Whilst stuying geometry, I tried to find the intersection point of two projective lines.
The first projective line $a$ goes through the homogeneous coordinates $[1:0:1]$ and $[1:1:1]$.
The second projective line $b$ goes through the homogeneous coordinates $[1:1:0]$ and $[0:1:0]$.
Now I would like to find their point of intersection. What I tried so far:
$x[1:0:1]+y[1:1:1] = (x+y,y,x+y)$
$z[1:1:0]+w[0:1:0] = (z,z+w,0)$
When I solve this system I get: $x=-y=-w$ and $z=0$.
This would result in the coordinate $(-y,w,0,0)$.
Now I am unsure what to do, or whether this is correct at all. It seems like the intersection point of two projective lines has 4 coordinates now? That doesn't make sense to me. Could someone maybe help me with this problem?
You did the right thing in solving the system of equations, but what you need to do now is substitute back in the point you are looking for:
$$[x+y:y:x+y] = [-y+y:y:-y+y] = [0:y:0] = [0:1:0]$$
or equivalently
$$[z:z+w:0] = [0:w:0]=[0:1:0]$$