Analytical and numerical aspects on motion of polygonal curves with constant area speed

TL;DR

This paper develops an analytical and numerical framework for studying the motion of polygonal curves that preserve area at a constant speed, using a system of ODEs and implicit time discretization.

Contribution

It introduces a novel formulation of area-preserving polygonal curve motion as ODEs and proposes an implicit discretization method maintaining the polygonal class.

Findings

The method accurately preserves constant area speed during evolution.

Polygonal curves remain within a prescribed class throughout the motion.

The approach provides a stable numerical scheme for polygonal curve evolution.

Abstract

General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while solution polygonal curves keep belonging to the polygonal class.