I am trying to find the appropriate coordinates for a plane with a handle (of topology $\mathbb{R}^2 \# \mathbb{T}^2$), without having to use several coordinate patches.
My current intuition is to use two cyclical coordinates, $\theta$ corresponding to some angle around the half torus embedded in the surface (such that $\theta = 0$ corresponds to the outside of the torus and $\theta = \pi$ is the inside), and another angle $\phi$ to run the whole conic section for that angle $\theta$. For instance, $\theta = \pi$ will correspond to a circle, $\theta = \frac{\pi}{2}$ is the transition from the ellipse to the parabola, and so on.
Basically the parametric equation of a conic section such that
\begin{equation} r = \frac{l}{1 - e(\theta) \cos(\phi)} \end{equation}
With $l$ a constant and $e$ a function of $\theta$, such that it is periodic, probably something of the form $e(\sin(\theta))$, with $e(0) = 0$ and $e(1) \rightarrow \infty$.
Is this intuition correct? Do those coordinates indeed span the entire plane with a handle? And what would be the line element of such a surface in that coordinate system?