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Arc-length and curvature
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Consider a road and a car. If you are inside, you can track the direction, i.e., the tangent, and how fast you turn.
From calc 3, how fast is measured by the curvature, $\frac{df}{ds}$.
In the discrete case, the turn is easy to see -- it's sudden. So curvature = the angle.
Here $\frac{\pi}{2}$ is the curvature at $a$: