How to define a path of vehicle

I would like to find an approach to define a path of vehicle in path-length coordinate.

As input I have an array of Cartesian coordinates (X, Y). I need to convert it to some function (object) which allow me to get a curvature of path, X, Y as functions of path length. Also there is a issue that I have a closed path like the following:

What can I use for such task?