Fixed space is defined relative to origin $(0,0,0)$. Fixed space has an origin with angles $(0,0,0)$ as well.
A particle $P$ has $(x,y,z)$ coordinates relative to the origin: $P_x$, $P_y$, and $P_z$.
An observer $O$ has $(x,y,z)$ coordinates relative to the origin: $O_x$, $O_y$, and $O_z$. $O$ also has angles $\theta_{Ox}$, $\theta_{Oy}$, and $\theta_{Oz}$, defined as the rotation about the given axis in fixed space.
The particle and the observer have a restriction in their relationship. $P_z$ is guaranteed to exist on the far half of observers $z$ axis, such that if the observer is facing the particle, the positive $x$ and $y$ axis of the observer are right and up respectively. (The particle is restricted to half the observer's frame of reference.) In other words, the particle has a 0-180 degree domain relative to the observer about the $z$ and $x$ axes.
What I'm trying to find is this: If the observer has given coordinates and rotation relative to the origin and a particle exists with the specified restrictions, what are the three relative ($\theta_{R(x,y,z)}$) angles between the observer and the particle? In other words, a function {$P_x$, $P_y$, $P_z$, $O_x$, $O_y$, $O_z$, $\theta_{Ox}$, $\theta_{Oy}$, $\theta_{Oz}$,} -> {$\theta_{Rx}$, $\theta_{Ry}$, $\theta_{Rz}$} I've come up with several equations that seemed right, only to be quickly disproved. I'm coming to my wit's end on this.