On discrete version of curve shortening flow

Question

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?

Answer 1

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.

Answer 2

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)

 

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

Answer 3

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

Answer 4

I think this is a very interesting question. However, my answer is that the Gage-Hamilton-Grayson theorem probably not holds for your curvature flow.

I just write a program in metapost, which indicates under your flow, the vertex will move "linearly" (on a line), I believe you can prove that strictly, and from which we can conclude that for some simple curve will have intersections when time goes to infinity.

Here is an example of the output:

The SVG (which can be scaled) can be download from my cloud storage.

In the graph, the initial polygon is draw by blue, and the final polygon is draw by red. I distinguish each step by dashed line.

The initial 20-polygon is given by

(luamplib)               {randomseed:=164.13048854794684}
(luamplib)               >> "z[1]= (7.1663813354355792,2.115410943910577)"
(luamplib)               {randomseed:=1282.7618269274324}
(luamplib)               >> "z[2]= (6.1198547032965198,3.1294415485934794)"
(luamplib)               {randomseed:=403.39625352345581}
(luamplib)               >> "z[3]= (5.3874308800111868,4.8324648551767311)"
(luamplib)               {randomseed:=3339.1733235391926}
(luamplib)               >> "z[4]= (3.5465950228343885,6.1858621289233282)"
(luamplib)               {randomseed:=1839.1381135784272}
(luamplib)               >> "z[5]= (1.2672314185221965,7.8164049682362586)"
(luamplib)               {randomseed:=3420.0974717036256}
(luamplib)               >> "z[6]= (-3.7256314873696939,6.8671219483403938)"
(luamplib)               {randomseed:=510.18410242830026}
(luamplib)               >> "z[7]= (-5.6650842541666693,6.4155586555366675)"
(luamplib)               {randomseed:=12.746562895524596}
(luamplib)               >> "z[8]= (-7.3959720323999152,5.2194560696781931)"
(luamplib)               {randomseed:=1303.1675809171504}
(luamplib)               >> "z[9]= (-6.7246625396621438,1.5797575825552226)"
(luamplib)               {randomseed:=1827.3952890813218}
(luamplib)               >> "z[10]= (-9.1673328058487815,-0.45325023466673714)"
(luamplib)               {randomseed:=7.8023681259649997}
(luamplib)               >> "z[11]= (-9.510793507129911,-3.9190636005348356)"
(luamplib)               {randomseed:=1418.1949241054936}
(luamplib)               >> "z[12]= (-7.425895076183326,-4.2579680419470947)"
(luamplib)               {randomseed:=1480.3974041179326}
(luamplib)               >> "z[13]= (-3.3223499681157214,-6.4744847060536879)"
(luamplib)               {randomseed:=876.68583251129962}
(luamplib)               >> "z[14]= (-1.8601946468263932,-6.6736668112858331)"
(luamplib)               {randomseed:=2646.6707481852013}
(luamplib)               >> "z[15]= (-0.45124071673288363,-9.4417326012844018)"
(luamplib)               {randomseed:=132.57237941859191}
(luamplib)               >> "z[16]= (2.9170180427407999,-8.5078926786590099)"
(luamplib)               {randomseed:=1047.8092142440955}
(luamplib)               >> "z[17]= (4.2494809218946745,-6.7105559485829076)"
(luamplib)               {randomseed:=2021.5000778674312}
(luamplib)               >> "z[18]= (8.0277937535304655,-3.4644864211184609)"
(luamplib)               {randomseed:=2055.7357077356655}
(luamplib)               >> "z[19]= (10.187937026289996,-3.2670946608207374)"
(luamplib)               {randomseed:=3756.315583007492}
(luamplib)               >> "z[20]= (8.0942754644586294,-0.78244053305555517)"