Geography Reference
In-Depth Information
as the conventional usage in mathematical cartography.
Through the tedious expansion process, Yang (1989, 2000) gave a power series of the
eccentricity
e
for the conformal latitude
as
(13)
B
sin 2
B
sin 4
B
sin 6
B
sin 8
B
2
4
6
8
Due to that (11) is a very complicated transcendental function, the coefficients
,
,
,
in (13) derived by hand are only expanded to eighth-order terms of
e
. They are not accurate
as expected and there are some mistakes in the eighth-order terms of
e
.
In fact, Mathematica works perfectly in solving derivatives of any complicated functions. By
means of Mathematica, the new derived forward expansion expanded to tenth-order terms
of
e
reads
B
sin 2
B
sin 4
B
sin 6
B
sin 8
B
sin 10
B
(14)
2
4
6
8
10
The derived coefficients in (13) and (14) are listed in Table 1 for comparison.
Coefficients derived by Yang(1989, 2000)
Coefficients derived by the author
1
5
3
281
7
2
4
6
8
10
e
e
e
e
e
2
1
5
3
1399
2
24
32
5760
240
2
4
6
8
e
e
e
e
2
5
7
697
93
2
24
32
53760
4
6
8
10
e
e
e
e
4
5
7
689
48
80
11520
2240
4
6
8
e
e
e
4
13
461
1693
48
80
17920
6
8
10
e
e
e
6
13
1363
480
13440
53760
6
8
e
e
6
1237
131
480
53760
8
10
e
e
8
677
17520
161280
10080
8
e
8
367
161280
10
e
10
Table 1.
The comparison of coefficients of the forward expansion of conformal latitude derived by
Yang (1989, 2000) and the author
Table 1 shows that the eighth order terms of
e
in coefficients given by Yang(1989, 2000) are
erroneous.
2.3. The forward expansion of the authalic latitude
From the knowledge of mapping projection theory, the area of a section of a lune with a
width of a unit interval of longitude
F
is
cos
B
sin
B
1
1
e
sin
B
B
2
2
2
2
(15)
Fa
(1
e
)
dBa
(1
e
)
l
41 in
2
2
2
2
2
0
e
e
B
(1
e
sin
B
)
2(1
e
sin
B
)
where
F
is also called authalic latitude function.
