Homogeneous Cartesian Coordinates represent Unique Finite Point

Theorem

Let $\CC$ denote the Cartesian plane.

Let $P$ be an arbitrary point in $\CC$ which is not the point at infinity.

Let $Z \in \R_{\ne 0}$ be fixed.

Then $P$ can be expressed uniquely in homogeneous Cartesian coordinates as:

$P = \tuple {X, Y, Z}$

Proof

Let $P$ be expressed in (conventional) Cartesian coordinates as $\tuple {x, y}$.

As $Z$ is fixed and non-zero, there exists a unique $X \in \R$ and a unique $Y \in \R$ such that:

\(\ds x\)

\(=\)

\(\ds \dfrac X Z\)

\(\ds y\)

\(=\)

\(\ds \dfrac Y Z\)

Then by definition of homogeneous Cartesian coordinates, $\tuple {X, Y, Z}$ uniquely represents $P$.

$\blacksquare$

Sources

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