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. 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.
Günter Rote
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