Talk:Curve-shortening flow

Normal and curvature not well-defined

The definition of the differential equation does not specify what is meant by "normal" (there are of course two options). Assuming a consistent definition of normal is used, then I believe it should be the "signed curvature", not the "curvature", to allow concave deformations to push "out".

Geometric heat flow for vector fields

There seems also to be the concept of a geometric heat flow on vector fields, as described in the arxiv paper A GEOMETRIC HEAT FLOW FOR VECTOR FIELDS. It implies that geometric heat flow is not a 1-1 synonym for curve-shortening flow, as there can be geometric heat flows on more than just curves or manifolds. Geometric heat flow on vector fields might also be worth mentioning in the Beyond curves section, but I'm not confident enough with the scope of the topic to say if it belongs. --Mark viking (talk) 04:33, 4 December 2015 (UTC)

I added a footnote remarking that the geometric heat flow name has other uses. —David Eppstein (talk) 22:37, 19 April 2016 (UTC)

Looks good, thanks. --Mark viking (talk) 23:25, 19 April 2016 (UTC)

Definition correct?

In the differential equation, the curve C is parameterized by arc length. Is that correct? I think it should be parameterized by the angle of the tangent vector; otherwise, the derivative with respect to t is not perpendicular to the tangent vector of C. I've tried to solve that differential equation for a circle, it seems to be wrong. — Preceding unsigned comment added by 92.218.115.27 (talk) 12:26, 2 February 2020 (UTC)

GA Review

This review is transcluded from Talk:Curve-shortening flow/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: Mark viking (talk · contribs) 23:12, 20 April 2016 (UTC)

GA review – see WP:WIAGA for criteria

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I plan to review this article in the coming days. As this is my first GA review, comments and corrections are welcome. Per the review process, a quick skim of the article shows it to have no glaring failures of good article criteria and it easily passes onto the detailed review. Below, I number the issues to be fixed or to check my fixes.

1. Is it well written?

   A. The prose is clear and concise, and the spelling and grammar are correct:

   I found no spelling or grammatical errors. Figure captions are sentence fragments, but this is allowed by MOS:CAPTION.

   B. It complies with the manual of style guidelines for lead sections, layout, words to watch, fiction, and list incorporation:

   No issues: the layout conforms to MOS:LAYOUT, fiction doesn't apply, and the embedded lists conform to MOS:EMBED

   1. The lead section looks well structured according to MOS:LEAD and of appropriate length. All sections except Numerical approximations are summarized in the lead. There should be something about numerical approximations here.

   I added approximations to lead and condensed some other material there to keep the lead down to four paragraphs. —David Eppstein (talk) 21:22, 21 April 2016 (UTC)

   Looks good. --Mark viking (talk) 03:28, 22 April 2016 (UTC) Y

   2. Definitions section second para: This definition is clearly invariant... should be replaced with the reasons why it is invariant.

   Yes, that adverb is a bit of a red flag. Replaced by an explanation. —David Eppstein (talk) 21:22, 21 April 2016 (UTC)

   Good explanation. --Mark viking (talk) 03:28, 22 April 2016 (UTC) Y

   3. The are a number of uses of the word also in the article that don't need to be there. For instance, in the Beyond curves subsection, It is also possible to extend the definition... could be replaced with It is possible to extend the definition...

   Eleven alsos removed. —David Eppstein (talk) 21:22, 21 April 2016 (UTC)

   Great. --Mark viking (talk) 03:28, 22 April 2016 (UTC) Y

   4. I removed an unneeded "Note, however" from the first note.

2. Is it verifiable with no original research?

   A. It contains a list of all references (sources of information), presented in accordance with the layout style guideline:

   The Notes and References sections look conformant to the guideline. The citations use a mixed format with Harvard citations for what look like historical documents and numerical footnotes for citations verifying facts. The mixed citation format has been used in other good math articles, such as Reuleaux triangle, so seems fine to me.

   B. All in-line citations are from reliable sources, including those for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons—science-based articles should follow the scientific citation guidelines:

   The author has done a wonderful job of citing mathematical facts, historical sources and dates. The citations more than satisfy the scientific citation guidelines, with every paragraph being cited to one or more sources. I could identify no passages unsupported by a source. I spot-checked several citations and found them accurate. The page numbers for the books were appreciated.

   5. Among the sources, four stand out as not quite reliable: the (Allen 2012) arxiv paper, the (Ilmanen 2014) arxiv paper, the (Lam 2016) arxiv paper, and the (You 2014) PhD thesis. All four seem unpublished. Arxiv papers undergo sanity checks and professors usually don't let students put nonsense in their theses, but they seem to fail the RS criteria for independent peer review or editorial oversight.

   The Allen paper appears to be an undergraduate research project supervised by a new assistant professor, included in an MAA poster session but otherwise unpublished; I agree that that's not reliable enough, and have removed it from the article. The Ilmanen paper has André Neves as a co-author, and so I would argue that it passes the "established expert" clause of WP:SPS. Similarly, the You thesis was supervised by Sigurd Angenent, passing the "supervised by recognized specialists in the field" clause of WP:SCHOLARSHIP. That leaves Lam & Lauer. Lauer is a 2012 Ph.D. with work in this area, and a temporary position at MIT (or maybe FU Berlin?), and I don't know who Lam is, so the case for being an established expert is weak. Can we consider https://plus.google.com/+PaulBryanguy/posts/34m6LEgUm46 to be a form of open peer review? It's by Paul Bryan, who has a tenure-track position at UCSD and also works in this area. —David Eppstein (talk) 21:22, 21 April 2016 (UTC)

   You raise a good point. I agree that Neves and Angenent can be considered experts in the field, so those refs become reliable. A Google+ endorsement is a pretty weak form of peer review, but it does show that someone in the field has read it and believes the result. I personally don't think the result is contentious, so I think keeping this marginally reliable source is OK. --Mark viking (talk) 04:15, 22 April 2016 (UTC) Y

   C. It contains no original research:

   6. I was unable to find any original research or synthesis in the prose; it is really tight. The only original elements I found were the illustrations created by the author. I have no reason to doubt the accuracy of any of the illustrations, but if there are sources these were drawn from, it would be useful to cite them.

   I didn't record the detailed provenance of the illustrations because I didn't think it was important, but I can explain here if you like. File:Convex curve shortening.png was made by taking many screenshots from a web curve-shortening applet (whose url I no longer remember) and then choosing a subset of them, compositing them together, and coloring them in Photoshop; that's why the gaps between the levels are a little uneven, because the only control I had over that was how often I took a screenshot. File:Curve-shortening ambiguity.svg was drawn in Adobe Illustrator, and is not geometrically precise for that reason, but I think that's not important (it's more about qualitative behavior than the exact shapes of the curves). File:Curve-shortening self-similar lens.svg was also drawn in Illustrator but is exact. For File:Grim reaper curve.svg I used some graphing software (that I didn't record) to make a plot of the function, and then colored it and added the repetitions in Illustrator, so that one is precise up to the resolution of the plotting software. And File:Anneal CA.png is a screenshot from Golly, cropped in Photoshop. —David Eppstein (talk) 21:32, 21 April 2016 (UTC)

   Thanks for the descriptions. The usual illustrator's tools don't really need attributions, but for specialized tools like the web applet, it's good to give credit where it is due and to check for licensing restrictions. The one applet I know of is at http://a.carapetis.com/csf/--is this the one? The associated web site and source code seem to have no restrictions on use of the applet or its generated results. Golly is unencumbered with license restrictions and looks good. --Mark viking (talk) 04:45, 22 April 2016 (UTC)

   That's the applet, yes. —David Eppstein (talk) 04:56, 22 April 2016 (UTC)

   OK, I'm satisfied there is no original research and that there are no licensing problems with the software used. --Mark viking (talk) 05:08, 22 April 2016 (UTC) Y

   D. It contains no copyright violations nor plagiarism:

   No copyvio or plagiarism was found and none of the prose has a whiff of copied text.

3. Is it broad in its coverage?

   A. It addresses the main aspects of the topic:

   Not an issue: Often math articles will have a dedicated history section; in this article historical information is distributed across sections. For this topic I think distributed is OK as the topic developed concurrently among disparate groups so there isn't a central story. There are sources such as Gage's Texas talk that talk of some of the history of the subject, but I haven't seen any sources presenting a unified view of historical development across math, physics, and computation.

   7. Are there important open questions or areas of particularly active research in the field that should be mentioned?

   If so I don't know what they are; I didn't see anything like that in the references I read. I get the vague impression that by now this subject is treated more as the simple well-understood toy problem that you use to develop your intuition before moving on to the higher-dimensional flows where the real problems are. —David Eppstein (talk) 21:50, 21 April 2016 (UTC)

   I was not able to find any such list of major open problems either. Chou mentions some open problems with respect to anisotropic CSF, but they don't seem notable enough for an encyclopedia article. OK, no such prose needed. --Mark viking (talk) 04:54, 22 April 2016 (UTC) Y

   8. Anisotropic curve shortening flow isn't mentioned. This variant has been used in describing crystal growth and in some kinds of image processing. Some sources are

   • Dziuk, G., 1999. Discrete anisotropic curve shortening flow.SIAM journal on numerical analysis, 36(6), pp.1808-1830.
   • Haußer, F. and Voigt, A., 2006. A numerical scheme for regularized anisotropic curve shortening flow. Applied mathematics letters, 19(8), pp.691-698.
   • Chapter 6 of Chou and Zhu's book
   • G. Unal, H. Krim and A. Yezzi, "Stochastic differential equations and geometric flows," in IEEE Transactions on Image Processing, vol. 11, no. 12, pp. 1405-1416, Dec 2002. (an interesting stochastic approach, but not as widely cited as the first three)

   If I read them correctly, Dziuk/Haußer/Voigt are doing something different than Chou/Zhu, although all are anisotropic. Anyway I added them to the "Related flows" section. —David Eppstein (talk) 06:51, 23 April 2016 (UTC)

   The added section looks good to me and seems of due weight. I agree mentioning both is best, as these are among the major sources for anisotropic work. This issue is resolved. --Mark viking (talk) 18:37, 23 April 2016 (UTC) Y}

   B. It stays focused on the topic without going into unnecessary detail (see summary style):

   A strength of this article is the concise manner in which subtopics are summarized with little beyond the main results or the most important facts. I don't see any fat to trim or off-topic prose. There is more detail (e.g., update rules) in the cellular automata section than the others, but it is not excessive.

4. Is it neutral?

   It represents viewpoints fairly and without editorial bias, giving due weight to each:

   The material all looks neutrally presented with no one approach to the problem dominant. Relative to the Chou and Zhu book, the various mathematical topics look of due weight (the anisotropic variant above being the exception). I don't know of secondary sources comprehensive across mathematical, physical, and computational aspects of the topic, but based on the sources out there, the article gives due weight to the different aspects of the problem.

5. Is it stable?

   It does not change significantly from day to day because of an ongoing edit war or content dispute:

   There has been naught but steady improvement, with no edit wars or instability present.

6. Is it illustrated, if possible, by images?

   A. Images are tagged with their copyright status, and valid fair use rationales are provided for non-free content:

   All illustrations are 'own work' by the author and are appropriately licensed.

   B. Images are relevant to the topic, and have suitable captions:

   The images included are all relevant to the topic that are illustrating.

   9. In the first illustration, there seems to be something missing. The curves are all of different colors, presumably to indicate their order in the time evolution, but no mention is made of the colors in the caption.

   I expanded the caption, I think more importantly to mention that the spacing of the curves is uneven. —David Eppstein (talk) 21:43, 21 April 2016 (UTC)

   I hadn't noticed the spacing, but you're right. The caption looks good. --Mark viking (talk) 04:57, 22 April 2016 (UTC) Y

7. Overall:

   Pass or Fail:

   I've finished my first pass through the GA criteria and listed all issues I found. Overall, this is a really strong, well written article with only minor issues found. Excellent work!

   Final verdict: All issues I found have been resolved and the article meets all GA criteria. I commend David for responding quickly and with patience as I worked my way through my first GA review. It seems traditional for GA reviews to last 7 days or longer, but with no other issues raised by other editors, I see no reason to keep this review open. Hence, this article gets an overall pass and is a good article. --Mark viking (talk) 19:08, 23 April 2016 (UTC)

Expansion on 1A: While the prose is clear, for technical topics like this, there should also be some emphasis on accessibility. I am not sure if this is normally part of GA criteria, hence it is outside the main list. If so, these are a few more small issues. If not, these are suggestions for improvement.

The author has done a wonderful job of summarizing complex results in prose with a minimum of formulas, but there can always be improvements. As a person relatively new to this topic, here are a couple of things that occurred to me as I began to read the article.

10. In curve-shortening flow, what is flowing? It would be good to explain more explicitly what this means. In the Muller work, it is the flow of metal inside the curve that is inducing a change in grain boundary. But for the pure math approach, my guess is that the normal vectors around the curve, over time, describe a 2-D vector field, the integral curves of which show the trajectories of points on the curve. Then one associates trajectories as analogous to streamlines in a fluid, so that the points on the curve can be said to flow. But the analogy is flawed because the the curve shortening system doesn't have the usual mass or volume conservation laws.

Nice explanation. --Mark viking (talk) 03:12, 23 April 2016 (UTC) Y

11. In the front tracking section, why is it called a front? Weather front? Reaction-diffusion front? What is elliptic regularization?

Looks good. --Mark viking (talk) 03:12, 23 April 2016 (UTC) Y

12. I didn't get a good intuition of of curve shortening dynamics in the region of three curve junctions. There are multiple infinite curvatures that don't get resolved immediately to analytic curves and break the continuity conditions on the PDE. I guess I can see how the more acute he angle, the stronger the curvature infinity, so that junctions move toward equi-angular, but there must be some sort of regularization scheme behind it to make the PDE defined. Or one could resolve it by ignoring the PDE and just calculating the gradient flow of the length functional.

I'm closing this one. I tried again to find an RS that had an intuitive explanation and failed. Improving the explanation here would be nice, but not at the cost of OR. Best to leave this to future work. --Mark viking (talk) 03:12, 23 April 2016 (UTC) Y

I worry that handling any of these would verge on original research, since I don't think there's much about them in the sources (or for #12 there's a lot but very technical rather than trying to work at an intuitive level). My own intuition is that in #10 it's the points on the curve that are flowing, and that in #12 one can mostly just let the curves flow and let the junctions follow. The elliptic regularization explanation should be possible but not immediately so (it will take me more time to understand that part well enough to provide more depth). —David Eppstein (talk) 05:01, 22 April 2016 (UTC)

Ah, some progress. For issue 10 we already have an article Flow (mathematics) that explains the formal concept of flow (I was on the right track) and could be linked to from the article. For issue 11, front tracking comes from multiphase fluid and free fluid simulation [1], [2]. In this context, a front is a boundary across which there is a solution discontinuity, like grain boundaries or flame propagation. Front tracking is the simulation of the boundary dynamics rather than the underlying bulk materials. This paper gives the basic idea of elliptic regularization. The idea is to take a first order differential equation in time over a continuous field like a curve or surface and add a small spatial laplacian term to the equation. I believe this term would penalize large curvature fluctuations and thus regularize or smooth out the curve. I can see where this would be useful for simulating front tracking. --Mark viking (talk) 06:37, 22 April 2016 (UTC)

I added some more material to the start of the "Definitions" section (#10, re what is a flow) and to the "Front tracking" subsection (#11, re why front tracking and what is elliptic regularization). —David Eppstein (talk) 00:37, 23 April 2016 (UTC)

The new prose is a nice addition. I think it helps understanding and provides a little more context. Issues 10,11 and 12 are resolved. Good progress with only issue 8, saying something about anisotropic curve shortening, remaining. --Mark viking (talk) 03:12, 23 April 2016 (UTC)

Ok, I think I've handled all the requests. It occurs to me that I should point out a deliberate mathematical inaccuracy: at several points I write that a flow is invariant under some kind of transformation of the plane when really what I mean is equivariant. I think that it's significantly less technical that way (and easily enough understood) but if you disagree please let me know. —David Eppstein (talk) 07:10, 23 April 2016 (UTC)

I took another read through the article and it all looks good with no new issues found.

Regarding invariant vs equivariant, I understand your point. The normal vector, for instance, is contravariant under rotations. But looking at the literature that deals with analyses of symmetries of curve-shortening flow, Altschuler 2012, Qu and Huang 2008, Broadbridge and Vassilou 2011, even a paper by Olver on invariant submanifold flows, all use the terminology "invariant" and none use "equivariant". So I think this broad use of the term invariant is backed by the sources and is fine by me. --Mark viking (talk) 18:55, 23 April 2016 (UTC)

Inaccuracy?

This is a great article but I'm confused by this paragraph:

The avoidance principle implies that any smooth curve eventually either reaches a singularity (such as a point of infinite curvature) or collapses to a point. For, if a given smooth curve C is surrounded by a circle, both will remain disjoint until one or the other collapses or reaches a singularity. But the enclosing circle shrinks under the curvature flow, remaining circular, until it collapses, and by the avoidance principle C must remain contained within it. By the same reasoning, the radius of the smallest circle that encloses C must decrease at a rate that is at least as fast as the decrease in radius of a circle undergoing the same flow.

For one thing, I'm not sure exactly what is being concluded, as every smooth curve reaches a singularity, and the given argument says nothing about the specific subcase of collapsing to a point. It seems that what is being proved is that it is impossible for the flow to exist for all time. If so, the last sentence of the paragraph seems to be irrelevant, as it is sufficient to consider any surrounding circle. Gumshoe2 (talk) 20:05, 20 September 2020 (UTC)

The last sentence gives an explicit bound on how quickly the singularity/collapse happens. —David Eppstein (talk) 20:22, 20 September 2020 (UTC)

I see. The way I had read it, I'd thought that the previous sentence was making a simple observation which was setting up the last sentence, which was proving the claim. My mistake. I'm still confused by what I said about the first sentence though. Is the point just that infinite-time existence is impossible? Gumshoe2 (talk) 20:28, 20 September 2020 (UTC)

More or less. Either it collapses to a point (which is what always actually happens) or it becomes non-smooth (not ruled out by this argument but also not possible). —David Eppstein (talk) 21:22, 20 September 2020 (UTC)

I made a small edit to clarify. I think it was phrased a little confusingly. Gumshoe2 (talk) 05:19, 8 October 2020 (UTC)

Ok, thanks. I don't see anything to object to in your changes. —David Eppstein (talk) 05:27, 8 October 2020 (UTC)