Arc-length and curvature

Curvature

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 $a$: