Reflection in an affine hyperplane ($x\mapsto x-2\frac{n^{T}x-d}{n^{T}n}n$)

For a nonzero real normal vector $n$, orthogonal reflection in $n^{T}x=d$ has the displayed affine map. It is $x\mapsto Hx+t$ with $H=I-2n{n}^T/(n^{T}n)$ and $t=2dn/(n^{T}n)$. In homogeneous coordinates its matrix isEvery point on the hyperplane is fixed, and the signed normal displacement changes sign. Choosing any point $p$ on the hyperplane gives $t=p-Hp$, so translating by $-p$, applying the reflection matrix $H$ and translating back gives the same transformation.