Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

Details Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

2006-05-16

arxiv.org/abs/cs/0605070v1

Computer Science

Robotics

15 pages, 18 figures

Scientific paper

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.

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