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.

Examples

Parallel Lines

Two parallel lines are said to intersect at the point at infinity.

Parallel Planes

Two parallel planes are said to intersect at a line at infinity, which is a generalisation of the concept of a point at infinity.

Asymptote

The asymptote to a curve can be said to intersect the curve at the point at infinity.

Also see

• Results about the point at infinity can be found here.

Historical Note

The idea of a point at infinity was introduced by Johannes Kepler, who suggested that a parabola might be considered as an ellipse with one focus at infinity.

The idea was developed by Girard Desargues when he put his ideas of projective geometry together.

This assumed the existence of an ideal point 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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): infinity $(2)$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): infinity $(2)$