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
0 1 2 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-
{
}
 
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