When you considered translating the camera to position $\left[\begin{array}{c}p\\ q\\ q\end{array}\right],$ you should have found that the

formulas expressing the $(u,v,w)$ coordinates in terms of the $(x,y,z)$ coordinates were not purely

linear. To fix this, we introduce the notion of the camera matrix $P.$ This is a $34$ matrix $P$

such that

$\left[\begin{array}{c}u\\ v\\ w\end{array}\right]=P\left[\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right].$

Problem $5$ Compute the camera matrices for the two scenarios in the previous problems: The

camera translated to $\left[\begin{array}{c}p\\ q\\ r\end{array}\right],$ and the camera translated to that position and then rotated $$ around a

vertical axis.

Reconstructing the camera matrix. Suppose we know the coordinates of a number of points

in space, and we would like to determine where the camera was placed.

$\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right],\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

and that the camera plots them in positions

$\left[\begin{array}{c}0\mathrm{.}5\\ 0\mathrm{.}7\end{array}\right],\left[\begin{array}{c}0\mathrm{.}25\\ 0\mathrm{.}35\end{array}\right],\left[\begin{array}{c}1\mathrm{.}3\\ 1\mathrm{.}3\end{array}\right],\left[\begin{array}{c}0\mathrm{.}65\\ 0\mathrm{.}65\end{array}\right],\left[\begin{array}{c}1\mathrm{.}1\\ -0\mathrm{.}1\end{array}\right],\left[\begin{array}{c}0\mathrm{.}55\\ -0\mathrm{.}05\end{array}\right],\left[\begin{array}{c}1\mathrm{.}9\\ 0\mathrm{.}5\end{array}\right],\left[\begin{array}{c}0\mathrm{.}95\\ 0\mathrm{.}25\end{array}\right]$

respectively.

Problem $6$ Find the camera matrix. You should need to solve $16$ linear equations in $12$ variables.

Is there a unique solution? Why or why not? You should use software to solve these equations

$($Maple$,$ MATLAB, Mathematica, num. py$,$ etcetera$),$ your solution should make it clear what equations

you solved, how you solved them, and what answer you got.

Problem $7$ Find the location of the camera in $(x,y,z)$ coordinates.