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edit approved Sep 2, 2017 at 13:15
C.F.G
edit approved Sep 2, 2017 at 13:15
C.F.G
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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.

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.

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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Ryan Budney
Ryan Budney
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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.