On the equilibria of finely discretized curves and surfaces

Domokos, G.; Lángi, Zsolt; Szabó, Tímea

doi:10.1007/s00605-011-0361-x

Cited by 13 publications

(24 citation statements)

References 19 publications

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“…k = 2), then there are constants C 1 , C 2 > 0 such that C 1 < N ∆ (o) < C 2 for all values of ∆. A stronger version of this result was proven in [16].…”

Section: Dynamic Theory: Local Equilibria On Finely Discretized Evolmentioning

confidence: 77%

“…In [16] we provided a mathematical justification for this observation. We studied the inverse phenomenon: namely, we were seeking the numbers and types of static equilibrium points of the families of polyhedra r ∆ arising as equidistant ∆discretizations on an increasingly refined grid of a smooth curve r with N generic equilibrium points, denoted by m i (i = 1, 2, .…”

Section: 22mentioning

confidence: 89%

“…Thus, the formulas in (3) do not necessarily provide the numbers of all equilibrium points of P ∆ . This led in [16] to the following definition:…”

Section: 1mentioning

confidence: 99%

“…In [16] it is remarked that if the coordinate functions of r(τ ) are polynomials, then the curve has no irregular equilibrium sequence, and asked (Question 1, [16]) whether the same holds if the two coordinate functions are analytic. Here we prove that the answer to this question is affirmative not only for every analytic, but for every C 3 -class curve.…”

Section: 1mentioning

confidence: 99%

“…Equilibria on random meshes. In this subsection we prove a probabilistic version of the formulae (3) which are the main planar result in [16], and deals with an equidistant, n-element partition of the interval [τ , τ ]. Our goal is to show that even if the discretization is non-uniform, those formulae provide good estimates, thus the numbers predicted by the formulae may be observed in computer simulations.…”

Section: 2mentioning

confidence: 99%

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Tracking Critical Points on Evolving Curves and Surfaces

Domokos

Lángi

Sipos

2019

Experimental Mathematics

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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 ∆ of r, with N ∆ = N (r ∆ ) equilibria. There is strong intuitive evidence suggesting that under some particular evolution models (e.g. curvature-driven flows) N (t) and N ∆ (t) primarily evolve in the opposite manner (i.e. if one is increasing then the other is decreasing and vice versa). This observation appear to be a key factor in tracking geophysical abrasion. Here we create the mathematical framework necessary to understand these phenomenon 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 ∆ (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 ∆ are coupled to some extent: resonancelike phenomena in N ∆ (t) can be used to forecast downward jumps in N (t) (but not upward jumps). Beyond proving rigorous results for the ∆ → 0 limit on the nontrivial interplay between singularities in the discrete and continuum approximations we also show that our mathematical model is structurally stable, i.e. it may be verified by computer simulations.1991 Mathematics Subject Classification. 53A05, 53Z05.

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“…k = 2), then there are constants C 1 , C 2 > 0 such that C 1 < N ∆ (o) < C 2 for all values of ∆. A stronger version of this result was proven in [16].…”

Section: Dynamic Theory: Local Equilibria On Finely Discretized Evolmentioning

confidence: 77%

Section: 22mentioning

confidence: 89%

“…Thus, the formulas in (3) do not necessarily provide the numbers of all equilibrium points of P ∆ . This led in [16] to the following definition:…”

Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

See 3 more Smart Citations

Tracking Critical Points on Evolving Curves and Surfaces

Domokos

Lángi

Sipos

2019

Experimental Mathematics

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The Geometry of Abrasion

Domokos

Gibbons

2018

Bolyai Society Mathematical Studies

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Monotonicity of Spatial Critical Points Evolving Under Curvature-Driven Flows

Domokos

2014

J Nonlinear Sci

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We describe the variation of the number N (t) of spatial critical points of smooth curves (defined as a scalar distance r from a fixed origin O) evolving under curvature-driven flows. In the latter, the speed v in the direction of the surface normal may only depend on the curvature κ. Under the assumption that only generic saddle-node bifurcations occur, we show that N (t) will decrease if the partial derivative v κ is positive and increase if it is negative (Theorem 1). Justification for the genericity assumption is provided in Section 5. For surfaces embedded in 3D, the normal speed v under curvature-driven flows may only depend on the principal curvatures κ, λ. Here we prove the weaker (stochastic) Theorem 2 under the additional assumption that third-order partial derivatives can be approximated by random variables with zero expected value and covariance. Theorem 2 is a generalization of a result by Kuijper and Florack for the heat equation. We formulate a Conjecture for the case when the reference point coincides with the centre of gravity and we motivate the Conjecture by intermediate results and an example. Since models for collisional abrasion are governed by partial differential equations with v κ , v λ > 0, our results suggest that the decrease of the number of static equilibrium points is characteristic of some natural processes.

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On the equilibria of finely discretized curves and surfaces

Cited by 13 publications

References 19 publications

Tracking Critical Points on Evolving Curves and Surfaces

Tracking Critical Points on Evolving Curves and Surfaces

The Geometry of Abrasion

Monotonicity of Spatial Critical Points Evolving Under Curvature-Driven Flows

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