\[\mathbb{R}^n\]
are mappings that preserve distances, and by implication, areas and volumes. If \[f\]
is a rigid motion on \[\mathbb{R}^n\]
then for any \[\mathbf{x}, \; \mathbf{y} \in \mathbb{R}^n\]
, \[\left| f\mathbf{y} - \mathbf{x} \right| = \left| y-x \right|\]
.Any rigid motion in
\[\mathbb{R}^n\]
is an orthogonal mapping, that preserves angles and lengths, and a translation.Relative to an orthonormal basis, the matrix of any orthogonal mappings of
\[\mathbb{R}^2\]
has one of the forms \[A= \left( \begin{array}{cc} cos \theta & - sin \theta \\ sin \theta & cos \theta \end{array} \right), \; B= \left( \begin{array}{cc} cos \theta & sin \theta \\ sin \theta & -cos \theta \end{array} \right) \]
Matrix
\[A\]
represents rotation anticlockwise by an angle \[\theta\]
and matrix \[B\]
represents reflection in the line making an anticlockwise angle \[\frac{\theta}{2}\]
with the positive \[x\]
axis.Rotation about an arbitrary point
\[Q\]
can be represented as a translation \[\mathbf{QO}\]
followed by the rotation followed by the translation \[\mathbf{OQ}\]
. We can write this as \[F(P)=A(P-Q)+Q\]
.Reflection in an arbitrary line
\[\mathbf{r}=\mathbf{OQ} + t \mathbf{v}\]
can be represented as a translation \[\mathbf{QO}\]
followed by the rotation followed by the translation \[\mathbf{OQ}\]
. We can write this as \[F(P)=A(P-Q)+OQ\]
.Glide reflection along an arbitrary line
\[\mathbf{r}=\mathbf{OQ} + t \mathbf{v}\]
can be represented as a reflection in the line followed by a translation \[k \mathbf{v}\]
. We can write \[F(P)=BP+ k \mathbf{v}\]
.