Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 3 / 4A / c / Solution

This plane curve is an Archimedean spiral. Put ${\mathbf{e}}_r=(\cos u,\sin u,0)$ and ${\mathbf{e}}_{\theta }=(-\sin u,\cos u,0)$, so ${\mathbf{e}}_r^{\prime }={\mathbf{e}}_{\theta }$ and ${\mathbf{e}}_{\theta }^{\prime }=-{\mathbf{e}}_r$. ThenTherefore the curvature of a plane curve isThe parametrization is regular even at $u=0$, where the curvature of a plane curve is $2$.