Isometry: Difference between revisions
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{{Short description|Distance-preserving mathematical transformation}} |
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{{About|distance-preserving functions|other mathematical uses|isometry (disambiguation)|non-mathematical uses|Isometric (disambiguation){{!}}Isometric}} |
{{About|distance-preserving functions|other mathematical uses|isometry (disambiguation)|non-mathematical uses|Isometric (disambiguation){{!}}Isometric}} |
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{{Distinguish|Isometric projection}} |
{{Distinguish|Isometric projection}} |
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{{refimprove|date=June 2016}} |
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[[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>]] |
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In [[mathematics]], an '''isometry''' (or '''[[congruence (geometry)|congruence]]''', or '''congruent transformation''') is a [[distance]]-preserving transformation between [[metric spaces]], usually assumed to be [[Bijection|bijective]].<ref name=CoxeterIsometryDef>{{harvnb|Coxeter|1969|page=29}} <p>"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...<p>In particular, an ''isometry'' (or "congruent transformation," or "congruence") is a transformation which preserves length..."</ref> |
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[[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.''</ref>]] |
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==Introduction== |
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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;<ref>{{harvnb|Coxeter|1969|page=39}}<p>'''3.11''' ''Any two congruent triangles are related by a unique isometry.''</ref> 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).--> |
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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 ..." — Coxeter (1969) p. 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]]. |
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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 ''M'' involves an isometry from ''M'' into ''M''', a [[quotient set]] of the space of [[Cauchy sequence]]s on ''M''. The original space ''M'' 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]]. |
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== Introduction == |
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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. |
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In a two-dimensional or three-dimensional [[Euclidean space]], two geometric figures are [[Congruence (geometry)|congruent]] if they are related by an isometry;{{efn| |
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<p>'''3.11''' ''Any two congruent triangles are related by a unique isometry.''— Coxeter (1969) p. 39<ref>{{harvnb|Coxeter|1969|page=39}}</ref></p> |
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}} |
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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).--> |
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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> |
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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. |
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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]]. |
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An isometric surjective linear operator on a [[Hilbert space]] is called a [[unitary operator]]. |
An isometric surjective linear operator on a [[Hilbert space]] is called a [[unitary operator]]. |
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== |
== Definition == |
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Let |
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>, |
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:<math>d_Y\left(f(a),f(b)\right |
:<math>d_X(a,b)=d_Y\!\left(f(a),f(b)\right).</math><ref name=Beckman-Quarles-1953> |
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{{cite journal |
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| last1 = Beckman | first1 = F. S. |
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| |
| last1 = Beckman | first1 = F.S. |
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| last2 = Quarles | first2 = D.A. Jr. |
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| journal = [[Proceedings of the American Mathematical Society]] |
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| mr = 0058193 |
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| pages = 810–815 |
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| title = On isometries of Euclidean spaces |
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| volume = 4 |
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| year = 1953 |
| year = 1953 |
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| title = On isometries of Euclidean spaces |
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| doi=10.2307/2032415 |
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| journal = [[Proceedings of the American Mathematical Society]] |
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| url=http://www.ams.org/journals/proc/1953-004-05/S0002-9939-1953-0058193-5/S0002-9939-1953-0058193-5.pdf |
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| volume = 4 | issue = 5 | pages = 810–815 |
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| quote=<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> |
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| mr = 0058193 | jstor = 2032415 |
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| doi=10.2307/2032415 | doi-access = free |
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| url=https://www.ams.org/journals/proc/1953-004-05/S0002-9939-1953-0058193-5/S0002-9939-1953-0058193-5.pdf |
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}} |
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</ref>{{efn| |
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<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/> |
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}} |
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An isometry is automatically [[Injective function|injective]]; |
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]]. |
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A '''global isometry''', '''isometric isomorphism''' or '''congruence mapping''' is a [[bijective]] isometry. Like any other bijection, a global isometry has a [[function inverse]]. |
A '''global isometry''', '''isometric isomorphism''' or '''congruence mapping''' is a [[bijective]] isometry. Like any other bijection, a global isometry has a [[function inverse]]. |
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The inverse of a global isometry is also a global isometry. |
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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]]'''. |
Two metric spaces ''X'' and ''Y'' are called '''isometric''' if there is a bijective isometry from ''X'' to ''Y''. |
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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]]'''. |
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There is also the weaker notion of ''path isometry'' or ''arcwise isometry'': |
There is also the weaker notion of ''path isometry'' or ''arcwise isometry'': |
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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. This term is often abridged to simply ''isometry'', so one should take care to determine from context which type is intended. |
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. |
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This term is often abridged to simply ''isometry'', so one should take care to determine from context which type is intended. |
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;Examples |
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* Any [[reflection (mathematics)|reflection]], [[translation (geometry)|translation]] and [[rotation]] is a global isometry on Euclidean |
* 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}}. |
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*The map <math> |
* 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. |
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*The isometric [[linear map]]s from '''C'''<sup>''n''</sup> to itself are given by the [[unitary matrix|unitary matrices]].<ref>{{Cite journal | last1 = Roweis | first1 = S. T. | last2 = Saul | first2 = L. K. | 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 | year = 2000 | pmid = 11125150| pmc = }}</ref><ref>{{cite journal|last1=Saul |first1=Lawrence K. |last2=Roweis |first2=Sam T. | location=http://jmlr.org/papers/v4/saul03a.html| title= Think globally, fit locally: [[Unsupervised learning]] of [[Differentiable manifolds|nonlinear manifolds]] |journal=[[Journal of Machine Learning Research]]|volume=4|issue=June|pages=119-155|year=2003|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>{{Cite journal |last=Zhang |first=Zhenyue |last2=Zha|first2=Hongyuan |title=Principal Manifolds and [[nonlinear dimensionality reduction|Nonlinear Dimension Reduction]] via [[Local Tangent Space Alignment]] |journal=SIAM Journal on Scientific Computing |volume=26 |issue=1 |year=2004 |pages=313–338 |doi=10.1137/s1064827502419154}}</ref><ref>{{cite journal|last1=Zhang|first1=Zhenyue|last2=Wang|first2=Jing|title=MLLE: Modified Locally Linear Embedding Using Multiple Weights|journal=[[Advances in Neural Information Processing Systems]]|year=2006|volume=19|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> |
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== Isometries between normed spaces == |
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==Linear isometry== |
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The following theorem is due to Mazur and Ulam. |
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Given two [[normed vector space]]s ''V'' and ''W'', a '''linear isometry''' is a [[linear map]] ''f'' : ''V'' → ''W'' that preserves the norms: |
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:<math>\|f(v)\| = \|v\|</math> |
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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]]. |
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:'''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'')}}. |
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By the [[Mazur-Ulam theorem]], any isometry of normed vector spaces over '''R''' is [[Affine transformation|affine]]. |
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{{Math theorem|name=Theorem{{sfn | Narici|Beckenstein | 2011 | pp=275–339}}{{sfn | Wilansky | 2013 | pp=21–26}}|math_statement= |
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In an inner product space, the fact that any linear isometry is an orthogonal transformation can be shown by using polarization to |
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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. |
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prove ''<Ax, Ay> = <x, y>'' and then applying the Riesz representation theorem. |
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Then {{mvar|A}} maps midpoints to midpoints and is linear as a map over the real numbers <math>\mathbb{R}</math>. |
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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>. |
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}} |
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=== Linear isometry === |
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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: |
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:<math>\|Av\|_W = \|v\|_V </math> |
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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> |
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Linear isometries are distance-preserving maps in the above sense. |
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They are global isometries if and only if they are [[surjective]]. |
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In an [[inner product space]], the above definition reduces to |
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:<math>\langle v, v \rangle_V = \langle Av, Av \rangle_W </math> |
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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 |
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:<math>\langle A u, A v \rangle_W = \langle u, A^\dagger A v \rangle_V = \langle u, v \rangle_V</math>. |
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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]]). |
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By the [[Mazur–Ulam theorem]], any isometry of normed vector spaces over <math> \mathbb{R} </math> is [[Affine transformation|affine]]. |
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A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a [[conformal linear transformation]]. |
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;Examples |
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* 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> |
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{{cite journal |
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| last1 = Roweis | first1 = S.T. |
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| last2 = Saul | first2 = L.K. |
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| year = 2000 |
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| title = Nonlinear dimensionality reduction by locally linear embedding |
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| doi = 10.1126/science.290.5500.2323 |
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| journal = [[Science (journal)|Science]] |
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| volume = 290 | issue = 5500 | pages = 2323–2326 |
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| pmid = 11125150 | citeseerx = 10.1.1.111.3313 |
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}} |
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</ref><ref> |
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{{cite journal |
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|last1=Saul |first1=Lawrence K. |
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|last2=Roweis |first2=Sam T. |
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|date=June 2003 |
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| title= Think globally, fit locally: Unsupervised learning of nonlinear manifolds |
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|journal=[[Journal of Machine Learning Research]] |
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|volume=4 |issue=June |pages=119–155 |
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|quote=Quadratic optimisation of <math>\mathbf{M}=(I-W)^\top(I-W)</math> (page 135) such that <math>\mathbf{M}\equiv YY^\top</math> |
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}} |
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</ref><ref name=Zhang-Zha-2004/><ref name=Zhang-Wang-2006/> |
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== Manifold == |
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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. |
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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]]. |
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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. |
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=== Definition === |
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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 |
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:<math>g = f^{*} g', </math> |
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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>. |
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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>), |
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:<math>g(v, w) = g' \left( f_{*} v, f_{*} w \right).</math> |
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If <math>f </math> is a [[local diffeomorphism]] such that <math>g = f^{*} g',</math> then <math>f</math> is called a '''local isometry'''. |
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===Properties=== |
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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. |
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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]]. |
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[[Riemannian manifold]]s that have isometries defined at every point are called [[symmetric space]]s. |
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==Generalizations== |
==Generalizations== |
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* Given a positive real number ε, an '''ε-isometry''' or '''almost isometry''' (also called a '''[[Felix Hausdorff|Hausdorff]] approximation''') is a map <math>f |
* 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 |
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*# for |
*# for <math>x, x' \in X</math> one has <math>|d_Y(f(x),f(x')) - d_X(x,x')| < \varepsilon,</math> and |
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*# for any point |
*# 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> |
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: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. |
: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]]. |
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*The '''[[restricted isometry property]]''' characterizes nearly isometric matrices for sparse vectors. |
* The '''[[restricted isometry property]]''' characterizes nearly isometric matrices for sparse vectors. |
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*'''[[Quasi-isometry]]''' is yet another useful generalization. |
* '''[[Quasi-isometry]]''' is yet another useful generalization. |
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* One may also define an element in an abstract unital C*-algebra to be an isometry: |
* One may also define an element in an abstract unital C*-algebra to be an isometry: |
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*:<math> |
*:<math>a \in \mathfrak{A}</math> is an isometry if and only if <math>a^* \cdot a = 1.</math> |
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: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. |
: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. |
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*On a [[pseudo-Euclidean space]], the term ''isometry'' means a linear bijection preserving magnitude. See also [[Quadratic form#Quadratic spaces|Quadratic spaces]]. |
* On a [[pseudo-Euclidean space]], the term ''isometry'' means a linear bijection preserving magnitude. See also [[Quadratic form#Quadratic spaces|Quadratic spaces]]. |
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==See also== |
== See also == |
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{{div col begin|colwidth=15em}} |
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*[[Motion (geometry)]] |
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*[[ |
* [[Beckman–Quarles theorem]] |
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* {{annotated link|Conformal map}} |
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*[[Semidefinite embedding]] |
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* [[Dual norm#The double dual of a normed linear space|The second dual of a Banach space as an isometric isomorphism]] |
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*[[Flat (geometry)]] |
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*[[Euclidean plane isometry]] |
* [[Euclidean plane isometry]] |
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* [[Flat (geometry)]] |
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*[[Orthogonal group#3D isometries that leave the origin fixed|3D isometries that leave the origin fixed]] |
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*[[ |
* [[Homeomorphism group]] |
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*[[Involution (mathematics)|Involution]] |
* [[Involution (mathematics)|Involution]] |
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*[[ |
* [[Isometry group]] |
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* [[Motion (geometry)]] |
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*[[Homeomorphism group]] |
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* [[Myers–Steenrod theorem]] |
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*[[Partial isometry]] |
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* [[Orthogonal group#3D isometries that leave the origin fixed|3D isometries that leave the origin fixed]] |
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*[[Dual norm#The second dual of a Banach space| The second dual of a Banach space as an isometric isomorphism]] |
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* [[Partial isometry]] |
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* [[Scaling (geometry)]] |
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* [[Semidefinite embedding]] |
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* [[Space group]] |
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* [[Symmetry in mathematics]] |
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{{div col end}} |
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== |
== Footnotes == |
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{{ |
{{notelist}} |
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== References == |
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{{reflist|25em|refs= |
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<ref name=Zhang-Zha-2004> |
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{{cite journal |
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|last=Zhang |first=Zhenyue |
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|last2=Zha |first2=Hongyuan |
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|year=2004 |
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|title=Principal manifolds and nonlinear dimension reduction via local tangent space alignment |
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|journal=SIAM Journal on Scientific Computing |
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|volume=26 |issue=1 |pages=313–338 |
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|doi=10.1137/s1064827502419154 |citeseerx=10.1.1.211.9957 |
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}} |
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</ref> |
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<ref name=Zhang-Wang-2006> |
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{{cite conference |
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|last1=Zhang |first1=Zhenyue |
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|last2=Wang |first2=Jing |
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|year=2006 |
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|title=MLLE: Modified locally linear embedding using multiple weights |
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|editor1=Schölkopf, B. |
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|editor2=Platt, J. |
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|editor3=Hoffman, T. |
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|book-title=[[Advances in Neural Information Processing Systems]] |
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|series=NeurIPS Proceedings |volume=19 |
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|pages=1593–1600 |
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|conference=NIPS 2006 |
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|ISBN=9781622760381 |
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|url=https://papers.nips.cc/paper/3132-mlle-modified-locally-linear-embedding-using-multiple-weights |
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|quote=It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold. |
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}} |
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</ref> |
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}} <!-- end "refs=" --> |
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==Bibliography== |
==Bibliography== |
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{{div col begin|colwidth=25em}} |
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* {{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|ref=harv}} |
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* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} --> |
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* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} --> |
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* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} --> |
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* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Trèves | 2006 | p=}} --> |
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* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} |
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* {{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 }} |
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* {{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 }} |
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{{div col end}} |
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[[Category:Functions and mappings]] |
[[Category:Functions and mappings]] |
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[[Category:Metric geometry]] |
[[Category:Metric geometry]] |
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[[Category:Symmetry]] |
[[Category:Symmetry]] |
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[[Category:Equivalence (mathematics)]] |
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[[Category:Riemannian geometry]] |
Latest revision as of 20:34, 29 July 2024
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.[a] 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.
Introduction
[edit]Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a 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 congruent if they are related by an isometry;[b] the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection.
Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space involves an isometry from into a quotient set of the space of Cauchy sequences on The original space 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.
An isometric surjective linear operator on a Hilbert space is called a unitary operator.
Definition
[edit]Let and be metric spaces with metrics (e.g., distances) and A map is called an isometry or distance-preserving map if for any ,
An isometry is automatically injective;[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., if and only if . This proof is similar to the proof that an order embedding between partially ordered sets 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 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; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective. This term is often abridged to simply isometry, so one should take care to determine from context which type is intended.
- Examples
- Any reflection, translation and rotation is a global isometry on Euclidean spaces. See also Euclidean group and Euclidean space § Isometries.
- The map in 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
[edit]The following theorem is due to Mazur and Ulam.
- Definition:[5] The midpoint of two elements x and y in a vector space is the vector 1/2(x + y).
Theorem[5][6] — Let A : X → Y be a surjective isometry between normed spaces that maps 0 to 0 (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 . If X and Y are complex vector spaces then A may fail to be linear as a map over .
Linear isometry
[edit]Given two normed vector spaces and a linear isometry is a linear map that preserves the norms:
for all [7] 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
for all which is equivalent to saying that This also implies that isometries preserve inner products, as
- .
Linear isometries are not always unitary operators, though, as those require additionally that and (i.e. the domain and codomain coincide and defines a coisometry).
By the Mazur–Ulam theorem, any isometry of normed vector spaces over is affine.
A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a conformal linear transformation.
- Examples
- A linear map from to itself is an isometry (for the dot product) if and only if its matrix is unitary.[8][9][10][11]
Manifold
[edit]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 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 to another is a map which 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 Rm of Riemannian manifolds.
Definition
[edit]Let and be two (pseudo-)Riemannian manifolds, and let be a diffeomorphism. Then is called an isometry (or isometric isomorphism) if
where denotes the pullback of the rank (0, 2) metric tensor by . Equivalently, in terms of the pushforward we have that for any two vector fields on (i.e. sections of the tangent bundle ),
If is a local diffeomorphism such that then is called a local isometry.
Properties
[edit]A collection of isometries typically form a group, the isometry group. When the group is a continuous group, the infinitesimal generators of the group are the Killing vector fields.
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.
Riemannian manifolds that have isometries defined at every point are called symmetric spaces.
Generalizations
[edit]- 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 one has and
- for any point there exists a point with
- 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.
- One may also define an element in an abstract unital C*-algebra to be an isometry:
- is an isometry if and only if
- 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 spaces.
See also
[edit]- Beckman–Quarles theorem
- Conformal map – Mathematical function which preserves angles
- The second dual of a Banach space as an isometric isomorphism
- Euclidean plane isometry
- Flat (geometry)
- Homeomorphism group
- Involution
- Isometry group
- Motion (geometry)
- Myers–Steenrod theorem
- 3D isometries that leave the origin fixed
- Partial isometry
- Scaling (geometry)
- Semidefinite embedding
- Space group
- Symmetry in mathematics
Footnotes
[edit]- ^ a b "We shall find it convenient to use the word transformation in the special sense of a one-to-one correspondence 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]
- ^
3.11 Any two congruent triangles are related by a unique isometry.— Coxeter (1969) p. 39[3]
- ^
Let T be a transformation (possibly many-valued) of () into itself.
Let be the distance between points p and q of , and let Tp, Tq be any images of p and q, respectively.
If there is a length a > 0 such that whenever , then T is a Euclidean transformation of onto itself.[4]
References
[edit]- ^ Coxeter 1969, 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.
- ^ Coxeter 1969, p. 29
- ^ Coxeter 1969, p. 39
- ^ a b Beckman, F.S.; Quarles, D.A. Jr. (1953). "On isometries of Euclidean spaces" (PDF). Proceedings of the American Mathematical Society. 4 (5): 810–815. doi:10.2307/2032415. JSTOR 2032415. MR 0058193.
- ^ a b Narici & Beckenstein 2011, pp. 275–339.
- ^ Wilansky 2013, pp. 21–26.
- ^ Thomsen, Jesper Funch (2017). Lineær algebra [Linear Algebra]. Department of Mathematics (in Danish). Århus: Aarhus University. p. 125.
- ^ Roweis, S.T.; Saul, L.K. (2000). "Nonlinear dimensionality reduction by locally linear embedding". Science. 290 (5500): 2323–2326. CiteSeerX 10.1.1.111.3313. doi:10.1126/science.290.5500.2323. PMID 11125150.
- ^
Saul, Lawrence K.; Roweis, Sam T. (June 2003). "Think globally, fit locally: Unsupervised learning of nonlinear manifolds". Journal of Machine Learning Research. 4 (June): 119–155.
Quadratic optimisation of (page 135) such that
- ^ 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 10.1.1.211.9957. doi:10.1137/s1064827502419154.
- ^
Zhang, Zhenyue; Wang, Jing (2006). "MLLE: Modified locally linear embedding using multiple weights". In Schölkopf, B.; Platt, J.; Hoffman, T. (eds.). Advances in Neural Information Processing Systems. NIPS 2006. NeurIPS Proceedings. Vol. 19. pp. 1593–1600. ISBN 9781622760381.
It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold.
Bibliography
[edit]- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
- Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.
- Coxeter, H. S. M. (1969). Introduction to Geometry, Second edition. Wiley. ISBN 9780471504580.
- Lee, Jeffrey M. (2009). Manifolds and Differential Geometry. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4815-9.