PlanetPhysics/Euler 123 Sequence

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

$R_{123}(\varphi ,\theta ,\psi )=R_3(\psi )R_2(\theta )R_1(\varphi )$ \\

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

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

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

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

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

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

Finaly leaving us with the Euler 123 sequence

$R_3(\psi )R_2(\theta )R_1(\varphi )=\left[\begin{array}{ccc}{c}_{\psi }{c}_{\theta }& c_{\psi }{s}_{\theta }{s}_{\varphi }+s_{\psi }{c}_{\varphi }& -c_{\psi }{s}_{\theta }{c}_{\varphi }+s_{\psi }{s}_{\varphi }\\ -s_{\psi }{c}_{\theta }& -s_{\psi }{s}_{\theta }{s}_{\varphi }+c_{\psi }{c}_{\varphi }& s_{\psi }{s}_{\theta }{c}_{\varphi }+c_{\psi }{s}_{\varphi }\\ s_{\theta }& -c_{\theta }{s}_{\varphi }& c_{\theta }{c}_{\varphi }\end{array}\right]$