What is Relative motion; frame in translation, in rotation?

Let $\mathcal R$ be a frame (origin $O$) and $\mathcal R'$ another (origin $O'$) moving with respect to it. $\mathcal R'$ is in translation if its axes keep fixed directions in $\mathcal R$ (every point of $\mathcal R'$ has the velocity of $O'$); it is in rotation about a fixed axis $\Delta$ if $\Delta$ is fixed in both frames and $\mathcal R'$ turns about it at the angular velocity $\Omega(t)$. The velocity and acceleration of a point $M$ relative to $\mathcal R'$, $\vect v'$ and $\vect a'$, are those a stationary observer of $\mathcal R'$ measures.