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:

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:

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

When $l = 0$ we have:

When $m = 0$ we have:

The result follows.