Parameterization by arc length

Question

What happens if you reparameterized by arc length a curve that is already parameterized by arc length?

I just did it because I didn't read the statement right.

Answer 1

You get back the same parameter, because if $\gamma$ is a curve already parametrized by arc-length, say, $s$, then $\|\gamma'(s)\|=1$ for every $s$, hence $$p(s)=\int_0^{s}\|\gamma'(\sigma)\|\,d\sigma=s,$$ so the "new" parameterization by arc-length $p(s)$ is precisely the previous one, $s$.

Answer 2

Without loss of generality, suppose the curve $\ C \ $is a finite or infinite interval of real numbers. An arc length parametrization of $\ C \ $ is a bijection $\ f: I \to C \ $ where $\ I \ $ is also a finite or infinite interval of real numbers and such that $\ \forall x\in I \ $ then $\ |f'(x)| = 1. \ $ Any such function is of the form $\ f(x) = c \pm x \ $ where the constant $\ c \ $ and sign $\ \pm \ $ are such that endpoints get mapped to endpoints.

In the general case, where there are two arc length parametrizations of a curve, the composition of one of them followed by the inverse of the other reduces to the special case described above.