I am considering a given quadric
$ r^T Q r + b^T r + c = 0 $
and its intersection with a given plane
$ n^T (r - r_0) = 0 $
and I want to identify the intersection curve between the two.
My attempt:
From the equation of the plane, one can find two mutually orthogonal unit vectors $u_1, u_2$ that are also orthogonal to $n$, and thus the plane can parameterised by
$ p = r_0 + x u_1 + y u_2 = r_0 + V u $
where $V = [u_1, u_2] $ and $ u = [x, y]^T $
Plugging this into the equation of the quadric
$ (r_0 + V u) ^T Q (r_0 + V u) + b^T (r_0 + V u) + c = 0 $
which becomes
$ u^T Q_1 u + b_1^T u + c_1 = 0 $
Now it is a matter of identifying the curve described by this quadratic equation.
The possible curves are: Ellipse, Hyperbola, Parabola, Two intersecting lines, Two parallel lines, A single line, A single point, or the empty set.
I appreciate any additional input, or further details that I may have missed.