8-2 Special mapping equations 233
N
Z
T N E
2
A 1 ,A 2
r
A 2
−A 2
A 1
−A 1
p = π(P )
P
O
√
X 2 + Y 2
S
Fig. 8.4. Conformal mapping of the ellipsoid-of-revolution onto a tangential plane: normal aspect, P ∈ E
2
A 1 ,A 2 ,
p = π(P ), not UPS.
Lemma 8.9 (Meridian arc length, inverse computation).
A forward computation of the meridian arc length based upon a uniform series expansion is provided
by formula (8.39). Its inverse function can be represented by
Φ =
π
2
−
r
A 1 E 0
− F 2 sin 2
r
A 1 E 0
− F 4 sin 4
r
A 1 E 0
− F 6 sin 6
r
A 1 E 0
−
−F 8 sin 8
r
A 1 E 0
− F 10 sin 10
r
A 1 E 0
+ O(E
12 ) ,
(8.41)
subject to
E 0 = 1 −
1
4
E
2
−
3
64
E
4
−
5
256
E
6
−
175
16384
E
8
−
441
65536
E
10
(8.42)
and
F 2 =
3
8
E
2 +
3
16
E
4 +
213
2048
E
6 +
255
4096
E
8 +
166479
655360
E
10 ,
F 4 =
21
256
E
4 +
21
256
E
6 +
533
8192
E
8
−
120563
327680
E
10 ,
F 6 =
151
6144
E
6 +
155
4096
E
8 +
2767911
9175040
E
10 ,
F 8 =
1097
131072
E
8
−
273697
4587520
E
10 .
(8.43)
End of Lemma.