Euler angles is supposed to uniquely define the two differently oriented co-ordinate axes. Can someone help me in visualising how a set of three angle takes one co-ordinate axes system to the other co-ordinate system. It would be helpful if one can start with given two differently oriented co-ordinate system and one set of Euler angle and then define the sequence of rotations.
Thank you very much for answering. Please put a feedback if one needs further clarification
We want to bring the z axis to z’ axis,
I)
we rotate around the positive z axis with angle $\varphi$ . The rotation matrix is:
$$R_z(\varphi)=\left[ \begin {array}{ccc} \cos \left( \varphi \right) &-\sin \left( \varphi \right) &0\\ \sin \left( \varphi \right) &\cos \left( \varphi \right) &0\\ 0&0&1 \end {array} \right] $$
II)
we rotate around the y’ axis with angle $\vartheta$ . The rotation matrix is:
$$R_{y'}(\vartheta)=\left[ \begin {array}{ccc} \cos \left( \vartheta \right) &0&\sin \left( \vartheta \right) \\ 0&1&0 \\ -\sin \left( \vartheta \right) &0&\cos \left( \vartheta \right) \end {array} \right] $$
III)
thus we get to picture III, now we can rotate about the z' axis with the angle $\psi$ the rotation matrix is :
$$R_{z'}(\psi)=\left[ \begin {array}{ccc} \cos \left( \psi \right) &-\sin \left( \psi \right) &0\\ \sin \left( \psi \right) &\cos \left( \psi \right) &0\\ 0&0&1\end {array} \right] $$
thus the transformation matrix between body fixed coordinate system and inertial system is:
$$R=R_z(\varphi)\,R_{y'}(\vartheta)\,R_{z'}(\psi)\quad,R^{-1}=R^T$$
where $\varphi\,,\vartheta\,,\psi$ are the Euler angle
Remark
after the three rotation you get:
the x axis $\mapsto$ x' axis $\mapsto$ x'' axis.
the y axis $\mapsto$ y' axis $\mapsto$ y'' axis.
z axis $\mapsto$ z' axis
where
$x''\perp y''\perp z'$ are the body fixed coordinate system
