On the equilibria of finely discretized curves and surfaces

Details On the equilibria of finely discretized curves and surfaces On the equilibria of finely discretized curves and surfaces

2011-06-03

arxiv.org/abs/1106.0626v2

Mathematics

Differential Geometry

21 pages, 2 figures

Scientific paper

10.1007/s00605-011-0361-x

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.

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