Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots
<|reference_start|>Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots: 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.<|reference_end|>
@article{smith2006curve,
    title={Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots},
    author={Stephen L. Smith, Mireille E. Broucke, and Bruce A. Francis},
    journal={arXiv preprint arXiv:cs/0605070},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0605070},
    primaryClass={cs.RO cs.MA}
}