So I have a transformation matrix that performs a stereographic projection and I need to find the center of projection and the plane to which it maps the transformed points.
$P = \begin{bmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&1&0\end{bmatrix}$
I tried multiplying some points to see where they got moved, and I found that the y-value of the point after projection is always one since the homogenizing coordinate cancels it out. This leads me to think that the plane that the points are being projected upon is $y=1$. Is this correct?
$\begin{bmatrix}0&1&0&0\\0&0&1&0\\1&0&0&0\\0&0&1&0\end{bmatrix} \begin{bmatrix}x\\y\\z\\w\end{bmatrix} =\begin{bmatrix}y\\z\\x\\z\end{bmatrix} =\begin{bmatrix}y/z\\1\\x/z\\1\end{bmatrix}$
With this information, how do I find the center of projection?
EDIT: I read on Wikipedia that the projection is undefined at one point: the projection point. My question now is: when is the projection undefined? Is it undefined when all values of a projected point are zero? So would I solve the equation and get that the center of projection is the origin?