Definition:Homogeneous Cartesian Coordinates

This page is about Homogeneous Cartesian Coordinates. For other uses, see Homogeneous.

Definition

Let $\CC$ denote the Cartesian plane.

Let $P = \tuple {x, y}$ be an arbitrary point in $\CC$.

Let $x$ and $y$ be expressed in the forms:

\(\ds x\)

\(=\)

\(\ds \dfrac X Z\)

\(\ds y\)

\(=\)

\(\ds \dfrac Y Z\)

where $Z$ is an arbitrary real number.

$P$ is then determined by the ordered triple $\tuple {X, Y, Z}$, the terms of which are called its homogeneous Cartesian coordinates.

Also denoted as

This can also be presented as:

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

Also see

• Results about homogeneous Cartesian coordinates 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