I'm working on a personal JavaFX project, and I need to check if two sprites overlap. Originally, I modelled them as ellipses. I was then able to then simplify the problem into checking the intersection of a circle and a non-rotated ellipse centered at the origin. For extra context, what I did was center one of the ellipses at the origin via a translation of $\mathbb{R}^2$, then applied a transformation $\begin{pmatrix}k&0\\0&1\end{pmatrix}$ so that the centered ellipse became a circle. I then translated once again so that the ellipse was centered, and the rotated the plane so that the ellipse was no longer rotated.
However, no matter what I tried (system of equations, checking which curve intersected a ray passing through the origin first, minimizing inequalities, etc.), I always ended up with a quartic equation, which is just not viable to solve. So, I decided to use some other curves to represent the sprites. I first thought of Bezier curves, but I don't think using degree 3 or higher polynomials would improve the situation.
My question is, is there a type of curve that is roughly ovular, where calculating the intersection of such curves is easier?