Given a closed parametric curve $c(t)=(c_x(t),c_y(t))$, on some interval $t\in [a,b]$, is it possible to find a closed form function $f(x,y)$, such that the set of points on the curve $C$ is equal to the set $f$'s roots? In other words, $$ \{c(t) \ | \ t \in [a,b]\}=\{(x,y) \in \mathbb R^2 \ | \ f(x,y)=0\}. $$
If this is not possible in general, what properties could $c(t)$ satisfy that would allow finding a corresponding $f$ easy? Continuity, being a simple curve, etc.
This question is interesting computationally since it is easy to define pretty curves as parametric functions, but computing via shader programs is not possible. However, computing a distance function from each point on the plane to the nearest point on the curve is much easier, so a process to convert curves to distance functions is what I am looking for.
If the parametrization is given by polynomials, or rational functions, it is possible in principle to find a polynomial equation that every point on the curve will satisfy. However, it may not be very useful.
In other cases, finding a closed form function may be hard.
Example $$x \mapsto (x^3 + e^x, \sin x)$$
The function $\phi \colon x \mapsto x^3 + e^x$ is an analytic diffeomorphism of $\mathbb{R}$. So you could write the curve as the graph of the function
$$t \mapsto \sin \phi^{-1}(t)$$
But $\phi^{-1}$ does not have a "nice" formula. So here we are.