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If I have $n$ number of vectors $p \in \mathbb{R}^2$ on a surface of a rectangle, would it be possible to estimate the underlying rectangle orientation? The rectangle show in the figure is a virtual rectangle, we do not know the exact orientation of the virtual rectangle. Dimension of the rectangle is know, and the all the vectors are in same reference frame.

Initial thinking:

  1. use von Mises distribution to sample rotation value centered at the mean of $p$. Most of the sampled rotation are not valid.
  2. Take two vectors $p_1, p_2$ randomly, and find the angle between $p_1$ and $p_2$. Continue until the distance between all the vectors are know. Although this method seems to be shedding light on the orientation, however, I do not know how to proceed further with this method.

Is there any mathematical technique that can handle this?

example rectangle with sampled points on top of it

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  • $\begingroup$ I think the first thing to do would be to find the largest distance between any two of the points. If that distance is $\le$ the short side of the rectangle, then it can be in any orientation. But if that distance is close to the length of a diagonal, as in your example, then the rectangle has to be close to the two orientations with those points in diagonally opposite corners. $\endgroup$
    – Paul Sinclair
    Commented Mar 6, 2023 at 19:00
  • $\begingroup$ One other step would be to find the convex hull of the points. Only the points on the hull tell you anything useful about the orientation. If the hull points are in a rectangle, the interior points have to be as well, so there is nothing to learn from them. This will significantly cut down on the points you have to examine to arrive at an orientation. In your example, it looks like about 7 points are on the hull (estimated by eyeball, it might be only 5 or 6). The rest can be ignored. $\endgroup$
    – Paul Sinclair
    Commented Mar 6, 2023 at 20:01
  • $\begingroup$ Thank you. I will try your suggestions, and update the results here. $\endgroup$
    – goldfinch
    Commented Mar 6, 2023 at 20:44

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