Geography Reference
In-Depth Information
Denote
1
1 1
l
41
e
A
(16)
2
e
e
2(1
e
)
Suppose that there is an imaginary sphere with a radius
2
Ra
(1
eA
)
(17)
and whose area from the spherical equator
to spherical latitude with a width of a
unit interval of longitude is equal to the ellipsoidal area
F
, it holds
0
2
2
2
R
sin
a
(1
e
)
A
sin
F
(18)
Therefore, it yields
1
sin
B
1
1
e
sin
B
arcsin
l
41 in
(19)
2
2
A
e
e
B
2(1
e
sin
B
)
where is authalic latitude. Yang(1989, 2000) gave its series expansion as
B
sin 2
B
sin 4
B
sin 6
B
sin 8
B
(20)
2
4
6
8
(19) is a complicated transcendental function. It is almost impossible to derive its eighth-
order derivate by hand. There are some mistakes in the high order terms of coefficients ,
, , .The new derived forward expansion expanded to tenth-order terms of
e
by
means of Mathematica reads
B
sin 2
B
sin 4
B
sin 6
B
sin 8
B
sin 10
B
(21)
2
4
6
8
10
The derived coefficients in (20) and (21) are listed in Table 2 for comparison.
Coefficients derived by Yang(1989, 2000)
Coefficients derived by the author
1
31
59
42811
605399
2
4
6
8
10
1
31
59
126853
e
e
e
e
e
2
4
6
8
e
e
e
e
2
3
180
560
604800
11975040
2
3
180
560
518400
17
61
76969
215431
4
6
8
10
e
e
e
e
17
61
3622447
4
6
8
e
e
e
4
360
1260
1814400
5987520
4
360
1260
94089600
383
3347
1751791
6
8
10
e
e
e
383
6688039
6
6
8
e
e
45360
259200
119750400
6
43560
658627200
6007
201293
8
10
e
e
27787
23522400
8
8
3628800
59875200
e
8
5839
171
10
e
10
07200
Table 2.
The comparison of coefficients of the forward expansion of conformal latitude derived by
Yang (1989, 2000) and the author
