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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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