Reflection (geometry)

In analytic geometry, a reflection is a linear operation &sigma; on a vector space with &sigma;2 = 1, i.e., &sigma; is an involution. Reflecting twice an arbitrary vector brings  back the original vector :

\sigma( \vec{\mathbf{r}}\,) = \vec{\mathbf{r}}\,' \quad\hbox{and}\quad \sigma( \vec{\mathbf{r}}\,'\,) = \vec{\mathbf{r}}. $$

Reflection in a plane
If $$\hat{\mathbf{n}}$$ is a unit vector normal (perpendicular) to a plane&mdash;the mirror plane&mdash;then $$ (\hat{\mathbf{n}}\cdot\vec{\mathbf{r}})\hat{\mathbf{n}}$$ is the projection of $$\vec{\mathbf{r}}$$ on this unit vector. From the figure it is evident that

\vec{\mathbf{r}} - \vec{\mathbf{r}}\,' = 2 (\hat{\mathbf{n}}\cdot\vec{\mathbf{r}})\, \hat{\mathbf{n}} \;\Longrightarrow\; \vec{\mathbf{r}}\,' = \vec{\mathbf{r}} - 2 (\hat{\mathbf{n}}\cdot\vec{\mathbf{r}})\hat{\mathbf{n}} $$ If a non-unit normal $$\vec{\mathbf{n}}$$ is used then substitution of

\hat{\mathbf{n}} = \frac{\vec{\mathbf{n}}}{ |\vec{\mathbf{n}}|} \equiv \frac{\vec{\mathbf{n}}}{n} $$ gives the mirror image,

\vec{\mathbf{r}}\,' = \vec{\mathbf{r}} - 2 \frac{ (\vec{\mathbf{n}}\cdot\vec{\mathbf{r}})\vec{\mathbf{n}}}{n^2} $$ This relation can be immediately generalized to m-dimensional inner product spaces. Let the space Vm allow an orthogonal direct sum decomposition into a 1-dimensional and a (m&minus;1)-dimensional subspace,

V_m = V_1 \oplus V_{m-1} $$ and let v be an element of the one-dimensional space V1 then the involution

r \mapsto r - 2v\frac{ (v,r)}{(v,v)} $$ is a reflection of r in the hyperplane Vm&minus;1. (By definition a hyperplane is an m&minus;1-dimensional linear subspace  of a linear space of dimension m.) The inner product of two vectors v  and w is notated as (v, w), which is common for vector spaces of arbitrary dimension.