What if the ball's orbit isn't circular?

It doesn't matter-- we can describe any 2D trajectory using polar coordinates, just as we can describe it using Cartesian coordinates. The $r$ coordinate is the distance from the origin, and the $\phi$ coordinate is the angle between the $\vec{r}$ vector and the horizontal axis. You can always find $r$ and $\phi$ from $x$ and $y$ using $r=\sqrt{x^2+y^2}$ and $\phi = \tan^{-1}{\frac{y}{x}}$. If the trajectory is circular, then polar coordinates are especially convenient, since $r$ will be constant.