Geography Reference
In-Depth Information
Question: “How can we generate the mapping equations of
such a conformeomorphism?”
Answer: “Let us work out this in the following passage in
more detail.”
The forward computation of the meridian arc length
r
=
f
(
Δ
) supplies us with the radial coordi-
nate
r
of an equidistant mapping of a point of the ellipsoid-of-revolution to a corresponding point
on the tangential plane at the North Pole: compare with Lemma
8.8
. The central problem we
are left with can be formulated as follows: given the radial coordinate
r
, find the surface normal
ellipsoidal latitude
Φ
. Such a problem of generating the inverse function can be solved by series
inversion. For details, we here have to direct you to Appendix
B
, where the standard series
inversion of a homogeneous univariate polynomial is outlined, and where additional references of
how to do it are given. Basic formulae are supplied by Lemma
8.9
.
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
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
−
2
−
(8.41)
r
A
1
E
0
− F
10
sin 10
r
A
1
E
0
+O(
E
12
)
,
−F
8
sin 8
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