Talk:Centroid/Archive 1

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Archive 1

Centroid = intersection of bisecting planes (NOT)

In geometry, the centroid or barycenter of an object

X

in

n

-dimensional space is the intersection of all hyperplanes that divide

X

into two halves of equal measure.

This is wrong. The line parallel to one side of a triangle that divides it in half is sqrt(1/2) from the opposite corner, not 2/3, which is where the centroid is. -phma 21:06, 26 Jul 2004 (UTC)

Oops! Thanks... Jorge Stolfi 23:26, 15 September 2004 (UTC)

Centroid of a circle

Does anyone know what the formula for the centroid of a circle is? I tried to derive it, but it didn't work. I can't find it online, either. Can anyone tell me what it is?

--Gscshoyru 17:42, 17 September 2005 (UTC)

Isn't it just the center?.... and if it isn't just use the formula in the article, with the circle centered at the origin. the y value is given by int(1/2*f(x)^2 from a to b) divided by the area. Xunflash 04:28, 30 October 2005 (UTC)

Yes, at the center, compare Point_groups_in_three_dimensions#Center_of_symmetry.--Patrick 11:07, 30 October 2005 (UTC)

Suspected Excessive Promotion of Herve Abdi

An anon at 129.110.8.39 (pc0839.utdallas.edu) added the reference:

==References==

*\{\{cite paper | author=Abdi, H | title = [1] ((2007). Centroid, center of gravity, center of mass, barycenter. In N.J. Salkind (Ed.): Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.| year = 2007 |\}\}

Someone might want to check whether this is a worthwhile addition. Lunch 22:32, 1 September 2006 (UTC)

Another reference to Herve Abdi! Inserted by an anonymous user with ip address 129.110.8.39 which seems to belong to the University of Texas at Dallas. Apparently the only editing activity so far has been to insert excessive references to publications by Herve Abdi (of the University of Texas at Dallas). The effect is that many Wikipedia articles on serious scientific topics currently are citing numerous rather obscure publications by Abdi et al, while ignoring much more influential original publications by others. I think this constitutes an abuse of Wikipedia. As a matter of decency, I suggest to 129.110.8.39 to remove all the inappropriate references in the numerous articles edited by 129.110.8.39 , before others do it. Truecobb 20:20, 14 April 2007 (UTC)

I have waited for several months; now I'll delete the obscure reference. Truecobb 21:23, 15 July 2007 (UTC)

missing pic

I didn't feel comfortable editing this page but felt I should point out that one image on this page does not appear for me.

The image Triangle_centroid_1.svg is on the page but appears not to contain anything. Clicking on the image yields http://en.wikipedia.org/wiki/Image:Triangle_centroid_1.svg which also appears not to contain anything. However, clicking on the image on that page yields a URL which does have a non-empty image http://upload.wikimedia.org/wikipedia/commons/8/85/Triangle_centroid_1.svg

--128.89.80.117 20:00, 29 October 2007 (UTC) Dan B.

I modified the svg file on commons to work better, I hope. It may take a while for the modified image to propagate back here through the various levels of caching we have. —David Eppstein 20:29, 29 October 2007 (UTC)

Proof that the centroid of a triangle divides each median in the ratio 2:1

This proof is not a proof. It is based on the statement that GBOC is a parallelogram, which is not proved.

Paolo.dL (talk) 09:38, 19 February 2008 (UTC)

I've fixed the hole in the proof. Occultations (talk) 02:57, 20 February 2008 (UTC)

Not to disparage the work, but Wikipedia math articles are not supposed to include proofs (except, of course, for articels on theorems and proofs). Perhaqps it can be moved to a textbook wiki?

 === Proof that the centroid of a triangle divides each median in the ratio 2:1 ===
 
 Let the medians AD, BE and CF of the triangle ABC intersect at G, the centroid of the triangle, and let the straight line   AD be extended up to the point O such that 
 : $AG=GO.\,$ 
 
 Then the triangles AGE and AOC are similar (common angle at A, AO is twice AG, AC is twice AE), and so OC is parallel to GE.  But GE is BG extended, and so OC is parallel to BG. Similarly, OB is parallel to CG.
 
 The figure GBOC is therefore a parallelogram. Since the diagonals of a parallelogram bisect one another, the point of intersection D between the diagonals GO and BC is such that GD = DO, and
 
 :  $GO=GD+DO=2GD.\,$ 
 
 So,  $AG=GO=2GD,\,$ 
 
 or  $AG:GD=2:1.\,$ 
 
 This is true for every other median.

--Jorge Stolfi (talk) 03:49, 24 October 2008 (UTC)

Centroid = center of mass?

From WordNet (r) 2.0 (August 2003) [wn]:
  centroid
      n : the center of mass of an object of uniform density

WordNet says, the center of mass is called centroid when the object has uniform density distribution. Is it ture?

Not exactly. When the object has an uniform density distribution, the centre of mass coincides with the object's geometric centre. Yet, the names of the concepts remain the same, which means that a centroid is still a centroid and a centre of mass is still a centre of mass --Maciel 11:35, 25 Sep 2004 (UTC)

I agree with both of you. On a very slightly different note I've never seen a text that equates the physical center of mass with its "centroid", unlike the current version of the introduction: "In physics, the words centroid and barycenter may mean either the center of mass or the center of gravity of an object, depending on context." Weisstein's physicsworld page partially supports this usage, but he cites no sources, and I'm more inclined to think he's just wrong. I'll change the intro to be more conservative. Melchoir 04:06, 19 April 2006 (UTC)

I have restored the more "liberal" physics definition, that admits "centroid"="barycenter" in some cases. While I have no textbook citation to support this alternative, it does seem to occur in many less formal documents found over the net [2] [3] [4]. Methinks that Wikipedia should inform readers of all common names and usages, not just the "officially correct" ones. (On the other hand, if all physics textbooks *do* define "centroid"

\neq

"barycenter", then the wording of the head parag should make it clear that "centroid"="barycenter" is only colloquial.) All the best, --Jorge Stolfi (talk) 15:43, 24 October 2008 (UTC)

Programming the centroid computation

Just a minor edit how to calculate centroid of a triangle using programming. I am sure someone will find it useful. SnegoviK 12:22, 24 February 2006 (UTC)

Good try, but unfortunately that whole section seems quite pointless. The previous section already says that the centroid of a triangle is obtained by averaging the coordinates of the corners; that is all one needs to know in order to solve your programming exercise. Why would the reader want to know other (incorrect!) ways of doing it? Besides, this is a geometry article; computer language details do not belong here.
But please do not get discouraged, surely you will find many other ways to help Wikipedia.
All the best, Jorge Stolfi 17:06, 24 February 2006 (UTC)

I just realised that you are absolutely right! I am sorry I didn't manage to come with a resourceful article, I will try harder next time. Your comment is very helpful to me, thanks. ;) SnegoviK 19:33, 24 February 2006 (UTC)

The formula for finding a polygon's centroid is not precise. It may give a negative where a positive is called for. For a better formula, see: http://tog.acm.org/resources/GraphicsGems/gemsiv/centroid.c — Preceding unsigned comment added by 212.143.119.137 (talk) 10:26, 27 September 2009 (UTC)

Object which is a regular base?

there is no example of median in any object which is a regular base using object. —Preceding unsigned comment added by 59.94.188.39 (talk) 13:55, 4 January 2010 (UTC)
- I cannot understand the preceding comment. Can anyone clarify it? (From the way it was formatted, I suspect that ist was just a test edit.) Thanks, --Jorge Stolfi (talk) 15:36, 4 January 2010 (UTC)

Contradiction in article

The first section distinguishes, clearly, between "centroid" (also called "barycenter" in geometry), and "barycenter" as used in physics. Then the section "Locating the centroid" describes a method which finds not the centroid (as defined at the start of the article), but the physics-style barycenter.

I realise that different people use the term "centroid" in different ways. But the article ought to be consistent. It should not define "centroid" in one way then use it in another. Maproom (talk) 09:32, 9 July 2010 (UTC)

I agree. I've made a change in the "Locating the centroid" section that I hope addresses your concern. I don't think we need to specify a uniform gravitational field do we? The method can be extended, in theory, for any uniform solid, but the practicalities of recording lines through solids are questionable. Dbfirs 15:52, 9 July 2010 (UTC)

Your improvement certainly helps - but I was mistaken, I had missed the word "moment" in the first paragraph of the article. I am considering changing "average" to "mean" in that paragraph - what do you think? Maproom (talk) 16:40, 9 July 2010 (UTC)

Yes, I agree that arithmetic mean is the required average here, but we need to explain in simple words that can be understood by all. How about "average" (arithmetic mean)? Dbfirs 19:42, 9 July 2010 (UTC)

I think that would be an improvement. Maproom (talk) 08:23, 10 July 2010 (UTC)

Of a finite set of points

Casually flipping through, and I noticed that for this equation, there is no mention of what variable K is. Sum of masses? Size of left foot? Poor form, define your variables, people... —Preceding unsigned comment added by 67.40.180.206 (talk) 23:49, 6 August 2010 (UTC)

Anyone familiar with the notation would notice that k is just a count - the number of points. I realise, however, that Wikipedia articles are read by those unfamiliar with mathematics, so I've added the meaning of k. Dbfirs 07:24, 7 August 2010 (UTC)

Centroids in GIS systems

In Geographic information systems, the term centroid may refer to other points than the geometric centre, primarily because of the desire that the centroid is inside the object. See for example [5]. Apus 08:24, 11 October 2006 (UTC)

Different applications will define their own twist on "centre" but they should not use the term "centroid" without qualification for these quirky points. I don't think we need to include every application's modified "centroids" in the general article. Dbfirs 22:25, 20 April 2011 (UTC)

use of "centroid" to denote military test range center

I understand the argument that this is a different application of the term and I agree that perhaps a seperate page could be made for it, but is that really necessary? There are other examples in Wikipedia where a minor-use of a term is denoted on the same page where the majority-use of the term is explained. My understanding is that the test-range use of the term also stems from the maths term itself, so it has the same root. — Preceding unsigned comment added by Oceanbourne (talk • contribs) 07:18, 20 April 2011 (UTC)

WP:NOTDEF (and in particular the "homograph" line in the table of major differences between an encyclopedia and a dictionary) is very clear: when the same word is used to mean different things then they go in different articles. —David Eppstein (talk) 07:48, 20 April 2011 (UTC)

Fine, when I get time I will create a seperate entry for the military use of the term. I feel it deserves a Wikipedia entry and not just a dictionary entry. Mike (talk) 18:30, 20 April 2011 (UTC)

How does the military usage differ from the mathematical meaning (or shall I wait for the article to find out)? We do have a separate article for geographical centre. Dbfirs 08:45, 26 March 2013 (UTC)

Clarification needed: "parts of equal moment"

Does anyone know what the phrase

straight lines that divide X into two parts of equal moment about the line.

in the first sentence of the lede means? I've marked it with a clarification needed tag. Duoduoduo (talk) 20:20, 25 March 2013 (UTC)

It doesn't mean "parts of equal area", which I thought at first, because e.g. for a triangle only three of the infinitude of lines through the centroid divide the area equally (see Bisection#Area bisectors and area-perimeter bisectors of a triangle). Duoduoduo (talk) 22:23, 25 March 2013 (UTC)

The moment of something about a line is the integral, over that something, of its distance from the line. Maproom (talk) 22:25, 25 March 2013 (UTC)

In this case, the "something" is area, but it would be useful to have a simpler explanation (if there is one) for the lead, with the technicalities being explained lower down. We could just say "two parts of equal area moment about the line" or "two parts of equal moment of area about the line" but neither clarifies much. Ought we to include the simple phrase "centre of area" in the lead? Dbfirs 22:47, 25 March 2013 (UTC)

The way I understand "centroid" is "the mean position of all the points in the shape". I find the first sentence of the article quite surprising. I am aware of a theorem that says "a line (or plane, in the case of a three-dimensional thing) divides it into two parts of equal moment about that line (plane) if and only if it passes through its centroid". I guess I could prove it if I had to. But to start the article with that theorem, before the reader has any understanding of what a centroid is, strikes me as unhelpful. Maproom (talk) 07:15, 26 March 2013 (UTC)

I've added an informal introduction to cater for readers without a mathematical background (though some mathematicians might not like the informality). Dbfirs 08:58, 26 March 2013 (UTC)

Several comments:

(1) The article talks about both the centroid of a continuous region and the centroid of a set of discrete points. Is the latter really conventional? If so, the lede ought to reflect that possible setting.

(2) A problem with the newly inserted informal sentence is that, in addition to not allowing for discrete sets of points, it also doesn't allow for three-dimensional objects.

(3) The new sentence uses the phrasing "centroid of an area...a cardboard cut-out of the area could be perfectly balanced". I think this is a misuse of the word "area", which has a technical meaning as a scalar value. I'll change it to "two-dimensional region".

(4) I agree with Maproom -- I think the part about equal moments about a line is a theorem about the centroid, not a definition, and as such should not be in the lede, or at least not before the definition.

I'll take a shot at revising the lede based on these points. Duoduoduo (talk) 14:23, 26 March 2013 (UTC)

Centroid of a finite set of points?

The short section "Of a finite set of points" has been here since 22 November 2007. It was put here by editor Zeroin23a, who only edited Wikipedia from 22-25 November 2007. I don't think I've ever heard of the centroid of a set of points except here, and I suspect that it is Zeroin23a's original research. Any objection to my removing the section? Duoduoduo (talk) 15:18, 26 March 2013 (UTC)

The average of a set of points is an extremely standard concept (one of the most commonly used central tendencies in statistics) and is essentially the same as the centroid of a finite set. Whether it's ever actually called the centroid is a different question, though. —David Eppstein (talk) 15:53, 26 March 2013 (UTC)

Yes, still called "centroid" (here for example). Dbfirs 22:18, 26 March 2013 (UTC)

Also in the Wikipedia article Nine-point circle#Other properties of the Nine-point circle: The nine-point center lies at the centroid of four points comprising the triangle's three vertices and its orthocenter. Duoduoduo (talk) 17:53, 29 March 2013 (UTC)

Smallest circle problem

I was wondering if someone can find a reputable source for the problem of finding the smallest circle which inscribes a polygon in general? I have heuristically worked out a solution for triangles, but that would count as original research, so this cannot go into the article! Just to satiate anyone's curiosity the triangle problem can be solved as follows.

First, determine if 2 or 3 points of the triangle will be on the circle's parameter. Two edges will be on the parameter when the second longest edge is shorter than half the longest edge. This can be clearly seen by selecting the midpoint of the longest edge, producing a circle with radius =1/2 the longest edge. This will inscribe the third vertex since the second longest edge is less than or equal to radius. Otherwise all three vertices will be on the circle's parameter. When that is the case you can analytically solve for some point (x,y) which produces the same distance from all three vertices, provided the triangle is not degenerate. Simply algebraically solve (y-y1)^2+(x-x1)^2=(y-y2)^2+(x-x2)^2=(y-y3)^2+(x-x3)^2 for x and y (although the solution is a bit ugly so I'm not going to post it here). Mouse7mouse9 20:31, 23 April 2013 (UTC)

See Smallest-circle problem. There are known algorithms that solve the problem in linear time for arbitrary polygons. —David Eppstein (talk) 20:38, 23 April 2013 (UTC)

Thank you! Mouse7mouse9 20:44, 23 April 2013 (UTC)

For an acute triangle, all three vertices will lie on the circle. So draw a circle centred on the point where the perpendicular bisectors of the sides meet. For an obtuse triangle, only two vertices will lie on the circle: draw a circle centred on the midpoint of the longest side. Maproom (talk) 21:11, 23 April 2013 (UTC)

Coding for centroid

I have moved the following coding here because it wasn't explained in the article. What language is it?

       public static function getCentroid(vertices:Array):Point
       {
           var i:int, j:int;
           var A:Number = 0;
           var Cx:Number = 0;
           var Cy:Number = 0;
           for(i=0;i<vertices.length;i++){
               j = (i+1)%vertices.length;
               A += (vertices[i].x * vertices[j].y) - (vertices[j].x * vertices[i].y);
               Cx += (vertices[i].x + vertices[j].x)*((vertices[i].x * vertices[j].y) - (vertices[j].x * vertices[i].y));
               Cy += (vertices[i].y + vertices[j].y)*((vertices[i].x * vertices[j].y) - (vertices[j].x * vertices[i].y));
           }
           A /= 2;
           Cx /= (6*A);
           Cy /= (6*A);
           return new Point(Cx,Cy); //it may be placed out of polygon
       }

Any comments about whether it should be included (with explanation and perhaps collapsed)? Dbfirs 17:37, 13 June 2013 (UTC)

I don't see the need for code here at all. There is a simple formula for this that can be expressed much more concisely in mathematical notation than in code, and there is nothing to code beyond implementing the formula. —David Eppstein (talk) 17:45, 13 June 2013 (UTC)

That's what I thought, but I didn't want to discourage the anon editor who added it by just reverting. Dbfirs 17:47, 13 June 2013 (UTC)

'Barycenter' redirects here

But there is no mention of celestial mechanics or orbits in the Centroid article. Lori (talk) 02:48, 15 October 2013 (UTC)

Yes, you are correct; it's a valid criticism. I've now redirected both Barycenter and Barycentre to the more appropriate Center of mass article. Would it be better to redirect to the subsection Center of mass#Astronomy where there is a further link to astronomical applications? Dbfirs 11:18, 15 October 2013 (UTC)