Perspective transformation - Deriving formula for single Camera

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I would like to solve an equation and of that formulate a formula that takes two inputs and gives one output:

Input:

• (u,v) - Pixel coordinates
• (t) - Translation of the camera with respect to the plane, in one dimension (z)

Output

• (x,y) - World coordinate

And for the rotation of the camera, this is set to static. So the only part that can alternate is the height of the camera with respect to the plane.

I've successfully solved an equation system, but for when the camera has fixed rotation and fixed height as described here: https://dsp.stackexchange.com/a/46591...

But now I want to express a formula but that takes one additional parameter (height in [mm]).

However, I'm not sure how this equation system would look like described here, https://dsp.stackexchange.com/a/46591... to reflect my additional parameter.

My goal is to have a camera mounted on a linear rail (that moves in the z-direction and is vertical to the plane) that can detect objects on the plane. To my help, I have a laser sensor that constantly measures the height from the plane to the camera, which can be given as an input to the transform.

Any help is appreciated!