Euclidean group

In mathematics, the Euclidean group E(n), also known as 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 motions.

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.

Direct and indirect isometries

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

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

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 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 determinant of the transformation cannot jump from +1 to −1.

The Euclidean groups are not only topological groups, they are Lie groups, so that calculus 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 semidirect product structure of both groups. This gives, a fortiori, two ways of writing elements in an explicit notation. These are:

  1. by a pair (A, b), with A an n × n orthogonal matrix, and b a real column vector of size n; or
  2. by a single square 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 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.

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(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:

where A is an orthogonal matrix

or an orthogonal transformation followed by a translation:


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


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(n). In other words, O(n) is (in the natural way) also the quotient group of E(n) by T(n):

O(n) ≅ E(n) / 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. 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) ≅ E+(n) / T(n).


Types of subgroups of E(n):

Examples in 3D of combinations:

Overview of isometries in up to three dimensions

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

Isometries of E(1)
Type of isometry Degrees of freedom Preserves orientation?
Identity 0 yes
Translation 1 yes
Reflection in a point 1 no
Isometries of E(2)
Type of isometry Degrees of freedom Preserves orientation?
Identity 0 yes
Translation 2 yes
Rotation about a point 3 yes
Reflection in a line 2 no
Glide reflection 3 no
Isometries of E(3)
Type of isometry Degrees of freedom Preserves orientation?
Identity 0 yes
Translation 3 yes
Rotation about an axis 5 yes
Screw displacement 6 yes
Reflection in a plane 3 no
Glide plane operation 5 no
Improper rotation 6 no
Inversion in a point3 no

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.

Commuting isometries

For some isometry pairs composition does not depend on order:

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:

See also


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