Spacecraft on an Asteroid with Unknown Shape

Question

The scientists at NASA are investigating an asteroid in space. From particulars transmitted by the Hubble Space Telescope, it is known to the scientists that the asteroid is either a ball or a cube, but they are in the dark about what it actually is. To determine the exact shape, the scientists are sending a spacecraft to the asteroid. Once the spacecraft lands, the rover is programmed to start moving at the landing site and finish at the point symmetric to the landing site with respect to the center of the asteroid. On its way, the rover will keep transmitting its exact location in terms of spatial co-ordinates with respect to the spacecraft, so that the trajectory of the rover movement can be known to the scientists at NASA. Using these data, they plan to determine whether the asteroid is spherical or cubical.

Is their plan sure to work?

_{Source: Tournament of Towns.}

Answer 1

The answer is

NO.

The question is equivalent to analyzing the intersection of a cube and a sphere which share a common center. Thus the question gets reduced to figuring out whether such intersection, which is a curve, can connect two opposite points on the sphere/cube.

Let the edge of the cube has length $1$. Then if you pick the radius of the sphere $1$, the intersection will be a set of $6$ points. If you start gradually increasing it, you can see that when it becomes equal to $\frac{\sqrt{2}}{2}$, the intersection consist of $6$ circles inscribed in the sides of the cube. Then the rover can just move along these circles from one point to its opposite and NASA won't be able to find out the exact shape.

Remark: From the arguments above you can see that $1:\frac{\sqrt{2}}{2}$ is the only edge-radius ratio, for which we can't figure out the shape.

P.S. I see now there is already a link towards a solution posted above in the comments, so I'm making this community Wiki.

Answer 2

Absolutely would work. If the spacecraft lands on a side of the cube, the rover will pass over at least 2 edges which will be seen spatially as going further away (over the edge) then getting closer again before going down that side. If it's a sphere/ball, there will be a constant shift in the distance between the rover and spacecraft until it reaches the far side.

Answer 3

I think it should work?
If the asteroid is spherical, the vector describing the rover's movement relative to the craft will be constantly changing, with no variance because spheres are infinity-dimensionally symmetric

If the astroid is a cube, there will certainly be a non-zero interval where the rover's movement is in a straight line, no matter where the craft lands.

EDIT: Proven wrong. Assumed (incorrectly) a straight line path is programmed into the rover.