On Planar Curves with Position-Dependent Curvature

Abstract

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 $$\kappa $$ 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 $$\kappa = ar^{b}$$ , with arbitrary real constants $$a,b$$ , is established.