Question

Real world problem: two cones capped by a hemisphere are given by the locations of their vertices, their directions, their half-angles, and the radius of the sphere capping it (which is centered at the vertex). Technically, these would be two conical sections of two different spheres. How do you accurately find out if the two cones are intersecting?

Answer 1

It's easy enough for uncapped, infinite cones (double check my math, but I think this is correct): if
\[a.b + \cos(\alpha+\beta) \ge 0\]
Then the two infinite cones are intersecting. It gets more complicated when the cones are capped by a sphere section, however:

Where A is the center of the sphere and r its radius, d a direction vector, and \[alpha]\ the half angle of the conical section. Imagine two of these with different parameters and locations, and find if they intersect :)