View source for Isometry

{{Short description|Distance-preserving mathematical transformation}}
{{About|distance-preserving functions|other mathematical uses|isometry (disambiguation)|non-mathematical uses|Isometric (disambiguation){{!}}Isometric}}
{{Distinguish|Isometric projection}}

{{Merge from|Isometry (disambiguation)|discuss=Talk:Isometry#Proposed merge of Isometry (disambiguation) into Isometry|date=November 2024}}
[[File:Academ Reflections with parallel axis on wallpaper.svg|thumb|upright=1.4|A [[Function composition|composition]] of two [[Euclidean group#Direct and indirect isometries|opposite]] isometries is a direct isometry. [[Reflection (mathematics)|A reflection]] in a line is an opposite isometry, like  {{math|''R''<sub> 1</sub>}} or {{math|''R''<sub> 2</sub>}}  on the image. [[Translation (geometry)|Translation]] {{math|''T''}} is a direct isometry: [[Rigid body|a rigid motion]].<ref>{{harvnb|Coxeter|1969|p=46}} <p>'''3.51''' ''Any direct isometry is either a translation or a rotation. Any opposite isometry is either a reflection or a glide reflection.''</p></ref>]]

In mathematics, an '''isometry''' (or '''congruence''', or '''congruent transformation''') is a [[distance]]-preserving transformation between [[metric space]]s, usually assumed to be [[Bijection|bijective]].{{efn| name=CoxeterIsometryDef|"We shall find it convenient to use the word ''transformation'' in the special sense of a one-to-one correspondence <math>P \to P'</math> 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 {{mvar|P}} and a second member {{mvar|P'}} and that every point occurs as the first member of just one pair and also as the second member of just one pair... {{pb}} In particular, an ''isometry'' (or "congruent transformation," or "congruence") is a transformation which preserves length&nbsp;..." — Coxeter (1969) p.&nbsp;29<ref>{{harvnb|Coxeter|1969|page=29}}</ref>}} The word isometry is derived from the [[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]] known as a [[motion (geometry)|motion]].

== Introduction ==

Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a [[Transformation (geometry)|transformation]] 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]], two geometric figures are [[Congruence (geometry)|congruent]] if they are related by an isometry;{{efn|
<p>'''3.11''' ''Any two congruent triangles are related by a unique isometry.''— Coxeter (1969) p.&nbsp;39<ref>{{harvnb|Coxeter|1969|page=39}}</ref></p>
}}
the isometry that relates them is either a rigid motion (translation or rotation), or a [[Function composition|composition]] of a rigid motion and a [[Reflection (mathematics)|reflection]].<!-- commentary: I presume "they" here means the geometric figures. Still commenting out because it doesn't seem to help. --><!--They are equal, up to an action of a rigid motion, if related by a [[Euclidean group#Direct and indirect isometries|direct isometry]] (orientation preserving).-->

Isometries are often used in constructions where one space is [[Embedding|embedded]] in another space. For instance, the [[Complete space#Completion|completion]] of a metric space <math>M </math> involves an isometry from <math>M </math> into <math>M',</math> a [[quotient set]] of the space of [[Cauchy sequence]]s on <math>M.</math> 
The original space <math>M </math> is thus isometrically [[isomorphism|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 set|closed subset]] of some [[normed vector space]] and that every complete metric space is isometrically isomorphic to a closed subset of some [[Banach space]].

An isometric surjective linear operator on a [[Hilbert space]] is called a [[unitary operator]].

== Definition ==

Let  <math>X</math> and <math>Y</math> be [[metric space]]s with metrics (e.g., distances)  <math display="inline">d_X </math> and <math display="inline">d_Y.</math> A [[function (mathematics)|map]] <math display="inline">f\colon X \to Y </math> is called an '''isometry''' or '''distance-preserving map''' if for any <math>a, b \in X</math>,

:<math>d_X(a,b)=d_Y\!\left(f(a),f(b)\right).</math><ref name=Beckman-Quarles-1953>
{{cite journal
 | last1 = Beckman | first1 = F.S.
 | last2 = Quarles | first2 = D.A. Jr.
 | year = 1953
 | title = On isometries of Euclidean spaces
 | journal = [[Proceedings of the American Mathematical Society]]
 | volume = 4 | issue = 5 | pages = 810–815
 | mr = 0058193  | jstor = 2032415
 | doi=10.2307/2032415 | doi-access = free
 | url=https://www.ams.org/journals/proc/1953-004-05/S0002-9939-1953-0058193-5/S0002-9939-1953-0058193-5.pdf
}}
</ref>{{efn|
<br />Let {{mvar|T}} be a transformation (possibly many-valued) of <math>E^n</math> (<math>2\leq n < \infty</math>) into itself.<br />Let <math>d(p,q)</math> be the distance between points {{mvar|p}} and {{mvar|q}} of <math>E^n</math>, and let {{mvar|Tp}}, {{mvar|Tq}} be any images of {{mvar|p}} and {{mvar|q}}, respectively.<br />If there is a length {{mvar|a}} > 0 such that <math>d(Tp,Tq)=a</math> whenever <math>d(p,q)=a</math>, then {{mvar|T}} is a Euclidean transformation of <math>E^n</math> onto itself.<ref name=Beckman-Quarles-1953/>
}}

An isometry is automatically [[Injective function|injective]];{{efn| name=CoxeterIsometryDef}} 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., <math>d(a,b) = 0</math> if and only if <math>a=b</math>. This proof is similar to the proof that an [[order embedding]] between [[partially ordered set]]s is injective. Clearly, every isometry between metric spaces is a [[topological embedding]].

A '''global isometry''', '''isometric isomorphism''' or '''congruence mapping''' is a [[bijective]] isometry. Like any other bijection, a global isometry has a [[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 (mathematics)|set]] of bijective isometries from a metric space to itself forms a [[group (mathematics)|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 [[Arc length#Definition|lengths of curves]]; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective.<ref>{{Cite journal |last=Le Donne |first=Enrico |date=2013-10-01 |title=Lipschitz and path isometric embeddings of metric spaces |url=https://link.springer.com/article/10.1007/s10711-012-9785-2 |journal=Geometriae Dedicata |language=en |volume=166 |issue=1 |pages=47–66 |doi=10.1007/s10711-012-9785-2 |issn=1572-9168}}</ref><ref>{{Cite book |last1=Burago |first1=Dmitri |title=A course in metric geometry |last2=Burago |first2=Yurii |last3=Ivanov |first3=Sergeï |date=2001 |publisher=Providence, RI: American Mathematical Society (AMS) |isbn=0-8218-2129-6 |series=Graduate Studies in Mathematics |volume=33 |pages=86–87 |chapter=3 Constructions, §3.5 Arcwise isometries}}</ref> This term is often abridged to simply ''isometry'', so one should take care to determine from context which type is intended.

;Examples
* Any [[reflection (mathematics)|reflection]], [[translation (geometry)|translation]] and [[rotation]] is a global isometry on [[Euclidean space]]s. See also [[Euclidean group]] and {{slink|Euclidean space|Isometries}}.
* The map <math>x \mapsto |x| </math> in <math>\mathbb R </math> 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''':{{sfn | Narici|Beckenstein | 2011 | pp=275–339}}  The '''midpoint''' of two elements {{mvar|x}} and {{mvar|y}} in a vector space is the vector {{math|{{sfrac|1|2}}(''x'' + ''y'')}}. 

{{Math theorem|name=Theorem{{sfn | Narici|Beckenstein | 2011 | pp=275–339}}{{sfn | Wilansky | 2013 | pp=21–26}}|math_statement=
Let {{math|''A'' : ''X'' → ''Y''}} be a surjective isometry between [[normed space]]s that maps 0 to 0 ([[Stefan Banach]] called such maps '''rotations''') where note that {{mvar|A}} is ''not'' assumed to be a ''linear'' isometry. 
Then {{mvar|A}} maps midpoints to midpoints and is linear as a map over the real numbers <math>\mathbb{R}</math>. 
If {{mvar|X}} and {{mvar|Y}} are complex vector spaces then {{mvar|A}} may fail to be linear as a map over <math>\mathbb{C}</math>. 
}} 

=== Linear isometry ===

Given two [[normed vector space]]s <math> V </math> and <math> W ,</math> a '''linear isometry''' is a [[linear map]] <math> A : V \to W </math> that preserves the norms:
:<math>\|Av\|_W = \|v\|_V </math>
for all <math>v \in V.</math><ref name="Thomsen 2017 p125">{{cite book |last=Thomsen |first=Jesper Funch |year=2017 |title=Lineær algebra |trans-title=Linear Algebra |page=125 |lang=da |location=Århus |publisher=Aarhus University |series=Department of Mathematics}}</ref> 
Linear isometries are distance-preserving maps in the above sense. 
They are global isometries if and only if they are [[surjective]].

In an [[inner product space]], the above definition reduces to 

:<math>\langle v, v \rangle_V = \langle Av, Av \rangle_W </math>

for all <math>v \in V,</math> which is equivalent to saying that <math>A^\dagger A = \operatorname{Id}_V.</math> This also implies that isometries preserve inner products, as

:<math>\langle A u, A v \rangle_W = \langle u, A^\dagger A v \rangle_V = \langle u, v \rangle_V</math>.

Linear isometries are not always [[unitary operator]]s, though, as those require additionally that <math> V = W </math> and <math> A A^\dagger = \operatorname{Id}_V</math> (i.e. the [[Domain_of_a_function|domain]] and [[codomain]] coincide and <math> A </math> defines a [[Unitary operator|coisometry]]).

By the [[Mazur–Ulam theorem]], any isometry of normed vector spaces over <math> \mathbb{R} </math> is [[Affine transformation|affine]].

A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a [[conformal linear transformation]].

;Examples

* A [[linear map]] from <math> \mathbb{C}^n </math> to itself is an isometry (for the [[dot product]]) if and only if its matrix is [[unitary matrix|unitary]].<ref>
{{cite journal
 | last1 = Roweis | first1 = S.T.
 | last2 = Saul | first2 = L.K.
 | year = 2000
 | title = Nonlinear dimensionality reduction by locally linear embedding
 | doi = 10.1126/science.290.5500.2323
 | journal = [[Science (journal)|Science]]
 | volume = 290 | issue = 5500 | pages = 2323–2326
 | pmid =  11125150 | bibcode = 2000Sci...290.2323R
 | citeseerx = 10.1.1.111.3313
}}
</ref><ref>
{{cite journal
 |last1=Saul   |first1=Lawrence K.
 |last2=Roweis |first2=Sam T.
 |date=June 2003
 | title= Think globally, fit locally: Unsupervised learning of nonlinear manifolds
 |journal=[[Journal of Machine Learning Research]]
 |volume=4 |issue=June |pages=119–155
 |quote=Quadratic optimisation of <math>\mathbf{M}=(I-W)^\top(I-W)</math> (page 135) such that <math>\mathbf{M}\equiv YY^\top</math>
}}
</ref><ref name=Zhang-Zha-2004/><ref name=Zhang-Wang-2006/>

== Manifold ==

An isometry of a [[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 (mathematics)|metric]] on the manifold; a manifold with a (positive-definite) metric is a [[Riemannian manifold]], one with an indefinite metric is a [[pseudo-Riemannian manifold]]. Thus, isometries are studied in [[Riemannian geometry]].

A '''local isometry''' from one ([[Pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]] to another is a map which [[pullback (differential geometry)|pulls back]] the [[metric tensor]] on the second manifold to the metric tensor on the first. When such a map is also a [[diffeomorphism]], such a map is called an '''isometry''' (or '''isometric isomorphism'''), and provides a notion of [[isomorphism]] ("sameness") in the [[category theory|category]] '''Rm''' of Riemannian manifolds.

=== Definition ===

Let <math>R = (M, g) </math> and <math>R' = (M', g') </math> be two (pseudo-)Riemannian manifolds, and let <math>f : R \to R' </math> be a diffeomorphism. Then <math>f </math> is called an '''isometry''' (or '''isometric isomorphism''') if

:<math>g = f^{*} g', </math>

where <math>f^{*} g' </math> denotes the [[pullback (differential geometry)|pullback]] of the rank (0, 2) metric tensor <math>g' </math> by <math>f</math>. 
Equivalently, in terms of the [[pushforward (differential)|pushforward]] <math>f_{*},</math> we have that for any two vector fields <math>v, w </math> on <math>M </math> (i.e. sections of the [[tangent bundle]] <math>\mathrm{T} M </math>),

:<math>g(v, w) = g' \left( f_{*} v, f_{*} w \right).</math>

If <math>f </math> is a [[local diffeomorphism]] such that <math>g = f^{*} g',</math> then <math>f</math> is called a '''local isometry'''.

===Properties===
A collection of isometries typically form a group, the [[isometry group]]. When the group is a [[continuous group]], the [[Lie group|infinitesimal generators]] of the group are the [[Killing vector field]]s. 

The [[Myers–Steenrod 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]].

[[Symmetric space]]s are important examples of [[Riemannian manifold]]s that have isometries defined at every point.

==Generalizations==
* Given a positive real number ε, an '''ε-isometry''' or '''almost isometry''' (also called a '''[[Felix Hausdorff|Hausdorff]] approximation''') is a map <math>f \colon X \to Y </math> between metric spaces such that
*# for <math>x, x' \in X</math> one has <math>|d_Y(f(x),f(x')) - d_X(x,x')| < \varepsilon,</math> and
*# for any point <math>y \in Y</math> there exists a point <math>x \in X</math> with <math>d_Y(y, f(x)) < \varepsilon </math>

:That is, an {{mvar|ε}}-isometry preserves distances to within {{mvar|ε}} and leaves no element of the codomain further than {{mvar|ε}} away from the image of an element of the domain. Note that {{mvar|ε}}-isometries are not assumed to be [[continuous function|continuous]].

* The '''[[restricted isometry property]]''' characterizes nearly isometric matrices for sparse vectors.
* '''[[Quasi-isometry]]''' is yet another useful generalization.
* One may also define an element in an abstract unital C*-algebra to be an isometry: 
*:<math>a \in \mathfrak{A}</math> is an isometry if and only if <math>a^* \cdot a = 1.</math>
: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]], the term ''isometry'' means a linear bijection preserving magnitude. See also [[Quadratic form#Quadratic spaces|Quadratic spaces]].

== See also ==
{{div col begin|colwidth=15em}}
* [[Beckman–Quarles theorem]]
* {{annotated link|Conformal map}}
* [[Dual norm#The double dual of a normed linear space|The second dual of a Banach space as an isometric isomorphism]]
* [[Euclidean plane isometry]]
* [[Flat (geometry)]]
* [[Homeomorphism group]]
* [[Involution (mathematics)|Involution]]
* [[Isometry group]]
* [[Motion (geometry)]]
* [[Myers–Steenrod theorem]]
* [[Orthogonal group#3D isometries that leave the origin fixed|3D isometries that leave the origin fixed]]
* [[Partial isometry]]
* [[Scaling (geometry)]]
* [[Semidefinite embedding]]
* [[Space group]]
* [[Symmetry in mathematics]]
{{div col end}}

== Footnotes ==
{{notelist}}

== References ==
{{reflist|25em|refs=

<ref name=Zhang-Zha-2004>
{{cite journal
 |last1=Zhang |first1=Zhenyue
 |last2=Zha  |first2=Hongyuan
 |year=2004
 |title=Principal manifolds and nonlinear dimension reduction via local tangent space alignment
 |journal=SIAM Journal on Scientific Computing
 |volume=26 |issue=1 |pages=313–338
 |doi=10.1137/s1064827502419154 |citeseerx=10.1.1.211.9957
}}
</ref>

<ref name=Zhang-Wang-2006>
{{cite conference
 |last1=Zhang |first1=Zhenyue
 |last2=Wang  |first2=Jing
 |year=2006
 |title=MLLE: Modified locally linear embedding using multiple weights
 |editor1=Schölkopf, B.
 |editor2=Platt, J.
 |editor3=Hoffman, T.
 |book-title=[[Advances in Neural Information Processing Systems]]
 |series=NeurIPS Proceedings |volume=19
 |pages=1593–1600
 |conference=NIPS 2006
 |isbn=9781622760381
 |url=https://papers.nips.cc/paper/3132-mlle-modified-locally-linear-embedding-using-multiple-weights
 |quote=It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold.
}}
</ref>

}} <!-- end "refs=" -->

==Bibliography==
{{div col begin|colwidth=25em}}
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} --> 
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}}
* {{cite book |last=Coxeter|first=H. S. M.|author-link1=Harold Scott MacDonald Coxeter|title=Introduction to Geometry, Second edition|year=1969|publisher=[[John Wiley & Sons|Wiley]]|isbn=9780471504580 }}
* {{cite book | last=Lee |first= Jeffrey M. | title=Manifolds and Differential Geometry |location=Providence, RI |publisher=American Mathematical Society | year=2009 |isbn=978-0-8218-4815-9 |url=https://books.google.com/books?id=QqHdHy9WsEoC }}
{{div col end}}

{{Metric spaces}}

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[[Category:Functions and mappings]]
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[[Category:Riemannian geometry]]
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