Geography Reference
In-Depth Information
Table 22.13
Zero order integral of arc length of an elliptic meridian, power series expansion, powers of 1
,E
2
,E
4
,
etc. and
sin
2
B
,
sin
4
B
,etc.
y
0
=
B
0
0
M
(
B
)
dB
=
A
1
[
E
0
B
0
+
E
2
sin2
B
0
+
E
4
sin
4
B
0
+
E
6
sin
6
B
0
+
E
8
sin
8
B
0
+
E
10
sin
10
B
10
+
O
(
sin
12
B
0
)]
(22.86)
1
4
E
2
−
3
64
E
4
−
5
64
E
6
−
175
16384
E
8
−
441
65536
E
10
E
0
=1
−
3
8
E
2
−
3
32
E
4
−
45
1024
E
6
−
105
4096
E
8
−
2205
131072
E
10
E
2
=
−
15
256
E
4
+
45
1024
E
6
+
16384
E
8
+
1575
525
65536
E
10
E
4
=
35
3072
E
6
−
175
12288
E
8
−
3675
262144
E
10
E
6
=
−
315
131072
E
8
+
2205
524288
E
10
E
8
=
693
1310720
E
10
E
10
=
−
Table 22.14
Polynomial representation of the equidistant mapping of the Reference Meridian
“
x
is called
Easting
(Rechtswert) and y
Northing
(Hochwert)”
“the
North component
”
ansatz
l
=0:
y
=
η
(
q,
0) =
γ
0
+
γ
1
q
+
γ
2
q
2
+
O
q
3
=
x
h
=0
γ
h
q
h
(22.87)
“
recurrence
relation”
d
h
y
dQ
h
[
Q
0
(
B
0
)]
γ
h
=
1
h
!
(22.88)
d
h
y
dQ
h
=
dB
(
d
h−
1
y
d
)
dB
dQ
(22.89)
dQ
h−
1
E
2
sin
2
B
1
dB
dQ
=cos
B
1
−
(22.90)
−
E
2
{δ
0
,δ
1
,δ
2
,δ
3
,...δ
N
}∈O
. Accordingly Table
22.16
contains the final form of the harmonic
map for the
Northern function y
(
q,l
) based on the solution of the first boundary value prob-
lem. Such a representation has the disadvantage that any
Geodetic Data Base
contains only
Gauss ellipsoidal coordinates
(
L, B
)and
not
ellipsoidal isometric coordinates (
L, Q
)or(
l,q
)of
Mercator type.
It is for this reason that we have transformed the powers
{
q,q
2
,...,q
N
}
into pow-
b, b
2
,...,b
N
ers
in Tables
22.16
,
22.17
,
22.18
and
22.19
: The series expansion Eqs.(
22.102
)-
(
22.104
)ofTable
22.16
leaves us with the problem to compute the coecients
q
r
of type
Eqs.(
22.106
)-(
22.107
) explicitly done in Table
22.17
,Eqs.(
22.116
)-(
22.126
): All computations
have been “
Mathematical
” controlled. The concluding representation of the “
Northern func-
{
}
Search WWH ::
Custom Search