How to get the intersection of two parallel lines on projective plane $\mathbb{P}^{2}$? On a real projective plane $\mathbb{P}^2$, say we have two parallel lines; namely, $2x+y=0$ and $4x+2y+1=0$. What would be the equations of the projective lines, and how to find the point of their intersection?
Thanks a lot!
 A: There are various approaches, but it is common to go to homogeneous coordinates.  So the projective lines have homogeneous equations $2x+y=0$ and $4x+2y+z=0$. They meet where $z=0$ and $2x+y=0$, so at $(1,-2,0)$. 
Because we are using homogeneous coordinates, each component of $(1,-2,0)$ can be multiplied by the same non-zero constant.
Remark: I do not know what notation is used in your course, so  used standard old-fashioned notation. Your version may use equivalence classes. If so, it should not be difficult to translate. 
A: The point of intersection can be read directly from the equations of the lines. Rewrite them in normal form as $(2,1)\cdot(x,y)=0$ and $(4,2)\cdot(x,y)=-1$. The two normals are non-zero scalar multiples of each other, so the lines are indeed parallel. By definition parallel lines intersect at  their common point at infinity, while a line’s point at infinity and its direction vector are identical in homogeneous coordinates, so we can see immediately that the two lines intersect at $(-1,2,0)$.
