Suppose $\beta_1(t)$ and $\beta_2(t)$ are two parametric curves defined on $[0,1]$. Let $\beta_1^*(t)$ and $\beta_2^*(t)$ are two re-parametrized of the above curves.
Now, I looking for a re-parametrization, where the sum of two re-parametrized curve also have the same properties as the component curves (reparametrized curve).
To clarify the above question: let us consider the arc-length parametrization (unit speed) of $\beta_1(t)$ and $\beta_2(t)$ as $\beta_1^*(t)$ and $\beta_2^*(t)$, respectively, than the sum function $\beta_1^*(t)+\beta_2^*(t)$ is not arc-length parametrized (not having unit speed). So arc length parametrization is not that I am looking for.
Any suggestions! Thanks.