How to find the coordinates of the fourth vertex of a square in a projective perspective
By Sophia Bowman
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If I suppose that I know the coordinate of three points (eg. A'B'C') in a 2D coordinate system of A'B'C'D', my question is how can I find the coordinate of the fourth point D' (the same question for A''B''C''D'')? Is it even possible? Do I need more information?
For the rectangle ABCD would be easy to get the coordinates of D knowing ABC, because it is a normal square in a euclidean geometry, using the simple formula: $$A+B-C,\ A+C-B~or \ B+C-A$$
The projection used in here is a perspective projection like a pinhole model used by cameras. I do know the intrinsic parameters of the camera (focal length, pixels size..etc) and also the size of the square in real world, however, I don't know the distance Z of that square from the camera. That's parameters matrix:
$$P = \begin{pmatrix} \alpha_u & 0 & u_0 \\ 0 & \alpha_v & v_0 \\ 0 & 0 & 1 \end{pmatrix}$$
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