If you try to show this formula in comments, it fails.
$$M:\left[\begin{array}{cc}x&y\end{array}\right]\mapsto\left[\begin{array}{ccc}x&y&1\end{array}\right]M=\left[\begin{array}{ccc}u&v&w\end{array}\right]\mapsto\left[\begin{array}{cc}\frac{u}{w}&\frac{v}{w}\end{array}\right]$$
Source here.