The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n), for inhomogeneous special orthogonal group.
The Euclidean group E(n) comprises all translations, rotations, and reflections of
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, often denoted SE(n) and E+(n), whose elements are called rigid motions or Euclidean motions.
They comprise arbitrary combinations of translations and rotations, but not reflections.
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).
The isometries that reverse handedness are called indirect, or opposite.
It turns out that the special Euclidean group SE(n) = E+(n) is connected in this topology.
On the other hand, the group E(n) as a whole is not connected: there is no continuous trajectory that starts in E+(n) and ends in E−(n).
The continuous trajectories in E(3) play an important role in classical mechanics, because they describe the physically possible movements of a rigid body in three-dimensional space over time.
The position and orientation of the body at any later time t will be described by the transformation f(t).
For that reason, the direct Euclidean isometries are also called "rigid motions".
This gives, a fortiori, two ways of writing elements in an explicit notation.
The origin of Euclidean geometry allows definition of the notion of distance, from which angle can then be deduced.
Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way:
with c = Ab T(n) is a normal subgroup of E(n): for every translation t and every isometry u, the composition
Together, these facts imply that E(n) is the semidirect product of O(n) extended by T(n), which is written as
In other words, O(n) is (in the natural way) also the quotient group of E(n) by T(n):
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.
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).
Types of subgroups of E(n): Examples in 3D of combinations: 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.
For some isometry pairs composition does not depend on order: The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances.
Glide reflections with translation by the same distance are in the same class.