Geographische Koordinaten auf der Kugel
Parametrisiert wie in Geodätische Koordinatensysteme Geographische Koordinaten geschildert.
(
x
y
z
)
=
(
R
⋅
c
o
s
(
λ
)
⋅
s
i
n
(
ϕ
)
R
⋅
s
i
n
(
λ
)
⋅
s
i
n
(
ϕ
)
R
⋅
c
o
s
(
ϕ
)
)
{\displaystyle {\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}R\cdot cos{(\lambda )}\cdot sin{(\phi )}\\R\cdot sin{(\lambda )}\cdot sin{(\phi )}\\R\cdot cos{(\phi )}\end{pmatrix}}}
ϕ
∈
(
0
;
π
)
λ
∈
[
−
π
;
π
)
{\displaystyle \phi \in (0;\pi )\,\lambda \in [-\pi ;\pi )}
Siehe Gaußsches Dreibein
Tangentialraum :
T
S
2
(
λ
,
ϕ
)
=
{
x
→
λ
,
x
→
ϕ
}
=
{
(
−
R
⋅
sin
(
λ
)
⋅
sin
(
ϕ
)
R
⋅
cos
(
λ
)
⋅
sin
(
ϕ
)
0
)
,
(
R
⋅
cos
(
λ
)
⋅
cos
(
ϕ
)
R
⋅
sin
(
λ
)
⋅
cos
(
ϕ
)
−
R
⋅
sin
(
ϕ
)
)
}
{\displaystyle T_{S^{2}}(\lambda ,\phi )=\lbrace {\vec {x}}_{\lambda },{\vec {x}}_{\phi }\rbrace ={\bigg \lbrace }{\begin{pmatrix}-R\cdot \sin {(\lambda )}\cdot \sin {(\phi )}\\R\cdot \cos {(\lambda )}\cdot \sin {(\phi )}\\0\end{pmatrix}},{\begin{pmatrix}R\cdot \cos {(\lambda )}\cdot \cos {(\phi )}\\R\cdot \sin {(\lambda )}\cdot \cos {(\phi )}\\-R\cdot \sin {(\phi )}\end{pmatrix}}{\bigg \rbrace }}
Flächennormale
N
S
2
(
λ
,
ϕ
)
=
(
x
→
λ
∧
x
→
ϕ
)
|
x
→
λ
∧
x
→
ϕ
|
=
(
−
c
o
s
(
λ
)
⋅
sin
(
ϕ
)
−
s
i
n
(
λ
)
⋅
sin
(
ϕ
)
−
c
o
s
(
ϕ
)
)
{\displaystyle N_{S^{2}}(\lambda ,\phi )={\frac {({\vec {x}}_{\lambda }\wedge {\vec {x}}_{\phi })}{\vert {\vec {x}}_{\lambda }\wedge {\vec {x}}_{\phi }\vert }}={\begin{pmatrix}-cos{(\lambda )}\cdot \sin {(\phi )}\\-sin{(\lambda )}\cdot \sin {(\phi )}\\-cos{(\phi )}\end{pmatrix}}}
x
→
λ
λ
=
(
−
R
⋅
cos
(
λ
)
⋅
sin
(
ϕ
)
−
R
⋅
sin
(
λ
)
⋅
sin
(
ϕ
)
0
)
{\displaystyle {\vec {x}}_{\lambda \lambda }={\begin{pmatrix}-R\cdot \cos {(\lambda )}\cdot \sin {(\phi )}\\-R\cdot \sin {(\lambda )}\cdot \sin {(\phi )}\\0\end{pmatrix}}}
x
→
λ
ϕ
=
(
−
R
⋅
sin
(
λ
)
⋅
cos
(
ϕ
)
R
⋅
cos
(
λ
)
⋅
cos
(
ϕ
)
0
)
=
x
→
ϕ
λ
{\displaystyle {\vec {x}}_{\lambda \phi }={\begin{pmatrix}-R\cdot \sin {(\lambda )}\cdot \cos {(\phi )}\\R\cdot \cos {(\lambda )}\cdot \cos {(\phi )}\\0\end{pmatrix}}={\vec {x}}_{\phi \lambda }}
x
→
ϕ
ϕ
=
(
−
R
⋅
cos
(
λ
)
⋅
sin
(
ϕ
)
−
R
⋅
sin
(
λ
)
⋅
sin
(
ϕ
)
−
R
⋅
cos
(
ϕ
)
)
{\displaystyle {\vec {x}}_{\phi \phi }={\begin{pmatrix}-R\cdot \cos {(\lambda )}\cdot \sin {(\phi )}\\-R\cdot \sin {(\lambda )}\cdot \sin {(\phi )}\\-R\cdot \cos {(\phi )}\end{pmatrix}}}
Hier nachschauen!
b
11
=
⟨
x
→
ϕ
ϕ
,
N
S
2
(
λ
,
ϕ
)
⟩
=
−
R
⋅
sin
(
ϕ
)
2
{\displaystyle b_{11}=\langle {\vec {x}}_{\phi \phi },N_{S^{2}}(\lambda ,\phi )\rangle =-R\cdot \sin {(\phi )}^{2}}
b
12
=
b
21
=
⟨
x
→
ϕ
λ
,
N
S
2
(
λ
,
ϕ
)
⟩
=
0
{\displaystyle b_{12}=b_{21}=\langle {\vec {x}}_{\phi \lambda },N_{S^{2}}(\lambda ,\phi )\rangle =0}
b
22
=
⟨
x
→
λ
λ
,
N
S
2
(
λ
,
ϕ
)
⟩
=
−
R
{\displaystyle b_{22}=\langle {\vec {x}}_{\lambda \lambda },N_{S^{2}}(\lambda ,\phi )\rangle =-R}
B
=
(
b
11
b
12
b
21
b
22
)
=
(
−
R
⋅
sin
(
ϕ
)
2
0
0
−
R
)
=
−
1
R
⋅
G
{\displaystyle \mathbf {B} ={\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}={\begin{pmatrix}-R\cdot \sin {(\phi )}^{2}&0\\0&-R\end{pmatrix}}={\frac {-1}{R}}\cdot \mathbf {G} }
Definition
Bemerkung: Dies ist eine Variante, die jeweiligen Hauptkrümmungen mittels erster und zweiter Fundamentalform zu berechnen. Es führen aber viele Wege nach Rom.
A
:=
(
a
11
a
12
a
21
a
22
)
{\displaystyle \mathbf {A} :={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}}
so dass:
−
G
⋅
A
=
B
⇔
A
=
−
G
−
1
⋅
B
{\displaystyle -\mathbf {G} \cdot \mathbf {A} =\mathbf {B} \Leftrightarrow \mathbf {A} =-\mathbf {G^{-1}} \cdot \mathbf {B} }
⇒
A
=
−
(
1
R
2
⋅
sin
2
(
ϕ
)
0
0
1
R
2
)
⋅
(
−
R
⋅
sin
(
ϕ
)
2
0
0
−
R
)
=
(
1
R
0
0
1
R
)
{\displaystyle \Rightarrow \mathbf {A} =-{\begin{pmatrix}{\frac {1}{R^{2}\cdot \sin ^{2}{(\phi )}}}&0\\0&{\frac {1}{R^{2}}}\end{pmatrix}}\cdot {\begin{pmatrix}-R\cdot \sin {(\phi )}^{2}&0\\0&-R\end{pmatrix}}={\begin{pmatrix}{\frac {1}{R}}&0\\0&{\frac {1}{R}}\end{pmatrix}}}
Daraus lässt sich die Gaußsche und Mittlere Krümmung berechnen:
K
=
a
11
⋅
a
22
=
1
R
2
{\displaystyle K=a_{11}\cdot a_{22}={\frac {1}{R^{2}}}}
H
=
a
11
+
a
22
2
=
1
R
{\displaystyle H={\frac {a_{11}+a_{22}}{2}}={\frac {1}{R}}}
Siehe hier . Mit u1 =
λ
.
{\displaystyle \lambda .}
, u2 =
ϕ
{\displaystyle \phi }
α
=
1
{\displaystyle \alpha =1}
,
β
=
1
{\displaystyle \beta =1}
,
γ
=
1
{\displaystyle \gamma =1}
Γ
11
1
:=
1
2
g
11
(
∂
g
11
∂
u
1
+
∂
g
11
∂
u
1
−
∂
g
11
∂
u
1
)
+
1
2
g
12
(
∂
g
12
∂
u
1
+
∂
g
21
∂
u
1
−
∂
g
11
∂
u
2
)
=
0
{\displaystyle \Gamma _{11}^{1}:={\frac {1}{2}}g^{11}({\frac {\partial g_{11}}{\partial u^{1}}}+{\frac {\partial g_{11}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{1}}})+{\frac {1}{2}}g^{12}({\frac {\partial g_{12}}{\partial u^{1}}}+{\frac {\partial g_{21}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{2}}})=0}
α
=
2
{\displaystyle \alpha =2}
,
β
=
1
{\displaystyle \beta =1}
,
γ
=
1
{\displaystyle \gamma =1}
Γ
12
1
:=
1
2
g
11
(
∂
g
11
∂
u
2
+
∂
g
12
∂
u
1
−
∂
g
21
∂
u
1
)
+
1
2
g
12
(
∂
g
12
∂
u
2
+
∂
g
22
∂
u
1
−
∂
g
21
∂
u
2
)
=
cos
(
ϕ
)
sin
(
ϕ
)
{\displaystyle \Gamma _{12}^{1}:={\frac {1}{2}}g^{11}({\frac {\partial g_{11}}{\partial u^{2}}}+{\frac {\partial g_{12}}{\partial u^{1}}}-{\frac {\partial g_{21}}{\partial u^{1}}})+{\frac {1}{2}}g^{12}({\frac {\partial g_{12}}{\partial u^{2}}}+{\frac {\partial g_{22}}{\partial u^{1}}}-{\frac {\partial g_{21}}{\partial u^{2}}})={\frac {\cos {(\phi )}}{\sin {(\phi )}}}}
α
=
1
{\displaystyle \alpha =1}
,
β
=
2
{\displaystyle \beta =2}
,
γ
=
1
{\displaystyle \gamma =1}
Γ
11
2
:=
1
2
g
21
(
∂
g
11
∂
u
1
+
∂
g
11
∂
u
1
−
∂
g
11
∂
u
1
)
+
1
2
g
22
(
∂
g
12
∂
u
1
+
∂
g
21
∂
u
1
−
∂
g
11
∂
u
2
)
=
−
sin
(
ϕ
)
⋅
cos
(
ϕ
)
{\displaystyle \Gamma _{11}^{2}:={\frac {1}{2}}g^{21}({\frac {\partial g_{11}}{\partial u^{1}}}+{\frac {\partial g_{11}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{1}}})+{\frac {1}{2}}g^{22}({\frac {\partial g_{12}}{\partial u^{1}}}+{\frac {\partial g_{21}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{2}}})=-\sin {(\phi )}\cdot \cos {(\phi )}}
α
=
1
{\displaystyle \alpha =1}
,
β
=
1
{\displaystyle \beta =1}
,
γ
=
2
{\displaystyle \gamma =2}
Γ
21
1
=
Γ
12
1
{\displaystyle \Gamma _{21}^{1}=\Gamma _{12}^{1}}
α
=
2
{\displaystyle \alpha =2}
,
β
=
1
{\displaystyle \beta =1}
,
γ
=
2
{\displaystyle \gamma =2}
,
Γ
12
2
:=
1
2
g
11
(
∂
g
21
∂
u
2
+
∂
g
12
∂
u
2
−
∂
g
22
∂
u
1
)
+
1
2
g
12
(
∂
g
22
∂
u
2
+
∂
g
22
∂
u
2
−
∂
g
22
∂
u
2
)
=
0
{\displaystyle \Gamma _{12}^{2}:={\frac {1}{2}}g^{11}({\frac {\partial g_{21}}{\partial u^{2}}}+{\frac {\partial g_{12}}{\partial u^{2}}}-{\frac {\partial g_{22}}{\partial u^{1}}})+{\frac {1}{2}}g^{12}({\frac {\partial g_{22}}{\partial u^{2}}}+{\frac {\partial g_{22}}{\partial u^{2}}}-{\frac {\partial g_{22}}{\partial u^{2}}})=0}
α
=
1
{\displaystyle \alpha =1}
,
β
=
2
{\displaystyle \beta =2}
,
γ
=
2
{\displaystyle \gamma =2}
Γ
21
2
=
Γ
12
2
{\displaystyle \Gamma _{21}^{2}=\Gamma _{12}^{2}}
α
=
2
{\displaystyle \alpha =2}
,
β
=
2
{\displaystyle \beta =2}
,
γ
=
2
{\displaystyle \gamma =2}
Γ
22
2
:=
1
2
g
21
(
∂
g
21
∂
u
2
+
∂
g
12
∂
u
2
−
∂
g
22
∂
u
1
)
+
1
2
g
22
(
∂
g
22
∂
u
2
+
∂
g
22
∂
u
2
−
∂
g
22
∂
u
2
)
=
0
{\displaystyle \Gamma _{22}^{2}:={\frac {1}{2}}g^{21}({\frac {\partial g_{21}}{\partial u^{2}}}+{\frac {\partial g_{12}}{\partial u^{2}}}-{\frac {\partial g_{22}}{\partial u^{1}}})+{\frac {1}{2}}g^{22}({\frac {\partial g_{22}}{\partial u^{2}}}+{\frac {\partial g_{22}}{\partial u^{2}}}-{\frac {\partial g_{22}}{\partial u^{2}}})=0}
Zurück zum Inhaltsverzeichnis