A shape evolution model under affine transformations

TL;DR

This paper introduces a discrete dynamical system modeling shape evolution of convex bodies under affine transformations, identifying invariants and orbit structures, with applications to pebble abrasion processes.

Contribution

It presents a novel affine-invariant shape evolution model and analyzes the invariant elements and orbit structures within this framework.

Findings

Identified invariant elements in all affine classes.

Described the orbits of bodies in certain classes.

Applied the model to pebble shape abrasion processes.

Abstract

In this note we describe a discrete dynamical system acting on the similarity classes of a plane convex body within the affine class of the body. We find invariant elements in all affine classes, and describe the orbits of bodies in some special classes. We point out applications with abrasion processes of pebble shapes.