Editing Isometry (section)

== 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.