An isometry is a map between two metric spaces that preserves distances, i.e.,
 |
where 
is the map and
is the distance
function. Every isometry is injective but need not
be surjective. A surjective isometry is an isometric
isomorphism, and isometries from a metric space into
another are also called isometric mappings or isometric embeddings.
Isometries of a space onto itself are sometimes called congruence transformations. Two figures that can be transformed into each other by such an isometry are said to be congruent (Coxeter and Greitzer 1967, p. 80).
An isometry of the plane is a linear transformation which preserves length. Isometries include rotation, translation, reflection, glides, and the identity map. Two geometric figures related by an isometry are said to be geometrically congruent (Coxeter and Greitzer 1967, p. 80).
If a plane isometry has more than one fixed point, it must be either the identity transformation or a reflection. Every isometry of period two (two applications of the transformation preserving lengths in the original configuration) is either a reflection or a half-turn rotation. Every isometry in the plane is the product of at most three reflections (at most two if there is a fixed point). Every finite group of isometries has at least one fixed point.
