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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