Euclidean group

A Euclidean isometry can be direct or indirect, depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, whose elements are called rigid motions or Euclidean motions. They comprise arbitrary combinations of translations and rotations, but not reflections.

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 invented.

The number of degrees of freedom 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 translational symmetry, and the remaining n(n − 1)/2 to rotational symmetry.

The direct isometries (i.e., isometries preserving the handedness of chiral subsets) comprise a subgroup of E(n), called the special Euclidean group and usually denoted by E+(n) or SE(n). They include the translations and rotations, and combinations thereof; including the identity transformation, but excluding any reflections.

The isometries that reverse handedness are called indirect, or opposite. For any fixed indirect isometry R, such as a reflection about some hyperplane, every other indirect isometry can be obtained by the composition of R with some direct isometry. Therefore, the indirect isometries are a coset of E+(n), which can be denoted by E(n). It follows that the subgroup E+(n) is of index 2 in E(n).

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

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[clarification needed] groups. This gives, a fortiori, two ways of writing elements in an explicit notation. These are:

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 origin of Euclidean geometry allows definition of the notion of distance, from which angle can then be deduced.

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

It has as subgroups the 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:

T(n) is a normal subgroup of E(n): for every translation t and every isometry u, the composition

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).

Countably infinite groups without arbitrarily small translations, rotations, or combinations
Countably infinite groups with arbitrarily small translations, rotations, or combinations
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(1), E(2), and E(3) can be categorized as follows, with degrees of freedom:

Chasles' theorem asserts that any element of E+(3) is a screw displacement.

See also 3D isometries that leave the origin fixed, space group, involution.

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 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.