3-1 Coordinates relating to manifolds 121
Example 3.5 (Circle S
1
r , minimal atlas: I = 2).
Earlier, we generated a local coordinate system of the “one-sphere” S
1
r by the orthogonal projection
p = π 1 (P ) of a point P of the “one-sphere” S
1
r onto the x axis. Alternatively, we project the point
P orthogonally by q = π 2 (P ) onto the 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 < 2π}.
1st chart, 1st parameter set:
2nd chart, 2nd parameter set:
Φ 1 (x) = λ(x) ,
Φ 2 (x) = α(x) ,
Φ
−1
1 (x) = r
cos λ
sin λ
∼
Φ
−1
2 (x) = r
cos α
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 .
(3.19)
I = 2: {Φ
−1
1 (U)} ∪ {Φ
−1
2 (U)} covers the “one-sphere” completely in the sense of a minimal atlas.
Formally, in Fig. 3.5 such a minimal atlas is illustrated.
End of Example.
Example 3.6 (Sphere S
2
r , minimal atlas: I = 2).
Beforehand, a first coordinate system of the “two-sphere” S
2
r had been introduced by an orthogonal
projection p = π 1 (P ) of a point P of the “two-sphere” S
2
r onto the equatorial plane. Alternatively, let
us make an orthogonal projection q = π 2 (P ) of a point P of the “two-sphere” S
2
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 <
2π}. 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 ∈ {β ∈ R − π/2 < β < +π/2}.
The orientation of the meta-equatorial plane is conventionally denoted as transverse. Here, we only
introduce the 1st and 2nd charts.
1st chart, 1st parameter set:
2nd chart, 2nd parameter set:
Φ 1 (x) =
λ(x)
φ(x)
,
Φ 2 (x) =
α(x)
β(x)
,
Φ
−1
1 (x) = r
⎡
⎣
cos λ cos φ
sin λ cos φ
sin φ
⎤
⎦ ∼
Φ
−1
2 (x) = r
⎡
⎣
cos α cos β
sin α cos β
sin β
⎤
⎦ ∼
∼ x 1 (λ, φ) =
∼ 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 .
(3.20)
I = 2: {Φ
−1
1 (U)} ∪ {Φ
−1
2 (U)} covers the “two-sphere” as a minimal atlas. Let us identify the sets of
exceptional points, both in the chart {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 exceptional points. In the
right chart {α, β} ∈ Φ 2 (x), the meta-North Pole (“West Pole”), the meta-South Pole (“East Pole”),
and the α = 0 meridian define the set of right exceptional points.
End of Example.
Example 3.5 (Circle S
1
r , minimal atlas: I = 2).
Earlier, we generated a local coordinate system of the “one-sphere” S
1
r by the orthogonal projection
p = π 1 (P ) of a point P of the “one-sphere” S
1
r onto the x axis. Alternatively, we project the point
P orthogonally by q = π 2 (P ) onto the 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 < 2π}.
1st chart, 1st parameter set:
2nd chart, 2nd parameter set:
Φ 1 (x) = λ(x) ,
Φ 2 (x) = α(x) ,
Φ
−1
1 (x) = r
cos λ
sin λ
∼
Φ
−1
2 (x) = r
cos α
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 .
(3.19)
I = 2: {Φ
−1
1 (U)} ∪ {Φ
−1
2 (U)} covers the “one-sphere” completely in the sense of a minimal atlas.
Formally, in Fig. 3.5 such a minimal atlas is illustrated.
End of Example.
Example 3.6 (Sphere S
2
r , minimal atlas: I = 2).
Beforehand, a first coordinate system of the “two-sphere” S
2
r had been introduced by an orthogonal
projection p = π 1 (P ) of a point P of the “two-sphere” S
2
r onto the equatorial plane. Alternatively, let
us make an orthogonal projection q = π 2 (P ) of a point P of the “two-sphere” S
2
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 <
2π}. 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 ∈ {β ∈ R − π/2 < β < +π/2}.
The orientation of the meta-equatorial plane is conventionally denoted as transverse. Here, we only
introduce the 1st and 2nd charts.
1st chart, 1st parameter set:
2nd chart, 2nd parameter set:
Φ 1 (x) =
λ(x)
φ(x)
,
Φ 2 (x) =
α(x)
β(x)
,
Φ
−1
1 (x) = r
⎡
⎣
cos λ cos φ
sin λ cos φ
sin φ
⎤
⎦ ∼
Φ
−1
2 (x) = r
⎡
⎣
cos α cos β
sin α cos β
sin β
⎤
⎦ ∼
∼ x 1 (λ, φ) =
∼ 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 .
(3.20)
I = 2: {Φ
−1
1 (U)} ∪ {Φ
−1
2 (U)} covers the “two-sphere” as a minimal atlas. Let us identify the sets of
exceptional points, both in the chart {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 exceptional points. In the
right chart {α, β} ∈ Φ 2 (x), the meta-North Pole (“West Pole”), the meta-South Pole (“East Pole”),
and the α = 0 meridian define the set of right exceptional points.
End of Example.