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For an assignment in computer graphics, i need to be able to determine a plane's rotation by just holding it in front of my webcam. So basically I only got 2D coords of the plane's points. I searched the internet to find how to determine the plane's rotation around the different axis but I found no information about it. Is there a good way to do this?

Image of the problem.

Thank you!

kennytm
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Diederik
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  • Do you know the camera's perspective angle, and the plane's distance from the camera? – kennytm May 05 '16 at 13:56
  • No, Like in my drawing, I only have information about the 4 corners, nothing else. I thought about there being a relation between the lengths of the sides but I can't find anything about it. – Diederik May 05 '16 at 13:57
  • Ok. Alternatively, do you know the plane's width and height in real life? – kennytm May 05 '16 at 14:01
  • Yes, I do know the length of the sides in real life. – Diederik May 05 '16 at 14:06

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When you project a 3D point to 2D, it will appear on the screen at $$ (x, y, z) \mapsto \left( -\frac{fx}z, -\frac{fy}z \right), \tag{*} $$ here $f$ is the focal length of the camera. Assume $x$ increases to the right, $y$ increases up and $z$ decreases far (right-hand rule).

Assume the plane's center is at $(0,0,-d)$ and the rotation angle is $\theta$ about the $x$-axis, and the plane's real size is $2W\times 2H$, then the four coordinates of the plane in 3D space would be \begin{gather} (-W, H\cos\theta,-d-H\sin\theta), \tag{top-left} \\ (W, H\cos\theta,-d-H\sin\theta), \tag{top-right} \\ (-W, -H\cos\theta,-d+H\sin\theta), \tag{bottom-left} \\ (W, -H\cos\theta,-d+H\sin\theta). \tag{bottom-right} \\ \end{gather} Performing transformation (*) to all 4 points, we can find the widths $w$ of the horizontal edges at the 2D projection, and also the distance between the two edges $h$: \begin{align} w_\text{top} &= f\cdot\frac{2W}{d+H\sin\theta} \\ w_\text{bottom} &= f\cdot\frac{2W}{d-H\sin\theta} \\ h &= f\cdot H\cos\theta\left(\frac1{d+H\sin\theta} + \frac1{d-H\sin\theta}\right) \end{align} There are 3 equations and 3 variables $f,d,\theta$, so we could get the solution.

Note that this assumes we know that the plane's center is $(0, 0)$. If we don't know where the center is, then there isn't enough constraints to find the angle.

kennytm
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