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I'm curious to what extent people have studied "curvature flows" on PL closed curves in the plane.

There's a paper by Gage and Hamilton from 1986 that describes the long-term behaviour of smooth curves in the plane under the curvature flow.

A discrete version of this flow could go like this. The ambient space for this will be the space of PL 1-dimensional compact connected submanifolds of $\mathbb R^2$ and for the sake of argument, let's fix the length of the intervals. So the curve consists of $n$ straight line segments, and the $i$-th interval has length $l_i$, and the set $\{l_1,l_2,\cdots,l_n\} \subset (0,\infty)$ is the data that describes this space of closed curves. The "curvature flow" would be the dynamical system given by placing a spring at each vertex of your curve (the spring acts on the angle), and you make the spring's "natural angle" to be $\pi$. Is this flow complete like the Gage-Hamilton flow?

More generally, has there been much study of finitary analogues to the Gage-Hamilton flow, in the spirit of my initial question? I imagine there has, I'm not sure which terms to search for on MathSciNet.

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This is not a complete answer, but in my book with Satyan Devadoss, Discrete and Computational Geometry, we include (following Gage-Hamilton) a discussion of the delightful paper by Bennett Chow and David Glickenstein,

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

Here is an image I made to illustrate their "discrete flow":


DiscreteFlow

They prove that every simple polygon evolves under their discrete flow so that it converges to a point whose shape is asymptotically an affine transformation of a regular polygon.

It is now clear that Ryan is seeking linkage reconfigurations. The key paper is this:

Robert Connelly, Erik D. Demaine, and Günter Rote, “Straightening Polygonal Arcs and Convexifying Polygonal Cycles”, Discrete & Computational Geometry, volume 30, number 2, September 2003, pages 205–239. (author link)
Fig1 CDR

There have been quite a few papers following this one in the last decade. I wrote a short news-article-like summary of this great result here (arXiv link).

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  • $\begingroup$ Thanks. Have you seen an analysis of any flows that keep the edge lengths constant? $\endgroup$
    – Ryan Budney
    Commented Sep 16, 2013 at 23:37
  • $\begingroup$ Ah, interesting question! There is work on morphings of arbitrary polygons to convex polygons, while all edges remain fixed in length, so it is a reconfiguration of a closed linkage. Perhaps this is what you mean? $\endgroup$
    – Joseph O'Rourke
    Commented Sep 17, 2013 at 0:16
  • $\begingroup$ Yes, exactly, I'm looking for a "linkage" version of the papers you cite. $\endgroup$
    – Ryan Budney
    Commented Sep 17, 2013 at 0:18
  • $\begingroup$ @RyanBudney: I added a linkage reference. Hope that helps! $\endgroup$
    – Joseph O'Rourke
    Commented Sep 17, 2013 at 0:52
  • $\begingroup$ Unfortunately the Connelly, Demaine and Rote paper takes as input to the motion more information than just the polygon -- they make a choice of "struts". So their motion does not make sense as a continuous function on the space of all polygons, and in that regard it's not analogous to the Gage-Hamilton result. Do you know if anyone has done work that does not involve making choices -- something that defines dynamics on the space of all polygons (as a space)? Connelly-Demaine-Rote basically just show that the polygon space is connected. I want a flow that gives contractibility. $\endgroup$
    – Ryan Budney
    Commented Sep 17, 2013 at 21:00
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There is a lot of (by now not so) recent work on the Carpenter's Rule Problem, which is the natural PL version of curve shortening (and indeed, Streinu's algorithm (as well as CDR) use some of the same ideas.

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