270 9 “Ellipsoid-of-revolution to sphere and from sphere to plane”
The term authalic latitude φ has been introduced by O. S. Adams (1921, p. 65) or J. P. Snyder (1982,
p. 19). In Box 9.8, the two interesting concepts are summarized.
Box 9.8 (The two interesting concepts).
First case:
A 1 = r , a = 1 .
(9.72)
Second case
(“identical surface area”):
4πr
2 = 4πA
2
1
„
1
2
+
1 − E
2
4E
ln
1 + E
1 − E
«
⇒
r
2 =
1
2
A
2
1
„
1 +
1 − E
2
2E
ln
1 + E
1 − E
«
⇒
sin φ =
(1 − E
2 )
a
sin Φ
(1−E 2 sin 2 Φ)
+
1
2E
ln
1+E sin Φ
1−E sin Φ
1 +
1−E 2
2E
ln
1+E
1−E
.
(9.73)
Example:
a = 1 .
(9.74)
Λ 1 =
ar cos φ
A 1 cos Φ
(1 − E
2 sin
2 Φ)
1/2 ,
Λ 2 =
rdφ/dΦ
A 1 (1 − E 2 )
(1 − E
2 sin
2 Φ)
3/2 ,
(9.75)
dφ
dΦ
=
1
a cos φ
1
r 2
A
2
1 (1 − E
2 )
(1 − E 2 sin
2 Φ) 2 cos Φ
⇒
(9.76)
Λ 2 = Λ
−1
1 =
=
A 1 cos Φ
a cos φ
1
r
(1 − E
2 sin
2 Φ)
−1/2 =
1
a
A 1 cos Φ
r cos φ
1
(1 − E 2 sin
2 Φ) 1/2 .
(9.77)
We here also note the remarkable representations for sin φ(sin Φ). Furthermore, note the remarkable
representations for the cases A 1 = r versus 4πr
2 = 4πA
2
1
1/2 + [(1 − E
2 )/4E] ln[(1 + E)/(1 − E)]
and Λ 1 and Λ 2 , respectively.