Point of Intersection with Line at Infinity

Definition
Let $\LL$ be a straight line embedded in a cartesian plane $\CC$ given in homogeneous Cartesian coordinates by the equation:


 * $l X + m Y + n_1 Z = 0$

Then $\LL$ intersects the line at infinity at the point:
 * $\tuple {-m, l, 0}$

Proof
By definition, the line at infinity is the line whose equation in homogeneous Cartesian coordinates is:


 * $n_2 Z = 0$

for some $n_2$.

Let $\tuple {X, Y, Z}$ be the intersection of $\LL$ with the line at infinity.

From Intersection of Straight Lines in Homogeneous Cartesian Coordinate Form:
 * $\tuple {X, Y, Z} = \tuple {m n_2 - 0 n_1, n_1 0 - l n_2, l 0 - 0 m}$

from which the result follows by dividing the homogeneous Cartesian coordinates by $-n2$.