Geography Reference
In-Depth Information
is oriented with
respect to the equatorial plane (
x, y
) by the angular parameter “spherical latitude”
φ
,anelement
of the open interval
φ
of the open interval
λ
∈{
R
|
0
<λ<
2
π
}
. In contrast, the straight line
P
-
O
∈{
R
|−
π/
2
<φ <
+
π/
2
}
. Again, we emphasize the
open domain
2
(
λ, φ
)
∈{
R
|
0
<λ<
2
π,
−
π/
2
<φ <
+
π/
2
}
.
Φ
(
x
)=
λ
(
x)
Φ
−
1
(
x
):
x
(
λ, φ
)=
e
1
r
cos
φ
cos
λ
⇔
φ
(
x
)
+
e
2
r
cos
φ
sin
λ
+
e
3
r
sin
φ,
(3.9)
⎡
⎤
cos
φ
cos
λ
cos
φ
sin
λ
sin
φ
(
r
=
x
2
+
y
2
+
z
2
)
,
⎣
⎦
Φ
−
1
(
x
)=
r
(3.10)
d(
x
1
,
x
2
)=
=
r
(cos
φ
1
cos
λ
1
−
cos
φ
2
cos
λ
2
)
2
+(cos
φ
1
sin
λ
1
−
cos
φ
2
sin
λ
2
)
2
+(sin
φ
1
−
sin
φ
2
)
2
=
=
r
√
2
1
−
(cos
φ
1
cos
φ
2
cos
λ
1
cos
λ
2
+cos
φ
1
cos
φ
2
sin
λ
1
sin
λ
2
+sin
φ
1
sin
φ
2
) =
(3.11)
=
r
√
2
1
λ
2
)+sin
φ
1
sin
φ
2
=
r
√
2
√
1
−
cos
φ
1
cos
φ
2
cos(
λ
1
−
−
cos
Ψ.
We here again apologize for our sloppy notation
x
(
λ, φ
) meaning
x
=
κ
(
λ, φ
), but introduced
for shorthand writing.
⎧
⎨
arctan(
y/x
) for
x>
0
arctan (
y/x
)+
π
for
x<
0
(
π/
2) sgn
y
λ
(
x
)=
,
for
x
=0
,y
=0
⎩
for
x
=0
,y
=0
undefined
φ
(
x
)=
arctan
√
x
2
+
y
2
undefined
z
for
x
=
y
=
z
=0
.
(3.12)
Note that
is the exceptional point set, namely the
half
meridian
South-Pole-North-Pole passing the point
x
=
r, y
=0
,z
= 0, sometimes called
Greenwich Meridian
. Such a half meridian is not curved by the angular parameter set
{
λ
=0or2
π, φ
=
π/
2or
−
π/
2
}
{
λ, φ
}
!
2
(ii) Topology on
Φ
(
S
r
)
.
2
The topology on
Φ
(
S
r
) is defined by the Euclidean
metric
, namely the distance function
λ
2
)
2
+(
φ
1
φ
2
)
2
d(
y
1
,
y
2
):=
y
1
−
y
2
2
=
|
(
λ
1
−
−
|
.
(3.13)
End of Example.
You may have wondered why did we introduce
open sets
,an
open domain of parameters
to
coordinate a surface or a Riemann manifold of type
1
2
r
, respectively. Actually, we have
postulated that
Φ
(
x
) should be “one-to-one”. This is not guaranteed for the spherical
South Pole
or
North Pole
since
λ
(
x, y, z
)for
x
=0
,y
=0
,z
=
S
r
or
S
r
as a mapping is “one-to-infinity”,
for instance. Further arguments are given as soon as we equip the manifold with a differential
structure (differential topology). Indeed, you may have realized that in terms of open sets or
an open domain of parameters,
±
r
is
not completely
covered. We are therefore forced to
introduce more than one set of parameters, hoping that their union
∪
U
i
,i∈{
1
,...,I}
covers
totally
S
r
or
S
S
r
and
S
r
,and
M
2
, in general.
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