Saturday, September 15, 2018

geometry - Vector path length of a hypotenuse

Figure 1



Consider the red path from A that zigzags to B, which takes $n$ even steps of length $w$. The path length of the route $P_n$ will be equal to:




$ P_n = P_x + P_y = \frac{n}{2}\times w + \frac{n}{2}\times w = n \times w $



But $\frac{n}{2}\times w = 1$ beacuse it is the length of one of the sides of the triangle so:



$P_n = 2$



Which will be true no matter how many steps you take. However in the limit $n \to \infty, w \to 0$ the parth length $P_\infty$ suddenly becomes:



$P_\infty = \sqrt{1^2 + 1^2} = \sqrt{2}$




Due to Pythagoras. Why is this the case? It seems the path length suddenly decreases by 0.59!

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