How to find the curve of intersection of two surfaces?

Question

Let

$$ f(x,y,z)=y+x^{2}-z=0 $$ $$ g(x,y,z)=x^{3}+y^{2}-z=0 $$

How to find the curve of intersection of $f$ and $g$? Please don't work it out just tell me how to go about it?

Thanks.

Answer 1

I would look for a parametric equation $x=x(t)$, $y=y(t)$, $z=z(t)$. Since $z$ is by itself in both equations, I seems reasonable to choose $z(t)=t$. This leaves you with two equations in the unknowns $x,y$ to be solved in terms of $t$. This may lead to a nasty fourth order equation. After a little thought, it looks that choosing $x(t)=t$ will make things easier.