when is there a parametric curve of $g(x,y) =0$
Solution 1:
Given $g(x,y)=0$, we can define a heightmap $h(x)$, and implicit function theorem defines the heightmap when $g(x,h(x))=0$ is evaluated.
How this works, we need to choose g(x,y) as a circle: $g(x,y)=x^2+y^2-10^2$, then $h(x)$ heightmap can be calculated using the following pattern: $$ g(x,h(x)) = x^2 + h(x)^2-10^2 = 0 $$ $$ h(x)^2 = 10^2-x^2$$ $$ h(x) = \pm\sqrt{10^2-x^2}$$
Now once you have $h(x)$, you can define parametric curve as $t \mapsto (t,h(t))$.