An arbitrary conic section can be written in terms of a symmetric 3x3 matrix $A$ as the set of points $X^T=(x,y,1)$ satisfying $X^T A X=0$. This question is only concerned with cases where such conic sections are either ellipses (which in this context does not exclude circles as a special case) or parabolae.
This question is not concerned with singular ellipses (i.e. ellipses which have shrunk to a single point, or which are reduced to a finite line segment on account of the semi-minor axes being zero) and is not concerned with singular parabolae (e.g. parabolae which have collapsed to a semi-infinite ray such as would be seen if one took the limit as $k$ goes to infinity of the parabolae defined by $y=k x^2$, or which have collapsed to an infinite line in the manner of $y=0x^2$). Throughout this question all conics are thus assumed to define to a bona-fide non-degenerate ellipses (including circles) and nona-fide parabolae that would be recognised as such by young children.
With the above restriction in mind, one my define the interior, $I(A)$, of the conic $A$, to be the smaller of the two disjoint regions of the $(x,y)$ plane that have the conic as a boundary. (Technically, for this purposes of this question the boundary itself is included within the interior and not within the exterior. For example, the interior of the parabola $y=x^2$ is considered to the be the set of points having $y\ge x^2$ while the exterior would be those having $y<x^2$. The interior of the ellipse $x^2+y^2=1$ is the set of points having $x^2+y^2\le 1$ and the exterior are those having $x^2+y^2>1$.) The diagram at the foot of the question may help visualise the boundary/interior difference.
With those definitions in mind, we can now ask:
What is the condition on the matrices $A$ and $B$ such that there is a non-empty intersection $I(A)\cap I(B)$ between the interiors of the elliptic or parabolic conics these matrices define?
For the case that both conics defined by $A$ and $B$ are ellipses, this question is answered in F. Etayo et al. / Computer Aided Geometric Design 23 (2006) 324–350 (and summarised below).
I am aware of no published answer, however, that deals with the case where one or more of the conics defined by $A$ and $B$ is a parabola. In https://arxiv.org/pdf/1411.4312.pdf (declaration of interest: I was an author) it was conjectured that the test of Etayo et al is, in fact, valid not only for determining intersections of ellipse interiors, but is also valid for the case were one or both of the conics is a parabola.
My question for maths stack exchange, therefore, is can the above conjecture be proved? (Or be found to be an old result long known about.)
This feels to me like the sort of thing the Ancient Greeks must have answered long ago …
Happy musings!
Aside:
The test of Etayo et al says that the ellipses defined by $A$ and $B$ are separated if and only if either $$ a\ge 0, \\−3b+a^2 >0, −27c^2 +18cab + a^2b^2 − 4a^3c − 4b^3 >0,\\ 3ac+ba^2 −4b^2 <0 $$ or $$a < 0, \\−3b + a^2 > 0, −27c^2 + 18cab + a^2b^2 − 4a^3c − 4b^3 > 0$$ (in which the commas mean “and”) wherein $a$, $b$ and $c$ are defined to be the coefficients appearing in $$f(λ)=\lambda^3 +a\lambda^2 +b\lambda+c $$ if one defines $$f(λ)=\det(λA+B).$$
My question asks readers to show (or refute) the suggestion that this is a valid test for parabola+parabola or parabola+ellipse interior intersection in addition to its already proven relationship to ellipse+ellipse interior intersection.
