Definition:Point at Infinity/Homogeneous Cartesian Coordinates/Point on Line

Definition

Let $\LL$ be a straight line embedded in a cartesian plane $\CC$.

Let $\LL$ be given in homogeneous Cartesian coordinates by the equations:

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

The point at infinity is expressed in homogeneous Cartesian coordinates by an ordered triple in the form:

$\tuple {-m, l, n}$

Also see

• Point at Infinity of Intersection of Parallel Lines, which justifies this construction

Sources

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