266 9 “Ellipsoid-of-revolution to sphere and from sphere to plane”
Let us present the summary of the Gauss mapping “ellipsoid-of-revolution to sphere” based upon
the stretch equation (9.50) in form of Lemma 9.1.
ar cos φ
A 1 cos Φ
(1−E 2 sin 2 Φ) 1/2
=
ar cos φ
N (Φ) cos Φ
.
(9.50)
Lemma 9.1 (Gauss mapping “ellipsoid-of-revolution to sphere”).
The mapping equations from the ellipsoid-of-revolution adequately parameterized by {Λ e , Φ e } to the
sphere adequately parameterized by {λ s , φ s } of type conformal read
λ s = λ
s
0 + a(Λ e − Λ
e
0 ) ,
(9.51)
tan
π
4
+
φ s
2
= c
a
tan
π
4
+
Φ e
2
a
1 − E sin Φ e
1 + E sin Φ e
aE/2
,
(9.52)
and
a = cos Φ 0
N 0
M 0
+ tan
2 Φ 0 ,
c =
tan
π
4 +
φ 0
2
1/a
tan
π
4 +
Φ 0
2
1−E sin Φ 0
1+E sin Φ 0
E/2 =
=
exp q 0 /a
exp Q 0
,
ln c =
1
a
q 0 − Q 0 ,
tan φ 0 =
N 0
M 0
tan Φ 0 .
(9.53)
Relative to the equidistant mapping of the “fundamental point”, P 0 (Λ 0 , Φ 0 ) → p 0 = p(λ 0 , φ 0 ), there
hold the conditions
Λ 0 = 1 , Λ
0 = 0 , Λ
0 = 0 .
(9.54)
The mean spherical radius reads
r =
M 0 N 0 .
(9.55)
End of Lemma.
The following chains of calculations, on the one hand, supply us with proofs of the above relations,
and on the other hand, supply us with additional relations needed to understand the above relations.