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Isometry

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(Redirected fromOrthonormal transformation)

Distance-preserving mathematical transformation

This article is about distance-preserving functions. For other mathematical uses, seeisometry (disambiguation). For non-mathematical uses, seeIsometric.

Not to be confused withIsometric projection.

Acomposition of twoopposite isometries is a direct isometry. Areflection in a line is an opposite isometry, likeR₁ (reflection w.r.t the center diagonal line) orR₂ (reflection w.r.t the right diagonal line) on the image.TranslationT is a direct isometry:a rigid motion.^[1]

In mathematics, anisometry (orcongruence, orcongruent transformation) is adistance-preservingtransformation betweenmetric spaces, usually assumed to bebijective.^[a] The word isometry is derived from theAncient Greek: ἴσοςisos meaning "equal", and μέτρονmetron meaning "measure". If the transformation is from a metric space to itself, it is a kind ofgeometric transformation known as amotion.

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 atransformation 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-dimensionalEuclidean space, two geometric figures arecongruent if they are related by an isometry;^[b]the isometry that relates them is either a rigid motion (translation or rotation), or acomposition of a rigid motion and areflection.

Isometries are often used in constructions where one space isembedded in another space. For instance, thecompletion of a metric space $M {\displaystyle M}$ involves an isometry from $M {\displaystyle M}$ into $M^{'}, {\displaystyle M',}$ aquotient set of the space ofCauchy sequences on $M . {\displaystyle M.}$ The original space $M {\displaystyle M}$ is thus isometricallyisomorphic to a subspace of acomplete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to aclosed subset of somenormed vector space and that every complete metric space is isometrically isomorphic to a closed subset of someBanach space.

An isometric surjective linear operator on aHilbert space is called aunitary operator.

Definition

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Let $X {\displaystyle X}$ and $Y {\displaystyle Y}$ bemetric spaces with metrics (e.g., distances) ${\textstyle d_{X}}$ and ${\textstyle d_{Y}.}$ Amap ${\textstyle f\colon X\to Y}$ is called anisometry ordistance-preserving map if for any $a,b\in X$ ,

d_{X}(a,b)=d_{Y}\!\left(f(a),f(b)\right).

^[4]^[c]

An isometry is automaticallyinjective;^[a] otherwise two distinct points,a andb, could be mapped to the same point, thereby contradicting the coincidence axiom of the metricd, i.e., $d(a,b)=0$ if and only if $a=b$ . This proof is similar to the proof that anorder embedding betweenpartially ordered sets is injective. Clearly, every isometry between metric spaces is atopological embedding.

Aglobal isometry,isometric isomorphism orcongruence mapping is abijective isometry. Like any other bijection, a global isometry has afunction inverse. The inverse of a global isometry is also a global isometry.

Two metric spacesX andY are calledisometric if there is a bijective isometry fromX toY. Theset of bijective isometries from a metric space to itself forms agroup with respect tofunction composition, called theisometry group.

There is also the weaker notion ofpath isometry orarcwise isometry:

Apath isometry orarcwise isometry is a map which preserves thelengths 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.^[5]^[6] This term is often abridged to simplyisometry, so one should take care to determine from context which type is intended.

Examples

Anyreflection,translation androtation is a global isometry onEuclidean spaces. See alsoEuclidean group andEuclidean space § Isometries.
The map $x\mapsto |x|$ in $\mathbb {R}$ is apath isometry but not a (general) isometry. Note that unlike an isometry, this path isometry does not need to be injective.

Isometries between normed spaces

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The following theorem is due to Mazur and Ulam.

Definition:^[7] Themidpoint of two elementsx andy in a vector space is the vector⁠1/2⁠(x +y).

Theorem^[7]^[8]—LetA :X →Y be a surjective isometry betweennormed spaces that maps 0 to 0 (Stefan Banach called such mapsrotations) where note thatA isnot assumed to be alinear isometry. ThenA maps midpoints to midpoints and is linear as a map over the real numbers $\mathbb {R}$ . IfX andY are complex vector spaces thenA may fail to be linear as a map over $\mathbb {C}$ .

Linear isometry

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Given twonormed vector spaces $V {\displaystyle V}$ and $W, {\displaystyle W,}$ alinear isometry is alinear map $A:V\to W$ that preserves the norms:

\|Av\|_{W}=\|v\|_{V}

for all $v\in V.$ ^[9] Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they aresurjective.

In aninner product space, the above definition reduces to

\langle v,v\rangle _{V}=\langle Av,Av\rangle _{W}

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

\langle Au,Av\rangle _{W}=\langle u,A^{\dagger }Av\rangle _{V}=\langle u,v\rangle _{V}

Linear isometries are not alwaysunitary operators, though, as those require additionally that $V=W$ and $AA^{\dagger }=\operatorname {Id} _{V}$ (i.e. thedomain andcodomain coincide and $A {\displaystyle A}$ defines acoisometry).

By theMazur–Ulam theorem, any isometry of normed vector spaces over $\mathbb {R}$ isaffine.

A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is aconformal linear transformation.

Examples

Alinear map from $\mathbb {C} ^{n}$ to itself is an isometry (for thedot product) if and only if its matrix isunitary.^[10]^[11]^[12]^[13]

Manifold

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An isometry of amanifold 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 ametric on the manifold; a manifold with a (positive-definite) metric is aRiemannian manifold, one with an indefinite metric is apseudo-Riemannian manifold. Thus, isometries are studied inRiemannian geometry.

Alocal isometry from one (pseudo-)Riemannian manifold to another is a map whichpulls back themetric tensor on the second manifold to the metric tensor on the first. When such a map is also adiffeomorphism, such a map is called anisometry (orisometric isomorphism), and provides a notion ofisomorphism ("sameness") in thecategoryRm of Riemannian manifolds.

Definition

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Let $R=(M,g)$ and $R'=(M',g')$ be two (pseudo-)Riemannian manifolds, and let $f:R\to R'$ be adiffeomorphism. Then $f {\displaystyle f}$ is called anisometry (orisometric isomorphism) if

g=f^{*}g',

where $f^{*}g'$ denotes thepullback of the rank (0, 2) metric tensor $g^{'} {\displaystyle g'}$ by $f {\displaystyle f}$ . Equivalently, in terms of thepushforward $f_{*},$ we have that for any two vector fields $v, w {\displaystyle v,w}$ on $M {\displaystyle M}$ (i.e. sections of thetangent bundle $\mathrm {T} M$ ),

g(v,w)=g'\left(f_{*}v,f_{*}w\right).

If $f {\displaystyle f}$ is alocal diffeomorphism such that $g=f^{*}g',$ then $f {\displaystyle f}$ is called alocal isometry.

Properties

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A collection of isometries typically form a group, theisometry group. When the group is acontinuous group, theinfinitesimal generators of the group are theKilling vector fields.

TheMyers–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 aLie group.

Symmetric spaces are important examples ofRiemannian manifolds that have isometries defined at every point.

Generalizations

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Given a positive real number ε, anε-isometry oralmost isometry (also called aHausdorff approximation) is a map $f\colon X\to Y$ between metric spaces such that
1. for $x,x'\in X$ one has $|d_{Y}(f(x),f(x'))-d_{X}(x,x')|<\varepsilon ,$ and
2. for any point $y\in Y$ there exists a point $x\in X$ with $d_{Y}(y,f(x))<\varepsilon$

That is, anε-isometry preserves distances to withinε and leaves no element of the codomain further thanε away from the image of an element of the domain. Note thatε-isometries are not assumed to becontinuous.

Therestricted 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:
$a\in {\mathfrak {A}}$ is an isometry if and only if $a^{*}\cdot a=1.$

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 apseudo-Euclidean space, the termisometry means a linear bijection preserving magnitude. See alsoQuadratic spaces.

Footnotes

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^^a ^b"We shall find it convenient to use the wordtransformation in the special sense of a one-to-one correspondence $P\to P'$ 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 memberP and a second memberP' 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, anisometry (or "congruent transformation," or "congruence") is a transformation which preserves length ..." — Coxeter (1969) p. 29^[2]
^
3.11Any two congruent triangles are related by a unique isometry.— Coxeter (1969) p. 39^[3]
^
LetT be a transformation (possibly many-valued) of $E^{n}$ ( $2\leq n<\infty$ ) into itself.
Let $d(p,q)$ be the distance between pointsp andq of $E^{n}$ , and letTp,Tq be any images ofp andq, respectively.
If there is a lengtha > 0 such that $d(Tp,Tq)=a$ whenever $d(p,q)=a$ , thenT is a Euclidean transformation of $E^{n}$ onto itself.^[4]

References

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^Coxeter 1969, p. 46
3.51Any 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 ^bBeckman, 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.
^Le Donne, Enrico (2013-10-01)."Lipschitz and path isometric embeddings of metric spaces".Geometriae Dedicata.166 (1):47–66.doi:10.1007/s10711-012-9785-2.ISSN 1572-9168.
^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 0-8218-2129-6.
^^a ^bNarici & 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.Bibcode:2000Sci...290.2323R.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 $\mathbf {M} =(I-W)^{\top }(I-W)$ (page 135) such that $\mathbf {M} \equiv YY^{\top }$
^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

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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.

Metric spaces (Category)

Basic concepts

Main results

Maps

Types of
metric spaces

Sets

Examples

Manifolds	Euclidean distance Riemannian
Functional analysis andMeasure theory	Chebyshev distance Inner product space Lévy metric Lévy–Prokhorov metric Metrizable topological vector space Normed space Taxicab geometry Wasserstein metric
General topology	Discrete space Intrinsic metric Laakso space Product metric