1
$\begingroup$

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 "$...+F=0$".

Improve this question
$\endgroup$
  • 1
    $\begingroup$ See en.wikipedia.org/wiki/… $\endgroup$ – lab bhattacharjee Feb 5 '18 at 5:31
  • $\begingroup$ @lab bhattacharjee from the link it says to evaluate $B^2 - 4AC$ - do I need to rotate the conic section first though? $\endgroup$ – Jbag1212 Feb 5 '18 at 5:37
  • $\begingroup$ Just check if $B^2-4AC<0$ $\endgroup$ – lab bhattacharjee Feb 5 '18 at 5:52
  • 1
    $\begingroup$ 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. $\endgroup$ – Lubin Feb 5 '18 at 5:52
  • $\begingroup$ Your ellipse necessarily passes through the origin? $\endgroup$ – Bernard Feb 5 '18 at 9:15
2
$\begingroup$

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:

  1. The quadratic part of the equation (associated to the matrix $Q$) has signature $(2,0)$;
  2. The quadratic form on $\mathbf R^3$ associated to $M$ has signature $(2,1)$.
Improve this answer
$\endgroup$
  • $\begingroup$ 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"? $\endgroup$ – Jbag1212 Feb 5 '18 at 21:46
  • $\begingroup$ 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. $\endgroup$ – Bernard Feb 5 '18 at 22:01
1
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.