Question

How to parameterize a folium curve (x^3 + y^3 - 9xy = 0)?

Answer 1

Consider if you set y=tx. Then plugging this in, you have \[x^3+x^3t^3-9tx^2=x^2(x(t^3+1)-9t)=0\] Thus since x is not identically zero, you have that \[x(t^3+1)-9t=0 \implies x=\frac{9t}{t^3+1}\] Since y=tx, this give you also \[y=tx=\frac{9t^2}{t^3+1}\] This parametrizes your folium.

Answer 2

Thanks, that's it.