Let $I_1, I_2$ are two non-degenerating intervals of $\mathbb{R}$ and, let $\gamma_j : I_j \to \mathbb{R}^n,\quad j=1,2$ be two parametrized regular $\mathcal{C}^r$-curves with same trace (image in $\mathbb{R}^n$) $$\gamma_1(I_1)=\gamma_2(I_2).$$ What are the necessary and sufficient conditions that guarantee the existence of a $\mathcal{C}^r$-function $\varphi: I_1\to I_2$ such that
- $\varphi'(t)\neq 0$
- $\gamma_1(t)=\gamma_2(\varphi(t))$
for all $t\in I_1$ ?
I am trying to understand this notion of equivalent curves.