where is the map and
is the distance
function. Isometries are sometimes also called congruence transformations. Two figures
that can be transformed into each other by an isometry are said to be 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.