Nice model.
Google search came up with "Intersection of quadrics"
There is a QI program and 4 research papers. http://hal.inria.fr/docs/00/18/60/89/PDF/JSC1.pdf
(Google each of the 4 research paper titles to find them.)

The little "parabolic" looking curve is not an ellipse, not a conic, and is not planar... it is a "second order curve...."