First let me introduce the definitions then I will come to my actual doubt.
Parametrized Curve- A parametrized curve is smooth map $\gamma : I \rightarrow \mathbb{R^3}$, where $I$ is open interval of $\mathbb{R}$.
Parametrized Surface- A Parametrized surface is smooth map $\sigma : U \rightarrow \mathbb{R^3}$ such that $\sigma : U \rightarrow \sigma(U) \subset \mathbb{R^3}$ is homeomorphism onto its image, such that $\{\frac{\partial \sigma (u,v)}{\partial u}, \frac{\partial \sigma (u,v)}{\partial v}\}$ is linearly independent for all $(u,v) \in U$.Then $S=\sigma(U)$ is called parametrized surface in $\mathbb{R^3}$, where $U$ is open subset of $\mathbb{R^3}$.
I want to know what is the motivation for doing this reparametrization of curve and surfaces ? In the case of curve, I have something, that is, due to the reparametrization the rate of it transverse. For example, Arc length parametrization gives the unit speed on the curve. But still need proper motivation of doing it for curves.
Also, What about in the case of reparametrization of surface? What is benifit/motivation doing the reparametrization of surfaces?
Note that I am not giving explanation of what do we mean by reparametrization of curves and surfaces.