Euclidean group

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 isometries associated with the Euclidean metric, are called Euclidean moves.

E⁺(n) is the subgroup of direct isometries, i.e., isometries preserving orientation, also called rigid motions; they are the rigid body moves.

The other are the indirect isometries.

The number of degrees of freedom for E(n) and E⁺(n) is triangular number n(n + 1)/2 which gives 3 in case n = 2, and 6 for n = 3.

These groups 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 known.

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 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:

x\mapsto A(x+b)

where A is an orthogonal matrix

or an orthogonal transformation followed by a translation:

x\mapsto Ax+b

.

T is a normal 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 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 quotient group of E(n) by T:

O(n)

\cong

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 rotation, 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 plane, which may be taken to include the origin, or in 3D, a rotoreflection).

We have:

SO(n)

\cong

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: O_h and I_h. The groups I_h 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 point group. This includes lattices. Examples more general than those are the discrete space 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 distances 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 discrete group 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:

all rotations about one fixed axis
ditto combined with reflection in planes through the axis and/or a plane perpendicular to the axis
ditto combined with discrete translation along the axis or with all isometries along the axis
a discrete point group, frieze group, or wallpaper group in a plane, combined with any symmetry group 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 helix; 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 dihedral group of R³, Dih(R³).

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 semidirect product structure of both groups. Instead of by a pair (A, b), Euclidean group elements can also be represented as square matrices of size n + 1, as explained for the affine group.

In the terms of the Erlangen programme, Euclidean geometry is therefore a specialisation of affine geometry. All affine theorems apply; the extra factor is the notion of distance, from which angle can be deduced.

Rigid body motions

Another use of a Euclidean group is for the kinematics of a rigid body, in classical mechanics. A rigid body motion is in effect the same as a curve in E⁺(3).

The Euclidean groups are Lie groups, so that calculus notions can be adapted immediately from this setting.

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 (glide reflection) - 3

Commuting isometries

For some isometry pairs composition does not depend on order:

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 conjugacy class; 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 (righthand or lefthand 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.