I found on Mathworld ( http://mathworld.wolfram.com/Ellipse.html ) that the quadratic equation:
$ax^2 + 2bxy + cy^2 + 2dx + 2fy + g = 0$
represent an ellipse only when, after defining:
$\Delta = \left|\begin{array}{ccc} a&b&d\\b&c&f\\d&f&g\end{array}\right|$ $J = ac-b^2$ $I = a+c$
the following conditions are met: $\Delta \neq 0$, $J > 0$ and $\Delta/I < 0$.
What I don't understand are the constraint other than $J > 0$. I've never seen these constraints before, and I really wonder why they are here.
Also, I noticed that the formulas to find the semi-axes and angle given after on the MathWorld article work only if these constraints are met.
The second part of my question is then: is there an other way to find the length and orientation of the axis of an ellipse given its quadratic equation? Or is it always possible to transform the quadratic equation of an ellipse into an equivalent one respecting the constraints described on MathWorld?