I have a highly non-trivial function, call it $f(x,y)$, in which I want to look at its zero set in the $(x,y)$ plane, and plot/evaluate a different function $g$ precisely along this zero set. I have a nice way of doing this when the zero set of $f$ can be parameterized by $x$, i.e. $y(x)$, but in general, this zero set will not be nicely parameterized by $x$.
Is there a nice way to parameterize zero sets by some parameter $t$, like length along the curve, and then save values and plot $g(x(t),y(t))$? Perhaps it might be helpful to take an easy example like
Unlike the case of the circle above, my zero set certainly does not have a nice, obvious parameterization, so instead perhaps someone can help me form a numerical recipe using the simple example above.
**If it helps, I can upload the case I actually have in mind, but it's rather messy and requires a lot of pre-amble.
Thanks in advance!