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Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

Stephen L. Smith·Mireille E. Broucke·Bruce A. Francis
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Abstract

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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Stephen L. Smith, Mireille E. Broucke, Bruce A. Francis. 2006-05-16. Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots. https://arxiv.org/abs/cs/0605070

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