Definition:Line at Infinity

Definition

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

$l X + m Y + n Z = 0$

Let $l = m = 0$.

Then from Intersection of Straight Line in Homogeneous Cartesian Coordinates with Axes, $\LL$ intersects both the $x$-axis and the $y$-axis at the point at infinity.

Such a straight line cannot exist on $\CC$, so such an $\LL$ is known as the line at infinity.

Sources

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