Given two infinite cylinders:
- radius $r_0$, centered at the origin and pointing $\hat u_1=(0,0,1)$
- radius $r_1$, centered at $(d,0,0)$ and pointing at $u_2$
For a given distance $d$, if $u_2$ is selected uniformly at random over all possible orientations, what is the probability $P(d; r_0,r_1)$ that the two infinite cylinders will overlap?
Trivial limits: $P(d \le r_0+r_1)=1$, $P(d \gg r_0+r1) \approx 0$.