Reflection in a line or plane

Definition

Let $a \in \mathbb{R}^n$ be a nonzero vector and $c$ a real number, and let $\cdot$ be the dot product. Let $H = \{\, x \in \mathbb{R}^n : x \cdot a = c \,\}$ be the hyperplane with the normal vector $a$. The reflection in $H$ is the map $$\sigma_H \colon \mathbb{R}^n \to \mathbb{R}^n, \qquad \sigma_H(x) = x - 2\, \frac{x \cdot a - c}{a \cdot a}\, a.$$ For $n = 2$, $H$ is a line; for $n = 3$, $H$ is a plane.

Remarks

The map $\sigma_H$ is an indirect Euclidean isometry. Its fixed points are the points of $H$, and $\sigma_H \circ \sigma_H$ is the identity. For $x \notin H$, the hyperplane $H$ is the perpendicular bisector of $x$ and $\sigma_H(x)$. The linear part of $\sigma_H$ is an orthogonal matrix with determinant $-1$ and the eigenvalues $-1, 1, \dots, 1$. Reflections in hyperplanes generate the Euclidean group.

Explanations

L1 human

A reflection is a mirror image. Each point goes to the point on the other side of the mirror, at the same distance from it. The reflection in the line $y = x$ in the plane sends $(x, y)$ to $(y, x)$.