Motivation: Develop some intuition on the solution of the intersecting points of two ellipsoids.
In two dimensions, two arbitrary ellipses can intersect in (at most) four points (excepting the pathological perfect overlap). If you move one ellipse away you can get two of the points to reduce to a single point. Moving further, you can remove this point leaving only two total points of intersection. Finally, you can reduce these two points the same way until you have zero points of intersection. By writing down the equations, it seems that this could be understood as the zeros of a quartic polynomial, where the number of real solutions correspond to the number of points of intersection $f(x)=0$.
In three dimensions, there are now parametrized curves of intersection rather then points. Is it correct to consider the solution of two ellipsoids as closed curves (or points) in a two dimensional plane of a two dimensional polynomial $f(x,y)=0$?