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## Is there any algorithm for determining 3d position in such case?

Suppose I have the following image (i.e. I have the coordinates of all points in 2d so I can regenerate lines and check where they cross each other)

Now suppose I have another image of what I know to be the same lines:

How can I determine plane rotation and Z depth on second image (asuming first one's center was in point (0,0,0) with no rotation)?

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Start here: en.wikipedia.org/wiki/Epipolar – Dan Piponi Apr 2 2010 at 23:11

What you see in the second image is a projection of the first, after a rotation. So treat it exactly like that. Meaning, you have 4 lines in $\mathbb{R}^3$. You know their equations. Furthermore, you have another 4 lines in $\mathbb{R}^2$ corresponding to the second image.
To solve, parameterize the family of possible 4 lines in $\mathbb{R}^3$ that project onto the 4 lines in $\mathbb{R}^2$. Find the matrix that takes the original 4 lines to generic quadruple in the mentioned family. Now write equations to ensure that matrix is actually a rotation. You should have enough information for there to be at most one quadruple that can actually be gotten from the original lines. If not, then your question has multiple answers.