Isometry

Distance-preserving mathematical transformation

A composition of two opposite isometries is a direct isometry. A reflection in a line is an opposite isometry, like R ₁ or R ₂ on the image. Translation T is a direct isometry: a rigid motion.^[1]

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

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 ${\displaystyle M}$ involves an isometry from ${\displaystyle M}$ into ${\displaystyle M',}$ a quotient set of the space of Cauchy sequences on ${\displaystyle M.}$ The original space ${\displaystyle M}$ is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.

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

Definition

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be metric spaces with metrics (e.g., distances) ${\textstyle d_{X}}$ and ${\textstyle d_{Y}.}$ A map ${\textstyle f\colon X\to Y}$ is called an isometry or distance-preserving map if for any ${\displaystyle a,b\in X}$,

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

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., ${\displaystyle d(a,b)=0}$ if and only if ${\displaystyle a=b}$. 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 ${\displaystyle x\mapsto |x|}$ in ${\displaystyle \mathbb {R} }$ 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:^[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 ${\displaystyle \mathbb {R} }$. If X and Y are complex vector spaces then A may fail to be linear as a map over ${\displaystyle \mathbb {C} }$.

Linear isometry

Given two normed vector spaces ${\displaystyle V}$ and ${\displaystyle W,}$ a linear isometry is a linear map ${\displaystyle A:V\to W}$ that preserves the norms:

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

for all ${\displaystyle v\in V.}$^[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

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

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

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

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

By the Mazur–Ulam theorem, any isometry of normed vector spaces over ${\displaystyle \mathbb {R} }$ is affine.

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

Examples

• A linear map from ${\displaystyle \mathbb {C} ^{n}}$ to itself is an isometry (for the dot product) if and only if its matrix is unitary.^{[8][9][10][11]}

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 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

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

${\displaystyle g=f^{*}g',}$

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

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

If ${\displaystyle f}$ is a local diffeomorphism such that ${\displaystyle g=f^{*}g',}$ then ${\displaystyle f}$ 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 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

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

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.

• 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:

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

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, the term isometry means a linear bijection preserving magnitude. See also Quadratic spaces.

See also

• 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

1. "We shall find it convenient to use the word transformation in the special sense of a one-to-one correspondence ${\displaystyle P\to P'}$ 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]

2. 3.11 Any two congruent triangles are related by a unique isometry.— Coxeter (1969) p. 39^[3]

3. 
   Let T be a transformation (possibly many-valued) of ${\displaystyle E^{n}}$ ( ${\displaystyle 2\leq n<\infty }$) into itself.
   Let ${\displaystyle d(p,q)}$ be the distance between points p and q of ${\displaystyle E^{n}}$, and let Tp, Tq be any images of p and q, respectively.
   If there is a length a > 0 such that ${\displaystyle d(Tp,Tq)=a}$ whenever ${\displaystyle d(p,q)=a}$, then T is a Euclidean transformation of ${\displaystyle E^{n}}$ onto itself.^[4]

References

1. 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.

2. Coxeter 1969, p. 29
3. Coxeter 1969, p. 39
4. 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.
5. Narici & Beckenstein 2011, pp. 275–339.
6. Wilansky 2013, pp. 21–26.
7. Thomsen, Jesper Funch (2017). Lineær algebra [Linear Algebra]. Department of Mathematics (in Danish). Århus: Aarhus University. p. 125.
8. 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.
9. 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 ${\displaystyle \mathbf {M} =(I-W)^{\top }(I-W)}$ (page 135) such that ${\displaystyle \mathbf {M} \equiv YY^{\top }}$
10. 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.
11. 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.

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