Re: [math-fun] discrete curve shortening flow
Thing is, the discrete approximation to a polygon's "integral curvature" (w.r.t. arclength) at a vertex is independent of which sequence of smooth curves is used to approach the polygon near the vertex. (The integral is [see below].)* But for the integral of *squared* curvature, that is not the case. On Feb. 19, 2017, at 8:55 PM, Veit Elser <ve10@cornell.edu> wrote: ----- 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.
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.
----- —Dan —————————————————————————— * It's the exterior angle.
On Feb 19, 2017, at 11:25 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Thing is, the discrete approximation to a polygon's "integral curvature" (w.r.t. arclength) at a vertex is independent of which sequence of smooth curves is used to approach the polygon near the vertex. (The integral is [see below].)*
But for the integral of *squared* curvature, that is not the case.
Of course, but I thought he was interested in flows that make polygons round. The energy you cook up to define the flow then should *not* be a topological invariant (i.e. independent of shape).
On Feb. 19, 2017, at 8:55 PM, Veit Elser <ve10@cornell.edu> wrote: ----- 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.
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.
-----
—Dan —————————————————————————— * It's the exterior angle.
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Veit Elser