9-3 The equal area mappings “ellipsoid-of-revolution to plane” 269
The principal stretches of the conformal mapping “ellipsoid- of- revolution to sphere” are explicitly
given by (9.66).
Λ 1 = Λ 2 = Λ =
a
√
M 0 N 0 cos φ
N (Φ) cos Φ
, a as given above ,
(9.66)
φ = 2 arctan
c
a
tan
π
4
+
Φ
2
a
1 − E sin Φ
1 + E sin Φ
aE/2
−
π
2
.
(9.67)
9-3 The equal area mappings “ellipsoid-of-revolution to plane”
The equal area mappings from the ellipsoid-of-revolution to the plane: the condition of equal area, the
standard integrals, authalic latitude.
First, we postulate the condition of eq ual area Λ 1 Λ 2 = 1 and subseq uently we take advantage of the
standard integrals that are collected in Box 9.7.
Λ 1 Λ 2 = 1
⇔
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 = 1
⇔
(9.68)
r
2 cos φ dφ =
A
2
1 (1 − E
2 )
a
cos Φ
dΦ
(1 − E 2 sin
2 Φ) 2 .
(9.69)
Box 9.7 (The standard integrals).
First standard integral
(conic mapping: Lambert conformal):
r
2 sin φ =
A
2
1 (1 − E
2 )
a
Z
cos Φ
dΦ
(1 − E 2 sin
2 Φ) 2 + c ,
∆ :=
π
2
− Φ ⇒ −d∆ = +dΦ , c := 0 ,
−
Z
sin ∆
(1 − E 2 cos 2 ∆) 2 d∆ =
cos ∆
2(1 − E 2 cos 2 ∆)
+
1
4E
ln
1 + E cos ∆
1 − E cos ∆
=
=
sin Φ
2(1 − E 2 sin
2 Φ)
+
1
4E
ln
1 + E sin Φ
1 − E sin Φ
.
(9.70)
Second standard integral
(equal area conic mapping):
ar
2 sin φ = A
2
1 (1 − E
2 )
»
sin Φ
2(1 − E 2 sin
2 Φ)
+
1
4E
ln
1 + E sin Φ
1 − E sin Φ
–
,
sin φ =
A
2
1 (1 − E
2 )
ar 2
»
sin Φ
2(1 − E 2 sin
2 Φ)
+
1
4E
ln
1 + E sin Φ
1 − E sin Φ
–
.
(9.71)