On planar curves with position-dependent curvature

TL;DR

This paper systematically studies planar curves with curvature depending on position, using dynamical systems, and classifies all simple closed solutions for specific curvature functions.

Contribution

It provides a dynamical systems framework for position-dependent curvature and classifies closed solutions for curvature functions of the form  r^b.

Findings

Complete classification of simple closed solutions for  r^b curvature functions.

Dynamical systems approach applicable to various position-dependent curvature problems.

Insights into homothetic solutions of curvature-driven flows.

Abstract

Motivated by homothetic solutions to curvature-driven flows of planar curves, as well as their many physical applications, this work carries out a systematic study of oriented 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 a distinguished point, the basic dynamical concepts are seen to apply in other situations as well. As an application, a complete classification of all simple closed solutions of $\kappa = ar^b$, with real constants $a,b$, is established.