I am trying to deduce an expression in terms of the $x$,$y$,$z$ coordinates of three points $P_1$, $P_2$ and $P_3$.
Given the relatively small motion of points $P_1$ and $P_2$, I want to find the expression for the "precession" angle $\theta$ (measured clockwise as from above, see the figure linked below), belonging to the interval $(-\pi,\pi)$, that the point $P_3$ has with respect to an axis passing through $P_2$. The zero would be reached when $P_1$ and $P_3$ are aligned with that axis.
Definition of the precession angle
I suppose that I could set $$ \theta=\operatorname{atan2}(y_3,x_3) $$ where $x_3$, $y_3$, $z_3$ are the coordinates of point $P_3$ in a reference frame where the $z$ axis passes through $P_2$ and the $x$ axis passes through $P_1$. But the problem is that these two points can also move, so I should obtain an expression that also depends on their coordinates treated as variables, and I am unsure how to do this...
In a generic frame, where the three points have coordinates $(x_i,y_i,z_i)$ with $i=1,2,3$, you could use $$ \theta = \operatorname{atan2}(y_3-y_2,x_3-x_2)- \operatorname{atan2}(y_1-y_2,x_1-x_2) \mod{2\pi} $$ if we can assume that the direction of the axis passing through $P_2$ is constant and parallel to the $z$ axis. Would that work?