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).