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There is a relationship between the control points configuration and the accuracy of homography and planar pose estimation methods. I am really interested in understanding the reason behind this for my Ph.D. For the moment I found that the condition number of the A matrix in the DLT transform can serve as metric to predict the accuracy of a given control point configuration and not surprisingly the most stable 4-point configuration is a square (we proved this empirically by minimizing the condition number using a gradient descent approach).
I think also the relative orientation of the points have an influence on the estimation and this is what is happening here. If you are having similar problems please contact me, I am looking for examples of these anomalies.
More information about my research here:
https://arxiv.org/abs/1803.03025
https://www.youtube.com/watch?v=a6lDrwgqNmY
