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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?

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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:
     Two spheres
     (Image from

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