Probability that two infinite cylinders overlap

Question

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: $P(d \le r_0+r_1)=1$, $P(d \gg r_0+r1) \approx 0$.

Answer 1

We can restrict our attention to the case where $u2$ is a unit vector, and write it as $u2 = (\cos\theta\cos\phi, \sin\theta\cos\phi, \sin\phi)$. Then Rahul's condition gives you a region in $(\theta, \phi)$ space in which overlap occurs. You can calculate the area of this region by doing some integration. On the other hand, the total area of $(\theta, \phi)$ space is $4\pi$ (the surface area of a unit sphere). The probability of overlap is the ratio of these two areas.