Glide reflection

Definition

Let $H = \{\, x \in \mathbb{R}^n : x \cdot a = c \,\}$ be a hyperplane, with $a \neq 0$, and let $\sigma_H$ be the reflection in $H$. Let $w \in \mathbb{R}^n$ be a nonzero vector with $w \cdot a = 0$, and let $T_w$ be the translation by $w$. The glide reflection with the axis $H$ and the glide vector $w$ is the map $T_w \circ \sigma_H$.

Remarks

Because $w$ is parallel to $H$, the two maps commute: $T_w \circ \sigma_H = \sigma_H \circ T_w$. A glide reflection is an indirect isometry with no fixed point. Its square is the translation by $2w$. If the translation vector is perpendicular to $H$ in place of parallel, then $T_w \circ \sigma_H$ is the reflection in a hyperplane parallel to $H$. In the plane, the axis is the glide line. Glide reflections are symmetries of many frieze and wallpaper patterns.

Explanations

L1 human

First reflect the plane in a line, then slide it along that line. Neither step alone gives the result. The map $(x, y) \mapsto (x + 1, -y)$ is the glide reflection with the axis $y = 0$ and the glide vector $(1, 0)$.