In several branches of applied mathematics the problem arises to describe the intersection of two cones in three-space.
I have searched and found a few references that discuss the problem for cones with parallel axes. I am interested in the general case.
Assume that one cone has vertex at the origin with a certain "cone angle" (is that the phrase?) at the vertex and an axis some vector through the origin. Situate the second cone at point (b,0,0) with a (perhaps) different cone angle and axis some arbitrary vector through (b,0,0). Describe the locus of points where they intersect.
The answer ought to be a polynomial in the various parameters. I want a fully symbolic answer, no numerical methods. It will probably be a fairly large polynomial. Of course, the intersection might be empty, which simply means that when certain values are plugged in for the parameters, the polynomial has no real solution.