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.

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Reflection concept
Translation concept
Glide reflection definition

Tag	`0P42JB4E3M`
URI	`m2://0P42JB4E3M`
Kind	definition
Writing	W0 stub
MSC	51F20
Defines	Glide reflection
Uses	Reflection, Translation

Glide reflection

Definition

Remarks

Reading path

Further reading

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