I have the following 3D geometry question from a camera positioning problem:
(Related to this question which I got an answer to, but I was not able to extrapolate to 3D)
Point $P_1$ (parent) can only be on a spherical surface about (given) point C with given radius $R$. Point $P_2$ (child)'s position is calculated by moving (given) distance $d_r$ from parent in a direction outwards from the sphere center and then (given) distances $d_x$ and $d_y$ on the plane tangent to the sphere where $P_1$ is. I drew this as a vector $d$ on the diagram.
The question is, given any point in the Cartesian space, say $P_3$, where should I place $P_1$ along the sphere such that the line drawn between $P_3$ and $P_2$ is parallel to the line between $C$ and $P_1$?
Below is a picture that should help hopefully!
Things I understand so far: These points must all by coplanar, and $h = \sqrt{{d_x}^2 + {d_y}^2}$. Then $\theta$ is fixed by $h$ and $CP_3$, but the plane is not fixed since it can be any plane through $CP_3$.... how do I choose the plane?