- For the mechanical engineering and architecture usage, see isometric projection. For isometry in differential geometry, see isometry (Riemannian geometry).
In mathematics, an isometry is a distance-preserving map between metric spaces. Given a metric space, or a set and scheme for assigning distances between elements of the set, an isometry is a transformation which maps elements to another metric space such that the distance between the elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional space, two geometric figures are congruent if they are related by an isometry: related by either a rigid motion, or a composition of a rigid motion and a reflection. Up to a relation by a rigid motion, they are equal if related by a direct isometry.
Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space M involves an isometry from M into M', a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.
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 one should determine from context which one is intended.
An isometry is automatically injective. Clearly, every isometry between metric spaces is a topological embedding.
A global isometry, isometric isomorphism or congruence mapping is a bijective isometry.
Two metric spaces X and Y are called isometric if there is a bijective isometry from X to Y. The set of bijective isometries from a metric space to itself forms a group with respect to function composition, called the isometry group.
There is also the weaker notion of path isometry or arcwise isometry:
A path isometry or arcwise isometry is a map which preserves the lengths of curves (not necessarily bijective).
This is often called just isometry and one should determine from context which one is intended.
- Any reflection, translation and rotation is a global isometry on Euclidean spaces. See also Euclidean group.
- The map RR defined by is a path isometry but not an isometry.
 Linear isometry
for all v in V. Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are surjective.
- Given a positive real number ε, an ε-isometry or almost isometry (also called a Hausdorff approximation) is a map between metric spaces such that
- for x,x′ ∈ X one has |dY(ƒ(x),ƒ(x′))−dX(x,x′)| < ε, and
- for any point y ∈ Y there exists a point x ∈ X with dY(y,ƒ(x)) < ε
- That is, an ε-isometry preserves distances to within ε and leaves no element of the codomain further than ε away from the image of an element of the domain. Note that ε-isometries are not assumed to be continuous.
- The Restricted isometry property characterizes nearly isometric matrices for sparse vectors.
- Quasi-isometry is yet another useful generalization.
 See also
- Isometric projection
- Congruence (geometry)
- Euclidean plane isometry
- 3D isometries that leave the origin fixed
- Space group
- Isometries in physics
- Isometry group
- Homeomorphism group
- Partial isometry
- F. S. Beckman and D. A. Quarles, Jr., On isometries of Euclidean space, Proc. Amer. Math. Soc., 4 (1953) 810-815.