Suppose you're given an image of a cuboid like the one below, and you're told that this image was taken by a primitive pinhole camera that has a known world projection center coordinates and known orientation with respect to a world coordinate frame.
Your task now is to determine the dimensions of this cuboid, knowing only the given position and orientation of the camera, and the fact that $ABCD$ represents a rectangle that lies in the $xy$ plane of the world coordinate system.
For the image given below, you are given that the camera position is
$ E = (90, -60, 45) $
And, the orientation of the camera is given by the rotation matrix
$ R = \begin{bmatrix} 0.5547&& -0.3196&&0.768221\\0.83205&&0.213066 &&-0.51215\\0&&0.923287&&0.384111 \end{bmatrix} $
And measuring the pixel coordinates of the vertices in the image referenced by the upper left corner, where the first coordinate is the horizontal and the second the vertical, the vertices coordinates are as follows:
$\begin{array}{|c|c|c|} \hline \text{Vertex} & x & y \\ \hline A & 303 & 495 \\ \hline B & 498 &425 \\ \hline C & 219 & 296 \\ \hline D & 39 & 342 \\ \hline E & 328 & 168 \\ \hline F & 538 & 124 \\ \hline G & 229 & 45 \\ \hline H & 39 & 72 \\ \hline \end{array} $
How can you proceed to find the cuboid dimensions? And based on the above numbers, what are those dimensions?
Comment. I tried to simulate the figure you have. As can be seen in the figure we have cuboid ABCDEFGH thier coordinates in 3D shown. I constructed a plane passing points I (on edge GH), vertex B and vertex E. Then I dropped a perpendicular from point $F_1$ where diagonals of the cuboid meet. On this perpendicular I took the point $M(8.4, -5.6, 9)$ and constructed a plane passing M and parallel with the previous plane. I then drooped perpendiculars from all 8 vertices of the cuboid to this plane. Clearly the intersections of these perpendiculars with this plane are the projections of the vertices of the cuboid. Connecting these points gave a 2D figure you can see below:
[![enter image description here][2]][2]
In this figure similar to the cuboid the intersection of diagonals is the projection of that of the cuboid. The coordinates of this poin M in 2D on the plane for 2D figure is:
$x_M=\frac{x_1+x_2}2$
$y_M=\frac {y_1+y_2}2$
where $x_1, x_2$ and $y_1, y_2$ represent the coordinates of the opposite vertices of the 2D figure.
I calculated the coordinates of M from your figure, but they do not give a single value. This means your figure is not competant with the projection of a cuboid. So corrections are needed.
Now suppose you have correct 2D coordinates and the position of the plane this figure is on is known. To find the vertices of the cuboid you need to do the reverse procedure, i.e dropping perpendiculars from the vertices of the 2D figure on its plane These perpendiculars intersect with three planes in 3D which will be the vertices of the cuboid. From here you can see their coordinates and calculate the dimension of edges.
In figure below, where the edges of the 2D figure are exactly coincided with the ones of the cuboid, we can see that if the the coordinates given in 2D figure are acurate the dimensions found using these may be equal to those of the cuboid, but it must be checked. [2]: https://i.stack.imgur.com/LYQnY.png