added Alexander duality to generalisations and removed my earlier comments that are now obsolete |
|||
Line 22: | Line 22: | ||
:As for 2, note first that a homeomorphism is in this context the same thing as an injective continuous map (because ''S''<sup>1</sup> is compact, and ''R''<sup>2</sup> 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. — [[User:EmilJ|Emil]] [[User talk:EmilJ|J.]] 15:33, 23 September 2008 (UTC) |
:As for 2, note first that a homeomorphism is in this context the same thing as an injective continuous map (because ''S''<sup>1</sup> is compact, and ''R''<sup>2</sup> 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. — [[User:EmilJ|Emil]] [[User talk:EmilJ|J.]] 15:33, 23 September 2008 (UTC) |
||
== Applications in collision detection == |
== Applications in collision detection == |
||
Line 31: | Line 30: | ||
Example implementation: |
Example implementation: |
||
http://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Oticon6|Oticon6]] ([[User talk:Oticon6|talk]] • [[Special:Contributions/Oticon6|contribs]]) 14:14, 1 April 2009 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot--> |
http://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Oticon6|Oticon6]] ([[User talk:Oticon6|talk]] • [[Special:Contributions/Oticon6|contribs]]) 14:14, 1 April 2009 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot--> |
||
== Unsourced generalisation removed from the article == |
|||
I have consulted a number of standard textbooks in algebraic topology, and none of them seems to contain the following statement that is a considerable strengthening of the Jordan–Brouwer separation theorem (''n''-sphere replaced with a compact connected ''n''-manifold without boundary): |
|||
<!-- Is this true? --> |
|||
: Another generalisation of the Jordan curve theorem states that if ''M'' is any [[compact set|compact]] [[connected set|connected]] ''n''-dimensional submanifold of '''R'''<sup>n+1</sup> [[closed manifold|without boundary]], then ''M'' separates '''R'''<sup>n+1</sup> into precisely two regions, one of which is compact and the other is non-compact. |
|||
If someone can reliably source it, please, put it back, together with a precise citation. [[User:Arcfrk|Arcfrk]] ([[User talk:Arcfrk|talk]]) 21:11, 3 August 2010 (UTC) |
Revision as of 03:46, 4 August 2010
Mathematics Start‑class Mid‑priority | ||||||||||
|
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)
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!
- 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)
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)