Euclidean group

Under the action of the Euclidean group, these forms transform as follows.

A rigid body motion is in effect the same as a curve in the Euclidean group.

See also subgroups of the Euclidean group.

The Euclidean group acts on F(n) as follows.

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

For example, the Euclidean group acts affinely upon Euclidean space.

It is precisely the subgroup of the Euclidean group that fixes the line at infinity pointwise.

The set consisting of all point reflections and translations is Lie subgroup of the Euclidean group.

Here, generate the Euclidean group, acting within each planar wavefront, which justifies the name plane wave for this solution.

The set of proper rigid transformation is called special Euclidean group, denoted SE(n).

Thus the conjugacy class within the Euclidean group E(n) of a translation is the set of all translations by the same distance.

A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries.

E(n) is also called a special Euclidean group, and denoted SE(n).

The Euclidean group for SE(3) is used for the kinematics of a rigid body, in classical mechanics.

The smallest subgroup of the Euclidean group containing all translations by a given distance is the set of all translations.

The set of Euclidean plane isometries forms a group under composition: the Euclidean group in two dimensions.

It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections.

With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups.

The simplest motions follow a one-parameter subgroup of a Lie group, such as the Euclidean group of three-dimensional space.

The laws describing a solid are invariant under the full Euclidean group, but the solid itself spontaneously breaks this group down to a space group.

An example is the action of the Euclidean group E(n) upon the Euclidean space E.

The Poincaré group includes also translations and plays the same role as Euclidean groups of ordinary Euclidean spaces.

Thus there is a partition of the Euclidean group with in each subset one isometry that keeps the origin fixed, and its combination with all translations.

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

Therefore a symmetry group of rotational symmetry is a subgroup of E(m) (see Euclidean group).