I'm trying to parametrize the projective variety of $f(x,y,z)=x^2+y^2+z^2-2xy-2yz-2xz$ in $\mathbb{P}^2$.
Progress:
I know it'll need to be in two pieces, an affine piece and the piece at infinity (which will be in $\mathbb{P}^1$).
Piece at infinity: I think the piece at infinity will end up of the form $[x,y,0]$ but I'm not sure how to find those first two coordinates.
Affine piece: I'm not sure how to parametrize the affine piece aside from it (I think) being of the form $[x,y,1]$.
You complete the square and you dont have to do the affine and infinity line separately, this is unnatural.
So complete the $x$, square: $$=[x^2-2(y+z)x]+y^2+z^2-2yz=(x-y-z)^2-4yz$$ So you have $$(x-y-z)^2=4yz$$
This is parameterized as $$x-y-z=2st$$ $$y=t^2$$ $$z=s^2$$ and solving the first for $x$, $$x=t^2+2st+s^2=(t+s)^2.$$