264 9 “Ellipsoid-of-revolution to sphere and from sphere to plane”
9-2 The conformal mappings “ellipsoid-of-revolution to plane”
The conformal mappings from the ellipsoid-of-revolution to the plane: the conditions of conformality, the
standard integrals, spherical isometric latitude, ellipsoidal isometric latitude.
First, we postulate the condition of conformality Λ 1 = Λ 2 and subsequently we tak e advantage of the
standard integrals that are collected in B
ox 9.6.
Λ 1 = Λ 2
⇔
ar cos φ
A 1 cos Φ
(1 − E
2 sin
2 Φ)
1/2 =
r
A 1 (1 − E 2 )
dφ
dΦ
(1 − E
2 sin
2 Φ)
3/2
⇔
(9.3 7)
dφ
cos φ
=
1 − E
2
1 − E 2 sin
2 Φ
a
cos Φ
dΦ .
(9.3 8)
W
e here note in passing that via the first standard integral, we introduce spherical isometric latitude.
B y integration-by-parts, we split the second standard integral into three parts, namely by introducing
ellipsoidal isometric latitude.
Box 9.6 (The standard integrals).
First standard integral:
Z dφ
cos φ
= I first = ln tan
„
π
4
+
φ
2
«
:= q
(“spherical isometric latitude”).
(9.39)
Second standard integral:
Z
1 − E
2
1 − E 2 sin
2 Φ
dΦ
cos Φ
= I second .
(9.40)
1 − E
2
1 − E 2 sin
2 Φ
1
cos Φ
=
1
cos Φ
−
E
2
„
E cos Φ
1 + E sin Φ
−
E cos Φ
1 − E sin Φ
«
,
(9.41)
Z
1 − E
2
1 − E 2 sin
2 Φ
dΦ
cos Φ
=
= ln tan
„
π
4
+
Φ
2
«
−
E
2
ln
1 + E sin Φ
1 − E sin Φ
= ln
"
tan
„
π
4
+
Φ
2
« „
1 − E sin Φ
1 + E sin Φ
« E/2
#
:= Q
(“ellipsoidal isometric latitude”).
(9.42)
The combination of the first and the second standard integral leads us to the celebrated relation in
terms of the integration constants c and k, namely
q = a(Q + k) , k = ln c , c = exp k ;
ln tan
π
4
+
φ
2
= a ln tan
π
4
+
Φ
2
−
aE
2
ln
1 + E sin Φ
1 − E sin Φ
+ a ln c .
(9.43 )