Definition:Point at Infinity/Homogeneous Cartesian Coordinates

Definition

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

$\tuple {X, Y, Z}$

where:

$Z = 0$

$X$ and $Y$ are arbitrary.

Point on Line

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:

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

Sources

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