Tracking the critical points of curves evolving under planar curvature flows

Abstract

We are concerned with the evolution of planar, star-like curves and associated shapes under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flow. This family of flows has two parameters that control one constant and one curvature-dependent component for the velocity in the direction of the normal to the curve. The Andrews-Bloore flow includes as special cases the well known Eikonal, curve-shortening and affine shortening flows, and for positive parameter values its evolution shrinks the area enclosed by the curve to zero in finite time. A question of key interest has been how various shape descriptors of the evolving shape behave as this limit is approached. Star-like curves (which include convex curves) can be represented by a periodic scalar polar distance function \begin{document}$ r(\varphi) $\end{document} measured from a reference point, which may or may not be fixed. An important question is how the numbers and the trajectories of critical points of the distance function \begin{document}$ r(\varphi) $\end{document} and of the curvature \begin{document}$ \kappa(\varphi) $\end{document} (characterized by \begin{document}$ dr/d\varphi = 0 $\end{document} and \begin{document}$ d\kappa /d\varphi = 0 $\end{document}, respectively) evolve under the Andrews-Bloore flows for different choices of the parameters.

We present a numerical method that is specifically designed to meet the challenge of computing accurate trajectories of the critical points of an evolving curve up to the vicinity of a limiting shape. Each curve is represented by a piecewise polynomial periodic radial distance function, as determined by a chosen mesh; different types of meshes and mesh adaptation can be chosen to ensure a good balance between accuracy and computational cost. As we demonstrate with test-case examples and two longer case studies, our method allows one to perform numerical investigations into subtle questions of planar curve evolution. More specifically — in the spirit of experimental mathematics — we provide illustrations of some known results, numerical evidence for two stated conjectures, as well as new insights and observations regarding the limits of shapes and their critical points.