Solution of the Roth-Marques-Durian Rotational Abrasion Model

TL;DR

This paper provides an exact solution to the Roth-Marques-Durian rotational abrasion model, a complex PDE with a step function coefficient, and introduces a planar tree graph model to track the evolution of the contour's coarse features.

Contribution

It offers a novel analytical solution to a complex PDE and develops a new graph-based model for contour evolution in rotational abrasion.

Findings

Exact solution to the rotational abrasion PDE.

Development of a planar tree graph model.

Insights into the evolution of coarse features.

Abstract

We solve the rotational abrasion model of Roth, Marques and Durian (arXiv:1009.3492), a one-dimensional quasilinear partial differential equation resembling the inviscid Burgers equation with the unusual feature of a step function factor as a coefficient. The complexity of the solution is primarily in keeping track of the cases in the piecewise function that results from certain amputation and interpolation processes, so we also extract from it a model of an evolving planar tree graph that tracks the evolution of the coarse features of the contour.