Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

TL;DR

This paper introduces a polygon shortening flow inspired by Euclidean curve shortening, demonstrating that polygons can be made to shrink while maintaining convexity and decreasing perimeter, with applications to autonomous robot rendezvous.

Contribution

It proposes a linear polygon shortening scheme that preserves convexity and decreases perimeter, extending Euclidean curve shortening concepts to polygons.

Findings

Polygons shrink to an elliptical point.

Convex polygons remain convex during evolution.

Perimeter decreases monotonically.

Abstract

If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular point. This curve evolution is called Euclidean curve shortening and the result is known as the Gage-Hamilton-Grayson Theorem. Motivated by the rendezvous problem for mobile autonomous robots, we address the problem of creating a polygon shortening flow. A linear scheme is proposed that exhibits several analogues to Euclidean curve shortening: The polygon shrinks to an elliptical point, convex polygons remain convex, and the perimeter of the polygon is monotonically decreasing.