I have a polar function
$$ r(\theta)=\left(r+\epsilon\right)\cos(\theta)-\sqrt{r^{2}-\left(r+\epsilon\right)^{2}\sin^{2}(\theta)} $$
Is it possible to methodically conjure another polar function that has the same curvature (second derivative), and the same value at $0$?
This might help me approximate $r(\theta)$. I need, ultimately, a function $h(\theta)$ which approximates $r(\theta)$ for small $\theta$, accurate at $\theta=0$, and the function $h(\theta)$ should ideally take the form $h(\theta)=\sqrt{\ln\left(g(\theta)\right)}$; this would aid later integration as part of a Gaussian.