Parametrization of a curve and by arclength I am working on differential geometry and am having quite the blast, it's mathematics so what else can you have?
Enough about that, I am working on the curves and we have various parametrizations and one comes about that is by arclength, which involves integrals which can get very nasty very quickly.
My question is quite simply this, is there any benifit to doing parametrization by arclength over anything else? Is it because the length of the curve is always the same if constants and intervals are adjusted to match? Or does it provide benifits in certain formulas that justifies this specific one?
 A: When a smooth curve $x:[a,b]\to R^n$ is given by the arclength parameter, the tangent vector $x'(s)$ is already normalized,
that is, has length one for all $s$, and the curvature is given simply by the length of the vector $x''(s)$. This provides a particularly
elegant exposition of the Frenet-Serret equations:
$$\left(\begin{matrix} T^{'}\\ N^{'} \\ B^{'}\end{matrix}\right)=
\left(\begin{matrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\
0 & -\tau & 0 \end{matrix}\right)\left(\begin{matrix} T\\ N \\ B\end{matrix}\right)$$
Here $T,N,B$ stand for the tangent, normal and bi-normal vectors of a three-dimensional path, and $\kappa,\tau$ are the curvature and torsion of the curve, respectively. If you had used the time-parameter, the equations would get much nastier. Theoretically, having the normalized tangent vector obtained by one differentiation is very useful, because it is then much easier to construct a moving -frame at every point of the curve - that is, an orthonormal basis that consists of three vectors directly linked to the geometry of the curve - and these are the unit tangent vector, the unit normal vector, and the unit bi-normal vector (i.e., the cross-product of the first two). 
Although the arclength parameter is convenient from the theoretical point of view, sometimes specific calculations with specific curves are easier to perform with the parameter in which they are given, and so general formulas can be deduced for any parameter, but it turns out that in order to write them down it is useful to already have the Frenet-Serret equations as written above, which did require the arclength parametrization.
A: If you wanted to work out a line integral things can be easier using the arc length parametrisation say r(t) as |r'(t)|=1
