The most natural, diameter based, definition I can think of is, for any given point on the circle, the distance to the farthest point is a constant, D. You prob

The most natural, diameter based, definition I can think of is, for any given point on the circle, the distance to the farthest point is a constant, D. You probably need to add additional restrictions to make this rigorous (such as requiring the circle to be form a continuous, closed loop). You could also imagine rotating a line segment about its center. This has a nice sense of symmetry, as we are rotating about the center, not an arbitrary end. This also implies a generalization to off center rotations.