232 8 “Ellipsoid-of-revolution to tangential plane”
The hard work of the series expansion of the kernel representing the meridian arc length has finally
led us to Lemma 8.8, where an elegant version of the meridian arc length up to the order O(E
12 ) has
been achieved.
Lemma 8.8 (Meridian arc length, forward computation).
f (∆) = A 1
∆
0
(1 − E
2 )(1 − E
2 cos
2 ∆
)
−3/2 d∆
= A 1 (1 − E
2 )
π/2
Φ
(1 − E
2 sin
2 Φ
)
−3/2 dΦ
⇒
f (Φ) = A 1
E 0
π
2
− Φ
− E 2 sin 2Φ − E 4 sin 4Φ − E 6 sin 6Φ − E 8 sin 8Φ−
−E 10 sin 10Φ + O(E
12 )
,
(8.39)
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 ,
E 2 = −
3
8
E
2
−
3
32
E
4
−
45
1024
E
6
−
105
4096
E
8
−
2205
131072
E
10 ,
E 4 =
+
15
256
E
4 +
45
1024
E
6 +
525
16384
E
8 +
1575
65536
E
10 ,
E 6 =
−
35
3072
E
6
−
175
12288
E
8
−
3675
262144
E
10 ,
E 8 =
+
315
131072
E
8 +
2205
524288
E
10 ,
E 10 =
−
693
1310720
E
10 .
(8.40)
End of Lemma.
8-22 Conformal mapping
Let us postulate a conformal mapping of the ellipsoid-of-revolution onto a tangential plane at the North
Pole by means of the canoncial measure of conformality, i. e. Λ 1 = Λ 2 . Such a conformal mapping is
illustrated by a vertical section of Fig. 8.4.
Question.
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 (∆) supplies us with the radial coordinate
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.
The hard work of the series expansion of the kernel representing the meridian arc length has finally
led us to Lemma 8.8, where an elegant version of the meridian arc length up to the order O(E
12 ) has
been achieved.
Lemma 8.8 (Meridian arc length, forward computation).
f (∆) = A 1
∆
0
(1 − E
2 )(1 − E
2 cos
2 ∆
)
−3/2 d∆
= A 1 (1 − E
2 )
π/2
Φ
(1 − E
2 sin
2 Φ
)
−3/2 dΦ
⇒
f (Φ) = A 1
E 0
π
2
− Φ
− E 2 sin 2Φ − E 4 sin 4Φ − E 6 sin 6Φ − E 8 sin 8Φ−
−E 10 sin 10Φ + O(E
12 )
,
(8.39)
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 ,
E 2 = −
3
8
E
2
−
3
32
E
4
−
45
1024
E
6
−
105
4096
E
8
−
2205
131072
E
10 ,
E 4 =
+
15
256
E
4 +
45
1024
E
6 +
525
16384
E
8 +
1575
65536
E
10 ,
E 6 =
−
35
3072
E
6
−
175
12288
E
8
−
3675
262144
E
10 ,
E 8 =
+
315
131072
E
8 +
2205
524288
E
10 ,
E 10 =
−
693
1310720
E
10 .
(8.40)
End of Lemma.
8-22 Conformal mapping
Let us postulate a conformal mapping of the ellipsoid-of-revolution onto a tangential plane at the North
Pole by means of the canoncial measure of conformality, i. e. Λ 1 = Λ 2 . Such a conformal mapping is
illustrated by a vertical section of Fig. 8.4.
Question.
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 (∆) supplies us with the radial coordinate
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.