Geography Reference
In-Depth Information
Fig. 3.3.
“One-sphere”
S
r
, complete atlas built on four charts
U
4
:=
{
[
x, y
]
∈
S
r
|x<
0
},Φ
4
[
x, y
]:=
y
=
t
4
.
The sets
U
i
and their maps
Φ
(
U
i
)
∈
]
−
1
,
+1 [are open with respect to the chosen topology.
I
=4:
x
1
(
t
1
)=
e
1
t
1
+
e
2
r
2
Φ
−
1
(
t
1
)=
t
1
,
+
r
2
t
1
t
1
,
−
∼
−
x
2
(
t
2
)=+
e
1
r
2
Φ
−
2
(
t
2
)=
+
r
2
t
2
,t
2
−
∼
−
t
2
+
e
2
t
2
,
∼
x
3
(
t
3
)=
e
1
t
3
−
e
3
r
2
Φ
−
3
(
t
3
)=
t
3
,
t
3
−
r
2
−
−
t
3
,
(3.15)
e
1
r
2
Φ
−
4
(
t
4
)=
t
4
,t
4
r
2
t
4
+
e
4
t
4
.
−
−
∼
x
4
(
t
4
)=
−
−
1
Indeed, the union of the patches (“Umgebungsraume”)
U
∪
U
∪
U
∪
U
4
=
S
r
, which is the
1
2
3
r
is covered by the four charts
Φ
1
“one-sphere”
S
∈
V
1
,Φ
2
∈
V
2
,Φ
3
∈
V
3
,Φ
4
∈
V
4
,and
V
i
:=]
−
1
,
+1[
,
(
i
∈{
1
,
2
,
3
,
4
}
) completely. We have generated a complete atlas: consult Fig.
3.3
for animation.
End of Example.
r
, complete atlas:
I
=6).
Example 3.4 (Sphere
S
By means of an orthogonal projection
p
=
π
(
P
), we already introduced a first coordinate set of the
“two-sphere”
S
r
in terms of spherical longitude
λ
and spherical latitude
φ
. As local coordinates,
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