Reflection in a line or plane

Builds on Normal vector Euclidean isometry Perpendicular bisector

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.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Euclidean isometry concept
Normal vector concept
Perpendicular bisector concept
Reflection in a line or plane definition

Tag	`027MBSVPY3`
URI	`m2://027MBSVPY3`
Aliases	`reflection`, `mirror-isometry`
Kind	definition
Writing	W1 stated
MSC	51F15
Defines	Reflection
Uses	Normal vector, Euclidean isometry, Perpendicular bisector

Reflection in a line or plane

Definition

Remarks

Reading path

Further reading

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