I have a vector $\tilde{p} = (x,y,z)$ (homogenous coordinates). The corresponding non-homogenous vector is $p = (x/z, y/z)$.
Now the $\tilde{p}$ is a result of some linear transform $R(\theta)$ of another vector $\tilde{u}$:
$\tilde{p} = R\tilde{u}$
I need to compute this derivative:
$\frac{\partial p}{\partial \theta}$
According to chain rule:
$\frac{\partial p}{\partial \theta} = \frac{\partial p}{\partial \tilde{p}} \frac{\partial \tilde{p}}{\partial \theta}$
The second partial derivative from the product on the right side is pretty obvious to me:
$\frac{\partial \tilde{p}}{\partial \theta} = \frac{\partial}{\partial \theta}R\tilde{u}$
But I don't understand the first one which should be (according to textbook):
$\frac{\partial p}{\partial \tilde{p}} = \frac{\partial [x/z\quad y/z]}{\partial [x\quad y\quad z]} = \begin{bmatrix} 1/z & 0 & -x/z^{2} \\ 0 & 1/z & -y/z^{2} \end{bmatrix}$
and what are the $x,y,z$ here? Does it come from $\tilde{u}$ or from $\frac{\partial}{\partial \theta}R\tilde{u}$ ? How is the matrix built?