I am not sure I understand the question, see the remarks of Qfwfq. Here is something related which might nevertheless be interesting for some branches of applied mathematics: an algorithm for **parameterizing** the intersection of two *arbitrary* quadrics, given in implicit form with integer coefficients. There are several research reports and even an [implementation][1]. You can even input your coefficients on-line on the web and see the result. Maybe there is something easier for the special case of two circular cones.


  [1]: http://vegas.loria.fr/qi/