The identity is an isometry; nothing changes, so distance cannot change.

The corresponding map is an isometry but in general not onto.

Only the trivial isometry group leaves the whole line fixed.

It is the only isometry which belongs to more than one of the types described above.

If the isometry is odd, use the mirror; otherwise do not.

Consider now an isometry A whose range is not necessarily dense.

See also fixed points of isometry groups in Euclidean space.

In geometry, a motion is an isometry of a metric space.

The concept of partial isometry can be defined in other equivalent ways.

The idea of the proof is to create a global isometry between and .

Suppose for the moment that U is an isometry.

This shift operator is an isometry, therefore bounded below by 1.

That means that is a local isometry between and .

This explains why weights are introduced (to make the mapping an isometry).

An isometry of the cube can be identified in various ways:

Let be the subgroup of the isometry group generated by .

It is a subgroup of the isometry group of the space concerned.

An isometry is completely determined by its effect on three independent (not collinear) points.

In a Euclidean space, any translation is an isometry.

Differently put, the correspondence provides an isometry, locally, between the two surfaces.

Both are often called just isometry and one should determine from context which one is intended.

The 2-4 tree isometry was described in 1978 by Sedgewick.

For some isometry pairs composition does not depend on order:

A unitary transformation is an isometry, as one can see by setting in this formula.

This defines an isometry onto a dense subspace, as required.