Geography Reference
In-Depth Information
X
ae
2
(1
)
sin 2
K
0
(52)
B
a
a
sin 4
a
sin 6
a
sin 8
a
sin 10
2
4
6
8
10
B
sin 2
B
sin 4
B
sin 6
B
sin 8
B
sin 10
B
2
4
6
8
10
q
arctanh(sin
)
Since the coefficients
i
) are expressed in a power series of the eccentricity,
one could expand
q
as a power series of the eccentricity
e
at
a
,
2
(
, ,
5
2
i
e
in order to obtain the
direct expansion of the transformation from
X
to
q
. It is hardly completed by hand, but
could be easily realized by means of Mathematica. Omitting the main operations by means
of Mathematica yields the direct expansion of the transformation from meridian arc to
isometric latitude
0
X
ae
2
(1 )
arctanh(sin
K
(53)
0
q
)
sin
sin 3
sin 5
sin 7
sin 9
1
3
5
7
9
where
1
1
1
33
2363
2
4
6
8
10
e
e
e
e
e
1
4
64
3072
16384
1310720
1
13
13
1057
4
6
8
10
e
e
e
e
3
96
3072
8192
1966080
11
29
2897
6
8
10
e
e
e
(54)
5
7680
24576
3932160
25
727
8
10
e
e
7
86016
1966080
53
737280
10
e
9
4.1.2. The direct expansion of the transformation from isometric latitude to meridian arc
The whole formulas for the transformation from isometric latitude to meridian arc are as
follows:
arcsin(tanh
q
)
B
b
sin 2
b
sin 4
b
sin 6
b
sin 8
b
sin 10
2
4
6
8
10
(55)
B
sin 2
B
sin 4
B
sin 6
B
sin 8
B
sin 10
B
2
4
6
8
10
2
Xa
(1
eK
)
0
Expanding
X
as a power series of the eccentricity
e
at
e
by means of Mathematica
yields the direct expansion of the transformation from isometric latitude to meridian arc
0
