# Curvature flows for PL closed curves in the plane?

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.

• MR0840401 (87m:53003) Gage, M.(1-RCT); Hamilton, R. S.(1-UCSD) The heat equation shrinking convex plane curves. J. Differential Geom. 23 (1986), no. 1, 69–96. 53A04 (35K05 52A40 58E99 58G11)

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":

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)

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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Thanks. Have you seen an analysis of any flows that keep the edge lengths constant? –  Ryan Budney Sep 16 '13 at 23:37
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? –  Joseph O'Rourke Sep 17 '13 at 0:16
Yes, exactly, I'm looking for a "linkage" version of the papers you cite. –  Ryan Budney Sep 17 '13 at 0:18
@RyanBudney: I added a linkage reference. Hope that helps! –  Joseph O'Rourke Sep 17 '13 at 0:52
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. –  Ryan Budney Sep 17 '13 at 21:00