I have two rays, one of length $L_1$ and one of length $L_2$. I anchor these rays, each at one end, on the 3-space points $p_1$ and $p_2$. Assuming that the Euclidean distance between $p_1$ and $p_2$ is $\leq (L_1+L_2)$, what solution space defines the points where these rays can be oriented to touch at their free ends? How might I sample uniformly from this solution space?
Sample points uniformly from the circle of intersection between the two spheres of radii
$L_1$ and $L_2$ centered on $p_1$ and $p_2$ respectively:
(Image from Mathalino.com.)