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.

Examples

Arbitrary Example

Consider the polynomial equation $\map P {x, y}$:

$2 x^2 + x + 7 = y$

This can be expressed in homogeneous Cartesian coordinates $\map P {X, Y, Z}$ as:

$2 X^2 + X Z + 7 Z^2 = Y Z$

Also denoted as

A point $P$ represented in homogeneous Cartesian coordinates can also be denoted as:

$P = \tuple {X : Y : Z}$

Also known as

Homogeneous Cartesian coordinates are also known just as homogeneous coordinates.

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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): homogeneous coordinates
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): homogeneous coordinates