# How to parametrize the curve of intersection of two surfaces in $\Bbb R^3$?

I have to parametrize the curve of intersection of two surfaces. The surfaces are:

$$y^2 = z \text{ and } x + y = 4$$

Could someone please show me how to do this step by step? Thanks.

• Where are you stuck?
– Pedro
Nov 9, 2013 at 0:46
• My guess isyou don't even know what you're supposed to do. You need to find a map $\gamma\colon \Bbb R\to \Bbb R^3$ such that $\{\gamma(t)\colon t\in \Bbb R\}=\{(x,y,z)\in \Bbb R^3\colon y^2=z\land x+y=4\}$. Can you now do this? Nov 9, 2013 at 0:47

Let $y=t$. Then $x=4-t$ and $z=t^2$ and so $\vec r(t)= (x(t), y(t), z(t))= (4-t, t, t^2)$, $t\in\mathbf R$.