I have the ellipse given by $-3x^2-3y^2+4xy=-2$.
First question is, how do I know that it is really an ellipse? Wolfram alpha says so, but how could I tell in advance?
But more importantly, how can I find a suitable parametrization for this ellipse? I'm aware o similar questions in this forum, where rather convoluted expressions are given for the parameters of a rotated ellipse. But I don't see how to match this to my problem of the search of the parametrization.
For example, in non-tilted ellipses you have to manipulate the equation trying to get the canonical form $\left(\frac{x-x_0}{a}\right)^2 + \left(\frac{y-y_0}{a}\right)^2=1$. Once you have $a, b, x_0$ and $y_0$, you just apply:
$x=x_0+a\cos \theta$
$y=y_0+b\sin\theta$
with $0\le\theta\le2\pi$ and you have it.
Now, how can I do something similar to find the $x(\theta), y(\theta)$ for a tilted ellipse?
Best Answer
The equation is symmetric in $x$ and $y$, so it is natural to try the change of variable
$$x:=u+v,\\y:=u-v.$$ (This is a rotation by $45°$ combined with a scaling by $\sqrt2$, i.e. a similarity transform.)
Plugging in the equation, we get
$$-6(u^2+v^2)+4(u^2-v^2)=-2$$ or
$$u^2+5v^2=1.$$
This is an ellipse centered at the origin, with axes of length $1$ and $\dfrac1{\sqrt5}$ (after scaling).