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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.

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

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

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

A und B bzw. E je nach Ellipsoid.

Siehe Gaußsches Dreibein

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

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

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

Siehe hier:

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

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

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

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

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

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

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

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

Hier nachschauen!

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

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

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

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

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

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

${\displaystyle \alpha =2}$, ${\displaystyle \beta =1}$, ${\displaystyle \gamma =1}$

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

${\displaystyle \alpha =1}$, ${\displaystyle \beta =2}$, ${\displaystyle \gamma =1}$

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

${\displaystyle \alpha =1}$, ${\displaystyle \beta =1}$, ${\displaystyle \gamma =2}$

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

${\displaystyle \alpha =2}$, ${\displaystyle \beta =1}$, ${\displaystyle \gamma =2}$,

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

${\displaystyle \alpha =1}$, ${\displaystyle \beta =2}$, ${\displaystyle \gamma =2}$

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

${\displaystyle \alpha =2}$, ${\displaystyle \beta =2}$, ${\displaystyle \gamma =2}$

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

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