---
tag: 0P42JB4E3M
kind: definition
name: Glide reflection
msc: [51F20]
defines: 0167F9B344
uses: [06BRDF3F0E, 087VJM5Q77]
sources:
  - url: https://en.wikipedia.org/wiki/Glide_reflection
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Euclidean_plane_isometry
    locator: Glide reflections
---

## 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|06BRDF3F0E]] in $H$. Let $w \in \mathbb{R}^n$ be a nonzero vector with $w \cdot a = 0$, and let $T_w$ be the [[translation|087VJM5Q77]] 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 $(1, 0)$.