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| In [[mathematics]], the '''Euclidean group''' E(''n''), sometimes called ISO(''n'') or similar, is the [[symmetry group]] of ''n''-dimensional [[Euclidean space]]. Its elements, the [[isometry|isometries]] associated with the Euclidean [[Metric (mathematics)|metric]], are called '''Euclidean moves'''.
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| These [[group (mathematics)|group]]s are among the oldest and most studied, at least in the cases of dimension 2 and 3 – implicitly, long before the concept of group was invented.
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| ==Overview==
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| ===Dimensionality===
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| The number of [[degrees of freedom (physics and chemistry)|degrees of freedom]] for E(''n'') is
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| :''n''(''n'' + 1)/2,
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| which gives 3 in case {{nowrap|1=''n'' = 2}}, and 6 for {{nowrap|1=''n'' = 3}}. Of these, ''n'' can be attributed to available [[translational symmetry]], and the remaining {{nowrap|''n''(''n'' − 1)/2}} to [[rotational symmetry]].
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| ===Direct and indirect isometries===
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| <!-- Articles with one or more link to this section:
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| "Anisohedral tiling";
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| "Charts on SO(3)";
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| "Orientation (geometry)";
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| "Orthogonal group";
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| "Point groups in three dimensions";
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| "Point groups in two dimensions";
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| "Rigid body";
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| "Rotation group";
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| "Rotational symmetry";
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| "SE(2)";
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| "SE(3)";
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| "SE(n)";
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| "SO(5)";
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| "Spin tensor";
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| "Symmetry";
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| -->
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| There is a subgroup E<sup>+</sup>(''n'') of the '''direct isometries''', i.e., isometries preserving [[orientation (mathematics)|orientation]], also called '''rigid motions'''; they are the [[rigid body]] moves. These include the [[translation (mathematics)|translation]]s, and the [[rotation]]s, which together generate E<sup>+</sup>(''n''). E<sup>+</sup>(''n'') is also called a '''special Euclidean group''', and denoted SE(''n'').
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| The others are the '''indirect isometries'''. The subgroup E<sup>+</sup>(''n'') is of [[index of a subgroup|index]] 2. In other words, the indirect isometries form a single [[coset]] of E<sup>+</sup>(''n''). [[Universal quantification|Given any]] indirect isometry, for example a given [[Reflection (linear algebra)|reflection]] ''R'' that reverses orientation, all indirect isometries are given as ''DR'', where ''D'' is a direct isometry.
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| The Euclidean group for SE(3) is used for the kinematics of a [[rigid body]], in [[classical mechanics]]. A ''rigid body motion'' is in effect the same as a [[curve]] in the Euclidean group. Starting with a body ''B'' oriented in a certain way at time {{nowrap|1=''t'' = 0}}, its orientation at any other time is related to the starting orientation by a Euclidean motion, say ''f''(''t''). Setting {{nowrap|1=''t'' = 0}}, we have {{nowrap|1=''f''(0) = ''I''}}, the [[identity transformation]]. This means that the curve will always lie inside E<sup>+</sup>(3), in fact: starting at the identity transformation ''I'', such a continuous curve can certainly never reach anything other than a direct isometry. This is for simple topological reasons: the [[determinant]] of the transformation cannot jump from +1 to −1.
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| The Euclidean groups are not only [[topological group]]s, they are [[Lie group]]s, so that [[calculus]] notions can be adapted immediately to this setting.
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| ===Relation to the affine group===
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| The Euclidean group E(''n'') is a subgroup of the [[affine group]] for ''n'' dimensions, and in such a way as to respect the [[semidirect product]] structure of both groups. This gives, ''a fortiori'', two ways of writing down elements in an explicit notation. These are:
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| # by a pair {{nowrap|(''A'', ''b'')}}, with ''A'' an {{nowrap|''n'' × ''n''}} [[orthogonal matrix]], and ''b'' a real [[column vector]] of size ''n''; or
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| # by a single [[square matrix]] of size {{nowrap|''n'' + 1}}, as explained for the [[affine group]].
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| Details for the first representation are given in the next section.
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| In the terms of [[Felix Klein]]'s [[Erlangen programme]], we read off from this that [[Euclidean geometry]], the geometry of the Euclidean group of symmetries, is therefore a specialisation of [[affine geometry]]. All affine theorems apply. The extra factor in Euclidean geometry is the notion of [[distance]], from which [[angle]] can then be deduced.
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| ==Detailed discussion==
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| ===Subgroup structure, matrix and vector representation===
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| The Euclidean group is a subgroup of the group of [[affine transformation]]s.
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| It has as subgroups the [[translation (geometry)|translational]] group T(''n''), and the [[orthogonal group]] O(''n''). Any element of E(''n'') is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way:
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| :<math>x \mapsto A (x + b)</math>
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| where ''A'' is an [[orthogonal matrix]]
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| or an orthogonal transformation followed by a translation:
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| :<math>x \mapsto A x + b</math>.
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| T(''n'') is a [[normal subgroup]] of ''E''(''n''): for any translation ''t'' and any isometry ''u'', we have
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| :''u''<sup>−1</sup>''tu''
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| again a translation (one can say, through a displacement that is ''u'' acting on the displacement of ''t''; a translation does not affect a displacement, so equivalently, the displacement is the result of the linear part of the isometry acting on ''t'').
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| Together, these facts imply that E(''n'') is the [[semidirect product]] of O(''n'') extended by T(''n''). In other words, O(''n'') is (in the natural way) also the [[quotient group]] of E(''n'') by T(''n''):
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| :O(''n'') ≅ E(''n'') / T(''n'').
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| Now SO(''n''), the [[special orthogonal group]], is a subgroup of O(''n''), of [[index of a subgroup|index]] two. Therefore E(''n'') has a subgroup E<sup>+</sup>(''n''), also of index two, consisting of ''direct'' isometries. In these cases the determinant of ''A'' is 1.
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| They are represented as a translation followed by a [[rotation]], rather than a translation followed by some kind of [[Reflection (mathematics)|reflection]] (in dimensions 2 and 3, these are the familiar reflections in a [[mirror]] line or plane, which may be taken to include the [[Origin (mathematics)|origin]], or in 3D, a [[Improper rotation|rotoreflection]]).
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| We have:
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| :SO(''n'') ≅ E<sup>+</sup>(''n'') / T(''n'').
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| ===Subgroups===<!-- This section is linked from [[Symmetry group]] -->
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| Types of subgroups of E(''n''):
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| *[[Finite group]]s. They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) among the finite groups: O<sub>''h''</sub> and I<sub>''h''</sub>. The groups I<sub>''h''</sub> are even maximal among the groups including the next category.
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| *Countably infinite groups without arbitrarily small translations, rotations, or combinations, i.e., for every point the set of images under the isometries is topologically [[discrete space|discrete]]. E.g. for {{nowrap|1 ≤ ''m'' ≤ ''n''}} a group generated by ''m'' translations in independent directions, and possibly a finite point group. This includes [[lattice (group)|lattice]]s. Examples more general than those are the discrete [[space group]]s.
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| *Countably infinite groups with arbitrarily small translations, rotations, or combinations. In this case there are points for which the set of images under the isometries is not closed. Examples of such groups are, in 1D, the group generated by a translation of 1 and one of √2, and, in 2D, the group generated by a rotation about the origin by 1 radian.
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| *Non-countable groups, where there are points for which the set of images under the isometries is not closed. E.g. in 2D all translations in one direction, and all translations by rational distances in another direction.
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| *Non-countable groups, where for all points the set of images under the isometries is closed. E.g.
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| **all direct isometries that keep the origin fixed, or more generally, some point (in 3D called the [[rotation group SO(3)|rotation group]])
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| **all isometries that keep the origin fixed, or more generally, some point (the [[orthogonal group]])
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| **all direct isometries E<sup>+</sub>(''n'')
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| **the whole Euclidean group E(''n'')
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| **one of these groups in an ''m''-dimensional subspace combined with a discrete group of isometries in the orthogonal (''n''−''m'')-dimensional space
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| **one of these groups in an ''m''-dimensional subspace combined with another one in the orthogonal (''n''−''m'')-dimensional space
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| Examples in 3D of combinations:
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| *all rotations about one fixed axis
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| *ditto combined with reflection in planes through the axis and/or a plane perpendicular to the axis
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| *ditto combined with discrete translation along the axis or with all isometries along the axis
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| *a discrete point group, frieze group, or wallpaper group in a plane, combined with any symmetry group in the perpendicular direction
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| *all isometries which are a combination of a rotation about some axis and a proportional translation along the axis; in general this is combined with ''k''-fold rotational isometries about the same axis ({{nowrap|''k'' ≥ 1}}); the set of images of a point under the isometries is a ''k''-fold [[helix]]; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a ''k''-fold helix of such axes.
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| *for any point group: the group of all isometries which are a combination of an isometry in the point group and a translation; for example, in the case of the group generated by inversion in the origin: the group of all translations and inversion in all points; this is the generalized [[dihedral group]] of R<sup>3</sup>, Dih(R<sup>3</sup>).
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| ===Overview of isometries in up to three dimensions=== <!-- This section is linked from [[Isometry]] --> | |
| E(1), E(2), and E(3) can be categorized as follows, with [[degrees of freedom (physics and chemistry)|degrees of freedom]]:
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| E(1) – 1:
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| *E<sup>+</sup>(1):
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| **identity – 0
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| **translation – 1
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| *those not preserving orientation:
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| **reflection in a point – 1
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| E(''2'') – 3:
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| *E<sup>+</sup>(2):
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| **identity – 0
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| **translation – 2
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| **rotation about a point – 3
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| *those not preserving orientation:
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| **reflection in a line – 2
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| **reflection in a line combined with translation along that line ([[glide reflection]]) – 3
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| See also [[Euclidean plane isometry]].
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| E(3) – 6:
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| *E<sup>+</sup>(3):
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| **identity – 0
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| **translation – 3
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| **rotation about an axis – 5
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| **rotation about an axis combined with translation along that axis ([[screw theory|screw operation]]) – 6
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| *those not preserving orientation:
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| **reflection in a plane – 3
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| **reflection in a plane combined with translation in that plane ([[glide plane]] operation) – 5
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| **rotation about an axis by an angle not equal to 180°, combined with reflection in a plane perpendicular to that axis ([[improper rotation|roto-reflection]]) – 6
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| **[[inversion in a point]] – 3
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| See also [[Orthogonal_group#3D isometries that leave the origin fixed|3D isometries that leave the origin fixed]], [[space group]], [[Involution (mathematics)|involution]].
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| ===Commuting isometries===
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| For some isometry pairs composition does not depend on order:
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| *two translations
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| *two rotations or screws about the same axis
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| *reflection with respect to a plane, and a translation in that plane, a rotation about an axis perpendicular to the plane, or a reflection with respect to a perpendicular plane
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| *glide reflection with respect to a plane, and a translation in that plane
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| *inversion in a point and any isometry keeping the point fixed
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| *rotation by 180° about an axis and reflection in a plane through that axis
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| *rotation by 180° about an axis and rotation by 180° about a perpendicular axis (results in rotation by 180° about the axis perpendicular to both)
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| *two rotoreflections about the same axis, with respect to the same plane
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| *two glide reflections with respect to the same plane
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| ===Conjugacy classes===
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| The translations by a given distance in any direction form a [[conjugacy class]]; the translation group is the union of those for all distances.
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| In 1D, all reflections are in the same class.
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| In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class.
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| In 3D:
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| *Inversions with respect to all points are in the same class.
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| *Rotations by the same angle are in the same class.
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| *Rotations about an axis combined with translation along that axis are in the same class if the angle is the same and the translation distance is the same.
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| *Reflections in a plane are in the same class
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| *Reflections in a plane combined with translation in that plane by the same distance are in the same class.
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| *Rotations about an axis by the same angle not equal to 180°, combined with reflection in a plane perpendicular to that axis, are in the same class.
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| ==See also==
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| *[[Fixed points of isometry groups in Euclidean space]]
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| *[[Euclidean plane isometry]]
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| *[[Poincaré group]]
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| *[[Coordinate rotations and reflections]]
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| *[[Reflection through the origin]]
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| *[[Plane of rotation]]
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| {{DEFAULTSORT:Euclidean Group}}
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| [[Category:Lie groups]]
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| [[Category:Euclidean symmetries|*]]
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