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Yes, there is a way to tell that the bucket is off-center.
The shape of the surface will be a paraboloid of revoution wherever
the bucket is placed on the turntable. The vertex of the paraboloid
will be on the axis of revolution. If that axis does not pass
through the bucket then the surface is called an off-axis paraboloid.
Why is it possible to tell if the bucket is off-center but it isn't
possible to tell (if one is stuck inside a box) where one is relative to
the axis of rotation (without more information given)?