Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

Smith, Stephen L.; Broucke, Mireille E.; Francis, Bruce A.

Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

Authors

Smith, Stephen L.

Broucke, Mireille E.

Francis, Bruce A.

Description

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.
15 pages, 18 figures