I have a unit sphere, on the surface of which are defined spherical caps. I typically characterize the caps by the unit vector $n$ from the center of the sphere to the top of the cap, and the angle $\theta$.
My question is: given a pair of spherical caps, how can I determine whether they intersect?
The angle between $n_1$ and $n_2$ is the analogue of the "distance" between the centres of the two caps. The caps intersect if this angle is less than the sum of $\theta_1$ and $\theta_2$, assuming the $\theta_i$ play the role of "radius" and not "diameter" in your notation.