1 − E 2 cos
2 B
l
2 + O S (l
4 ) ,
Λ 2 = 1 .
(20.145)
Normal coordinates of type Riemann:
Λ 1 = 1 +
cos
2 B 0
6
1 − E
2 sin
2 B 0
1 − E 2
l
2 +
1
6
1 − E
2
1 − E 2 sin
2 B 0
b
2 + O R (l
3 , b
3 ) ,
Λ 2 = 1 .
(20.146)