Editing Euclidean space (section)

== Isometries ==
An [[isometry]] between two [[metric space]]s is a [[bijection]] preserving the distance,{{efn|If the condition of being a bijection is removed, a function preserving the distance is necessarily injective, and is an isometry from its domain to its image.}} that is

<math display="block">d(f(x), f(y))= d(x,y).</math>

In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm

<math display="block">\|f(x)\| = \|x\|,</math>

since the norm of a vector is its distance from the zero vector. It preserves also the inner product

<math display="block">f(x)\cdot f(y)=x\cdot y,</math>

since

<math display="block">x \cdot y=\tfrac 1 2 \left(\|x+y\|^2 - \|x\|^2 - \|y\|^2\right).</math>

An isometry of Euclidean vector spaces is a [[linear isomorphism]].{{efn|Proof: one must prove that <math>f(\lambda x+ \mu y) - \lambda f(x)-\mu f(y)=0</math>. For that, it suffices to prove that the square of the norm of the left-hand side is zero. Using the bilinearity of the inner product, this squared norm can be expanded into a linear combination of <math>\|f(x)\|^2,</math> <math>\|f(y)\|^2,</math> and <math>f(x)\cdot f(y).</math> As {{mvar|f}} is an isometry, this gives a linear combination of <math>\|x\|^2,\|y\|^2,</math> and <math>x\cdot y,</math> which simplifies to zero.}}{{sfn|Berger|1987|loc=Proposition 9.1.3}}

An isometry <math>f\colon E\to F</math> of Euclidean spaces defines an isometry <math>\overrightarrow f \colon \overrightarrow E \to \overrightarrow F</math> of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if {{mvar|E}} and {{mvar|F}} are Euclidean spaces, {{math|''O'' ∈ ''E''}}, {{math|''O''{{prime}} ∈ ''F''}}, and <math>\overrightarrow f\colon \overrightarrow E\to \overrightarrow F</math> is an isometry, then the map <math>f\colon E\to F</math> defined by

<math display="block">f(P)=O' + \overrightarrow f\Bigl(\overrightarrow{OP}\Bigr)\vphantom{\frac({}}</math>

is an isometry of Euclidean spaces.

It follows from the preceding results that an isometry of Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces.

===Isometry with prototypical examples===
If {{mvar|E}} is a Euclidean space, its associated vector space <math>\overrightarrow E</math> can be considered as a Euclidean space. Every point {{math|''O'' ∈ ''E''}} defines an isometry of Euclidean spaces

<math display="block">P\mapsto \overrightarrow {OP},\vphantom{\frac({}}</math>

which maps {{mvar|O}} to the zero vector and has the identity as associated linear map. The inverse isometry is the map

<math display="block">v\mapsto O+v.</math>

A Euclidean frame {{tmath|(O, e_1, \dots, e_n)}} allows defining the map

<math display="block">\begin{align}
E&\to \R^n\\
P&\mapsto \Bigl(e_1\cdot \overrightarrow {OP}, \dots, e_n\cdot\overrightarrow {OP}\Bigr),\vphantom{\frac({}}
\end{align}</math>

which is an isometry of Euclidean spaces. The inverse isometry is

<math display="block">\begin{align}
\R^n&\to E \\
(x_1\dots, x_n)&\mapsto \left(O+x_1e_1+ \dots + x_ne_n\right).
\end{align}</math>

''This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension.''

This justifies that many authors talk of <math>\R^n</math> as ''the'' Euclidean space of dimension {{mvar|n}}.

=== Euclidean group {{anchor|Rotations and reflections}} ===
{{main article|Euclidean group|Rigid transformation}}
An isometry from a Euclidean space onto itself is called ''Euclidean isometry'', ''Euclidean transformation'' or ''rigid transformation''. The rigid transformations of a Euclidean space form a group (under [[function composition|composition]]), called the ''Euclidean group'' and often denoted {{math|E(''n'')}} of {{math|ISO(''n'')}}.

The simplest Euclidean transformations are [[Translation (geometry)|translations]]

<math display="block">P \to P+v.</math>

They are in bijective correspondence with vectors. This is a reason for calling ''space of translations'' the vector space associated to a Euclidean space. The translations form a [[normal subgroup]] of the Euclidean group.

A Euclidean isometry {{mvar|f}} of a Euclidean space {{mvar|E}} defines a linear isometry <math>\overrightarrow f</math> of the associated vector space (by ''linear isometry'', it is meant an isometry that is also a [[linear map]]) in the following way: denoting by {{math|''Q'' − ''P''}} the vector <math>\overrightarrow {PQ},\vphantom{\frac({}}</math> if {{mvar|O}} is an arbitrary point of {{mvar|E}}, one has

<math display="block">\overrightarrow f\Bigl(\overrightarrow {OP}\Bigr)= f(P)-f(O).\vphantom{\frac({}}</math>

It is straightforward to prove that this is a linear map that does not depend from the choice of {{mvar|O.}}

The map <math>f \to \overrightarrow f</math> is a [[group homomorphism]] from the Euclidean group onto the group of linear isometries, called the [[orthogonal group]]. The kernel of this homomorphism is the translation group, showing that it is a normal subgroup of the Euclidean group.

The isometries that fix a given point {{mvar|P}} form the [[stabilizer subgroup]] of the Euclidean group with respect to {{mvar|P}}. The restriction to this stabilizer of above group homomorphism is an isomorphism. So the isometries that fix a given point form a group isomorphic to the orthogonal group.

Let {{mvar|P}} be a point, {{mvar|f}} an isometry, and {{mvar|t}} the translation that maps {{mvar|P}} to {{math|''f''(''P'')}}. The isometry <math>g=t^{-1}\circ f</math> fixes {{mvar|P}}. So <math>f= t\circ g,</math> and ''the Euclidean group is the [[semidirect product]] of the translation group and the orthogonal group.''

The [[special orthogonal group]] is the normal subgroup of the orthogonal group that preserves [[orientation (vector space)|handedness]]. It is a subgroup of [[index (group theory)|index]] two of the orthogonal group. Its inverse image by the group homomorphism <math>f \to \overrightarrow f</math> is a normal subgroup of index two of the Euclidean group, which is called the ''special Euclidean group'' or the ''displacement group''. Its elements are called ''rigid motions'' or ''displacements''.

Rigid motions include the [[identity function|identity]], translations, [[rotation]]s (the rigid motions that fix at least a point), and also [[screw axis|screw motions]].

Typical examples of rigid transformations that are not rigid motions are [[reflection (mathematics)|reflection]]s, which are rigid transformations that fix a hyperplane and are not the identity. They are also the transformations consisting in changing the sign of one coordinate over some Euclidean frame.

As the special Euclidean group is a subgroup of index two of the Euclidean group, given a reflection {{mvar|r}}, every rigid transformation that is not a rigid motion is the product of {{mvar|r}} and a rigid motion. A [[glide reflection]] is an example of a rigid transformation that is not a rigid motion or a reflection.

All groups that have been considered in this section are [[Lie group]]s and [[algebraic group]]s.