I'd like to get the parameters (rotation angle,$\Theta$, and translation coefficients, $x_0$ and $y_0$)) of a transformation for translating and rotating points in a coordinate system to another. As data I have a set of points [$P_1$,...$P_N$] of which I know their coordinates in the first system ([$x_1, y_1],[ x_2, y_2]$, ...) and the angles they are at in the second system ($\alpha_1, \alpha_2$, ...), instead of their coordinates ([$x'_1, y'_1],[ x'_2, y'_2]$, ...)

My problem in the real world is: I have a video analysis system that gets me the location of an object relative to its own coordinate system (in meters), and then I want to point a PTZ camera(which can rotate and zoom) to those points. For a number of reasons, let's suppose it is not feasible to obtain the GPS coordinates or the relative location the cameras. Instead, I thought of asking the user to manually point to some objects whose location relative to the analysis camera is known. The PTZ camera can give me its current pan (horizontal rotation), tilt and zoom values. For now I just need to do the calculations on the ground plane in 2D. So, based on the "pan" value for two or more points and their coordinates relative to the analysis camera, I'd like to get the transformation matrix in order to point to any arbitrary location given its coordinates.

Using the equations from the usual transformation matrix (like here or here) I haven't been able to deduce a formula for getting the parameters. Is it possible? How many points do I need at least?

Thank you.


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