Editing Isometry (section)

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