best way to find missing corner on photo of rubiks cube

Question

I've got the following photo of a rubiks cube and edited it so far that I've found out the coordinates of the outer 6 corners:

cube

My problem is: How do I find the coordinate of the last corner, which is located in the front of the image? Assume, that the image is practically y-axis symmetrical, but not x-axis symmetrical (distance between top and front corner is not the same as the distance between front and bottom)

An additional question would be, how to position the camera to take a photo of the cube as seen above, but where all 3 faces have the same area on the photo. What angle do I have to take the photo from?

Thanks for your help!

Are you after an exact answer based on projective geometry, or a strategy that takes into account the finite resolution of your image and numerical pitfalls dealing with measuring inaccuracies (but on the other hand accepts that the final answer my be a few pixels off)? — hmakholm left over Monica, Jul 25 '17 at 12:31
Do you mean the $2$-$D$ coordinates (i.e. the pixel coordinates) of the front corner? — Ben Grossmann, Jul 25 '17 at 12:33
I removed the (rubiks-cube) tag, since this question is about cubes in general, not really Rubik's cube. — Hans Lundmark, Jul 25 '17 at 16:12

score 1 · Accepted Answer · answered Jul 25 '17 at 12:46

If we answer the second question first, we end up with a convenient answer to the first question.

Suppose that the cube is positioned such that $3$-$D$ coordinates of its corners are $$ (0,0,0),(1,0,0),(0,1,0,),(0,0,1),(1,1,0),(1,0,1),(0,1,1),(1,1,1) $$ Then the camera should be looking down the $(1,1,1)$ vector. If $0$ degrees is straight down and $90^\circ$ is horizontal, then your camera's angle should be $$ \arccos\left(\frac{1}{\sqrt{3}} \right) \approx 54.74^\circ $$ If you have that perspective, then we have a convenient answer to the location of the front corner: it will be the centroid of the bottom, upper-left, and upper-right corners. That is, if $v_1,v_2,v_3$ are the vectors containing the coordinates of these three corners, then the position of the front corner will be $\frac{1}{3}(v_1 + v_2 + v_3)$.

best way to find missing corner on photo of rubiks cube

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