Curvature at a point of a curve

Question

For a smooth regular curve, there is the notion of curvature at any point. But if this curve is not differentiable at a point, then how to measure the curvature at that point? In the paper

Cufí, Julià; Reventós, Agustí; Rodríguez, Carlos J., Curvature for polygons, Am. Math. Mon. 122, No. 4, 332-337 (2015). ZBL1353.51009, the notion of curvature is defined at the vertex of a polygon. But how to define the curvature at a non-differentiable point of a non-polygonal curve?
Thank you.

Answer 1

Curve of circle of radius $R$ is has a curvature $\frac{1}{R}$.

Hence we can define a curvature as follows : If $c$ is a rectifiable curve, then let $p\in c$. If $a,\ b\in c$ s.t. $|p-a|=|p-b|=l$, then $a',\ b'$ are mid points of $[ap],\ [bp]$. Define $o$ s.t. $[po]$ divides $\angle\ a'pb'$ equally and triangles $\Delta poa',\ \Delta pob'$ are right triangles.

Hence curvature at $p$ is a limit of $\frac{1}{|p-o|}$ when $l$ goes to $0$.