# Intersection of Straight Line in Homogeneous Cartesian Coordinates with Axes

## Theorem

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

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

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

such that $l$ and $m$ are not both zero.

Then $\LL$ intersects:

- the $x$-axis $Y = 0$ at the point $\tuple {-n, 0, l}$
- the $y$-axis $X = 0$ at the point $\tuple {0, -n, m}$

When $l = 0$, $\LL$ is parallel to the $x$-axis with its point at infinity at $\tuple {-n, 0, 0}$

When $m = 0$, $\LL$ is parallel to the $y$-axis with its point at infinity at $\tuple {0, -n, 0}$.

## Proof

The intersection of $\LL$ with the $x$-axis is the point $\tuple {X, Y, Z}$ satisfied by:

\(\ds l X + m Y + n Z\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds Y\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds l X + n Z\) | \(=\) | \(\ds 0\) |

which is satisfied by setting $X = -n$ and $Z = l$, while $Y = 0$.

The intersection of $\LL$ with the $y$-axis is the point $\tuple {X, Y, Z}$ satisfied by:

\(\ds l X + m Y + n Z\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds X\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds m Y + n Z\) | \(=\) | \(\ds 0\) |

which is satisfied by setting $Y = -n$ and $Z = m$, while $X = 0$.

When $l = 0$ we have:

\(\ds m Y + n Z\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds Y\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds n Z\) | \(=\) | \(\ds 0\) |

When $m = 0$ we have:

\(\ds l X + n Z\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds X\) | \(=\) | \(\ds 0\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds n Z\) | \(=\) | \(\ds 0\) |

The result follows.

$\blacksquare$

## Sources

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