How to draw surface intersection with axis plane

Question

Suppose I have some equation of surface $S$ (let's take $f(u,v) = (uv,u+v,u^2 + v^2)$ for example). How do I draw intersection of surface $S$ with axis planes $x = 0$ and $y=0$, generally? I am looking for some general technique of how to draw those intersections, not how to draw this one exactly.

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$To find the intersection of the plane $x = 0$ with the image of a mapping $f:\Reals^{2} \to \Reals^{3}$ given in Cartesian coordinates by $$ f(u, v) = \bigl(x(u, v), y(u, v), z(u, v)\bigr), $$ set $x(u, v) = 0$ to get a set in the $(u, v)$-plane, then calculate the image of this set under $f$.

The intersection with the plane $y = 0$ (or $z = 0$, or in fact any plane whatsoever) is in principle found the same way.

Generally that's difficult. In your example, $x = 0$ if and only if $u = 0$ or $v = 0$.