An [[../Math/Affine Transformations|Affine Transformations]] that brings [[View Space]] [[../Math/Homogeneous Coordinates|Coordinates]] into [[World Space]] [[../Math/Coordinate System|Coordinate System]]. $\huge \begin{align} \text{Camera Data} &\begin{cases} C :& \text{Centre of Camera}\\ w :& \text{Width of Camera}\\ h :& \text{Height of Camera}\\ \vecn u :& \text{Right Dir.}\\ \vecn v :& \text{Up Dir.}\\ \huge \vecn u \parallel \vecn v \end{cases} \\ \end{align} $ We assume that $\vec u$, $\vec v$ form a [[../Math/Chirality|Right Handed]] coordinate system. $\huge \begin{align} M_{c} &= \mat{ \tilde u & \tilde v & \tilde C }\\ &=\mat{ u_{x} & v_{x} & C_{x}\\ u_{y} & v_{y} & C_{y}\\ 0&0&1\\ }\\ \end{align}$ #### [[Viewing Transform]] To get solve $V = M_{c}^{-1}$, >[!example] Example for [[../Math/Homogeneous Coordinates|Homogeneous]] $\Rn 2$ > [[Digression]]: [[../Math/Inverse Matrices|Inverse]] of an [[../Math/Affine Transformations|Affine Transformation]]: > >$\huge \begin{align} >M_{c} &= T_{\vec v} \cdot L \\ >M_{c}^{-1} &= L^{-1} \cdot T_{\vec v}^{-1} \\ >&= L^{-1} \cdot T_{-\vec v} \\ > >\\ > >L &= \mat{ a & b \\ c & d} \\ >L^{-1} &= \underbrace{\frac{1}{ad-bc}\mat{d&-b\\-c&a}}_{ >ad-bc \, \ne \, 0 >} \\ >\\ > > >M_{c}^{-1} &= > >\boxed{ >{\frac{1}{ad-bc}\mat{d&-b&0\\-c&a&0\\0&0&ad-bc}} >\cdot >\mat{ >1 & 0 & -C_{x} \\ >0 & 1 & -C_{y} \\ >0 & 0 & 1 \\ >}} >\\ >\end{align}$