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Euclidean group From Wikipedia, the free encyclopedia

In , the E(n), sometimes called ISO( n) or similar, is the group of n-dimensional . Its elements, the associated with the Euclidean metric, are called Euclidean moves .

These groups are among the oldest and most studied, at least in the cases of 2 and 3 — implicitly, long before the concept of group was known. Contents

 1 Overview  1.1 Dimensionality  1.2 Direct and indirect isometries  1.3 Relation to the

 2 Detailed discussion  2.1 structure, and vector representation  2.2  2.3 Overview of isometries in up to three  2.4 Commuting isometries  2.5 Conjugacy classes

 3 See also

Overview

Dimensionality

The number of for E(n) is

n(n + 1)/2,

which gives 3 in case n = 2, and 6 for n = 3. Of these, n can be attributed to available , and the remaining n(n − 1)/2 to .

Direct and indirect isometries

There is a subgroup E+(n) of the direct isometries , i.e., isometries preserving , also called rigid motions ; they are the moves. These include the translations, and the , which together generate E+(n). E+(n) is also called a special Euclidean group , and denoted SE (n).

The others are the indirect isometries . The subgroup E+(n) is of index 2. In other words, the indirect isometries form a single of E+(n). Given any indirect , for example a given R that reverses orientation , all indirect isometries are given as DR , where D is a direct isometry.

The Euclidean group for n = 3 is used for the of a rigid body , in . A rigid

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body is in effect the same as a in the Euclidean group. Starting with a body B oriented in a certain way at time t = 0, its orientation at any other time is related to the starting orientation by a Euclidean motion, say f(t). Setting t = 0, we have f(0) = I, the identity transformation. This means that the curve will always lie inside E+(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 of the transformation cannot jump from +1 to −1.

The Euclidean groups are not only topological groups, they are Lie groups, so that notions can be adapted immediately to this setting.

Relation to the affine group

The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the structure of both groups. This gives, a fortiori , two ways of writing down elements in an explicit notation. These are:

1. by a pair ( A, b), with A an n×n , and b a real column vector of size n; or 2. by a single matrix of size n + 1, as explained for the affine group.

Details for the first representation are given in the next section.

In the terms of 's Erlangen programme, we read off from this that Euclidean , the geometry of the Euclidean group of , is therefore a specialisation of . All affine theorems apply. The extra factor in is the notion of , from which angle can then be deduced. Detailed discussion

Subgroup structure, matrix and vector representation

The Euclidean group is a subgroup of the group of affine transformations.

It has as subgroups the translational group T, and the O(n). Any element of E(n) is a followed by an orthogonal transformation (the linear part of the isometry), in a unique way:

where A is an orthogonal matrix or an orthogonal transformation followed by a translation:

.

T is a subgroup of E(n): for any translation t and any isometry u, we have

u−1tu 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

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part of the isometry acting on t).

Together, these facts imply that E(n) is the semidirect product of O(n) extended by T. In other words O (n) is (in the natural way) also the of E(n) by T:

O(n) E(n) / T .

Now SO (n), the special orthogonal group, is a subgroup of O(n), of index two. Therefore E(n) has a subgroup E+(n), also of index two, consisting of direct isometries. In these cases the determinant of A is 1.

They are represented as a translation followed by a , rather than a translation followed by some kind of reflection (in dimensions 2 and 3, these are the familiar reflections in a mirror line or , which may be taken to include the origin, or in 3D, a rotoreflection).

We have:

SO (n) E+(n) / T .

Subgroups

Types of subgroups of E(n) :

 Finite groups. 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: Oh and Ih. The groups Ih are even maximal among the groups including the next category.  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. E.g. for 1 ≤ m ≤ n a group generated by m translations in independent directions, and possibly a finite . This includes lattices. Examples more general than those are the groups.  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.  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 in another direction.  Non-countable groups, where for all points the set of images under the isometries is closed. E.g.  all direct isometries that keep the origin fixed, or more generally, some point (in 3D called the rotation group)  all isometries that keep the origin fixed, or more generally, some point (the orthogonal group) +  all direct isometries E (n)  the whole Euclidean group E( n)  one of these groups in an m-dimensional subspace combined with a of isometries in the orthogonal n-m-dimensional space  one of these groups in an m-dimensional subspace combined with another one in the orthogonal n-m-dimensional space

Examples in 3D of combinations:

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 all rotations about one fixed axis  ditto combined with reflection in planes through the axis and/or a plane to the axis  ditto combined with discrete translation along the axis or with all isometries along the axis  a discrete point group, , or in a plane, combined with any in the perpendicular direction  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 ( k ≥ 1); the set of images of a point under the isometries is a k-fold ; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k-fold helix of such axes.  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 of R 3, Dih(R 3).

Overview of isometries in up to three dimensions

E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom:

E(1) - 1:

+  E (1):  identity - 0  translation - 1

 those not preserving orientation:  reflection in a point - 1

E(2) - 3:

+  E (2):  identity - 0  translation - 2  rotation about a point - 3

 those not preserving orientation:  reflection in a line - 2  reflection in a line combined with translation along that line () - 3

See also Euclidean plane isometry.

E(3) - 6:

+  E (3):  identity - 0  translation - 3  rotation about an axis - 5  rotation about an axis combined with translation along that axis (screw operation) - 6

 those not preserving orientation:  reflection in a plane - 3  reflection in a plane combined with translation in that plane ( operation) - 5  rotation about an axis by an angle not equal to 180°, combined with reflection in a plane

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perpendicular to that axis ( roto -reflection ) - 6  inversion in a point - 3

See also 3D isometries which leave the origin fixed, , .

Commuting isometries

For some isometry pairs composition does not depend on :

 two translations  two rotations or screws about the same axis  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  glide reflection with respect to a plane, and a translation in that plane  inversion in a point and any isometry keeping the point fixed  rotation by 180° about an axis and reflection in a plane through that axis  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)  two rotoreflections about the same axis, with respect to the same plane  two glide reflections with respect to the same plane

Conjugacy classes

The translations by a given distance in any direction form a ; the translation group is the union of those for all distances.

In 1D, all reflections are in the same class.

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.

In 3D:

 Inversions with respect to all points are in the same class.  Rotations by the same angle are in the same class.  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, and in corresponding direction (right-hand or left-hand screw).  Reflections in a plane are in the same class  Reflections in a plane combined with translation in that plane by the same distance are in the same class.  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. See also

 fixed points of isometry groups in Euclidean space  Euclidean plane isometry  Poincaré group  Coordinate rotations and reflections

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