Diffgeo: Beispiele: Rotationsellipsoid

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Die Fläche

Geodätische Koordinaten auf dem Rotationsellipsoid

Das Rotationsellipsoid mit Kreisform der Breitenkreise (in der Äquatorebene Radius der großen Halbachse A) und Ellipsenform bezüglich der Längenkreise. Kleine Halbachse an den Polen ist B.

Die w:Exzentrizität (Mathematik) gibt die Abplattung aufgrund der unterschiedlichen Länge von A und B an.

Parametrisierung

${\begin{pmatrix}X\\Y\\Z\end{pmatrix}}={\begin{pmatrix}N\cdot \cos {U}\cdot \cos {V}\\N\cdot \sin {U}\cdot \cos {V}\\N\cdot \sin {V}(1-E^{2})\end{pmatrix}}$

$V\in (-{\frac {\pi }{2}};{\frac {\pi }{2}})\,U\in [-\pi ;\pi )$

$N={\frac {A}{\sqrt {1-E^{2}\sin ^{2}V}}}$

A und B bzw. E je nach Ellipsoid.

(Gaußsche) Tangentenvektoren

Siehe Gaußsches Dreibein

${\vec {g}}_{1}={\vec {x}}_{U}=$

${\vec {g}}_{2}={\vec {x}}_{V}=$

${\vec {g}}_{3}={\frac {{\vec {x}}_{U}(u)\times {\vec {x}}_{V}(v)}{||{\vec {x}}_{U}(u)\times {\vec {x}}_{V}(v)||}}=$

erste Fundamentalform

erste Fundamentalgrößen

Siehe hier:

$g_{11}={\vec {x}}_{U}\cdot {\vec {x}}_{U}=N^{2}\cos ^{2}V$

$g_{12}=G_{21}={\vec {x}}_{U}\cdot {\vec {x}}_{V}=0$

$g_{22}={\vec {x}}_{V}\cdot {\vec {x}}_{V}=M^{2}$

erster Fundamentaltensor

$\mathbf {G} ={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix}}={\begin{pmatrix}N^{2}\cos ^{2}V&0\\0&M^{2}\end{pmatrix}}={\begin{pmatrix}N^{2}\cos ^{2}V&0\\0&N^{2}{\frac {(1-E^{2})^{2}}{(1-E^{2}\sin ^{2}V)^{2}}}\end{pmatrix}}$

Inverser erster Fundamentaltensor

$\mathbf {G} ^{-1}={\begin{pmatrix}g^{11}&g^{12}\\g^{21}&g^{22}\end{pmatrix}}={\frac {1}{1}}{\begin{pmatrix}0&0\\0&0\end{pmatrix}}$

zweite Fundamentalform

zweifache Ableitungen

${\vec {x}}_{uu}=$

${\vec {x}}_{uv}=$

${\vec {x}}_{vv}=$

zweite Fundamentalgrößen

Hier nachschauen!

$b_{11}={\vec {x}}_{uu}\cdot {\vec {n}}=$

$b_{12}=b_{21}={\vec {x}}_{uv}\cdot {\vec {n}}$

$b_{21}={\vec {x}}_{vv}\cdot {\vec {n}}=$

zweiter Fundamentaltensor

$\mathbf {B} ={\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}={\begin{pmatrix}L&M\\M&N\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}$

Christoffelsymbole

Siehe hier. Mit u¹ = u, u² = v. $\alpha =1$, $\beta =1$, $\gamma =1$

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

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

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

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

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

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

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

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

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

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

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

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

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

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