Geography Reference
In-Depth Information
conformal mapping or
universal transverse Mercator projection
conformal mapping. The arc
length of the coordinate line
L
0
= const., namely the
meridian
, between latitude
B
0
and
B
is computed by (
15.74
) as soon as we set up uniformly convergent Taylor series of type (
15.75
)
and integrate term-wise.
y
(0
,b
)=
B
B
0
G
22
(
B
∗
)d
B
∗
=
B
B
0
∞
M
(
B
∗
)d
B
∗
=
y
0
n
b
n
,
(15.74)
n
=1
∞
G
22
(
B
)=
M
(
B
)=
A
1
(1
− E
2
)
1
n
!
G
(
n
)
22
(
B
0
)
b
n
.
E
2
sin
2
B
)
3
/
2
=
(15.75)
(1
−
n
=1
Box
15.2
, which follows subsequently, contains a list of resulting coecients
y
0
n
, which establish
the setup of the constraints defined in Definition
15.3
.
Definition 15.3 (Constraints to the Korn-Lichtenstein equations of conformal mapping).
Let there be given the ellipsoidal Korn-Lichtenstein equations (
15.76
), subject to the integrability
condition, the Laplace-Beltrami equations (
15.77
), which generate a conformal mapping via a
polynomial representation of type (
15.29
)-(
15.32
) and the coecient constraints given by (
15.69
)-
(
15.73
).
G
11
/G
22
x
b
,
b
=
G
22
/G
11
x
l
,
y
l
=
−
(15.76)
sx
ll
+(
rx
b
)
b
=0
, sy
ll
+(
ry
b
)
b
=0
.
(15.77)
The
equidistant mapping
of the meridian of reference
L
0
establishes by means of constraints of
type (
15.78
)
the conformal mapping
of type
Gauss-Krueger
or
UTM
.
x
(0
,b
)=0
,
(0
,b
)=
∞
y
0
n
b
n
.
(15.78)
n
=1
End of Definition.
Box 15.2 (The equidistant mapping of the meridian of reference
L
0
,y
(0
,b
)=
n
=1
y
0
n
b
n
,coecients
y
01
,...,y
04
).
y
01
=
G
22
B
0
A
1
(1
−
E
2
)
=
E
2
sin
2
B
0
)
3
/
2
,
(1
−
2
G
22
/
G
22
B
0
2
[
G
22
]
=
1
y
02
=
1
=
3
2
A
1
E
2
(1
−
E
2
)cos
B
0
sin
B
0
(1
,
E
2
sin
2
B
0
)
5
/
2
−
B
0
1
G
22
2
]
/G
3
/
2
24
[2
G
22
G
22
−
y
03
=
(15.79)
22
A
1
E
2
(1
E
2
)
=
1
2
−
2sin
2
B
0
+4
E
2
sin
2
B
0
−
3
E
2
sin
2
B
0
)
,
E
2
sin
2
B
0
)
7
/
2
(1
−
(1
−
B
0
1
6
G
22
G
22
G
22
+3
G
22
3
]
/G
5
/
2
192
[4
G
22
G
y
04
=
−
22
22
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