# Definition:Point at Infinity

Jump to navigation
Jump to search

## 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