Re: [math-fun] discrete curve shortening flow
If one considers a polygon as a discrete approximation to a continuous closed curve, then each vertex has influence over a whole section of the curve. So it is not unreasonable to be less local when assigning a direction and 'curvature' at the vertex, dependent on the neighboring vertices. Whether there's a natural way to do this (and perhaps determine a 'best" continuous curve being approximated) might be an interesting question. Not that I disagree with your notion of fairness. --ms On 18-Feb-17 20:23, Warren D Smith wrote:
Your rectangle example seems damning. ANY vector "curvature" which always has the direction of the angle-bisector, is going to be killed by your rectangle example.
And I really do not think it is fair to use any other direction! -- Unless you abandon all pretense this has anything to do with "motion of the boundary in the 'inward normal direction' (which is a locally-defined notion) with velocity proportional to 'curvature' (which is a locally-defined notion)."
So that's kind of cute, actually. I mean, the thing works for smooth curves, but depends upon smoothness, and it works for simplices, which are not smooth, but that is seen to be special.
If you’re mostly interested in discrete approximations to continuous curves, I would use the integral of the squared curvature as the energy. When applied to 2D surfaces it is called the “Willmore energy”, and topological spheres always flow to the round sphere. I suspect the same happens to curves, although the curve will expand while it becomes round (the energy is not scale invariant in 1D as it is in 2D). There has been quite a lot of research on finding good approximations of curvature for triangulated surfaces. Brakke’s Surface Evolver does a nice job for 2D surfaces and I believe it also can be used for curves. -Veit
On Feb 18, 2017, at 9:09 PM, Mike Speciner <ms@alum.mit.edu> wrote:
If one considers a polygon as a discrete approximation to a continuous closed curve, then each vertex has influence over a whole section of the curve. So it is not unreasonable to be less local when assigning a direction and 'curvature' at the vertex, dependent on the neighboring vertices. Whether there's a natural way to do this (and perhaps determine a 'best" continuous curve being approximated) might be an interesting question.
Not that I disagree with your notion of fairness.
--ms
On 18-Feb-17 20:23, Warren D Smith wrote:
Your rectangle example seems damning. ANY vector "curvature" which always has the direction of the angle-bisector, is going to be killed by your rectangle example.
And I really do not think it is fair to use any other direction! -- Unless you abandon all pretense this has anything to do with "motion of the boundary in the 'inward normal direction' (which is a locally-defined notion) with velocity proportional to 'curvature' (which is a locally-defined notion)."
So that's kind of cute, actually. I mean, the thing works for smooth curves, but depends upon smoothness, and it works for simplices, which are not smooth, but that is seen to be special.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
-
Mike Speciner -
Veit Elser