Definition:Euclidean Plane

Definition
For any real number $a$ let:


 * $L_a = \left\{{ \left({x, y}\right) \in \R^2: x = a }\right\}$

Furthermore, define:


 * $L_A = \left\{{L_a: a \in \R }\right\}$

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


 * $L_{m,b} = \left\{{ \left({x, y}\right) \in \R^2: y = m x + b }\right\}$

Furthermore, define:


 * $L_{M,B} = \left\{{ L_{m,b}: m,b \in \R }\right\}$

Finally let:


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

The abstract geometry $\left({\R^2, L_E}\right)$ is called the Euclidean Plane.

This is shown to be an abstract geometry in Euclidean Plane is Abstract Geometry.

Also known as
Some authors use the term Cartesian plane instead of Euclidean plane.