Tracking Critical Points on Evolving Curves and Surfaces

Abstract

In recent years it became apparent that geophysical abrasion can be well characterized by the time evolution N(t) of the number N of static balance points of the abrading particle. Static balance points correspond to the critical points of the particle’s surface represented as a scalar distance function r, measured from the center of mass of the particle, so their time evolution can be expressed as $N(r(t))$. The mathematical model of the particle can be constructed on two scales: on the macro (global) scale the particle may be viewed as a smooth, convex manifold described by the smooth distance function r with $N=N(r)$ equilibria, while on the micro (local) scale the particle’s natural model is a finely discretized, convex polyhedral approximation r^Δ of r, with $N^{\Delta }=N(r^{\Delta })$ equilibria. There is strong intuitive evidence suggesting that under some particular evolution models (e.g., curvature-driven flows) N(t) and N^Δ(t) primarily evolve in the opposite manner (i.e. if one is increasing then the other is decreasing and vice versa). This observation appears to be a key factor in tracking geophysical abrasion. Here we create the mathematical framework necessary to understand these phenomena more broadly, regardless of the particular evolution equation. We study micro and macro events in one-parameter families of curves and surfaces, corresponding to bifurcations triggering the jumps in N(t) and N^Δ(t). Based on this analysis we show that the intuitive picture developed for curvature-driven flows is not only correct, it has universal validity, as long as the evolving surface r is smooth. In this case, bifurcations associated with r and r^Δ are coupled to some extent: resonance-like phenomena in N^Δ(t) can be used to forecast downward jumps in N(t) (but not upward jumps). Beyond proving rigorous results in the case of evolving planar curves for the $\Delta \to 0$ limit on the nontrivial interplay between singularities in the discrete and continuum approximations we also show that our mathematical model is structurally stable. This property serves as the basis for the second, experimental part of our research where we demonstrate via computer simulations that the phenomena on evolving surfaces appear to be closely analogous to the planar case, however, they also show additional geometric features which are still not completely understood.

Keywords:

• Equilibrium
• convex surface
• Poincaré–Hopf formula
• polyhedral approximation
• curvature-driven evolution

1991 MATHEMATICS SUBJECT CLASSIFICATION:

• 53A05
• 53Z05

Acknowledgment

The authors are most indebted to Phil Holmes for reading the initial manuscript and giving essential advice on several aspects. We also thank László Székelyhidi for drawing our attention to the analogy with tygers. The support of NKFIH grant K 119245 and grant BME FIKP-VÍZ by EMMI is gratefully acknowledged. ZL has been supported by grant UNKP-18-4 New National Excellence Program of EMMI and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.