78
1 From Riemann manifolds to Riemann manifolds
Continuation of Box.
Boundary conditions:
Φ = 0 ⇔ φ = 0 ⇒ c l = c r = 0 .
(1.280)
Equiareal map of E
2
A 1 ,A 1 ,A 2 → S
2
r :
r
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
–
;
case 1: A 1 = r ; case 2:
j left global surface element coincides
with right global surface element
;
S l = area
`
E
2
A 1 ,A 1 ,A 2
´
= area
` S
2
r
´
= S r ;
4πA
2
1
»
1
2
+
1 − E
2
2E
ln
1 + E
1 − E
–
= 4πr
2
⇒
r
2 =
1
2
A
2
1
»
1 +
1 − E
2
2E
ln
1 + E
1 − E
–
.
(1.281)
Authalic latitude (O. S. Adams (1921), p. 65; J. P. Snyder (1982), p. 19):
φ = f (Φ) ,
sin φ = sin f (Φ S l = S r ) .
(1.282)
Box 1.42 (The authalic equiareal map: E
2
A 1 ,A 1 ,A 2 → S
2
r ).
Authalic equiareal map:
λ = Λ ,
sin φ = (1 − E
2 )
»
sin Φ
1 − E 2 sin
2 Φ
+
1
2E
1 + E sin Φ
1 − E sin Φ
–
/
»
1 +
1 − E
2
2E
ln
1 + E
1 − E
–
.
(1.283)
Left and right principal stretches:
Λ 1 =
r cos φ
A 1 cos Φ
p
1 − E 2 sin
2 Φ ,
Λ 2 =
r
A 1 (1 − E 2 )
f
(Φ)(1 − E
2 sin
2 Φ)
3/2 ,
φ = f (Φ) ⇒ φ
= f
(Φ) =
1
r 2 cos φ
A
2
1 (1 − E
2 )
(1 − E 2 sin
2 Φ) 2 cos Φ ,
Λ 1 = λ
−1
1 =
r cos φ
A 1 cos Φ
p
1 − E 2 sin
2 Φ ,
Λ 2 = λ
−1
2 =
A 1 cos Φ
r cos φ
1
p
1 − E 2 sin
2 Φ
.
(1.284)