This question was inspired by this thread I just saw on Space Exploration Stack Exchange:
where an anecdote is mentioned regarding how that the famous physicist Richard Feynman realized that part of the fault for the 1986 Space Shuttle Challenger disaster lie in not realizing that "a circle is not the only figure which has the same width in all directions".
Basically, it seems, the inspectors had tried to verify that the fuel tank sections, which are tubes and need to have cross-sections which are almost exactly perfect circles at the points at which they join together, in order to ensure a good fitting-together thereof with no leaks, by naively measuring the diameter over and over in different places and seeing that all the diameter results seemed to agree, and hence concluded the tanks were "circular".
And the problem with this is that there are non-circular shapes which have the same "diameter" at all points - a famous example being the so-called Reuleaux triangle which, if used to form the cross-section of a drill bit and then such suitably mounted, can be used to quickly and usefully make squarish holes with rounded corners.
And this, to me, raises a question: is there any way to, using only the ability to measure between points on the object's circumference, and no other points, determine if that object is or is not circular?
In more formal mathematical terms, if we have a closed plane curve $C$, then what are the conditions on the distances
between points $a, b \in C$, that are necessary and sufficient for circularity of the curve?