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Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

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3 Author(s)
Smith, S.L. ; Univ. of California, Santa Barbara ; Broucke, M.E. ; Francis, B.A.

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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Automatic Control, IEEE Transactions on  (Volume:52 ,  Issue: 6 )