On the equilibria of finely discretized curves and surfaces

TL;DR

This paper investigates the static equilibrium points of finely discretized convex surfaces, deriving formulas that relate these points to surface curvature and center of gravity, with applications to natural pebble shapes.

Contribution

It introduces the concept of imaginary equilibrium indices and provides explicit formulas linking equilibrium points to surface geometry and discretization.

Findings

Equilibrium points fluctuate around specific indices as discretization increases.

Formulas relate equilibrium counts to principal curvatures and radial distances.

Results match observations on natural pebble surfaces.

Abstract

Our goal is to identify the type and number of static equilibrium points of solids arising from fine, equidistant $n$-discretrizations of smooth, convex surfaces. We assume uniform gravity and a frictionless, horizontal, planar support. We show that as $n$ approaches infinity these numbers fluctuate around specific values which we call the imaginary equilibrium indices associated with the approximated smooth surface. We derive simple formulae for these numbers in terms of the principal curvatures and the radial distances of the equilibrium points of the solid from its center of gravity. Our results are illustrated on a discretized ellipsoid and match well the observations on natural pebble surfaces.