1-13 One example: mapping from an ellipsoid-of-revolution to the sphere 77
Box 1.40 (Left Cauchy–Green matrix, left eigenspace: E
2
A 1 ,A 1 ,A 2 → S
2
r ).
Left manifold ({Λ, Φ} coordinates):
Right manifold ({λ, φ} coordinates):
dS
2 =
A
2
1 cos
2 Φ
1 − E 2 sin
2 Φ
dΛ
2 +
A
2
1 (1 − E
2 )
2
(1 − E 2 sin
2 Φ) 3 dΦ
2 .
ds
2 = r
2 cos
2 φ dλ
2 + r
2 dφ
2 .
(1.273)
“Ansatz”:
E
2
A 1 ,A 1 ,A 2 → S
2
r ; λ = Λ , φ = f (Φ) .
(1.274)
Left Cauchy–Green matrix:
C l = J
T
l G r J l =
"
r
2 cos
2 φ
0
0
r
2 f
2 (Φ)
#
,
J l =
"
D Λ λ D Λ φ
D Φ λ D Φ φ
#
=
"
1
0
0 f
(Φ)
#
, G r =
"
r
2 cos
2 φ 0
0
r
2
#
.
(1.275)
Left principal stretches, left eigenspace:
˛
˛ C l − Λ
2
l G l
˛
˛ =
˛
˛
˛
˛
˛
c 11 − G 11 Λ
2
l
0
0
c 22 − G 22 Λ
2
l
˛
˛
˛
˛
˛
= 0 ⇔
⇔
" c 11 − G 11 Λ
2
l = 0
c 22 − G 22 Λ
2
l = 0
#
⇔
2
6
6
4
Λ
2
1 =
c 11
G 11
=
r
2 cos
2 φ
A
2
1 cos 2 Φ
(1 − E
2 sin
2 Φ)
Λ
2
2 =
c 22
G 22
=
r
2 f
2 (Φ)
A
2
1 (1 − E 2 ) 2 (1 − E
2 sin
2 Φ)
3
3
7
7
5 .
(1.276)
Box 1.41 (Equiareal mapping: E
2
A 1 ,A 1 ,A 2 → S
2
r , φ = f (Φ)).
Area preserving postulate:
Λ 1 Λ 2 = 1 ⇔
r cos φ
A 1 cos Φ
p
1 − E 2 sin
2 Φ
rf
(Φ)
A 1 (1 − E 2 )
(1 − E
2 sin
2 Φ)
3/2 = 1 .
(1.277)
Equation of variables:
r
2 cos φ dφ =
dΦ
(1 − E 2 sin
2 Φ) 2 A
2
1 (1 − E
2 ) cos Φ .
(1.278)
Standard integrals:
∆ := π/2 − Φ ⇒ −d∆ = dΦ ,
Z
cos Φ
(1 − E 2 sin
2 Φ) 2 dΦ = −
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 ∆
+ c r =
=
sin Φ
2(1 − E 2 sin
2 Φ)
+
1
4E
ln
1 + E sin Φ
1 − E sin Φ
+ c r ,
R
cos φ dφ = sin φ + c l .
(1.279)