Rigid transformations are mappings from one reference frame to another reference frame in the Euclidean space $\mathbb R^n$. They comprise translation, rotation (and sometimes reflection).
More formally, rigid transformation are elements of the matrix Lie group SE($n$), the specially Euclidean group. If reflection is included, these proper rigid transformations are elements of the matrix Lie group E($n$), the Euclidean group.
Rigid transformations are mappings from one reference frame to another reference frame in the Euclidean space $\mathbb R^n$. They comprise translation, rotation (and sometimes reflection).
More formally, rigid transformation are elements of the matrix Lie group SE($n$), the specially Euclidean group. The group SE$(n)$ is the semi-direct product of the group of rotation SO$(n)$ and the Euclidean space $\mathbb R^n$. If reflection is included, these proper rigid transformations are elements of the matrix Lie group E($n$), the Euclidean group. The group E$(n)$ is the semi-direct product of the orthogonal group O$(n)$ and the Euclidean space $\mathbb R^n$.
Every rigid transformation $\boldsymbol\varphi:\mathbb{R}^n \to \mathbb{R}^n$ can be described as the sum of a translation motion and a special orthogonal matrix $$ \boldsymbol\varphi(\mathbf x) = \mathbf x_0 + \boldsymbol\Phi \mathbf{x} $$
