On global equilibria of finely discretized curves and surfaces

TL;DR

This paper investigates the static equilibrium points of finely discretized convex surfaces, linking local and global equilibria to the original smooth shape, and provides methods to interpret and compute these global equilibria.

Contribution

It extends previous work by defining and computing the global equilibrium points of discretized surfaces, connecting them to the smooth surface and offering practical methods for analysis.

Findings

Number of local equilibria fluctuates around imaginary equilibrium indices.

Global equilibria can be interpreted, defined, and computed on discretizations.

Results applicable to natural pebble surfaces and shape analysis.

Abstract

In an earlier work we identified the types and numbers of static equilibrium points of solids arising from fine, equidistant $n$-discretrizations of smooth, convex surfaces. We showed that such discretizations carry equilibrium points on two scales: the local scale corresponds to the discretization, the global scale to the original, smooth surface. In that paper we showed that as $n$ approaches infinity, the number of local equilibria fluctuate around specific values which we call the imaginary equilibrium indices associated with the approximated smooth surface. Here we show how the number of global equilibria can be interpreted, defined and computed on such discretizations. Our results are relevant from the point of view of natural pebble surfaces, they admit a comparison between field data based on hand measurements and laboratory data based on 3D scans.