On the Oval Shapes of Beach Stones

This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of single short partial integro-differential equation, to show how alone stone on planar beach might lead oval shapes observed empirically. The underlying idea this theory is intuitive observation that rate ablation at point surface proportional product curvature and likelihood contact with point. Specifically, key roles model are played by both random wave process global (non-local) shape stone, i.e., its away from beach. physical mechanism for conversion energy into potential stone. No closed-form or even asymptotic solution known basic which non-linear non-local. On other hand, preliminary numerical experiments presented deterministic continuous-time setting using standard curve-shortening algorithms discrete-time polyhedral-slicing Monte Carlo simulation.