UC Santa Barbara
Curve shortening and the rendezvous problem for mobile autonomous robots
- Author(s): Smith, Stephen L
- Broucke, Mireille E.
- Francis, Bruce. A.
- et al.
Published Web Locationhttps://doi.org/10.1109/TAC.2007.899024
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.