Point of Intersection with Line at Infinity

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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 $-n2$.

$\blacksquare$


Sources