I have a bit of a tricky problem that has come up in my engineering research, but I haven't quite got the brains to figure it out, though I've gotten pretty far.
Suppose that there is an unknown ellipse centered at the origin. I have a function that will generate random other ellipses that are also centered at the origin and that touch the unknown ellipse and are contained within it (in other words they are tangent to the unknown ellipse and contained within it). How many ellipses must I generate before I can uniquely determine the unknown ellipse? Assume no unlucky degenerate cases, like the function generating the same ellipse twice.
I believe the answer is 3, but I have no idea how to show it. It seems obvious that the answer is >2, and I think I can show that the answer is <=4.
If it helps, I can force the function to return a circle as the first ellipse it outputs, so it is possible to determine the length of the semi-minor diameter of the unknown ellipse , but I'm almost sure this information doesn't change the answer. Using a simulation and numerical optimization, I find that this actually reduces the required number to 2! I believe knowing the semi-minor diameter removes a degree of freedom.
Equivalently: There exists a stretching matrix that scales the space along a single dimension so that all of the ellipses outputed are circumscribed by a circle centered on the origin. If one eigenvalue of this matrix is 1 and the other is positive and <= 1, how many ellipses are needed to make this stretch matrix unique?
It would be a nice bonus if you can figure out how to determine this matrix from the ellipses.
Edit: The stretch matrix in the matrix version has to be an orthogonal stretch matrix.