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A simple analogy would be to take a pencil and draw a free-hand circle on a piece of paper, going round-and-round a number of times. Then ask, “what is the EXACT radius of that circle?

April 26, 2017analogy Circle draw free-hand going paper pencil piece round-and-round simple

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A simple analogy would be to take a pencil and draw a free-hand circle on a piece of paper, going round-and-round a number of times. Then ask, “what is the EXACT radius of that circle?

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It is impossible to give an answer; the curve that you have drawn is not exactly a circle. One may define an “osculating” radius, for example: the radius of curvature at any given point on the curve. However, this value is exact at that given point only. The value will change for a different place on the curve; or, if averaged over some portion of the curve; or, if averaged over some other portion of the curve.