I'm trying to understand the concept of angular velocity. I read [this](https://en.wikipedia.org/wiki/Angular_velocity#Addition_of_angular_velocity_vectors) paragraph on Wikipedia, which asserts that if a point $p$ has angular velocity $u$ within a coordinate frame $F_1$ which itself has angular velocity $v$ within some other frame $F_2$, then the point has angular velocity $u+v$ with respect to the second frame.

I want to set aside the specific formula $u+v$ and just focus on the abstract form $f(u, v)$ - in other words, just the fact that the angular velocity of $p$ with respect to $F_2$ can be deduced from the angular velocities of $p$ with respect to $F_1$ and of $F_1$ with respect to $F_2$. That seems like it ought to be impossible to me. In the diagram below, $C$ has the same angular velocity $u$ with respect to $B$ in both figures and $B$ has the same angular velocity $v$ with respect to $A$ in both figures, yet in the first figure $C$ has angular velocity $u+v$ with respect to $A$, whereas in the second figure $C$ has angular velocity $u-v$, zero if we set $u=v$.

[![enter image description here][1]][1]

What am I failing to understand?


  [1]: https://i.sstatic.net/6ak6V.png