Definition:Point at Infinity

Definition

Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, given by the equations:

\(\ds \LL_1: \ \ \)

\(\ds l_1 x + m_1 y + n_1\)

\(=\)

\(\ds 0\)

\(\ds \LL_2: \ \ \)

\(\ds l_2 x + m_2 y + n_2\)

\(=\)

\(\ds 0\)

Let $l_1 m_2 = l_2 m_1$, thus by Condition for Straight Lines in Plane to be Parallel making $\LL_1$ and $\LL_2$ parallel.

In this case the point of intersection of $\LL_1$ and $\LL_2$ does not exist.

However, it is convenient to define a point at infinity at which such a pair of parallel lines hypothetically "intersect".

Homogeneous Cartesian Coordinates

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.

Also see

• Results about point at infinity can be found here.

Sources

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