[[Affine Transformations|Affine]] equivalent for [[Orthogonal Projection Transformation]]. $\huge A= \frac{1}{\norms{\vec v}^{2}} \vec v {v}^{\intercal} = \hat v \hat v ^{\intercal} $ #### Using [[Homogeneous Coordinates]] With $\vec{n}^{\intercal} \vec{\mathbf{x}}=c$: $\huge\begin{align} \text{Line Normal }&\cases{\tilde{n}} = \mat{n_{x} \\ n_{y} \\ \vdots \\ 0} \\ \text{Augmented Normal }&\cases{\tilde{N}} = \mat{ \vec{n} \\ -c} \\ \end{align}$ $\huge A=I- \frac{1}{\tilde{N}^\intercal \tilde v} \tilde{v}\tilde N^{\intercal} $ $\huge A=I- \frac{1}{\tilde{N}^\intercal \tilde n} \tilde{n}\tilde N^{\intercal} $ $\huge A=I- \frac{1}{\norms{\vec n}^2} \tilde{n}\tilde N^{\intercal} $ >[!example] [[Orthogonal Projection]] into $l: x-2y=-5$ > >$\huge \vec n = \mat{1\\-2}, \tilde n = \mat{1\\-2\\0}, \tilde N =\mat{1\\-2\\5} >$ > >$\huge A= \mat{1&0&0\\0&1&0\\0&0&1} - \frac{1}{ \mat{1&-2&5}\mat{1\\-2\\0} } \mat{1\\-2\\0}\mat{1&-2&5} >$ >$\huge A= \frac{1}{5}\mat{4&2&-5\\2&-4&10\\0&0&5} $ >[!example] Project into: >$\begin{align} l: \vec{\mathbf{x}} &= \mat{3\\4\\-3} + t\mat{1\\0\\2} \\ \\ A &= \frac{1}{5} \mat{1&0&2\\0&0&0\\2&0&4} \\ \tilde{A} &= \begin{bmatrix} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \frac{1}{5} & 0 & \frac{2}{5} & 0 \\ 0 & 0 & 0 & 0\\ \frac{2}{5} & 0 & \frac{4}{5} &0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 1 \end{bmatrix} >\end{align}$