Proper fully defined notation for angular velocity

Question

Yeah, so, I know that objects rotate, about a point, in a chosen frame of reference. I was wondering if there is a mathematical notation that tells you the following :

• Point of rotation of a body
• Coordinate system describing the vector
• body of reference or the relative body which the body of interest's angular velocity is being measured with respect to ( I don't know if angular velocity of a body relative to another is a thing, if it is I want to know how its written mathematically)

I stumbled across the above notation ( found here Definition of angular velocity vector of $B$ in $A$ - Strange notation) but I don't think it states the coordinate system and the point of rotation.

I also stumbled across the above notation (found here Definition of angular velocity vector of $B$ in $A$ - Strange notation), it has more info but I don't understand it, I googled "angular velocity notations", nothing helpful came.

If you are wondering why I want the notation to specify the point of rotation, its because I noticed that determining relative angular velocity between to bodies is dependent on their points/axis of rotation, whether they coincide or are separate.

EDIT: Found another notation here Relating angular velocity between rotating frames

Answer 1

Notation often emphasises one aspect more than another. Its often an aspect that is under consideration. For example a tangent vector can be written as:

$v$ - it's position is left implicit

$v_p$ - it's position is explicit

And this is matched by the notation for a tangent bundle

$TM$ - we are considering the whole tangent bundle

$T_pM$ - we are considering the tangent space at $p$

Obviously, this is not an exact match, as $TM$ is not a fibre where the position of the fibre is left implicit.

And this is further matched by the notation of a bundle

$E$ - we are considering the whole fibre bundle

$E_p$ - we are considering the fibre of the bundle at $p$