Diffgeo: Beispiele: Kugel

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Die Fläche

Geographische Koordinaten auf der Kugel

Parametrisiert wie in Geodätische Koordinatensysteme Geographische Koordinaten geschildert.

Parametrisierung

${\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}R\cdot cos{(\lambda )}\cdot sin{(\phi )}\\R\cdot sin{(\lambda )}\cdot sin{(\phi )}\\R\cdot cos{(\phi )}\end{pmatrix}}$

$\phi \in (0;\pi )\,\lambda \in [-\pi ;\pi )$

(Gaußsche) Tangentenvektoren

Siehe Gaußsches Dreibein

Tangentialraum :

$T_{S^{2}}(\lambda ,\phi )=\lbrace {\vec {x}}_{\lambda },{\vec {x}}_{\phi }\rbrace ={\bigg \lbrace }{\begin{pmatrix}-R\cdot \sin {(\lambda )}\cdot \sin {(\phi )}\\R\cdot \cos {(\lambda )}\cdot \sin {(\phi )}\\0\end{pmatrix}},{\begin{pmatrix}R\cdot \cos {(\lambda )}\cdot \cos {(\phi )}\\R\cdot \sin {(\lambda )}\cdot \cos {(\phi )}\\-R\cdot \sin {(\phi )}\end{pmatrix}}{\bigg \rbrace }$

Flächennormale

$N_{S^{2}}(\lambda ,\phi )={\frac {({\vec {x}}_{\lambda }\wedge {\vec {x}}_{\phi })}{\vert {\vec {x}}_{\lambda }\wedge {\vec {x}}_{\phi }\vert }}={\begin{pmatrix}-cos{(\lambda )}\cdot \sin {(\phi )}\\-sin{(\lambda )}\cdot \sin {(\phi )}\\-cos{(\phi )}\end{pmatrix}}$

erste Fundamentalform

erste Fundamentalgrößen

Siehe hier:

$g_{11}=\langle {\vec {x}}_{\phi },{\vec {x}}_{\phi }\rangle =R^{2}\sin {(\phi )}^{2}$

$g_{12}=g_{21}=\langle {\vec {x}}_{\phi },{\vec {x}}_{\lambda }\rangle =0$

$g_{22}=\langle {\vec {x}}_{\lambda },{\vec {x}}_{\lambda }\rangle =R^{2}$

erster Fundamentaltensor

$\mathbf {G} ={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix}}={\begin{pmatrix}R^{2}\sin {(\phi )}^{2}&0\\0&R^{2}\end{pmatrix}}$

Inverser erster Fundamentaltensor

$\mathbf {G} ^{-1}={\frac {1}{g_{11}g_{22}-(g_{12})^{2}}}{\begin{pmatrix}g_{22}&-g_{21}\\-g_{12}&g_{11}\end{pmatrix}}={\begin{pmatrix}g^{11}&g^{12}\\g^{21}&g^{22}\end{pmatrix}}={\begin{pmatrix}{\frac {1}{R^{2}\cdot \sin ^{2}{(\phi )}}}&0\\0&{\frac {1}{R^{2}}}\end{pmatrix}}$

zweite Fundamentalform

zweifache Ableitungen

${\vec {x}}_{\lambda \lambda }={\begin{pmatrix}-R\cdot \cos {(\lambda )}\cdot \sin {(\phi )}\\-R\cdot \sin {(\lambda )}\cdot \sin {(\phi )}\\0\end{pmatrix}}$

${\vec {x}}_{\lambda \phi }={\begin{pmatrix}-R\cdot \sin {(\lambda )}\cdot \cos {(\phi )}\\R\cdot \cos {(\lambda )}\cdot \cos {(\phi )}\\0\end{pmatrix}}={\vec {x}}_{\phi \lambda }$

${\vec {x}}_{\phi \phi }={\begin{pmatrix}-R\cdot \cos {(\lambda )}\cdot \sin {(\phi )}\\-R\cdot \sin {(\lambda )}\cdot \sin {(\phi )}\\-R\cdot \cos {(\phi )}\end{pmatrix}}$

zweite Fundamentalgrößen

Hier nachschauen!

$b_{11}=\langle {\vec {x}}_{\phi \phi },N_{S^{2}}(\lambda ,\phi )\rangle =-R\cdot \sin {(\phi )}^{2}$

$b_{12}=b_{21}=\langle {\vec {x}}_{\phi \lambda },N_{S^{2}}(\lambda ,\phi )\rangle =0$

$b_{22}=\langle {\vec {x}}_{\lambda \lambda },N_{S^{2}}(\lambda ,\phi )\rangle =-R$

zweiter Fundamentaltensor

$\mathbf {B} ={\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}={\begin{pmatrix}-R\cdot \sin {(\phi )}^{2}&0\\0&-R\end{pmatrix}}={\frac {-1}{R}}\cdot \mathbf {G}$

Krümmung

Definition

Hauptkrümungen

Bemerkung: Dies ist eine Variante, die jeweiligen Hauptkrümmungen mittels erster und zweiter Fundamentalform zu berechnen. Es führen aber viele Wege nach Rom.

$\mathbf {A} :={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}$

so dass:

$-\mathbf {G} \cdot \mathbf {A} =\mathbf {B} \Leftrightarrow \mathbf {A} =-\mathbf {G^{-1}} \cdot \mathbf {B}$

$\Rightarrow \mathbf {A} =-{\begin{pmatrix}{\frac {1}{R^{2}\cdot \sin ^{2}{(\phi )}}}&0\\0&{\frac {1}{R^{2}}}\end{pmatrix}}\cdot {\begin{pmatrix}-R\cdot \sin {(\phi )}^{2}&0\\0&-R\end{pmatrix}}={\begin{pmatrix}{\frac {1}{R}}&0\\0&{\frac {1}{R}}\end{pmatrix}}$

Daraus lässt sich die Gaußsche und Mittlere Krümmung berechnen:

$K=a_{11}\cdot a_{22}={\frac {1}{R^{2}}}$

$H={\frac {a_{11}+a_{22}}{2}}={\frac {1}{R}}$

Christoffelsymbole

Siehe hier. Mit u¹ =$\lambda .$, u² = $\phi$ $\alpha =1$, $\beta =1$, $\gamma =1$

$\Gamma _{11}^{1}:={\frac {1}{2}}g^{11}({\frac {\partial g_{11}}{\partial u^{1}}}+{\frac {\partial g_{11}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{1}}})+{\frac {1}{2}}g^{12}({\frac {\partial g_{12}}{\partial u^{1}}}+{\frac {\partial g_{21}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{2}}})=0$

$\alpha =2$, $\beta =1$, $\gamma =1$

$\Gamma _{12}^{1}:={\frac {1}{2}}g^{11}({\frac {\partial g_{11}}{\partial u^{2}}}+{\frac {\partial g_{12}}{\partial u^{1}}}-{\frac {\partial g_{21}}{\partial u^{1}}})+{\frac {1}{2}}g^{12}({\frac {\partial g_{12}}{\partial u^{2}}}+{\frac {\partial g_{22}}{\partial u^{1}}}-{\frac {\partial g_{21}}{\partial u^{2}}})={\frac {\cos {(\phi )}}{\sin {(\phi )}}}$

$\alpha =1$, $\beta =2$, $\gamma =1$

$\Gamma _{11}^{2}:={\frac {1}{2}}g^{21}({\frac {\partial g_{11}}{\partial u^{1}}}+{\frac {\partial g_{11}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{1}}})+{\frac {1}{2}}g^{22}({\frac {\partial g_{12}}{\partial u^{1}}}+{\frac {\partial g_{21}}{\partial u^{1}}}-{\frac {\partial g_{11}}{\partial u^{2}}})=-\sin {(\phi )}\cdot \cos {(\phi )}$

$\alpha =1$, $\beta =1$, $\gamma =2$

$\Gamma _{21}^{1}=\Gamma _{12}^{1}$

$\alpha =2$, $\beta =1$, $\gamma =2$,

$\Gamma _{12}^{2}:={\frac {1}{2}}g^{11}({\frac {\partial g_{21}}{\partial u^{2}}}+{\frac {\partial g_{12}}{\partial u^{2}}}-{\frac {\partial g_{22}}{\partial u^{1}}})+{\frac {1}{2}}g^{12}({\frac {\partial g_{22}}{\partial u^{2}}}+{\frac {\partial g_{22}}{\partial u^{2}}}-{\frac {\partial g_{22}}{\partial u^{2}}})=0$

$\alpha =1$, $\beta =2$, $\gamma =2$

$\Gamma _{21}^{2}=\Gamma _{12}^{2}$

$\alpha =2$, $\beta =2$, $\gamma =2$

$\Gamma _{22}^{2}:={\frac {1}{2}}g^{21}({\frac {\partial g_{21}}{\partial u^{2}}}+{\frac {\partial g_{12}}{\partial u^{2}}}-{\frac {\partial g_{22}}{\partial u^{1}}})+{\frac {1}{2}}g^{22}({\frac {\partial g_{22}}{\partial u^{2}}}+{\frac {\partial g_{22}}{\partial u^{2}}}-{\frac {\partial g_{22}}{\partial u^{2}}})=0$

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