Talk:Jordan curve theorem

Formal proof

I have just clarified the formal proof priority, sorry for doing it anonymously - have not noted I was logged out. JosefUrban (talk) 15:41, 4 January 2009 (UTC)

Illustration

An illustration could really improve the article and give credence to the claim that the concept is intuitive.--Cronholm144 10:14, 25 May 2007 (UTC)

This has been done.
98.67.108.12 (talk) 16:23, 25 August 2012 (UTC)

Question

Hello, I have a question. A book that I am reading says that "A Jordan curve is an equivalence class of homeomorphisms of I into R² (or of S¹ into R² in the case of closed curves)." This article defines a Jordan curve as "a simply closed curve."

1. The book I am reading is implying an arbitrary equivalence relation? And what is the purpose of this equivalence class? (I assume that it is just a way of saying 'all the identical homeomorphisms', so that it gets all the same shapes that are represented differently?)

2. This article says that a Jordan curve is a simple closed curve, but I think the definition my book gave says that it doesn't have to be simply closed, though it may, i.e. 'a homeomorphism of S¹ in the case of closed curves'. But I know that this article is right, because a Jordan curve is defined identically in Curve.

Sorry for the stupid questions. Great article!

Veritas (talk) 16:37, 22 September 2008 (UTC)

As for 1, the book certainly refers to some specific equivalence relation. You have to look backwards for a definition of equivalent curves. It is hard to guess what they mean by it without reading it, though one possibility is that curves are equivalent if they differ by a homeomorphic change of parametrization.

As for 2, note first that a homeomorphism is in this context the same thing as an injective continuous map (because S¹ is compact, and R² is Hausdorff), thus the two definitions agree on what closed curves are Jordan curves. As far as I am aware, allowing Jordan curves to be non-closed is highly unusual. At any rate, the Jordan curve theorem only applies to closed curves. — Emil J. 15:33, 23 September 2008 (UTC)

To the original questioner: You do need to allow for the same thing having more than one name. Sometimes, there is even a very technical name and a common name. For those of us who have studied complex analysis and/or real calculus of two dimensions, "simple closed curve" tells it all. We know what those are.

In some kinds of mathematics on the Ph.D. level, some more precise wording might be necessary. Do not get "wrapped around the axle" about it.
98.67.108.12 (talk) 16:36, 25 August 2012 (UTC)

Applications in collision detection

Hi guys.... I think this article needs to be expanded to encompass one of its most useful applications: 2D collision detection. These sites give a good explanation of what the Jordan Curve Theorem means in a practical sense, and would help explain the concept to the less math-inclined. The strategy: http://tog.acm.org/editors/erich/ptinpoly/ Example implementation: http://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html —Preceding unsigned comment added by Oticon6 (talk • contribs) 14:14, 1 April 2009 (UTC)

suggested addition

This paper seems like a nice reference. It has nice pictures in it. I hope one of you experts will include it somehow :)

https://facultystaff.richmond.edu/~wross/PDF/Jordan-revised.pdf — Preceding unsigned comment added by 98.249.0.238 (talk) 00:37, 31 March 2012 (UTC)

OK, I did. :-) Boris Tsirelson (talk) 17:44, 31 March 2012 (UTC)

So, what about "M. Reeken"?

Now, he or she was "M. Reeken", and I added the "M", which is given in the references. So, "Who Reeken"? Please give the whole name.
Just calling someone "Reeken" or "Jordan" or "Jones" or "Schwartz" on first appearace is actually quite rude. If this is a person who wants to be referred to as "M. Reeken", then please give her or him that much. As for the present, we do not know if this was "Marion Reeken", "Michael Reeken", "Michelle Reeken", or "Madison Reeken".
98.67.108.12 (talk) 16:29, 25 August 2012 (UTC)

Michael, per the reference. Not that I know anything about him.—Emil J. 12:48, 27 August 2012 (UTC)

Contribution of A.S. Zoch - is it notable?

An anon "IP 67.226.22.24" just added two phrases:

• The difficult part of the proof of the Jordan Curve theorem lies in the fact that the two disjoint components of the plane cut by the given curve are open and the curve itself is closed.
• The Jordan Polygonal theorem was presented at ICM 1998 by A. S. Zoch and soon afterwards a simple proof of the Jordan Curve theorem was presented to AMS by the same author using linear lemmas.

About the latter phrase. As far as I know, this work by Zoch was presented on several conferences (including ICM 1998 and ICM 2002) in the form of short communication or contributed paper, but was never published in mathematical literature. Was it refereed? Is it notable?

About the former phrase. Does it make sense?

Boris Tsirelson (talk) 20:06, 28 February 2013 (UTC)

I’d say the answers are no and no.—Emil J. 13:07, 1 March 2013 (UTC)

So you were not there otherwise you may know the difference. Perhaps A. Zoch would explain things to you if you cared more about mathematics. A. S. Zoch has publications in the areas of mathematics, colour theory, optics, encryption, and light. — Preceding unsigned comment added by 67.226.62.182 (talk) 19:16, 9 June 2016 (UTC)

I have question

In a course on Complex Analysis I attended this Jordan Curve theorem was referenced with using curve Γ, which divides the plane to two regions which are bordered by Γ and which one is interior region(bounded) and another unbounded region is exterior region( but these regions have NO formal notation like I show following ) And I know also that at least I have used for set A, interior region as I(A) and exterior region E(A). Is this right use? I mean is it wrong that you would not define interior region by the edge of A as closed jordan curve C? So Interior region would be I(A). And is it wrong to not to define exterior region by this C? So exterior region would be E(A). — Preceding unsigned comment added by Alvoutila (talk • contribs) 12:58, 15 August 2014 (UTC)

My question too! Phrased more simply: does the curve belong to the interior (a closed or compact set), the exterior (open set), or neither? The notion of crossing the boundary implies that both interior and exterior sets are open, and only the boundary is compact.220.240.251.114 (talk) 01:09, 4 May 2016 (UTC)

Sorry, I do not understand this question. The Jordan curve is compact. Its complement is open. This complement is the union of two disjoint connected open sets, one bounded ("the interior") and one unbounded ("the exterior").

Ah, well, maybe I do understand. This use of the word "interior" is not its use in the article "interior (topology)". Somewhat misleading terminology, indeed. Boris Tsirelson (talk) 17:55, 4 May 2016 (UTC)

Translation?

Does the cited 4-page proof of Filippov have an English translation anywhere?? (If not, could someone make students translate that proof on their math gradschool language exam and post the results online??) It is a horrible tease to be directed to an "elementary" proof, only to find that it is in Russian. — Preceding unsigned comment added by 173.66.241.122 (talk) 19:07, 8 May 2016 (UTC)

The Russian text is available here. A similar proof by A.I. Vol'pert is here. Boris Tsirelson (talk) 17:58, 2 April 2019 (UTC)

A five-page proof, in English, given by Tvelberg in 1980, is available online for free. Why translate Filippov? Boris Tsirelson (talk) 19:52, 2 April 2019 (UTC)

complement of jordan arc

Article says

Furthermore, the complement of a Jordan arc in the plane is connected.

How can that be right? Should it say "not connected"? 67.164.113.165 (talk) 16:05, 2 April 2019 (UTC)

 Fixed I have removed the sentence, as, if "connected" is changed into "disconnected", one gets an assertion that is weaker than the theorem (exactly two connected components). Moreover, "furthermore" is not a synonymous of "in particular". D.Lazard (talk) 16:45, 2 April 2019 (UTC)

In the section you are discussing, a Jordan arc is defined there to be the image of an interval. On the other hand, possibly this (true, not duplicative) comment does not belong in its present location; at least, not without a source that makes some connection with the statement of the theorem. --JBL (talk) 00:59, 3 April 2019 (UTC)

OK, my mistake. Thanks for reverting me. D.Lazard (talk) 07:31, 3 April 2019 (UTC)

Continuous curve

Could someone please create Continuous curve and redirect it here?

Or rather, redirect it to Curve#Topological curve? There we see

"A simple closed curve is also called a Jordan curve. The Jordan curve theorem states that the set complement in a plane of a Jordan curve consists of two connected components (that is the curve divides the plane in two non-intersecting regions that are both connected)."

Boris Tsirelson (talk) 06:39, 28 September 2019 (UTC)

Works for me. — Preceding unsigned comment added by 96.40.48.159 (talk) 06:55, 28 September 2019 (UTC)

Well, I did. Boris Tsirelson (talk) 20:10, 28 September 2019 (UTC)

Application

• 07:56, July 18, 2021 - «Reverted 1 edit by Alexander Davronov talk): WP:BURDEN: "The burden to demonstrate verifiability lies with the editor who adds or restores material".»
• 07:11, July 18, 2021 - «Undid revision 1034105162 by D.Lazard talk) You may want to WP:TAG it first.»
• 20:35, July 17, 2021 - «Reverted 1 edit by Alexander Davronov talk): Vague and unsourced»
• 20:30, July 17, 2021

• @D.Lazard: WP:BURDEN: "The burden to demonstrate ... ". Hi. WP:BURDEN is mine, but I can't fulfill it while you're removing what I'm working on. Please, refrain from removing content carelessly. In this particular case signified by WP:DIFFs above the deletion isn't justified (you have provided no provision of such). Refer to WP:PRESERVE. Except of WP:BLP/WP:CV NO known provision gives you a right to hastily remove anything. Use {{citation needed}} (WP:TAGGING) to mark unsourced information before taking action. See also:

• Wikipedia:Don't demolish the house while it's still being built.
• WP:CANTFIX

Regards. --AXONOV (talk) ⚑ 08:25, 18 July 2021 (UTC)

Your sandbox is here for preparing your work before adding it to an article. So the fact that your addition is still incomplete (fact that a reader is not supplosed to know of) is not a reason for not reverting it. You are invoking WP:CANTFIX. It applies here, as your edit is so vague that nobody (except you, maybe) can verify it. More, it seems unbelievable that Jordan curve theorem can be useful for computing with polygons, and even for studying them. So, WP:verifiability is fundamental here, simply for verifying that the assertion is correct. D.Lazard (talk) 09:08, 18 July 2021 (UTC)

@D.Lazard: Please see: Keep it, don't remove!. AXONOV (talk) ⚑ 11:28, 18 July 2021 (UTC)

Application - Image caption

10:12, August 14, 2021 - «‎Application: fixing the caption format and wording)»

@D.Lazard: I appreciate your efforts but this one made it inaccurate relative to the source I've used when I've inserted the image. It was specifically talking about domains rather than simple term of regions. I propose to bring domains back.

AXONOV (talk) ⚑ 19:57, 5 September 2021 (UTC)

The above reference is not a source for Wikipedia, as it is anonymous and no indication is given about a possible publication.

In any case, a caption must be mathematically correct and use the same terminology as the surrounding article. This was not the case of the caption that I have edited, which contained undefined notation (B→A) and nonsensical phrases (crossing points of a ray and non-parallel edges).

It is not convenient to use "domain" in the caption for two reasons: 1/ The article uses "region". 2/ "Domain" has several precise mathematical meanings, and must not be used without indication on its meaning; otherwise, the sentence is confusing. D.Lazard (talk) 08:04, 6 September 2021 (UTC)

@D.Lazard: The reference is in fact an excerpt from the What Is Mathematics? book, which is referenced by the article.
[...] and nonsensical phrases (crossing points of a ray [...] There was no such phrase in the diff at the top.
[...] and non-parallel edges [...] Intersections should not be counted if the ray intersecting the edge is parallel to the edge itself (intersecting through two joined vertices of the polygon). That's why it should be mentioned.
[...] "Domain" has several precise mathematical meanings [...] I propose to clarify that this meaning was derived from domain of mathematical analysis. We are talking specifically about set of all possible points in a given area (A or B) which comprise a domain. AXONOV (talk) ⚑ 15:04, 6 September 2021 (UTC)

References

Illustration (again)

I noticed someone changed the image to be the borders of a certain disputed parcel of land in the Middle East, and that made me realize the example image before that was the borders of Jordan, the country. I'm not an expert in math, so I would appreciate input from someone who actually understands the topic of the article, but I think those images only add confusion. I changed the image back to the original example image. Thesixthstaff (talk) 22:40, 22 January 2024 (UTC)