On Planar Curves with Position-Dependent Curvature

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