Now that this has been kicked to the front page,
let me adorn Günter Rote's answer with an image from the [web pages][1] to which he refers:
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="http://www.loria.fr/equipes/vegas/images/quadrics/intersections/cas_t4l9a.jpg" width="400" />
<br />
<sub>"Parameterization of intersections of quadrics: theory, algorithms and implementation"</sub>


  [1]: http://www.loria.fr/equipes/vegas/curvesAndSurfaces.php#invariants