Professor was a little fuzzy on this topic, so I just wanted to make sure I have this definition correct:
Given a function $\alpha : T \to C \mid t \in [a,b]$ , where $t$ is the parameter and $C$ is the curve and $\alpha$ is bijective, we can find it's arc length parametrization in the following way:
Choose an arbitrary $t_0 \in T$, we shall call this the reference point for the parametrization.
We define the arc length function for $\alpha$ to be $s \mid s : T \to L$ as a bijective function $\forall t \in [c,d]$ given by
$$ s = \int_{t_o}^{t} \| \hat{r}^{'} (u) \|\, \mathrm{d}u $$ if we evaluate the integral, we obtain $s = A(t)$ so taking the inverse gives $A^{-1}(t) \mid A^{-1}(t) : L \to T$
Now the composition $\beta = \alpha \circ A^{-1} (t)$ is a function $\beta : L \to T \to C$ and therefore whatever is inputted into the function $\beta$ outputs the curve $C$ but the input is precisely the arc length of $C$ from $t_o$ to $t$