Axial and glide symmetries

Axial symmetry (reflection) with respect to a line $l$ maps every point $P$ in the plane to a point $P'$ such that $l$ is the perpendicular bisector of the segment $PP'$. Glide symmetry (glide reflection) is the composition of a reflection over a line $l$ and a translation along $l$.

Important properties

• Axial symmetry preserves distances and angles (it is an isometry).

• The axis of symmetry is the set of points that remain unchanged under the reflection.

• Glide symmetry is also an isometry, but no point (except possibly on the axis if the translation is zero) remains fixed.

• A glide symmetry is not the same as a simple reflection or translation.