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.

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.