PlanetPhysics/Euler 313 Sequence

For more info on Euler Sequences, notation and convention see the generic entry on [[../EulerAngleSequence/|Euler angle sequences]]. \\

$R_{313}(\varphi ,\theta ,\psi )=R_3(\psi )R_1(\theta )R_3(\varphi )$ \\

The rotation [[../Matrix/|matrices]] are

$R_3(\psi )=\left[\begin{array}{ccc}{c}_{\psi }& s_{\psi }& 0\\ -s_{\psi }& c_{\psi }& 0\\ 0& 0& 1\end{array}\right]$

$R_1(\theta )=\left[\begin{array}{ccc}1& 0& 0\\ 0& c_{\theta }& s_{\theta }\\ 0& -s_{\theta }& c_{\theta }\end{array}\right]$

$R_3(\varphi )=\left[\begin{array}{ccc}{c}_{\varphi }& s_{\varphi }& 0\\ -s_{\varphi }& c_{\varphi }& 0\\ 0& 0& 1\end{array}\right]$

Carrying out the multiplication from right to left \\

$R_1(\theta )R_3(\varphi )=\left[\begin{array}{ccc}1& 0& 0\\ 0& c_{\theta }& s_{\theta }\\ 0& -s_{\theta }& c_{\theta }\end{array}\right]\left[\begin{array}{ccc}{c}_{\varphi }& s_{\varphi }& 0\\ -s_{\varphi }& c_{\varphi }& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}{c}_{\varphi }& s_{\varphi }& 0\\ -s_{\varphi }{c}_{\theta }& c_{\theta }{c}_{\varphi }& s_{\theta }\\ s_{\theta }{s}_{\varphi }& -s_{\theta }{c}_{\varphi }& c_{\theta }\end{array}\right]$ \\

Finaly leaving us with the Euler 313 sequence \\

$R_3(\psi )R_1(\theta )R_3(\varphi )=\left[\begin{array}{ccc}{c}_{\psi }{c}_{\varphi }-s_{\psi }{s}_{\varphi }{c}_{\theta }& c_{\psi }{s}_{\varphi }+s_{\psi }{c}_{\theta }{c}_{\varphi }& s_{\psi }{s}_{\theta }\\ -s_{\psi }{c}_{\varphi }-c_{\psi }{s}_{\varphi }{c}_{\theta }& -s_{\psi }{s}_{\varphi }+c_{\psi }{c}_{\theta }{c}_{\varphi }& c_{\psi }{s}_{\theta }\\ s_{\theta }{s}_{\varphi }& -s_{\theta }{c}_{\varphi }& c_{\theta }\end{array}\right]$