Geography Reference
In-Depth Information
4.3. The direct expansions of transformations between isometric latitude and
authalic latitude function
4.3.1. The direct expansion of the transformation from isometric latitude to authalic
latitude function
The whole formulas for the transformation from isometric latitude to authalic latitude
function are as follows:
arcsin(tanh )
sin 2
q
B
b
b
sin 4
b
sin 6
b
sin 8
b
sin 10
2
4
6
8
10
(64)
B
sin 2
B
sin 4
B
sin 6
B
sin 8
B
sin 10
B
2
4
6
8
10
2
FR
sin
Expanding
F
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 authalic latitude
function
0
arcsin(tanh
q
)
(65)
2
Fa
sin
sin 3
sin 5
sin 7
sin 9
sin11
1
3
5
7
9
11
where
1
1
7
113
7
2
4
6
8
10
1
e
e
e
e
e
1
4
12
192
5760
560
1
7
1
43
2
6
8
10
e
e
e
e
3
12
960
192
13440
1
1
1
13
4
6
8
10
e
e
e
e
5
60
192
20160
11520
(66)
31
7
41
6
8
10
e
e
e
7
6720
2304
40320
41
1
8
10
e
e
9
26880
640
167
295680
10
e
11
4.3.2. The direct expansion of the transformation from authalic latitude function to
isometric latitude
The whole formulas for the transformation from authalic latitude function to isometric
latitude are as follows:
