# Isometry

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Distance-preserving mathematical transformation

This article is about distance-preserving functions. For other mathematical uses, see [isometry (disambiguation)](/source/Isometry_(disambiguation)). For non-mathematical uses, see [Isometric](/source/Isometric_(disambiguation)).

Not to be confused with [Isometric projection](/source/Isometric_projection).

A [composition](/source/Function_composition) of two [opposite](/source/Euclidean_group#Direct_and_indirect_isometries) isometries is a direct isometry. A [reflection](/source/Reflection_(mathematics)) in a line is an opposite isometry, like *R* 1 (reflection w.r.t the center diagonal line) or *R* 2 (reflection w.r.t the right diagonal line) on the image. [Translation](/source/Translation_(geometry)) *T* is a direct isometry: [a rigid motion](/source/Rigid_body).[1]

In mathematics, an **isometry** (or **congruence**, or **congruent transformation**) is a [distance](/source/Distance)-preserving [transformation](/source/Transformation_(mathematics)) between [metric spaces](/source/Metric_space), usually assumed to be [bijective](/source/Bijection).[a] The word isometry is derived from the [Ancient Greek](/source/Ancient_Greek): ἴσος *isos* meaning "equal", and μέτρον *metron* meaning "measure". If the transformation is from a metric space to itself, it is a kind of [geometric transformation](/source/Geometric_transformation) known as a [*motion*](/source/Motion_(geometry)).

## Introduction

Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a [transformation](/source/Transformation_(geometry)) which maps elements to the same or another metric space such that the distance between the image 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 [Euclidean space](/source/Euclidean_space), two geometric figures are [congruent](/source/Congruence_(geometry)) if they are related by an isometry;[b] the isometry that relates them is either a rigid motion (translation or rotation), or a [composition](/source/Function_composition) of a rigid motion and a [reflection](/source/Reflection_(mathematics)).

Isometries are often used in constructions where one space is [embedded](/source/Embedding) in another space. For instance, the [completion](/source/Complete_space#Completion) of a metric space M {\displaystyle M} involves an isometry from M {\displaystyle M} into M ′ , {\displaystyle M',} a [quotient set](/source/Quotient_set) of the space of [Cauchy sequences](/source/Cauchy_sequence) on M . {\displaystyle M.} The original space M {\displaystyle M} is thus isometrically [isomorphic](/source/Isomorphism) to a subspace of a [complete metric space](/source/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](/source/Closed_set) of some [normed vector space](/source/Normed_vector_space) and that every complete metric space is isometrically isomorphic to a closed subset of some [Banach space](/source/Banach_space).

An isometric surjective linear operator on a [Hilbert space](/source/Hilbert_space) is called a [unitary operator](/source/Unitary_operator).

## Definition

Let X {\displaystyle X} and Y {\displaystyle Y} be [metric spaces](/source/Metric_space) with metrics (e.g., distances) d X {\textstyle d_{X}} and d Y . {\textstyle d_{Y}.} A [map](/source/Function_(mathematics)) f : X → Y {\textstyle f\colon X\to Y} is called an **isometry** or **distance-preserving map** if for any a , b ∈ X {\displaystyle a,b\in X} ,

- d X ( a , b ) = d Y ( f ( a ) , f ( b ) ) . {\displaystyle d_{X}(a,b)=d_{Y}\!\left(f(a),f(b)\right).} [4][c]

An isometry is automatically [injective](/source/Injective_function);[a] otherwise two distinct points, *a* and *b*, could be mapped to the same point, thereby contradicting the coincidence axiom of the metric *d*, i.e., d ( a , b ) = 0 {\displaystyle d(a,b)=0} if and only if a = b {\displaystyle a=b} . This proof is similar to the proof that an [order embedding](/source/Order_embedding) between [partially ordered sets](/source/Partially_ordered_set) is injective. Clearly, every isometry between metric spaces is a [topological embedding](/source/Topological_embedding).

A **global isometry**, **isometric isomorphism** or **congruence mapping** is a [bijective](/source/Bijective) isometry. Like any other bijection, a global isometry has a [function inverse](/source/Function_inverse). The inverse of a global isometry is also a global isometry.

Two metric spaces *X* and *Y* are called **isometric** if there is a bijective isometry from *X* to *Y*. The [set](/source/Set_(mathematics)) of bijective isometries from a metric space to itself forms a [group](/source/Group_(mathematics)) with respect to [function composition](/source/Function_composition), called the **[isometry group](/source/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](/source/Arc_length#Definition); such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective.[5][6] This term is often abridged to simply *isometry*, so one should take care to determine from context which type is intended.

**Examples**

- Any [reflection](/source/Reflection_(mathematics)), [translation](/source/Translation_(geometry)) and [rotation](/source/Rotation) is a global isometry on [Euclidean spaces](/source/Euclidean_space). See also [Euclidean group](/source/Euclidean_group) and [Euclidean space § Isometries](/source/Euclidean_space#Isometries).

- The map x ↦ | x | {\displaystyle x\mapsto |x|} in R {\displaystyle \mathbb {R} } is a *path isometry* but not a (general) isometry. Note that unlike an isometry, this path isometry does not need to be injective.

## Isometries between normed spaces

The following theorem is due to Mazur and Ulam.

- **Definition**:[7] The **midpoint** of two elements x and y in a vector space is the vector ⁠1/2⁠(*x* + *y*).

**Theorem[7][8]**—Let *A* : *X* → *Y* be a surjective isometry between [normed spaces](/source/Normed_space) that maps 0 to 0 ([Stefan Banach](/source/Stefan_Banach) called such maps **rotations**) where note that A is *not* assumed to be a *linear* isometry. Then A maps midpoints to midpoints and is linear as a map over the real numbers R {\displaystyle \mathbb {R} } . If X and Y are complex vector spaces then A may fail to be linear as a map over C {\displaystyle \mathbb {C} } .

### Linear isometry

Given two [normed vector spaces](/source/Normed_vector_space) V {\displaystyle V} and W , {\displaystyle W,} a **linear isometry** is a [linear map](/source/Linear_map) A : V → W {\displaystyle A:V\to W} that preserves the norms:

- ‖ A v ‖ W = ‖ v ‖ V {\displaystyle \|Av\|_{W}=\|v\|_{V}}

for all v ∈ V . {\displaystyle v\in V.} [9] Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are [surjective](/source/Surjective).

In an [inner product space](/source/Inner_product_space), the above definition reduces to

- ⟨ v , v ⟩ V = ⟨ A v , A v ⟩ W {\displaystyle \langle v,v\rangle _{V}=\langle Av,Av\rangle _{W}}

for all v ∈ V , {\displaystyle v\in V,} which is equivalent to saying that A † A = Id V . {\displaystyle A^{\dagger }A=\operatorname {Id} _{V}.} This also implies that isometries preserve inner products, as

- ⟨ A u , A v ⟩ W = ⟨ u , A † A v ⟩ V = ⟨ u , v ⟩ V {\displaystyle \langle Au,Av\rangle _{W}=\langle u,A^{\dagger }Av\rangle _{V}=\langle u,v\rangle _{V}} .

Linear isometries are not always [unitary operators](/source/Unitary_operator), though, as those require additionally that V = W {\displaystyle V=W} and A A † = Id V {\displaystyle AA^{\dagger }=\operatorname {Id} _{V}} (i.e. the [domain](/source/Domain_of_a_function) and [codomain](/source/Codomain) coincide and A {\displaystyle A} defines a [coisometry](/source/Unitary_operator)).

By the [Mazur–Ulam theorem](/source/Mazur%E2%80%93Ulam_theorem), any isometry of normed vector spaces over R {\displaystyle \mathbb {R} } is [affine](/source/Affine_transformation).

A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a [conformal linear transformation](/source/Conformal_linear_transformation).

**Examples**

- A [linear map](/source/Linear_map) from C n {\displaystyle \mathbb {C} ^{n}} to itself is an isometry (for the [dot product](/source/Dot_product)) if and only if its matrix is [unitary](/source/Unitary_matrix).[10][11][12][13]

## Manifold

An isometry of a [manifold](/source/Manifold) is any (smooth) mapping of that manifold into itself, or into another manifold that preserves the notion of distance between points. The definition of an isometry requires the notion of a [metric](/source/Metric_(mathematics)) on the manifold; a manifold with a (positive-definite) metric is a [Riemannian manifold](/source/Riemannian_manifold), one with an indefinite metric is a [pseudo-Riemannian manifold](/source/Pseudo-Riemannian_manifold). Thus, isometries are studied in [Riemannian geometry](/source/Riemannian_geometry).

A **local isometry** from one ([pseudo](/source/Pseudo-Riemannian_manifold)-)[Riemannian manifold](/source/Riemannian_manifold) to another is a map which [pulls back](/source/Pullback_(differential_geometry)) the [metric tensor](/source/Metric_tensor) on the second manifold to the metric tensor on the first. When such a map is also a [diffeomorphism](/source/Diffeomorphism), such a map is called an **isometry** (or **isometric isomorphism**), and provides a notion of [isomorphism](/source/Isomorphism) ("sameness") in the [category](/source/Category_theory) **Rm** of Riemannian manifolds.

### Definition

Let R = ( M , g ) {\displaystyle R=(M,g)} and R ′ = ( M ′ , g ′ ) {\displaystyle R'=(M',g')} be two (pseudo-)Riemannian manifolds, and let f : R → R ′ {\displaystyle f:R\to R'} be a [diffeomorphism](/source/Diffeomorphism). Then f {\displaystyle f} is called an **isometry** (or **isometric isomorphism**) if

- g = f ∗ g ′ , {\displaystyle g=f^{*}g',}

where f ∗ g ′ {\displaystyle f^{*}g'} denotes the [pullback](/source/Pullback_(differential_geometry)) of the rank (0, 2) metric tensor g ′ {\displaystyle g'} by f {\displaystyle f} . Equivalently, in terms of the [pushforward](/source/Pushforward_(differential)) f ∗ , {\displaystyle f_{*},} we have that for any two vector fields v , w {\displaystyle v,w} on M {\displaystyle M} (i.e. sections of the [tangent bundle](/source/Tangent_bundle) T M {\displaystyle \mathrm {T} M} ),

- g ( v , w ) = g ′ ( f ∗ v , f ∗ w ) . {\displaystyle g(v,w)=g'\left(f_{*}v,f_{*}w\right).}

If f {\displaystyle f} is a [local diffeomorphism](/source/Local_diffeomorphism) such that g = f ∗ g ′ , {\displaystyle g=f^{*}g',} then f {\displaystyle f} is called a **local isometry**.

### Properties

A collection of isometries typically form a group, the [isometry group](/source/Isometry_group). When the group is a [continuous group](/source/Continuous_group), the [infinitesimal generators](/source/Lie_group) of the group are the [Killing vector fields](/source/Killing_vector_field).

The [Myers–Steenrod theorem](/source/Myers%E2%80%93Steenrod_theorem) states that every isometry between two connected Riemannian manifolds is smooth (differentiable). A second form of this theorem states that the isometry group of a Riemannian manifold is a [Lie group](/source/Lie_group).

[Symmetric spaces](/source/Symmetric_space) are important examples of [Riemannian manifolds](/source/Riemannian_manifold) that have isometries defined at every point.

## Generalizations

- Given a positive real number ε, an **ε-isometry** or **almost isometry** (also called a **[Hausdorff](/source/Felix_Hausdorff) approximation**) is a map f : X → Y {\displaystyle f\colon X\to Y} between metric spaces such that 1. for x , x ′ ∈ X {\displaystyle x,x'\in X} one has | d Y ( f ( x ) , f ( x ′ ) ) − d X ( x , x ′ ) | < ε , {\displaystyle |d_{Y}(f(x),f(x'))-d_{X}(x,x')|<\varepsilon ,} and 1. for any point y ∈ Y {\displaystyle y\in Y} there exists a point x ∈ X {\displaystyle x\in X} with d Y ( y , f ( x ) ) < ε {\displaystyle d_{Y}(y,f(x))<\varepsilon }

- 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](/source/Continuous_function).

- The **[restricted isometry property](/source/Restricted_isometry_property)** characterizes nearly isometric matrices for sparse vectors.

- **[Quasi-isometry](/source/Quasi-isometry)** is yet another useful generalization.

- One may also define an element in an abstract unital C*-algebra to be an isometry: - a ∈ A {\displaystyle a\in {\mathfrak {A}}} is an isometry if and only if a ∗ ⋅ a = 1. {\displaystyle a^{*}\cdot a=1.}

- Note that as mentioned in the introduction this is not necessarily a unitary element because one does not in general have that left inverse is a right inverse.

- On a [pseudo-Euclidean space](/source/Pseudo-Euclidean_space), the term *isometry* means a linear bijection preserving magnitude. See also [Quadratic spaces](/source/Quadratic_form#Quadratic_spaces).

## See also

- [Angular velocity](/source/Angular_velocity)

- [Beckman–Quarles theorem](/source/Beckman%E2%80%93Quarles_theorem)

- [Conformal map](/source/Conformal_map) – Mathematical function that preserves angles

- [The second dual of a Banach space as an isometric isomorphism](/source/Dual_norm#The_double_dual_of_a_normed_linear_space)

- [Euclidean plane isometry](/source/Euclidean_plane_isometry)

- [Flat (geometry)](/source/Flat_(geometry))

- [Homeomorphism group](/source/Homeomorphism_group)

- [Involution](/source/Involution_(mathematics))

- [Isometry group](/source/Isometry_group)

- [Motion (geometry)](/source/Motion_(geometry))

- [Myers–Steenrod theorem](/source/Myers%E2%80%93Steenrod_theorem)

- [3D isometries that leave the origin fixed](/source/Orthogonal_group#3D_isometries_that_leave_the_origin_fixed)

- [Partial isometry](/source/Partial_isometry)

- [Scaling (geometry)](/source/Scaling_(geometry))

- [Semidefinite embedding](/source/Semidefinite_embedding)

- [Space group](/source/Space_group)

- [Symmetry in mathematics](/source/Symmetry_in_mathematics)

## Footnotes

1. ^ [***a***](#cite_ref-CoxeterIsometryDef_3-0) [***b***](#cite_ref-CoxeterIsometryDef_3-1) "We shall find it convenient to use the word *transformation* in the special sense of a one-to-one correspondence P → P ′ {\displaystyle P\to P'} among all points in the plane (or in space), that is, a rule for associating pairs of points, with the understanding that each pair has a first member P and a second member P' and that every point occurs as the first member of just one pair and also as the second member of just one pair... In particular, an *isometry* (or "congruent transformation," or "congruence") is a transformation which preserves length ..." — Coxeter (1969) p. 29[2]

1. **[^](#cite_ref-5)** **3.11** *Any two congruent triangles are related by a unique isometry.*— Coxeter (1969) p. 39[3]

1. **[^](#cite_ref-7)** Let T be a transformation (possibly many-valued) of E n {\displaystyle E^{n}} ( 2 ≤ n < ∞ {\displaystyle 2\leq n<\infty } ) into itself. Let d ( p , q ) {\displaystyle d(p,q)} be the distance between points p and q of E n {\displaystyle E^{n}} , and let Tp, Tq be any images of p and q, respectively. If there is a length a > 0 such that d ( T p , T q ) = a {\displaystyle d(Tp,Tq)=a} whenever d ( p , q ) = a {\displaystyle d(p,q)=a} , then T is a Euclidean transformation of E n {\displaystyle E^{n}} onto itself.[4]

## References

1. **[^](#cite_ref-1)** [Coxeter 1969](#CITEREFCoxeter1969), p. 46 **3.51** *Any direct isometry is either a translation or a rotation. Any opposite isometry is either a reflection or a glide reflection.*

1. **[^](#cite_ref-2)** [Coxeter 1969](#CITEREFCoxeter1969), p. 29

1. **[^](#cite_ref-4)** [Coxeter 1969](#CITEREFCoxeter1969), p. 39

1. ^ [***a***](#cite_ref-Beckman-Quarles-1953_6-0) [***b***](#cite_ref-Beckman-Quarles-1953_6-1) Beckman, F.S.; Quarles, D.A. Jr. (1953). ["On isometries of Euclidean spaces"](https://www.ams.org/journals/proc/1953-004-05/S0002-9939-1953-0058193-5/S0002-9939-1953-0058193-5.pdf) (PDF). *[Proceedings of the American Mathematical Society](/source/Proceedings_of_the_American_Mathematical_Society)*. **4** (5): 810–815. [doi](/source/Doi_(identifier)):[10.2307/2032415](https://doi.org/10.2307%2F2032415). [JSTOR](/source/JSTOR_(identifier)) [2032415](https://www.jstor.org/stable/2032415). [MR](/source/MR_(identifier)) [0058193](https://mathscinet.ams.org/mathscinet-getitem?mr=0058193).

1. **[^](#cite_ref-8)** Le Donne, Enrico (2013-10-01). ["Lipschitz and path isometric embeddings of metric spaces"](https://link.springer.com/article/10.1007/s10711-012-9785-2). *Geometriae Dedicata*. **166** (1): 47–66. [doi](/source/Doi_(identifier)):[10.1007/s10711-012-9785-2](https://doi.org/10.1007%2Fs10711-012-9785-2). [ISSN](/source/ISSN_(identifier)) [1572-9168](https://search.worldcat.org/issn/1572-9168).

1. **[^](#cite_ref-9)** Burago, Dmitri; Burago, Yurii; Ivanov, Sergeï (2001). "3 Constructions, §3.5 Arcwise isometries". *A course in metric geometry*. Graduate Studies in Mathematics. Vol. 33. Providence, RI: American Mathematical Society (AMS). pp. 86–87. [ISBN](/source/ISBN_(identifier)) [0-8218-2129-6](https://en.wikipedia.org/wiki/Special:BookSources/0-8218-2129-6).

1. ^ [***a***](#cite_ref-FOOTNOTENariciBeckenstein2011275–339_10-0) [***b***](#cite_ref-FOOTNOTENariciBeckenstein2011275–339_10-1) [Narici & Beckenstein 2011](#CITEREFNariciBeckenstein2011), pp. 275–339.

1. **[^](#cite_ref-FOOTNOTEWilansky201321–26_11-0)** [Wilansky 2013](#CITEREFWilansky2013), pp. 21–26.

1. **[^](#cite_ref-Thomsen_2017_p125_12-0)** Thomsen, Jesper Funch (2017). *Lineær algebra* [*Linear Algebra*]. Department of Mathematics (in Danish). Århus: Aarhus University. p. 125.

1. **[^](#cite_ref-13)** Roweis, S.T.; Saul, L.K. (2000). "Nonlinear dimensionality reduction by locally linear embedding". *[Science](/source/Science_(journal))*. **290** (5500): 2323–2326. [Bibcode](/source/Bibcode_(identifier)):[2000Sci...290.2323R](https://ui.adsabs.harvard.edu/abs/2000Sci...290.2323R). [CiteSeerX](/source/CiteSeerX_(identifier)) [10.1.1.111.3313](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.111.3313). [doi](/source/Doi_(identifier)):[10.1126/science.290.5500.2323](https://doi.org/10.1126%2Fscience.290.5500.2323). [PMID](/source/PMID_(identifier)) [11125150](https://pubmed.ncbi.nlm.nih.gov/11125150).

1. **[^](#cite_ref-14)** Saul, Lawrence K.; Roweis, Sam T. (June 2003). "Think globally, fit locally: Unsupervised learning of nonlinear manifolds". *[Journal of Machine Learning Research](/source/Journal_of_Machine_Learning_Research)*. **4** (June): 119–155. Quadratic optimisation of M = ( I − W ) ⊤ ( I − W ) {\displaystyle \mathbf {M} =(I-W)^{\top }(I-W)} (page 135) such that M ≡ Y Y ⊤ {\displaystyle \mathbf {M} \equiv YY^{\top }}

1. **[^](#cite_ref-Zhang-Zha-2004_15-0)** Zhang, Zhenyue; Zha, Hongyuan (2004). "Principal manifolds and nonlinear dimension reduction via local tangent space alignment". *SIAM Journal on Scientific Computing*. **26** (1): 313–338. [CiteSeerX](/source/CiteSeerX_(identifier)) [10.1.1.211.9957](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.211.9957). [doi](/source/Doi_(identifier)):[10.1137/s1064827502419154](https://doi.org/10.1137%2Fs1064827502419154).

1. **[^](#cite_ref-Zhang-Wang-2006_16-0)** Zhang, Zhenyue; Wang, Jing (2006). ["MLLE: Modified locally linear embedding using multiple weights"](https://papers.nips.cc/paper/3132-mlle-modified-locally-linear-embedding-using-multiple-weights). In Schölkopf, B.; Platt, J.; Hoffman, T. (eds.). *[Advances in Neural Information Processing Systems](/source/Advances_in_Neural_Information_Processing_Systems)*. NIPS 2006. NeurIPS Proceedings. Vol. 19. pp. 1593–1600. [ISBN](/source/ISBN_(identifier)) [9781622760381](https://en.wikipedia.org/wiki/Special:BookSources/9781622760381). It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold.

## Bibliography

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- Lee, Jeffrey M. (2009). [*Manifolds and Differential Geometry*](https://books.google.com/books?id=QqHdHy9WsEoC). Providence, RI: American Mathematical Society. [ISBN](/source/ISBN_(identifier)) [978-0-8218-4815-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-4815-9).

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