Today I read a paper titled “Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots”
The abstract is:
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.