Geography Reference
In-Depth Information
Table 22.25
Distortion energy of the special harmonic map
L
E
dL
B
N
B
S
M
N
coscos
B
∗
∗
x
01
+4
y
02
l
2
+8
y
02
y
12
bl
2
+
O
(4)
dB
+
1
2
1
2
∫
=
1
2
d
S
l
tr
C
l
G
−
1
l
L
W
L
E
dL
B
N
B
S
y
01
+4
y
10
y
20
b
+
N
coscos
B
M
L
W
(4
y
20
+6
y
10
y
30
)
b
2
+2
y
10
y
12
l
2
+
O
(3)
dB
(22.143)
“Taylor expansions”
E
2
E
2
M
cos
B
=
1
−
1
−
E
2
sin
2
B
=
E
2
sin
2
B
0
1
−
1
−
1!
1
E
2
sin
2
B
0
−
2
E
2
sin2
B
0
b
+
O
(2)
+
1
···
−
(22.144)
E
2
sin
2
B
1
E
2
sin
2
B
0
1
E
2
cos
B
M
=
1
−
=
1
−
−
sin2
B
0
b
+
O
(2)
(22.145)
−
E
2
−
E
2
1
−
E
2
dS
l
tr
C
l
G
−
1
l
E
=
1
2
B
S
)
l
E
+
e
21
(
B
N
−
B
S
)
l
E
+
e
13
(
B
N
−
B
S
)
l
E
+
O
(5)
=
e
11
(
B
N
−
(22.146)
l
E
:=
L
E
−
L
0
=
L
0
−
L
W
,L
E
−
L
W
=2
l
E
(22.147)
1
− E
2
1
− E
2
B
0
+
1
−
E
2
sin
2
B
0
1
x
01
E
2
sin
2
B
0
x
01
y
10
−
e
11
:=
+4
E
2
sin
2
B
0
E
2
sin
2
B
0
−
E
2
1
−
1
−
E
2
sin
2
B
0
1
E
2
sin
2
B
0
1
4
1
−
y
10
y
20
B
0
+
1
−
y
10
B
0
(22.148)
−
E
2
−
E
2
E
2
sin
2
B
0
1
E
2
sin
2
B
0
1
E
2
1
−
E
2
sin
2
B
0
x
01
+2
1
−
1
2
y
10
e
21
:=
−
y
10
y
20
−
(22.149)
− E
2
sin
2
B
0
−
E
2
−
E
2
1
E
2
1
−
2
9
y
02
y
12
B
0
e
13
:=
−
(22.150)
E
2
sin
2
B
0
1
−
22-5 Case Studies
Two case studies illustrate the operational approach to the new harmonic map including the
distortion analysis by means of
Tissot ellipses.
The
first example
is defined by a harmonic map with
L
0
=9
◦
as the longitude of the
Reference
Meridian
of the
International Reference Ellipsoid
.The
Reference Parallel Circle
with
B
0
=
±
30
◦
as the Gauss ellipsoidal latitude has been chosen. Figure
22.4
illustrates the harmonic map in the
range
B ∈
[
−
40
◦
,
+40
◦
]
andL ∈
[
−
31
◦
,
+49
◦
].
In order to compare the new harmonic map of the
International Reference Ellipsoid
with
Gauss-Krueger conformal maps
and
UTM we have computed the second example for the Ref-
erence Meridian
L
0
∈{
6
◦
,
9
◦
,
12
◦
,
15
◦
}
and the
Reference Parallel Circle B
0
=51
◦
in the range
[46
◦
,
56
◦
]
,L
[4
.
5
◦
,
7
.
5
◦
];[7
.
5
◦
,
10
.
5
◦
] ; [10
.
5
◦
,
13
.
5
◦
];[13
.
5
◦
,
16
.
5
◦
]
B
∈
∈{
}
. Indeed we have chosen
a
strip width
of 3
◦
with
l
1
.
5
◦
,
+1
.
5
◦
] around the Reference Meridian
L
0
. Obviously the
Tissot
distortion ellipses
vary slowly within the
Meridian Strip.
Only at the point (
L
0
,B
0
)wecanenjoy
isometry as illustrated by Fig.
22.5
.
∈
[
−
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