THE CAMERA MATRIX
This problem describes how to use linear algebra to describe the world as seen through a camera.
The camera matrix. When we describe coordinates ($x, y, z$) on 3-dimensional space, we take
the y-axis to point up. We start with the camera at the origin of the ($x, y, z$) coordinate system,
pointing along the positive z-axis (so parallel to the ground), and held in its ordinary orientation.
The camera will draw the point $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ at the point $\begin{bmatrix} cx/z \\ cy/z \end{bmatrix}$ on the image pane, for some constant $c$.
The camera starts at the origin, pointing along the z-axis
Problem 1 Explain why $\begin{bmatrix} cx/z \\ cy/z \end{bmatrix}$ is the correct formula.
From now on, we'll normalize $c = 1$, so the point will be drawn at position $\begin{bmatrix} x/z \\ y/z \end{bmatrix}$ on the image pane,
We now move the camera to position $\begin{bmatrix} p \\ q \\ r \end{bmatrix}$, while keeping it pointing in the same direction (parallel to the z axis) and held upright.
The camera has been displaced to $\begin{bmatrix} p \\ q \\ r \end{bmatrix}$, still pointing parallel to the z-axis
Problem 2 Where will the point $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ be drawn now? (Note that your answer will include the variables $p, q, r, x, y$ and $z$.)
