Consider the rotation $r_{\Omega,\alpha}$ of center $\Omega$ and angle $\alpha$.Furthermore let $t_{\vec{v}}$ be the translation by vector $\vec{v}$. Then $$t_{\vec{v}}\circ r_{\Omega,\alpha}=r_{\Omega',\alpha}.$$ I am wondering how $\Omega'$ is defined.
For $r_{\Omega,\alpha}\circ t_{\vec{v}}=r_{\Omega',\alpha}$ we have the relation $\Omega+\vec{v}=\Omega'$. Is there a similar relation for $$t_{\vec{v}}\circ r_{\Omega,\alpha}=r_{\Omega',\alpha}.$$ Thanks in advance!
Translation commutes with rotation....