In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping.

Table of contents

1 General definitions 2 Isometry group of Euclidean space 2.1 Glide reflections

General definitions

The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry or arcwise isometry. Both are often called just isometry and you should guess from context which one is used.

Let and be metric spaces with metrics and , a map is called distance preserving if it for any we have A distance preserving map is automatically injective.

A global isometry is a bijective distance preserving map. A path isometry or arcwise isometry is a map which preserve lengths of curves (not nesesury bijective).

As an example, the map RR defined by

Metric spaces X and Y are called isometric if there is an isometry . The set of isometries from a metric space to itself form a group with respect to compositon (called isometry group).

Isometry group of Euclidean space

In Euclidean space with the usual distance function, the (global) isometries can be characterized: there are no more than the 'expected' examples generated by rotations, reflections and translations. To put this more accurately, the isometries form a group, that is the semidirect product of the orthogonal group and the group of translations. (This group is sometimes called the Galilean group, at least for three dimensions and in relation with its role in Newtonian mechanics as expressed by permissible changes of frame of reference. See Galilean transformation.)

Glide reflections

Within the isometry group of the plane, the product of a rotation and a translation can always be expressed as a single rotation (or translation). On the other hand the product of a reflection and a translation is usually not a reflection, but can produce a transformation with no everyday name: a glide reflection.

For example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it. In co-ordinates, it takes (x,y) to (x+1,-y). It fixes a system of parallel lines, but is a combination of a reflection in a line and a translation parallel to that line. If one considers the effect of a reflection combined with any translation, it is a glide reflection with respect to a line parallel to the line of the reflection, as one sees by resolving the translation into components parallel and orthogonal to that line.

See also: congruence (geometry), similarity (mathematics).