Geography Reference
In-Depth Information
Example 3.5 (Circle
S
r
, minimal atlas:
I
=2).
Earlier, we generated a local coordinate system of the “one-sphere”
S
r
by the
orthogonal projection
p
=
π
1
(
P
) of a point
P
of the “one-sphere”
S
r
onto the
x
axis. Alternatively, we project the point
P
orthogonally by
q
=
π
2
(
P
)ontothe
x
axis, chosen as the
y
axis. Again, we introduce an
angular parameter by
α
=
(
e
x
,
e
2
), an element of the
open interval α
∈{
α
∈
R
|
0
<α<
2
π
}
.
1st chart, 1st parameter set:
2nd chart, 2nd parameter set:
Φ
1
(
x
)=
λ
(
x
)
,
Φ
2
(
x
)=
α
(
x
)
,
Φ
−
1
(
x
)=
r
cos
λ
Φ
−
2
(
x
)=
r
cos
α
∼
∼
(3.19)
sin
λ
sin
α
∼
x
1
(
λ
)=
e
1
r
cos
λ
+
e
2
r
sin
λ.
∼
x
2
(
α
)=
e
1
r
cos
α
+
e
2
r
sin
α.
Φ
−
1
(
Φ
−
2
(
I
=2:
covers the “one-sphere” completely in the sense of a
minimal atlas
.
Formally, in Fig.
3.5
such a minimal atlas is illustrated.
{
U
)
}∪{
U
)
}
End of Example.
2
Example 3.6 (Sphere
S
r
, minimal atlas:
I
=2).
r
had been introduced by an orthogonal
Beforehand, a first coordinate system of the “two-sphere”
S
projection
p
=
π
1
(
P
) of a point
P
of the “two-sphere”
S
r
onto the equatorial plane. Alternatively,
let us make an orthogonal projection
q
=
π
2
(
P
) of a point
P
of the “two-sphere”
r
onto the
(
x
,y
) plane, which coincides with the
Greenwich Meridian Plane
spanned by
{
e
2
,
e
3
|O}
.(The
name
meridian
is derived from the word
noon
. Here, it coincides with the coordinate plane
λ
=0.)
Within the (
x
,y
) plane spanned by
{e
1
, e
2
|O}
=
{
e
2
,
e
3
|O}
, the point
q
is coordinated by the
angular parameter
α
,namely
α
=
(
e
x
, e
1
), also called
meta-longitude
, an element of the open
interval
α ∈{α ∈
R
|
0
<α<
2
π}
. The elevation angle of the vector
O
-
P
with respect to the
(
x
,y
) plane is the angular parameter
β
, also called
meta-latitude
, an element of the open interval
β
S
. The orientation of the
meta-equatorial plane
is conventionally
denoted as
transverse
. Here, we only introduce the 1st and 2nd charts.
∈{
β
∈
R
|−
π/
2
<β<
+
π/
2
}
1st chart, 1st parameter set:
2nd chart, 2nd parameter set:
Φ
1
(
x
)=
λ
(
x
)
,
Φ
2
(
x
)=
α
(
x
)
,
φ
(
x
)
β
(
x
)
⎡
⎤
⎡
⎤
cos
λ
cos
φ
sin
λ
cos
φ
sin
φ
cos
α
cos
β
sin
α
cos
β
sin
β
⎣
⎦
∼
⎣
⎦
∼
Φ
−
1
(
x
)=
r
Φ
−
2
(
x
)=
r
(3.20)
x
2
(
α,β
)=
=
e
1
r
cos
λ
cos
φ
+
e
2
r
sin
λ
sin
φ
+
e
3
r
sin
φ.
=
e
1
r
cos
α
cos
β
+
e
2
r
sin
α
cos
β
+
e
3
r
sin
β.
∼
x
1
(
λ, φ
)=
∼
Φ
−
1
(
Φ
−
2
(
I
=2:
covers the “two-sphere” as a
minimal atlas
. Let us identify the
sets of exceptional points, both in the chart
{
U
)
}∪{
U
)
}
{
λ, φ
}
and in the chart
{
α,β
}
. In the left chart
{
λ, φ
}∈
Φ
1
(
x
), the North Pole, the South Pole, and the
λ
= 0 meridian define the set of
left
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