Let me take the opportunity of this year-old question being kicked to the front page to adorn Günter Rote's answer with an image from the web pagespage (broken link) (edit: but see Wayback Machine and the current page) to which he refers:
http://www.loria.fr/equipes/vegas/images/quadrics/intersections/cas_t4l9a.jpg
"Parameterization of intersections of quadrics: theory, algorithms and implementation"