On global equilibria of finely discretized curves and surfaces

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.