Tracking the critical points of curves evolving under planar curvature flows

Eszter Fehér; Gábor Domokos; Bernd Krauskopf

doi:10.3934/jcd.2021017

Article Contents

2021, Volume 8, Issue 4: 447-494. Doi: 10.3934/jcd.2021017

This issue Previous Article Computing Reeb dynamics on four-dimensional convex polytopes Next Article Classification with Runge-Kutta networks and feature space augmentation

Tracking the critical points of curves evolving under planar curvature flows

1.
MTA-BME Morphodynamics Research Group
2.
Department of Morphology and Geometric Modeling, Budapest University of Technology and Economics, Müegyetem rakpart 1-3., 1111 Budapest, Hungary
3.
Department of Mechanics, Materials and Structures, Budapest University of Technology and Economics, Müegyetem rakpart 1-3., 1111 Budapest, Hungary
4.
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

Received: June 2021

Revised: October 2021

Early access: November 2021

Published: October 2021

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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 $ r(\varphi) $ 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 $ r(\varphi) $ and of the curvature $ \kappa(\varphi) $ (characterized by $ dr/d\varphi = 0 $ and $ d\kappa /d\varphi = 0 $, 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.

Keywords:

Mathematics Subject Classification: Primary: 35Q86, 53A04; Secondary: 65M70.

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Figure 1. The piecewise polynomial central radial distance function $ r $ (a1) and its derivatives $ r' $ (a2) and $ r'' $ (a3), generated by the phase-uniform mesh (black dots) of size $ N = 20 $ on the standard example curve (b), as defined in (36)

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Figure 2. The piecewise polynomial central radial distance function $ r $ generated by a uniform mesh of size $ N = 20 $ on the standard ellipse (38) with $ a = 1 $ and $ b = 0.5 $, for a phase-uniform mesh in panels (a1) and (a2), and for an arclength-uniform mesh in panels (b1) and (b2)

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Figure 3. Illustration of step $ i $ in panels (a) and of step $ i+1 $ in panels (b) of the mesh propagation for a mesh of size $ N = 20 $. The radial distance function $ r_{i-1} $ in panel (a1) generates the parameterization of the curve $ \Gamma_{i-1} $ in panel (a2) with normals $ \mathbf{n}_{i-1} $ (red arrows) at the mesh points (black dots). Euler steps at each mesh point give the new mesh that defines the next radial distance function $ r_{i} $ shown in panel (b1). It generates the parameterization of the next curve $ \Gamma_{i} $ in panel (b2) and its normals $ \mathbf{n}_{i} $ (red arrows) at the mesh points (black dots). The figure is for the standard example curve from (36); compare with Fig. 1

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Figure 4. Evolution of the standard curve (36) under the curve-shortening flow, (5) with $ c = 0 $ and $ \alpha = 1 $, as computed for stepsize parameter $ S = 10^{-5} $ up to $ t = 0.1062 $ in 36,599 steps with a phase-uniform mesh of size $ N = 20 $ at each step. Panels (a1) and (b1) show the radial distance functions of the final curve for the centroid $ C_t $ and for $ O = (0.05, 0.1) $ as the reference point, respectively. Panels (a2) and (b2) show seven curves at equally distributed time steps from $ t = 0 $ to $ t = 0.1062 $, together with trajectories of the critical points, determined from the minima (green) and maxima (red) of the respective radial distance function

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Figure 5. Evolution under the curve-shortening flow, (5) with $ c = 0 $ and $ \alpha = 1 $, of the trajectories of the critical points (green for minima and red for maxima) of the radial distance function $ r_{O} $ for the fixed reference point $ O = (0.01,0.05) $ (black dot) for the standard curve (36) in column (a), the curve given by (40) in column (b), and the curve given by (41) in column (c). Each computation is for an arclength-uniform mesh with $ N = 80 $ points, $ S = 10^{-5} $ and remeshing at every 10th time step. The top row shows the trajectories of the critical points of $ r_{O} $ with the respective initial curve and mesh in $ \mathbb{R}^2 $, and the bottom row shows them on the evolution surface in $ \mathbb{R}^2 \times \mathbb{R}^+ $

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Figure 6. Evolution under the curve-shortening flow (1), (5) with $ c = 0 $ and $ \alpha = 1 $, of the trajectories of the critical points (green for minima and red for maxima) of the radial distance function $ r_{C} $, with reference point at the (moving) centroid $ C(t) $, for the standard curve (36) in column (a), the curve given by (40) in column (b), and the curve given by (41) in column (c). Also shown are the trajectories of the critical points (light blue for minima and dark blue for maxima) of the signed curvature function $ \kappa $. Each computation is for an arclength-uniform mesh with $ N = 80 $ points, $ S = 10^{-5} $ and remeshing at every 10th time step. The top row shows the trajectories of the critical points of $ r_{C} $ and of $ \kappa $ with the respective initial curve and mesh in $ \mathbb{R}^2 $, and the bottom row shows them on the evolution surface in $ \mathbb{R}^2 \times \mathbb{R}^+ $

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Figure 7. Evolution of the standard ellipse (38) with $ a = 1 $ and $ b = 0.5 $ under (5) with $ c = 0 $ for $ \alpha = 1, 0.8, 0.6, 0.4, \frac{1}{3}, 0.25, 0.125, 0.1 $, computed with an arclength-uniform mesh with $ N = 80 $, remeshing at every 10th time step and $ S = 10^{-5} $. Panel (a) shows the respective time series of the isoperimetric quotient $ Q $ and panel (b) presents the rescaled final curves

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Figure 8. Evolution of the standard curve (36) under (5) with $ c = 0 $ for $ \alpha = 1, 0.8, 0.6, 0.4, \frac{1}{3}, 0.25, 0.125, 0.1 $, computed with an arclength-uniform mesh with $ N = 80 $, remeshing at every 10th time step and $ S = 10^{-5} $. Panel (a) shows the respective time series of the isoperimetric quotient $ Q $ and panel (b) presents the rescaled final curves

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Figure 9. Evolutions under (5) with $ c = 0 $ and $ \alpha = 0.1 $ of the $ D_n $-symmetric curves (43) with $ n = 3 $ and $ \delta = 0.05 $ (a), and with $ n = 4 $ and $ \delta = 0.03 $ (b), computed with an arclength-uniform mesh with $ N = 60 $ for $ n = 3 $ and $ N = 64 $ for $ n = 4 $, remeshing at every 10th time step and $ S = 10^{-4} $. Shown are a selection of curves with their respective meshes and the trajectories of the stationary points (green for minima and red for maxima); the computed final shape of the respective evolution is at the bottom right of each panel

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Figure 10. Evolutions under (5) with $ c = 0 $ and $ \alpha = 0.05 $ of the $ D_n $-symmetric curves (43) with $ n = 3 $ and $ \delta = 0.05 $ (a), with $ n = 4 $ and $ \delta = 0.03 $ (b), with $ n = 5 $ and $ \delta = 0.02 $ (c), and with $ n = 6 $ and $ \delta = 0.01 $ (d), computed with an arclength-uniform mesh with $ N = 64 $ for $ n = 4 $ and $ N = 60 $ otherwise, remeshing at every 10th time step and $ S = 10^{-4} $; illustrated as in Fig.9

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Figure 11. Evolutions under (5) with $ c = 0 $ and $ \alpha = 0.01 $ of the $ D_n $-symmetric curves (43) with $ n = 3 $ and $ \delta = 0.01 $ (a), with $ n = 4 $ and $ \delta = 0.03 $ (b), with $ n = 5 $ and $ \delta = 0.02 $ (c), and with $ n = 6 $ and $ \delta = 0.01 $ (d), computed with an arclength-uniform mesh with $ N = 64 $ for $ n = 4 $ and $ N = 60 $ otherwise, remeshing at every 10th time step and $ S = 10^{-4} $; illustrated as in Fig.9

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Figure 12. Evolution under (5) with $ c = 0 $ and $ \alpha = 0.05 $ (a) and $ \alpha = 0.01 $ (b) of the isoperimetric quotient $ Q $ for the $ D_n $-symmetric curves from Fig. 10 and Fig. 11, respectively; the horizontal dashed lines are at the values $ Q_n $ of the regular $ n $-gons

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Figure 13. Evolutions under (5) with $ c = 0 $ and $ \alpha = 0.1 $ of the $ C_n $-symmetric curves (45) with $ n = 3 $, $ \delta = 0.05 $ and $ \xi = 0.01 $ (a), and with $ n = 4 $, $ \delta = 0.03 $ and $ \xi = 0.01 $ (b), computed with an arclength-uniform mesh with $ N = 60 $ for $ n = 3 $ and $ N = 64 $ for $ n = 4 $, remeshing at every 10th time step and $ S = 10^{-5} $; illustrated as in Fig. 9

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Figure 14. Discrete and integral error norms for the discretizations with a phase-uniform mesh of size $ N $ with the centroid as reference point of the standard curve (36) in column (a) and of the standard ellipse (38) with $ a = 1 $ and $ b = 0.5 $ in column (b)

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Figure 15. Evolution of the standard curve (36) under the curve-shortening flow, (5) with $ c = 0 $ and $ \alpha = 1 $, for a phase-uniform mesh with remeshing at every time step, for mesh size $ N = 10 $ (a), $ N = 60 $ (b), $ N = 80 $ (c) and $ N = 100 $ (d). Here $ S = 10^{-5} $, the reference point is the centroid, and the stationary points are determined from the minima (green) and maxima (red) of the central radial distance function

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Figure 16. Evolution of the standard curve (36) in row (a) and of the ellipse (38) with $ a = 1 $ and $ b = 0.5 $ in row (b) under the curve-shortening flow, (5) with $ c = 0 $ and $ \alpha = 1 $, computed with a phase-uniform mesh with $ N = 80 $ points. Panels (a1) and (b1) are computed without remeshing, panels (a2) and (b2) with remeshing at every 10th time step, and panels (a3) and (b3) with remeshing at every time step. Here $ S = 10^{-5} $, the reference point is the centroid, and the stationary points are determined from the minima (green) and maxima (red) of the central radial distance function

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Figure 17. Evolution of the standard curve (36) in row (a), of the ellipse (38) with $ a = 1 $ and $ b = 0.5 $ in row (b) and of the ellipse (38) with $ a = 2 $ and $ b = 0.5 $ in row (c) under the curve-shortening flow, (5) with $ c = 0 $ and $ \alpha = 1 $, computed with an initial arclength-uniform mesh with $ N = 80 $ points. Panels (a1), (b1) and (c1) are computed without remeshing, panels (a2), (b2) and (c2) with remeshing at every 10th time step, and panels (a3), (b3) and (c3) with remeshing at every time step; during the remeshing the number $ N $ of mesh points is reduced to keep the arclength distance $ L/N $ between mesh points approximately constant, down to a minimum of $ N_{\rm min} = 40 $ mesh points. Here $ S = 10^{-5} $, the reference point is the centroid, and the stationary points are determined from the minima (green) and maxima (red) of the central radial distance function

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Figure 18. Evolution under the curve-shortening flow, (5) with $ c = 0 $ and $ \alpha = 1 $, of three rotations of the standard curve (36) over $ \varphi_0 $ as indicated, computed with a phase-uniform mesh of $ N = 80 $ points with remeshing at every 10th time step. The left column shows the mesh on the standard curve and the trajectories of the stationary points, and the right column shows these trajectories on the evolution surface in $ \mathbb{R}^2 \times \mathbb{R}^+ $. Here $ S = 10^{-5} $, the reference point is the centroid, and the stationary points are determined from the minima (green) and maxima (red) of the central radial distance function

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Figure 19. Evolution under the curve-shortening flow, (5) with $ c = 0 $ and $ \alpha = 1 $, of three rotations of the standard ellipse (38) with $ a = 1 $ and $ b = 0.5 $ over $ \varphi_0 $ as indicated, computed with a phase-uniform mesh of $ N = 80 $ points with remeshing at every 10th time step. The left column shows the mesh on the standard curve and the trajectories of the stationary points, and the right column shows these trajectories on the evolution surface in $ \mathbb{R}^2 \times \mathbb{R}^+ $. Here $ S = 10^{-5} $, the reference point is the centroid, and the stationary points are determined from the minima (green) and maxima (red) of the central radial distance function

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Figure 20. Evolution under the affine shortening flow, (5) with $ c = 0 $ and $ \alpha = \frac{1}{3} $, of the standard ellipse (38), with $ a = 1 $ and $ b = 0.5 $ in column (a) and with $ a = 2 $ and $ b = 0.5 $ in column (b). Each computation is for an arclength-uniform mesh with $ N = 80 $ points, $ S = 10^{-4} $ and the centroid as the reference point. Panels (a1) and (b1) show selected curves during the computation with the trajectories of the stationary points, determined from the extrema of the central radial distance function. Panels (a2) and (b2) show the distance of the corresponding isoperimetric quotient $ Q $ from its value for the respective ellipse as a function of time $ t $, and panels (a3) and (b3) show the integral distance of the computed curves $ \Gamma_{t_i} $ from the ellipse with same area and axis ratio, determined for the functions $ r $, $ r' $, $ r'' $ and $ \kappa $

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Figure 21. A star-like curve whose centroid $ C $ is not a reference point (a), and a non-convex star-like curve whose centroid $ C $ can be chosen as a reference point (b). Also shown are the stationary points (red dots), the inflection points (blue dots) and their tangent lines (blue lines), which bound the set of reference points (shaded)

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Table 1. The critical point evolution algorithm with mesh propagation by an Euler step and remeshing

Require: a star-like closed curve $ \Gamma $; parameters $ c $ and $ \alpha $ of (5); number of mesh points $ N $; stepsize parameter $ S $; accuracy parameters $ h_{\rm min} $ and $ A_{\rm min} $;
Ensure: construct the piecewise polynomial radial distance function $ r_0 $ for a uniform mesh $ M_0 $ of size $ N $ with dual mesh $ M^\varphi_0 $ for a suitable reference point $ O_0 $; ensure that the associated curve $ \Gamma_0 $ with parameterization $ \gamma_0 $ is sufficiently close to $ \Gamma $; set the step counter $ i $ to 1;
While none of criteria (ⅰ)-(ⅳ) from Sec. 4.3 are satisfied do calculate the polynomial derivatives $ r_{i-1}' $ and $ r_{i-1}'' $ symbolically; determine the stepsize $ h_i $ from $ \kappa_{i-1} $; evaluate the associated normals $ \mathbf{n}_{i-1} $ and the curvature $ \kappa_{i-1} $ in the points of $ M^\varphi_{i-1} $; perform the Euler step (39) at the mesh points to obtain the propagated mesh $ M_{i} = E_{h_i}(M_{i-1}) $ with dual $ M^\varphi_i $; choose a suitable reference point $ O_i $ for the new curve $ \Gamma_{i} = E_{h_i}(\Gamma_{i-1}) $; construct the piecewise polynomial radial distance function $ r_i $ for $ M_i $ and $ O_i $; [when required: remesh the curve with a uniform mesh $ M^* $ by computing the associated piecewise polynomial radial distance function $ r^_i $ with respect to $ O_i $; set $ r_i = r^_i $]; increase the step counter $ i $ by 1;
end while

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