[[Affine Transformations|Affine]] Reflection across $\alpha$: $\huge\begin{align*} \alpha &\in \Rn{n}, n \in \set{2, 3} \\ \huge \alpha &: \vec n ^{\intercal} \vec x = c \\ \\ \text{Proj into } \vec n^{\intercal} \vec x &= 0: \\ A &= I - \frac{1}{\norms{\vec n}^{2}} \vec n \vec n^{\intercal} \\ \let \vec x_{0} &\in \alpha \\ \vec b &= (A - I) \vec x_{0}\\ &= \pa{I - \frac{1}{\norms{\vec n}^{2}}\vec n \vec n^{\intercal} -I} \vec x_{0} \\ \vec b &= \frac{1}{\norms{\vec n}^{2}}\vec n \vec n^{\intercal}\vec x_{0} \\ \end{align*}$ ### With [[Homogeneous Coordinates]] $\huge\begin{align} A=I- \frac{2}{\norms{\tilde n}^2} \tilde n \tilde N ^ \intercal \end{align}$