If a curve $\alpha(s)$ on a surface $S$ is parametrized by arc length the geodesic curvature can easily be found
$$k_g(s)=\alpha''(s)\cdot (N(\alpha(s))\times \alpha'(s))$$
where $N$ is the unit normal.
The problem is that there are sometimes when it is extremely hard to reparametrize a curve by arclenght. Sometimes it involves eliptic integrals and so on.
In that case, how could one find the geodesic curvature? How can I find one expression for the geodesic curvature without needing to reparametrize by arc length?