Let's assume we have a linear coordinate system on a plane. If we make a 2d perspective projection of that plane from 3d, we get a "skewed" coordinate system.
For example, one axis would look like this in the original plane:
|.........|.........|.........|.........|.........|
but will look something like this in the projection:
|.|...|.....|..........|..........................|
I don't have any information at all about the focal length, the viewing angle, the distances involved, anything. The only information I have are the distances between the "markings" of the new coordinate system, and the knowledge that it was linear before the projection. this is why I can't find a way to use the well-known formulas, matrix notations, etc. of the perspective projection.
If I knew what form this function can take, I would be able to calculate its parameters from the points I have. For example, if it were quadratic, in the form of $ax^2 + bx + c$ I could calculate $a$, $b$ and $c$ by using the data points I have. I know it's not quadratic, so what can it be?