Geography Reference
In-Depth Information
Along the orthonormal base
{
e
1
,e
2
|O}
attached to the origin
O
, the center of the “one-sphere”,
2
=
x
2
+
y
2
=
r
2
>
0. A point
P
of the “one-sphere” is orthogonally projected on the
x
axis such that
p
=
π
(
P
). Refer to Fig.
3.1
for an illustration. The unit vector
e
x
:=
x
/
2
we define a
Cartesian coordinate system
{
x, y
}
such that
x
x
2
and the unit base vector
e
1
include the angle
λ
=
(
e
x
,
e
1
), an element of the open interval
λ
∈{
R
|
0
<λ <
2
π
}
.
Φ
−
1
(
x
):
x
(
λ
)=
e
1
r
cos
λ
+
e
2
r
sin
λ,
Φ
(
x
)=
λ
(
x
)
⇔
(3.3)
Φ
−
1
(
x
)=
r
cos
λ
,
(3.4)
sin
λ
d(
x
1
,
x
2
)=
r
(cos
λ
1
−
cos
λ
2
)
2
+(sin
λ
1
−
sin
λ
2
)
2
=
=
r
√
2
1
−
(cos
λ
1
cos
λ
2
+sin
λ
1
sin
λ
2
) =
(3.5)
=
r
√
2
1
−
cos(
λ
1
−
λ
2
)
.
We apologize for our sloppy notation
x
(
λ
) meaning
x
=
κ
(
λ
), but introduced for economical
reason: save extra symbols.
⎧
⎨
arctan(
y/x
)
for
x>
0
arctan(
y/x
)+
π
for
x<
0
λ
(
x
)=
.
(3.6)
(
π/
2) sgn
y
for
x
=0and
y
=0
⎩
undefined
for
x
=0and
y
=0
Note that
{λ
=0or2
π}
or, equivalently,
{x
=
r, y
=0
}
is the exceptional point which is not
curved by the angular parameter
λ
!
1
(ii) Topology on
Φ
(
S
r
)
.
The topology on
Φ
(
S
r
) is defined by the Euclidean metric, namely the distance function
d(
y
1
,
y
2
):=
y
1
−
y
2
2
=
|
λ
1
−
λ
2
|
.
(3.7)
End of Example.
r
, two-dimensional manifold, topology).
Example 3.2 (Sphere
S
2
2
First, we present the topology on
S
r
. Second, we present the topology on
φ
(
S
r
). The “two-sphere”
2
S
r
(sphere of radius
r
) is defined as the manifold
2
3
x
2
+
y
2
+
z
2
=
r
2
,r
+
,r>
0
S
r
:=
{
x
∈
R
|
∈
R
}
.
(3.8)
2
(i) Topology on
S
r
.
2
The topology on
S
r
is defined by the
Euclidean metric
, namely the distance function of the
3
, i.e. d(
x
1
,
x
2
):=
ambient space
R
x
1
−
x
2
2
. Along the orthonormal base
{
e
1
,
e
2
,
e
3
|O}
attached to the origin
O
, the center of the “two-sphere”, we define a
Cartesian coordinate system
2
=
x
2
+
y
2
+
z
2
=
r
2
>
0. A point
P
of the “two-sphere” is
orthogonally projected on the (
x, y
) plane, which is also called the
equatorial plane
, such that
p
=
π
(
P
). Refer to Fig.
3.2
for an illustration. The straight line
p
-
O
is oriented with respect to
the unit vector
e
1
or the
x
axis by the angular parameter “spherical longitude”
λ
,anelement
{
x, y, z
}
in such a way that
x
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