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
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $9$. Parallel lines. Points at infinity