Geography Reference
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where
X
is the meridian arc;
B
is the geodetic latitude;
a
is the semi-major axis of the
reference ellipsoid;
(1) is an elliptic integral of the second kind and there is no analytical solution. Expanding
the integrand by binomial theorem and itntegrating it item by item yield:
2
X a
(1
e KBK
)(
sin 2
BK
sin 4
BK
sin 6
BK
sin 8
BK
sin 10
B
)
(2)
0
2
4
6
8
10
where
3
45
175
11025
43659
2
4
6
8
10
K
1
e
e
e
e
e
0
4
64
256
16384
65536
3
15
525
2205
72765
2
4
6
8
10
K
e
e
e
e
e
2
8
32
1024
4096
131072
15
105
2205
10395
4
6
8
10
K
e
e
e
e
4
256
1024
16384
65536
(3)
35
105
10395
6
8
10
K
e
e
e
6
3072
4096
262144
315
131072
3465
524288
8
10
K
e
e
8
693
1310720
10
K
e
10
The rectifying latitude
is defined as
X
(4)
2
()
2
X
Inserting (2) into (4) yields
B
sin 2
B
sin 4
B
sin 6
B
sin 8
B
sin 10
B
(5)
2
4
6
8
10
where
KK
KK
KK
KK
KK
/
/
/
/
2
2
0
4
4
0
(6)
6
6
0
8
8
0
/
10
10
0
Yang (1989, 2000) gave an expansion similar to (5) but expanded
up to sin 8
B
. For
simplicity and computing efficiency, it is better to expand (6) into a power series of the
eccentricity. This process is easily done by means of Mathematica. As a result, (6) becomes:
