Where does this identity come from (natural parametrization)

Question

Reading Analytical Mechanics by Fasano and Marmi, I bumped into this identity with natural parametrization.

Where does the identity \begin{equation} s=\int_0^s\left|\frac{d\boldsymbol{x}}{d\sigma}\right|d\sigma \end{equation} come from?

Answer 1

In the complex plane we can say that a differential length $ds$ is given by

$$ds = |dz| =\bigg| \frac{dz}{du}\bigg| du =|\dot z|du$$

and thus the arc length between two parametric points, say $u_{1,2}$ is thus

$$s=\int_{u_1}^{u_2}|\dot z|du$$