Reflection (mathematics)

Properties

The matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Every rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is the result of reflecting in an odd number. Thus reflections generate the orthogonal group, and this result is known as the Cartan–Dieudonné theorem.

Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes. In general, a group generated by reflections in affine hyperplanes is known as a reflection group. The finite groups generated in this way are examples of Coxeter groups.

Reflection across a line in the plane

Reflection across an arbitrary line through the origin in two dimensions can be described by the following formula

${\displaystyle {\mathrm{Ref}}_l(v)=2\frac{v\cdot l}{l\cdot l}l-v,}$

where ${\displaystyle v}$ denotes the vector being reflected, ${\displaystyle l}$ denotes any vector in the line across which the reflection is performed, and ${\displaystyle v\cdot l}$ denotes the dot product of ${\displaystyle v}$ with ${\displaystyle l}$. Note the formula above can also be written as

${\displaystyle {\mathrm{Ref}}_l(v)=2{\mathrm{Proj}}_l(v)-v,}$

saying that a reflection of ${\displaystyle v}$ across ${\displaystyle l}$ is equal to 2 times the projection of ${\displaystyle v}$ on ${\displaystyle l}$, minus the vector ${\displaystyle v}$. Reflections in a line have the eigenvalues of 1, and −1.

Reflection through a hyperplane in n dimensions

More generally, if ${\displaystyle S}$ is a linear subspace and ${\displaystyle P_S}$ is the orthogonal projection onto ${\displaystyle S}$, then reflection across ${\displaystyle S}$ is given by

${\displaystyle R_S=2P_S-I.}$

This fixes every vector in ${\displaystyle S}$ and negates every vector in ${\displaystyle S^{\perp }}$. Since

${\displaystyle P_S+P_{S^{\perp }}=I,}$

the same reflection may equivalently be written as

${\displaystyle R_S=I-2P_{S^{\perp }}.}$

Thus a reflection can be described either by the subspace that remains fixed or by its orthogonal complement, whose vectors are reversed. In the preceding two-dimensional section, ${\displaystyle l}$ spans the subspace that remains fixed; in the formula below, ${\displaystyle a}$ spans its orthogonal complement, whose vectors are reversed.

Given a vector ${\displaystyle v}$ in Euclidean space ${\displaystyle {\mathbb{R}}^n}$, the formula for the reflection in the hyperplane through the origin, orthogonal to ${\displaystyle a}$, is given by

${\displaystyle {\mathrm{Ref}}_a(v)=v-2\frac{v\cdot a}{a\cdot a}a,}$

where ${\displaystyle v\cdot a}$ denotes the dot product of ${\displaystyle v}$ with ${\displaystyle a}$. Note that the second term in the above equation is just twice the vector projection of ${\displaystyle v}$ onto ${\displaystyle a}$. One can easily check that

• Ref_a(v) = −v, if ${\displaystyle v}$ is parallel to ${\displaystyle a}$, and
• Ref_a(v) = v, if ${\displaystyle v}$ is perpendicular to a.

Using the geometric product, the formula is

${\displaystyle {\mathrm{Ref}}_a(v)=-\frac{ava}{a^2}.}$

Since these reflections are isometries of Euclidean space fixing the origin they may be represented by orthogonal matrices. The orthogonal matrix corresponding to the above reflection is the matrix

${\displaystyle R=I-2\frac{a{a}^T}{a^{T}a},}$

where ${\displaystyle I}$ denotes the ${\displaystyle n\times n}$ identity matrix and ${\displaystyle a^T}$ is the transpose of a. Its entries are

${\displaystyle R_{ij}={\delta }_{ij}-2\frac{a_i{a}_j}{{\Vert a\Vert }^2},}$

where δ_ij is the Kronecker delta.

The formula for the reflection in the affine hyperplane ${\displaystyle v\cdot a=c}$ not through the origin is

${\displaystyle {\mathrm{Ref}}_{a,c}(v)=v-2\frac{v\cdot a-c}{a\cdot a}a.}$