Definition
Suppose that you have a 3 dimensional plane defined by equation:
$$z = 0$$
This plane contains a number of horizontal and vertical lines that form a grid on a plane that is centred around the origin. All of the line parameters (positions) are known and the lines are arranged with fixed spacing.
The plane is being observed from a fixed point sufficiently above the plane:
Now from our fixed point of view if the plane were to slightly move or rotate around arbitrary vector the lines on a grid would distort and probably stop to appear orthogonal to each other or if plane movement was made towards our point of view grid would increase in size.
The aim is to estimate plane parameters from known (measured) line parameters. This needs to happen for both the simple case when line measurements are precise and also in case when line parameter measurements are performed with small errors.
The Question
I'm looking for solution or inspiration to find simple geometric solution to this problem. I'm aware of camera pose estimation algorithms and direct linear transform that would provide solution to this and even more general case. But that is not what I'm looking for as the problem discussed is simpler and so it feels that it has simpler geometric solution.