Definition:Euclidean Plane

Definition

For any real number $a$ let:

$L_a = \set {\tuple {x, y} \in \R^2: x = a}$

Furthermore, define:

$L_A = \set {L_a: a \in \R}$

For any two real numbers $m$ and $b$ let:

$L_{m, b} = \set {\tuple {x, y} \in \R^2: y = m x + b}$

Furthermore, define:

$L_{M, B} = \set {L_{m, b}: m, b \in \R}$

Finally let:

$L_E = L_A \cup L_{M, B}$

The abstract geometry $\struct {\R^2, L_E}$ is called the Euclidean plane.

Also see

• Euclidean Plane is Abstract Geometry where this is shown to be an abstract geometry.

Also known as

Some authors use the term Cartesian plane instead of Euclidean plane.

Sources

• 1991: Richard S. Millman and George D. Parker: Geometry: A Metric Approach with Models (2nd ed.) ... (previous) ... (next): $\S 2.1$