Fastest way to determine if a conic section is an ellipse?

Question

Given an arbitrary conic section in the form $$Ax^2 + Bxy + Cy^2 + Dx + Ey +F=0$$ (Where the coefficients are real valued) is there a simple test which can determine whether or not a particular conic is an ellipse? I know that if a conic section is an ellipse, then $A$ and $C$ will have the same sign, however I am not sure if this is a sufficient condition as well.

Edit: Forgot to include the "$...+F=0$".

See https://en.wikipedia.org/wiki/Rotation_of_axes#Rotation_of_conic_sections — lab bhattacharjee, Feb 05 '18 at 05:31
@lab bhattacharjee from the link it says to evaluate $B^2 - 4AC$ - do I need to rotate the conic section first though? — Jbag1212, Feb 05 '18 at 05:37
No, the quantity $B^2-4AC$ is invariant under rotations. You might want to check this: it’s messy, but very satisfying to come out with the result. — Lubin, Feb 05 '18 at 05:52
The origin is a point of the ellipse. This is the case if there's no constant term in the equation. — Bernard, Feb 05 '18 at 21:45
Ah, with the conics I have, this is not necessarily the case. — Jbag1212, Feb 05 '18 at 21:47

score 3 · Answer 1 · answered Feb 05 '18 at 09:30

3

Here is the answer for the normalised equation of a non-degenerate conic $$Ax^2+2Bxy+Cy^2+2Dx+2Ey+T=0.$$

Consider the matrix $$M=\begin{bmatrix} A & B & D \\ B & C &E \\ D &E & F \end{bmatrix} $$ and the matrix of the quadratic part of the equation $$Q=\begin{bmatrix} A & B \\ B & C \end{bmatrix}. $$ The conic is non-degenerate if and only if $\det M\ne 0$. Further, the conic is an ellipse if and only if:

The quadratic part of the equation (associated to the matrix $Q$) has signature $(2,0)$;
The quadratic form on $\mathbf R^3$ associated to $M$ has signature $(2,1)$.

answered Feb 05 '18 at 09:30

Bernard

175,478

Is there an advantage to using this method over checking $B^2 - 4AC$? Does checking $B^2 - 4AC$ only work if it "passes through the origin"? – Jbag1212 Feb 05 '18 at 21:46
The sign $B^2-4AC$ tests the possibility of points at infinity, i.e. the possibility of a hyperbola (2 points at infinity) or a parabola (az double ipoint at infinity. It's equivalent to the signature of $A$, but there may be degenerate cases which are handled by the study of the matrix $M$. In the ellipse case, the curve may be empty or reduced to a point. – Bernard Feb 05 '18 at 22:01

score 1 · Answer 2 · answered Feb 05 '18 at 15:19

1

If $ I_2=B^2-4AC$ is negative then it is an ellipse whether or not (RHS being zero ) it passes through origin as special case.

answered Feb 05 '18 at 15:19

Narasimham

40,495

Fastest way to determine if a conic section is an ellipse?

2 Answers2