Curve Shortening and its Application to Multi-Agent Systems

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 in multi-agent systems, we address the problem of creating a polygon shortening flow. A simple linear scheme is proposed that exhibits several properties similar to Euclidean curve shortening. The polygon shrinks to an elliptical point; convex polygons remain convex; and, the perimeter of the polygon is monotonically decreasing.