\[\begin{align} \textrm{Radius of curvature: } & R & = \frac{ds}{d\psi} \\ \textrm{Cartesian co-ordinates: } & \tan \psi & = \frac{dy}{dx} \\ & \sin \psi & = \frac{dy}{ds} \\ & \cos \psi & = \frac{dx}{ds} \\ \end{align}\]
Intrinsic coordinates are typically useful in physics problems that depend heavily on curvature and rate of change of position, but very little on absolute position, e.g. if you wanted to calculate how much an aeroplane needs to roll when turning a corner at a given speed and radius, you would find the calculations the simplest in intrinsic coordinates.