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.

Has this been done?