Saturday, 20 January 2018

calculus - Computing diagonal Length of a Square

While studying rectification of curves, I considered a curve and to measure its length in a different fashion, and arrived at a problem. I would like to clarify the confusion in my understanding.



Consider a unit square in plane with vertices (0,0), (0,1),(1,0), (1,1). The diagonal joining (1,0) and (0,1) has length 2, well known. Suppose I approach theis diagonal in the following way: first by the path P1:(1,0)(1/2,0)(1/2,1/2)(0,1/2)(1,0). This is like tow "L"s, with top end of one joined to bottom end of other- forming stairs. The length of this path is 2. Next, we form path P2 with four "L"'s, in a nice way to form stairs. Again the length of this path is 2.



We see that the sequence {Pn} of paths, which are piecewise differentiable functions (?), converges to the diagonal (1,0)(0,1). But the length of each path is 2, but we cant conclude that the diagonal should have length 2. Why such a contradiction arises?

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