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• W3208483075 abstract "<p style='text-indent:20px;'>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 <inline-formula><tex-math id=M1>begin{document}$ r(varphi) $end{document}</tex-math></inline-formula> measured from a reference point, which may or may not be fixed. An important question is how the numbers and the trajectories of critical points of the distance function <inline-formula><tex-math id=M2>begin{document}$ r(varphi) $end{document}</tex-math></inline-formula> and of the curvature <inline-formula><tex-math id=M3>begin{document}$ kappa(varphi) $end{document}</tex-math></inline-formula> (characterized by <inline-formula><tex-math id=M4>begin{document}$ dr/dvarphi = 0 $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=M5>begin{document}$ dkappa /dvarphi = 0 $end{document}</tex-math></inline-formula>, respectively) evolve under the Andrews-Bloore flows for different choices of the parameters.</p><p style='text-indent:20px;'>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.</p>" @default.
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• W3208483075 date "2021-01-01" @default.
• W3208483075 modified "2023-09-26" @default.
• W3208483075 title "Tracking the critical points of curves evolving under planar curvature flows" @default.
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• W3208483075 doi "https://doi.org/10.3934/jcd.2021017" @default.
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