Talk:Voronoi diagram

message: Another Example is the Wigner-Seitz cell in materials science.--149.220.16.110 15:30, 15 September 2006 (UTC)
thread_title:  Wigner-Seitz cell 

message: This article already has some nice graphics; are we sure its a good idea to also add ascii art to this? I don't think so ... 
Also: there's talk about a rectangular tesllation with points "not at the center" ... how can that be? its a metric polygon; is this a subtle statement that the center of mass doesn't align with the metric center? Besides the "center of mass", there are other types of "centers" e.g. center for diffusion processes, etc. How about other definitions of "center"? 
Also, for a rectangular array, the voronoi cell is not a rectangle any more ... so the "examples" section has several confusions in it ... ditto for the remarks about isocleles trinagles ... linas 20:28, 26 July 2005 (UTC)
thread_title: Artwork; unusual statements

message: How about adding some mention of the algorithms used to calculate a voronoi diagram?  IIRC, there is one that works in O(n*log(n))?
:If you can describe such algoritms accurately, then please add them. linas 16:07, 23 November 2005 (UTC)
thread_title: Algorithms?

message: Voronoi appears to be of Ukranian descent.--MinorEdits 08:45, 1 June 2006 (UTC)
thread_title:  pronunciation? 

message: It might be helpful to talk about the order of a voronoi diagram. E.g. something along the lines of, "a Voronoi cell can be generated from a set of points from S. The cardinality of this set is referred as as the order of the Voronoi diagram".
It might also help to give algorithms for constructing order-k diagrams and the complexity of generating an order-k diagram.  —Preceding unsigned comment added by Sunupsundown (talk • contribs) 19:45, 1 October 2008 (UTC)
thread_title:  Order of a Voronoi diagram 

message: The concept behind Voronoi Diagrams can be explained in non mathematical terms/notation (ok, you need points, lines and distance, but not much more then that). With the help of a picture, shouldn't that sort of explanation be in the introductory blurb?
Also, it would be interesting to know what "human algorithms" were used to draw 2D voronoi diagrams back before computers.
A non-mathy observation: doesn't the Voronoi iteration explain the approximate cultural and sometimes political boundaries for countries with land borders adjacent to others?  That is, the farther from the capital, the weaker the influence.  A real world example like this would be useful.  —Preceding unsigned comment added by Bobbyleeds (talk • contribs) 15:45, 13 August 2010 (UTC)
replies: text: I did a quick Google search for this. I couldn't find a national-level example, but I did find an example for the states of the USA. Not sure how (or whether) to include it in the page though. The applications is already a bit messy... Warrickball (talk) 10:23, 22 August 2011 (UTC)

text: I've removed the incomplete WP:AFD nomination of the article. From the above comment its clear you ment the nom more as a protest rather than an actual case for deletion.--Salix (talk): 09:28, 29 October 2012 (UTC)

text: I've simplified the lead section which should make it a little simpler.--Salix (talk): 09:45, 29 October 2012 (UTC)
thread_title: Non mathy?

message: If you have just three points do you get the Fermat point? --Salix alba (talk) 22:22, 24 September 2007 (UTC)
replies: text: Nope. The Voronoi point is the intersection of orthogonal bisectors, i.e., of lines perpendicular to the sides of the triangle pasing through their middles. So you may easily prove that it will be the Fermat point only for an equilateral triangle.`'Míkka 06:24, 25 September 2007 (UTC)

text: It is the circumcenter, though.  We could say, in the properties section, that the mutual boundary point shared by any three adjacent points is their circumcenter. PhiloMath (talk) 16:39, 10 December 2007 (UTC)
thread_title:  Fermat point 

message: I deleted the software section, since there is really a lot of software that can do it and I don't think we should be in the business of plugging a particular package.  If you do add it back, I would like to see, eg, qhull mentioned. --Dylan Thurston 17:43, 1 October 2007 (UTC)
thread_title:  Software 

message: An easy algorithm for creating a Voronoi diagram would seem to be, for each pair of points, to draw a line between them, and then to draw a boundary perpendicular to that line, crossing it at its exact centre. The segments created by these boundaries then make up the cells in the Voronoi diagram. But what happens if this algorithm creates segments without a point inside them? JIP | Talk 13:47, 8 October 2007 (UTC)
replies: text: You could take the set of perpendicular bisectors, which will form a super set of the Voronoi diagram, And then search for those segments which form a polygon around each vertex. Whether this will be efficient is a good question and there has been a lot of work on efficient algorithms. In practice I'd recommend the Qhull program (see links) which works by computing a convex hull in n-dimensions. --Salix alba (talk) 15:40, 8 October 2007 (UTC)

text: Delaunay triangulation (the dual problem) has some more details of algorithms. --Salix alba (talk) 16:11, 8 October 2007 (UTC)
thread_title:  Algorithm? 

message: In the Definition section of the other article it claims that "For any (topologically) discrete set S of points in Euclidean space and for almost any point x, there is one point of S closest to x". This is untrue as shown by the example in the Discrete space page {1, 1/2, 1/4, ...} using the usual euclidean metric: Any negative number has no closest point in the set. It seems to me that some additional restriction like "no accumulation points" is required. TomC phys (talk) 02:28, 12 February 2009 (UTC)
replies: text: Actually, S just needs to include all it's limit points. Including 0 in your example fixes it with no problems. Nekura (talk) 03:03, 25 June 2009 (UTC)

text: Thinking more about it, is it really a requirement, or does it just make the diagram look nicer? The original definition starts with finite sets, so "closest point" could be defined for every point c in the space. But is it "required" that every point c have a closest point, or is it just a relic from the finite case? Is it the set of points in the space, or the set of cells associated with each point in S? Nekura (talk) 03:54, 25 June 2009 (UTC)
thread_title:  Discrete set of points? 

message: If I have a general graph consisting in a set of points and links among them, can I associate to it a "triangulation" in some sense? I mean, regarding the graph as a Voronoi dual to the former triangulation. If yes, can I do it for any d-dimensional generalization of triangle cells or are there requirements to fulfill?
I'd like if someone is able to point me out an answer and eventually a reference. I'm posting this question also in Delaunay's triangulation page.
Omar.zanusso (talk) 10:33, 5 March 2009 (UTC)
thread_title:  Question about existence 

message: In the introduction, a Voronoi diagram is defined as a decomposition of a metric space (any), but in the "Definition" section, it is spoken only of Voronoi diagrams in Euclidean spaces. 00:29, 19 June 2009 (UTC)
thread_title:  Conflict between Introduction and Definition 

message: I'm looking for information on weighted Voronoi diagram. Google only gave me references to programs that do it. There should be a section on it, including the difference between Multiplicative and Additive. Nekura (talk) 00:41, 25 June 2009 (UTC)
replies: text: Here you are. - Altenmann >t 18:59, 26 June 2009 (UTC)
thread_title:  Weighted Voronoi Diagrams 

message: This article says:
: 
:: Although normal Voronoi cells are defined as the set of points closest to a single point in S, Nth-order Voronoi cells are defined as the set of points closest to n points in S.  Higher-order Voronoi diagrams also subdivide space. 
:: 
:: Higher-order Voronoi diagrams can be generated recursively.  To generate the nth-order Voronoi diagram from set S, start with the (n − 1)th-order diagram and replace each cell generated by X = {x1, x2, ..., xn−1} with a Voronoi diagram generated on the set S − X.
: 
I find this quite unclear.  How do you measure distance to a finite set of points?  Is it the sum of the separate distances?  If so, it should say so, and if not, it should say what it is.
: 
Next, suppose I want to generate the 2nd-order Voronoi diagram.  I have a set S of points and I draw their Voronoi diagram.  I presume that's the "1st-order" Voronoi diagram.  We have n = 2.  So I am told to "start with the 1st-order Voronoi diagram and replace each cell generated by X = {x1} with a Voronoi diagram generated on the set S − X."  Delete points one-at-a-time and draw separate Voronoi diagrams?  And this tesselates space?  This doesn't make sense. Michael Hardy (talk) 11:45, 11 August 2009 (UTC)
thread_title:  Higher-order Voronoi diagrams 

message: The link which was recently added "V*-Diagram: A Highly efficient technique for processing moving nearest neighbor queries" seems to be an author referencing their person research which is related to Voronoi articles but clearly outside the scope of a general article on the subject.  I will remove this link soon unless someone objects. RuppertsAlgorithm (talk) 01:23, 11 September 2010 (UTC)
thread_title:  Recently Added Link 

message: New external link:
* A Voronoi diagram on a sphere
I downloaded this but see no content other than a caption.  How about you?  —Tamfang (talk) 01:47, 12 July 2012 (UTC)
thread_title:  WorldMap.pdf 

message: Interesting maps appeared on Reddits /r/mapporn including two real life maps of the US and world as a Voronoi diagram. They here and here. They should be added ASAP in my opinion rather than duplicating the one you have now. I hope it is changed but I do not know how to do so. 92.23.66.70 (talk) 15:37, 3 July 2013 (UTC)
thread_title:  real life voronoi 

message: The site melbourneschoolzones.com has apparently shut down due to a price increase on the Google Maps API, and the version on archive.org doesn't really work without JS, suggest the link is removed, maybe along with the entire section as it's not very useful without the map.  — Preceding unsigned comment added by 129.194.48.105 (talk) 10:45, 27 January 2020 (UTC)
thread_title:  Dead link to Melbourne school zones 

message: The first paragraph feels kind of dense to me and I was about to edit it, but being new to the editor side of Wikipedia, I was unsure if I should actually submit it. Thoughts? 
"...or in other words, a Voronoi diagram is map shaded to indicate where the nearest significant point is from any location."  — Preceding unsigned comment added by NomadicVoxel (talk • contribs) 07:07, 5 May 2020 (UTC)
thread_title:  Suggestion to simplify the first paragraph 

message: The section Properties contains this passage:
"Assume the setting is the Euclidean plane and a discrete set of points is given. Then two points of the set are adjacent on the convex hull if and only if their Voronoi cells share an infinitely long side."
But what does it mean for two points of the set to be "adjacent on the convex hull" ???
I hope someone knowledgeable about this subject can clarify this for readers. 2601:200:C000:1A0:F447:6226:63A4:B204 (talk) 19:16, 11 May 2022 (UTC)
replies: text: What do you think it means for two vertices of a polygon to be adjacent? —David Eppstein (talk) 19:31, 11 May 2022 (UTC)
thread_title:  What are "adjacent" points on a convex hull?  Clarification is needed 

message: Today, I made an addition to the page that was removed because it cites my article. I believe the suggested modification is interesting and pertinent enough to be included, but I acknowledge my strong bias and so I leave this issue to be resolved by the community. Thanks, Ryan. Ryancots (talk) 15:41, 28 December 2022 (UTC)
replies: text: I have a few concerns besides COI about your proposed edit. WP:PREPRINT for one. Also, this property sounds like it might be a property of any partition into convex sets with marked points, and not a special property of Voronoi partitions. I would also want the property to be described in a style similar to the other properties, and not in the style of a math paper. Eigenbra (talk) 17:45, 28 December 2022 (UTC)
replies: text: Great, thanks for checking out the edit, and thanks for your feedback. Ryancots (talk) 10:29, 29 December 2022 (UTC)
thread_title:  Information about the surface area of cells relevant? 

message: :
Both parts of this statement are clearly true, but the applicability of "since" is not obvious to me. —Tamfang (talk) 03:07, 25 August 2023 (UTC)
replies: text: I agree. Edited to remove that clause entirely. Eigenbra (talk) 13:34, 25 August 2023 (UTC)
thread_title:  section 

message: This article uses an overly abstract definition that is rare in practice (by at least an order of magnitude I would guess) and overly technical for much of the intended audience. I think it should be refocused on the Euclidean plane case, with a more abstract / general definition deferred to a section about generalizations. Let me recommend a first paragraph along the lines of:
: In mathematics, a Voronoi diagram is a partition of the Euclidean plane into regions close to each of a given finite set of points (called seeds, sites, or generators). For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer (at smaller distance) to that seed than to any other. The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation. The concept of a Voronoi diagram has been generalized to use objects other than points as the seeds and to settings of higher dimension or more generally to arbitrary metric spaces with an alternative notion of distance.
This would then simplify much of the later presentation which repeatedly adds "in the Euclidean plane" or the like as a qualifier. –jacobolus (t) 20:10, 3 September 2023 (UTC)
thread_title:  fundamental definition should be points in the Euclidean plane, not arbitrary metric spaces