PlanetPhysics/Euler 212 Sequence

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

$R_{212}(\varphi ,\theta ,\psi )=R_2(\psi )R_1(\theta )R_2(\varphi )$ \\

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

$R_2(\psi )=\left[\begin{array}{ccc}{c}_{\psi }& 0& -s_{\psi }\\ 0& 1& 0\\ s_{\psi }& 0& c_{\psi }\end{array}\right]$

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

$R_2(\varphi )=\left[\begin{array}{ccc}{c}_{\varphi }& 0& -s_{\varphi }\\ 0& 1& 0\\ s_{\varphi }& 0& c_{\varphi }\end{array}\right]$

Carrying out the [[../Matrix/|matrix multiplication]] from right to left \\

$R_1(\theta )R_2(\varphi )=\left[\begin{array}{ccc}1& 0& 0\\ 0& c_{\theta }& s_{\theta }\\ 0& -s_{\theta }& c_{\theta }\end{array}\right]\left[\begin{array}{ccc}{c}_{\varphi }& 0& -s_{\varphi }\\ 0& 1& 0\\ s_{\varphi }& 0& c_{\varphi }\end{array}\right]=\left[\begin{array}{ccc}{c}_{\varphi }& 0& -s_{\varphi }\\ s_{\theta }{s}_{\varphi }& c_{\theta }& s_{\theta }{c}_{\varphi }\\ c_{\theta }{s}_{\varphi }& -s_{\theta }& c_{\theta }{c}_{\varphi }\end{array}\right]$ \\

Finaly leaving us with the Euler 212 sequence \\

$R_2(\psi )R_1(\theta )R_2(\varphi )=\left[\begin{array}{ccc}{c}_{\psi }{c}_{\varphi }-s_{\psi }{c}_{\theta }{s}_{\varphi }& s_{\psi }{s}_{\theta }& -c_{\psi }{s}_{\varphi }-s_{\psi }{c}_{\theta }{c}_{\varphi }\\ s_{\theta }{s}_{\varphi }& c_{\theta }& s_{\theta }{c}_{\varphi }\\ s_{\psi }{c}_{\varphi }+c_{\psi }{c}_{\theta }{s}_{\varphi }& -c_{\psi }{s}_{\theta }& -s_{\psi }{s}_{\varphi }+c_{\psi }{c}_{\theta }{c}_{\varphi }\end{array}\right]$