Geography Reference
In-Depth Information
(2nd)
g
11
g
22
q
φ
=
r,
g
22
g
11
q
λ
=0
⇒
Δ
λ,φ
q
=
g
11
g
22
q
φ
+
g
22
g
11
q
λ
= 0
(1.209)
φ
λ
q. e. d.
Right orientation:
r
2
cos
φ
>
0
p
λ
p
φ
q
λ
q
φ
=
p
λ
q
φ
−
p
φ
q
λ
=
(1.210)
due to
−
π/
2
<Φ<
+
π/
2
⇒
cos
Φ>
0
(1.211)
q. e. d.
Box 1.27 (Conformally flat left and right manifolds, ellipsoid-of-revolution versus sphere).
Left manifold,
Right manifold,
{
Λ, Φ
}
left coordinates:
{
λ, φ
}
right coordinates:
(1.212)
A
1
cos
2
Φ
A
1
(1
−E
2
)
2
(1
d
S
2
=
E
2
sin
2
Φ
d
Λ
2
+
E
2
sin
2
Φ
)
3
d
Φ
2
.
d
s
2
=
r
2
cos
2
φ
d
λ
2
+
r
2
d
φ
2
.
1
−
−
Conformally flat left manifold, Conformally flat right manifold,
{
P, Q
}
conformally
{
p, q
}
conformally
left coordinates
right coordinates
(1.213)
(isometric, isothermal):
(isometric, isothermal):
cos
2
Φ
1
−E
2
sin
2
Φ
d
S
2
=
d
s
2
=cos
2
φ
×
×
A
1
d
Λ
2
+
cos
2
Φ
d
Φ
2
.
r
2
d
λ
2
+
cos
2
φ
d
φ
2
.
E
2
)
2
(1
−E
2
sin
2
Φ
)
2
A
1
(1
−
r
2
1
×
×
Left differential form:
Right differential form:
d
P
:=
A
1
d
Λ,
d
p
:=
r
d
λ,
(1.214)
1
−E
2
1
−E
2
sin
2
Φ
1
1
d
Q
:=
A
1
cos
Φ
d
Φ.
d
q
:=
r
cos
φ
d
φ.
Left factor of conformality: Right factor of conformality:
λ
l
:=
1
−E
2
sin
2
Φ
,
d
S
2
cos
2
Φ
cos
2
φ
=:
λ
r
,λ
r
=
d
s
2
=
λ
l
,
,
d
s
l
d
s
r
d
s
l
=d
P
2
+d
Q
2
,
d
p
2
+d
q
2
=d
s
r
,
(1.215)
d
S
2
=
λ
l
(d
P
2
+d
Q
2
)
,
λ
r
(d
p
2
+d
q
2
)=d
s
2
,
cos
2
Φ
d
S
2
=
1
−E
2
sin
2
Φ
(d
P
2
+d
Q
2
)
.
cos
2
φ
(d
p
2
+d
q
2
)=d
s
2
.
Search WWH ::
Custom Search