# Euclidean plane

The **Euclidean plane** is the plane that is the object of study in Euclidean geometry (high-school geometry). The plane and the geometry are named after the ancient-Greek mathematician Euclid.

The Euclidean plane is a collection of points *P*, *Q*, *R*, ... between which a distance ρ is defined, with the properties,

- ρ(
*P*,*Q*) ≥ 0 and ρ(*P*,*Q*) = 0 if and only if*P*=*Q* - ρ(
*P*,*Q*) = ρ(*Q*,*P*) - ρ(
*P*,*Q*) ≤ ρ(*P*,*R*) + ρ(*R*,*Q*) (triangular inequality).

As is known from Euclidean geometry lines can be drawn between points and different geometric figures (triangles, squares, etc.) can be constructed. A geometric figure can be translated and rotated without change of shape. Such a map is called a *rigid motion* of the figure. The totality of rigid motions form a group of infinite order, the Euclidean group in two dimensions, often written as *E*(2).

Formally, the Euclidean plane is a 2-dimensional affine space with inner product.