Length of curves with same images

Question

From a geometrically intuitive point of view, it is obvious that if two injective $C^1$ curves $\gamma,\delta$ with values in $\mathbb R^n$ have the same images, then their lengths $\ell(\gamma)$ and $\ell(\delta)$ (as defined by the standard definition from differential geometry) are equal. This is a well-known resulut if $\gamma$ and $\delta$ are reparametrizations of one another; however, they need not necessarily be. Can it still be proved without this assumption?

Answer 1

How about $$ \gamma(t)=(\cos(t), \sin(t)) \ \ t\in[0,2\pi] $$ and $$ \delta(t)=(\cos(t), \sin(t)) \ \ t\in[0,4\pi] $$ They clearly have the same image, but $\ell(\gamma)=2\pi \neq 4\pi=\ell(\delta)$.