Euclidean Group

The Euclidean group $E(n)$ is the group of all isometries of the Euclidean Space ${\mathbb{R}}^n$. These are transformations that preserve the Euclidean distance between any two points.

Any element $g\in E(n)$ can be represented as a transformation $g:{\mathbb{R}}^n\to {\mathbb{R}}^n$ of the form:

$$g(\mathbf{x})=\mathbf{Rx}+\mathbf{c}$$

where $\mathbf{R}$ is an $n\times n$ orthogonal matrix (satisfying ${\mathbf{R}}^{\top }\mathbf{R}=I$) and $\mathbf{c}\in {\mathbb{R}}^n$ is a translation vector.

Definition (Special Euclidean Group)

The special Euclidean group $\mathrm{SE}(n)$ is the subgroup of $E(n)$ consisting of orientation-preserving isometries. These are the transformations where the rotation part has determinant $1$:

$$\mathrm{SE}(n)=\left\{(\mathbf{R},\mathbf{c})\mid \mathbf{R}\in \mathrm{SO}(n),\mathbf{c}\in {\mathbb{R}}^n\right\}$$

where $\mathrm{SO}(n)=\{\mathbf{R}\in O(n)\mid \det (\mathbf{R})=1\}$ is the Special Orthogonal Group.

Matrix Representation

Elements of $\mathrm{SE}(n)$ are often represented using homogeneous coordinates as $(n+1)\times (n+1)$ matrices:

$$\left[\begin{array}{cc}\mathbf{R}& \mathbf{c}\\ 0^{\top }& 1\end{array}\right]$$

The group operation (composition of transformations) then corresponds to matrix multiplication:

$$\left[\begin{array}{cc}{\mathbf{R}}_1& {\mathbf{c}}_1\\ 0^{\top }& 1\end{array}\right]\left[\begin{array}{cc}{\mathbf{R}}_2& {\mathbf{c}}_2\\ 0^{\top }& 1\end{array}\right]=\left[\begin{array}{cc}{\mathbf{R}}_1{\mathbf{R}}_2& {\mathbf{R}}_1{\mathbf{c}}_2+{\mathbf{c}}_1\\ 0^{\top }& 1\end{array}\right]$$

Semidirect Product Structure

The Euclidean group can be described as the semidirect product of the translation group and the orthogonal group:

$$E(n)\cong {\mathbb{R}}^n⋊O(n)$$

Similarly, $\mathrm{SE}(n)\cong {\mathbb{R}}^n⋊\mathrm{SO}(n)$.

Relation to Rigid Body Dynamics

In the context of rigid body dynamics, $\mathrm{SE}(3)$ represents the configuration space of a rigid body in 3D space. Each element of $\mathrm{SE}(3)$ describes a possible position and orientation of the body relative to a reference frame.