Geography Reference
In-Depth Information
3
3
111
141
1533
2
4
6
8
10
e
e
e
e
e
2
8
16
1024
2048
32768
15
15
405
165
4
6
8
10
e
e
e
e
4
256
256
8192
4096
35
35
4935
6
8
10
(7)
e
e
e
6
3072
2048
262144
315
315
8
10
e
e
8
131072
65536
693
1310720
10
e
10
2.2 The forward expansion of the conformal latitude
Omitting the derivation process, the explicit expression for the isometric latitude
q
is
e
/ 2
2
1
e
B
1
e
sin
B
B
q
dB
ln tan
(8)
2
2
421 in
e
B
0
(1
e
sin
B
) cos
B
arctan h( sin
Be
)-
arctan h( sin
e
B
)
For the sphere, putting
e
0
and rewriting the geodetic latitude as the conformal one
, (8)
becomes
q
ln tan(
)
arctan h(sin
)
(9)
42
Comparing (9) with (8) leads to
B
e B
eB
1 in
e
/ 2
tan(
)
tan(
)(
)
(10)
42
421 in
Therefore, it holds
B
e
1 in
B
e
/2
2arctan tan(
)( )
421 in
(11)
e
B
2
Since the eccentricity is small, the conformal latitude is close to the geodetic one. Though
(11) is an analytical solution of
, (11) is usually expanded into a power series of the
eccentricity
2
3
9
1
1
1
2
3
(,)
Be
(,0)
B
e
e
e
2
3
9
e
2!
3!
9!
e
e
e
e
0
e
0
e
0
e
0
(12)
10
1
10!
9
10
e
e
10
e
e
0
