Transformational Geometry Introduction

A transformation of the plane is an invertible function whose domain and range are the whole Euclidean plane. Using the term transformation instead of function or mapping indicates that geometric properties are of primary interest. An isometry is a special type of transformation of the plane that preserves distances. Two figures that can be transformed into each other by an isometry are congruent figures. For this reason, isometries are sometimes referred to as congruence mappings. Congruence mapping is a distance-preserving isomorphism between metric spaces.

There are four Euclidean isometries: reflection, translation, rotation, and glide reflection. The glide reflection is composed of a reflection followed by a translation along the axis of reflection. It is the only isometry that requires two steps.

Not all transformations of the plane are isometries, however. A dilation is a transformation that preserves shape but not size. So two figures that can be transformed into each other by a dilation are similar.