Let $C:\mathbb{R}\rightarrow\mathbb{R}^2$ be a differentiable, simple, regular parametric curve with an unbounded image. I need to construct a differentiable at least almost everywhere function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ whose zero level set is the image of $C$.
Having almost no experience with vector calculus or differential geometry, I intend to do this by considering a vector field $p:\mathbb{R}^2\rightarrow\mathbb{R}^2$ where $p(x)$ is the normal vector to the curve at each point $x$ in the image set of $C$ and $\nabla f=p$.
How can I find, if possible, $p(x)$ at all $x \in\mathbb{R}^2$ not in the image set of $C$? I thought of, given that $\mathbb{R}^2$ is path-connected, finding $p$ by finding a solution to PDE $\text{curl}(p(x,y))=0$ on $\mathbb{R}^2$ with the aforementioned constraint that for all $x$ on the image of $C$, $p(x)$ is the normal vector to the curve. If able to find a simple enough $p$, I would then recover $f$ from gradient field $p$ and pray that the image set of $C$ be some level curve of $f$.
For what curves can we end up with a closed-form expression for $f$? Are there other methods for finding a closed-form expression for $f$ given an expression for $C$?