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One can define an analogous version of the curve shortening flow for polygons in $\mathbb R^2$, namely defined by the differential equation $\dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}$, where $p_i$ is the $i$-th vertex of the polygon and and $v_i$ is the vector that goes from $p_i$ to the circumcenter of the triangle defined by $p_{i-1}$, $p_i$ and $p_{i+1}$.

The Gage-Hamilton-Grayson theorem states that simple curves remain simple under the curve-shortening flow.

Does this still hold for polygons under this analogous flow?

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This is not a direct answer, but may I point you to an earlier question of Ryan Budney, and to the paper by Bennett Chow and David Glickenstein,

"Semidiscrete Geometric Flows of Polygons." American Mathematical Monthly. April 2007. (MAA link)


 DiscreteFlow

Under their discrete flow, every simple polygon converges to a point whose shape is asymptotically an affine transformation of a regular polygon.

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  • $\begingroup$ The thing is that simple polygons do not remain simple under the flow studied by Bennett Chow and David Glickenstein as pointed in "Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots" arxiv.org/pdf/cs.RO/0605070.pdf $\endgroup$ – Gerardo Arizmendi Dec 10 '13 at 20:06
  • $\begingroup$ I think the problem relies in the fact that in this case you can split the differential equation in the x and y coordinates. $\endgroup$ – Gerardo Arizmendi Dec 10 '13 at 20:09
  • $\begingroup$ @GerardoArizmendi: Yes, you are correct. I am not aware of a definition of discrete flow that guarantees simplicity---I'll be interested to learn! Of course you could always round out the vertices and apply a smooth flow, but then it is no longer discrete. $\endgroup$ – Joseph O'Rourke Dec 10 '13 at 20:10
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Not exactly an answer to your question, but Peter Scott and I worked out a polygonal flow that is guaranteed to keep curves embedded in "Shortening Curves on Surfaces", Topology 33, (1994) 25-43.

A version of this is implemented in java at disk flow applet

The problem with the Birkhoff flow is that long segments move faster than short segments, and can overtake them, creating self-intersections. This can be overcome by getting the length of segments from the surface (using intersections with some sort of grid for example) rather than from a parametrization of the domain, as with Birkhoff.

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A related flow does keep simple, see Ileana Streinu's (Joseph's colleague at Smith!) work http://cs.smith.edu/~streinu/Research/robotics.html and related work by Connelly/Demaine/Rote

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    $\begingroup$ Yes, and see my pictures at an answer to Ryan's question here. But arguably, these are not discrete curve-shortening flows... $\endgroup$ – Joseph O'Rourke Dec 11 '13 at 0:56
  • $\begingroup$ I would argue that these are very close to curve-shortening flows. $\endgroup$ – Igor Rivin Dec 11 '13 at 1:04
  • $\begingroup$ Interesting links! $\endgroup$ – Gerardo Arizmendi Dec 11 '13 at 1:23

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