So I have the equation defined in homogeneous coordinates $[w; x, y]$ as $[1+t^2; 1-t^2, 2t]$
$$w = 1+t^2$$ $$x = 1-t^2$$ $$y = 2t$$
If I do $w+x-y$ I get $-2t+2$, so $t = -(w+x-y-2)/2$. I was then going to plug this back into one of the original parameters and set equal to $0$.
So $2t = 0 $
$${-(w+x-y-2)}/{2} = 0$$ $$-(w+x-y-2) = 0 $$ $$-w + x - y +2 = 0$$ $$ x - y - w +2 = 0$$
And that would be my final equation.
However I feel like this isn't right because I feel like I shouldn't have the 'w' term in there at all. Originally I was going to change back into Cartesian Coordinates and divide $x$ and $y$ by $w$ but I found the new equations to be more cumbersome so I decided to try it this way.
So am I doing something wrong or is this correct?
If the definition is as at the top of the question $[w,x,y]=[1+t^2,1-t^2,2t]$ then you can compute that $$x^2+y^2-w^2=0$$ by plugging in for $w,x,y$ their parametric expressions.
If the definition is really $[w,x,y]=1+t^2,1-t^2,2t+1]$ then the above equation doesn't work, because $x^2+y^2-w^2=4t+1$ in that case, and the parameter $t$ has not been eliminated.
I would guess it should be as at the top of the question with $2t$ for $y$, as then this is a standard way to give a parametric expression for $x^2+y^2=w^2$.