How to [[Reflection Transformations|mirror]] geometric entities? Let $\braket{a,b,c,d}$ be [[Planes|Plane]] coefficients of the reflective surface $P$, where $\hat{n}=\braket{a, b, c}$ is a [[Normal Vector]] of $P$. $\huge \begin{align} P(x,y,z): ax+by+cz+d &= w_{c} \end{align}$ ![[Planar Map .excalidraw.svg|500]] %%[[Planar Map .excalidraw.md|🖋 Edit in Excalidraw]]%% Let $c$ be the [[Vector]] from the origin of the plane to the [[Point]] you are reflecting. $w_{c}$ denotes the [[Vector Projection|projection]] of $c$ onto $\hat n$. $\huge \begin{align} c' &= \vec c - 2 w_{c} \hat{n} \\ w_{c} &= \proj{\hat{n}}\pa{\tilde{ c}} \\ &= \mat{x\\y\\z\\1} \cdot \mat{a\\b\\c\\d} \\ \end{align}$ You can also express this in [[Matrix]] form: $\large \begin{align} M &= I_{4} - 2\mat{a\\b\\c\\0}\mat{a&b&c&d} \\ M &= I_{4} - 2\mat{a\\b\\c\\0} \oplus \mat{a\\b\\c\\d} \\ \end{align}$