Similar to [[Camera Modeling Transform (2D)]].
$\huge
M_{C} = \mat{
{\vec u} & \vec v & \vec n & \tilde{e}
}
$
>[!tip] The variables here are defined [[Perspective Projection|here]]
To get the [[Viewing Transform]], you take [[../Math/Inverse Matrices|Inverse]] of $M_C$.
$\huge
V = M_{C}^{-1}
$
>[!note] [[Geometry]] of Perspective
> $M_c$ nor $V$ effect the [[Geometry]] of the [[Perspective Projection|Projection]], only rotations and translations are happening.
>
> The linear part of both of these are necessarily a rotation.
>[!example]-
>Camera with:
>$\begin{align}
>&\begin{matrix}
>E=\pa{3,1,-2}
>&\vec u = \mat{0\\1\\0}
>&\vec v = \mat{0\\0\\1}
>&\vec n = \mat{1\\0\\0}
>\end{matrix}
>\end{align}$
>$\begin{align}
>M_{c} &= \augmented{c|c|c|c}{\vec u & \vec v & \vec n & E} \\
>&= \mat{
>0&0&1&3\\
>1&0&0&1\\
>0&1&0&-2 \\
>0&0&0&-1
>} \\
>&= T_{\vec v} \circ L \\
>\end{align}$
>$\begin{align}
>V &= M_{c}^{-1} \\
>&=L^{-1} \circ T_{-\vec v} \\
>&= \underbrace{L^{\intercal}}_{\text{Rotation}} \circ T_{-\vec v} \\
>&=
>\mat{
>0&1&0&0\\
>0&0&1&0 \\
>1&0&0&0 \\
>0&0&0&1
>}\mat{
>1&0&0&-3\\
>0&1&0&-1\\
>0&0&1&2\\
>0&0&0&1
>}
>\\&=\color{pink} \boxed{\color{white}\mat{
>0&1&0&-1\\
>0&0&1&2\\
>1&0&0&-3\\
>0&0&0&1
>}}
>\end{align}
>$