This was a question that was asked during a tutorial which I didn't have an answer to... any explanation would be appreciated!
Suppose $g: \mathbb{R}^2 \to \mathbb{R}$ is a function and say I look at the surve $g(x,y) = 0$. Under what conditions can we guarantee that there exists a parametric curve (we need it to be differentiable) $(x(t), y(t))$ that describe this curve? Also is there a way to come up with $x(t)$ and $y(t)$ explicitly?
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 $t \mapsto (t,h(t))$.