The most general motion of a rigid body is a roto-traslation.
Firstly is it correct that any point (let's call it $O$) of the rigid body can be seen as the point through which passes a istantaneous rotation axis and the motion of any other point of the rigid body (let's call it $P$) can be seen as the composition of the traslation motion of $O$ plus a rotational motion about the istantaneous rotation axis passing through $O$?
I'll make an example of that, because I have another doubt. Consider the disk rolling and possibly slipping on the ground
First imagine that the rotation axis is passing through $A$. Then the velocity of $D$ and $C$ are
$$\vec v_D=\vec v_A+\vec \omega \times \overrightarrow{AD}$$
$$\vec v_C=\vec v_A+\vec \omega \times \overrightarrow{AC}$$
But the velocity of $D$ can be rewritten in terms of the velocity of $C$, indipendently from $A$
$$\vec v_D-\vec v_C=\vec \omega \times (\overrightarrow{AD}-\overrightarrow{AC})$$
$$\vec v_D=\vec v_C+\vec \omega \times \overrightarrow{CD}$$
So the $\vec{v_D}$ can be written in infinite different way but $\vec{v_D}$ is always the same.
Is it correct or does $\vec{v_D}$ change if calculated "with respect to" two different points (in the example $A$ and $C$)?
This confuses me because on textbook I read that, while $\vec{\omega}$ is indipendent from the chosen rotation axis, $\vec{v}$ depends on it. But in the example we just rewrited $\vec{v_D}$ in a different way, the vector $\vec{v_D}$ itself should be that same in any case.