Reparametrization of a curve Given a parametrized curve, we know that its arc length parametrization is its unit speed reparametrization. However, I wanted to know if there was any generic procedure to find any other reparametrizations of the same curve, which are not unit speed, and are non trivial?

Assume you have a curve $\gamma : [a,b] \to \mathbb R^d$ and $\varphi : [a,b] \to [a,b]$ is a reparametrization, i.e., $\varphi'(t) > 0$. Then you can prescribe any speed function for your parametrization. Given a function $\sigma: [a,b] \to \mathbb R_{>0}$, define $\varphi$ via the ODE
$$ \varphi'(t) = \frac{\sigma(t)}{|\gamma'(\varphi(t))|} \,. $$
Then the reparametrized curve $\tilde \gamma (t) = \gamma \circ \varphi(t)$ has speed $\sigma$, i.e.,
$$ |\tilde \gamma'(t)| = \sigma(t)\,. $$
The only restriction on $\sigma$ is that $\varphi(b) = b$.
