I have the following equation:
$$a(\alpha)\mathrm{sin}^2(\alpha)-b(\alpha)\mathrm{sin}(\alpha)\mathrm{cos}(\alpha)+c(\alpha)\mathrm{sin}^2(\alpha)=1$$
The authors state that this is the equation of an ellipse as $\alpha$ varies from 0 to 2$\pi$.
I want to plot this ellipse in Cartesian (x,y) coordinates, but I am not sure where to even begin and the authors of the paper do not elaborate. Note that $a$, $b$, and $c$ are known functions.
I've found some answers that talk about converting from polar to Cartesian form (e.g. here and here and here), but they generally involve simple cases which are either already in Cartesian coordinates, or do not have coefficients which are also a function of $\alpha$. I feel that one of my primary confusions is how $\alpha$ relates to the parametric form (if at all).
Any help is appreciated.