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I A :=
l E
B N
d B cos B { ( Λ 1 1) 2 +( Λ 2 1) 2
}
1
2 S E 2 A 1 A 2
d l
,
:=
(20.142)
E 2 sin 2 B ) 2
(1
l W
B S
= d S E 2 A 1 ,A 2
S E 2 A 1 ,A 2
:=
l E
B N
cos B
:= A 1 (1
E 2 )
d l
d B
E 2 sin 2 B ) 2 =
(20.143)
(1
−l E
B S
E 2 ) l E sin B N + 2
3 E 2 sin 3 B S )+O( E 4 ) .
(sin B S + 2
=2 A 1 (1
3 E 2 sin 3 B N
A 1 ,A 2
, the total deformation energy, the Airy measure, Gauss-Krueger coordi-
nates, Soldner coordinates, and Riemann coordinates).
Corollary 20.5 (
E
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
1+
E 2 l 2 +O GK ( l 4 ) .
E 2
(20.144)
2
1
Parallel coordinates of type Soldner:
Λ 1 =1+ cos 2 B
2
1+
E 2 cos 2 B l 2 +O S ( l 4 ) ,
E 2
(20.145)
1
Λ 2 =1 .
Normal coordinates of type Riemann:
Λ 1 =1+ cos 2 B 0
E 2 sin 2 B 0
1 − E 2
E 2
1
l 2 + 1
6
1
E 2 sin 2 B 0 b 2 +O R ( l 3 ,b 3 ) ,
(20.146)
6
1
Λ 2 =1 .
The total deformation energy (total distortion energy, total distance distortion) over the symmet-
ric 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:
l E
1
20
I AGK =
E 2 ) 2 ×
(1
1
sin 3 B N
sin 3 B S
sin 5 B N
sin 5 B S
2
3
+ 1
5
×
×
sin B N
sin B S
sin B N
sin B S
(20.147)
1
+O 2 ( E 4 ) +
3 E 2 sin 3 B N
sin 3 B S
sin B N sin B S
2
×
+O GK ( l E ) ,
 
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