How using the arc length will make the definition of curvature independent of parametrization?

Question

Here is the part of James stewart that says this:

How using the arc length will make the definition of curvature independent of parametrization? Could someone explain this to me please?

Answer 1

Briefly speaking, because although reparametrizations by arc-length are not unique, they are almost unique -- to the extent that the curvature will not depend on the choice of reparametrization.

In other words, the algorithm would work as follows: start with your curve, choose any reparametrization by arc-length, and compute the curvature there. The result you get depends only on the initial curve.

To justify these claims:

1. If $s_1$ and $s_2$ are two arc-length parameters for $r$, then ${\rm d}r/{\rm d}s_1$ and ${\rm d}r/{\rm d}s_2$ are unit vectors. But the chain rule gives us that $$\frac{{\rm d}r}{{\rm d}s_1} = \frac{{\rm d}s_2}{{\rm d}s_1}\frac{{\rm d}r}{{\rm d}s_2}\implies \left|\frac{{\rm d}s_2}{{\rm d}s_1}\right|=1,$$and so $s_2 = \pm s_1+c$.
2. If $T_1$ and $T_2$ are unit tangents computed with respect to the parameters $s_1$ and $s_2$, then $T_1 = \pm T_2$, and so ${\rm d}T_1/{\rm d}s_1 = \pm{\rm d}T_2/{\rm d}s_2$ have the same length.