On Planar Curves with Position-Dependent Curvature

Motivated by homothetic solutions in curvature-driven flows of planar curves, as well as their many physical applications, this article carries out a systematic study of oriented smooth curves whose curvature κ is a given function of position or direction. The analysis is informed by a dynamical systems point of view. Though focussed on situations where the prescribed curvature depends only on the distance r from one distinguished point, the basic dynamical concepts are seen to be applicable in other situations as well. As an application, a complete classification of all closed solutions of κ = a r b , with arbitrary real constants a , b , is established.