Talk:Barycentric coordinate system

Error in section on special points

I think the sentence

The centroid has barycentrics 1:1:1.[4]

is wrong, since these coordinates do not sum up to one. It should be 1/3:1/3:1/3.

I fixed this.ScientistBuilder (talk) 19:43, 15 June 2022 (UTC)

It's not wrong. From the article: "The use of colons in the notation of the tuple means that barycentric coordinates are a sort of homogeneous coordinates, that is, the point is not changed if all coordinates are multiplied by the same nonzero constant". Most of the other coordinates in that section don't necessarily add up to one; I think they've been chosen to have no denominators or common factors. I changed it back to 1:1:1 to have more consistency in that section. I also added an explicit note that that they're homogeneous so readers can know that they're not necessarily normalized. It seems like the larger section of Barycentric coordinates on triangles could use more clarification like this. FionaLovesCats (talk) 03:34, 5 May 2023 (UTC)

Barycentric coordinates on triangles

I would like to discuss the appropriateness of this example to illustrate the use of barycentric coordinates relative to a triangle. I have been using barycentric coordinates off-and-on for some time, and there are many geometric uses which are much more accessible. Moreover, the use of baycentric coordinates is independent of calculus. I feel the example does not really aid the understanding of barycentric coordinates relative to a triangle.

Converting to barycentric coordinates

The equation concluded in this section is wrong and doesn't satisfy the actual requirement of $0<\lambda _{i}<1\;\forall \;i{\text{ in }}1,2,3$; it satisfies only $0<\lambda _{i}\;\forall \;i{\text{ in }}1,2,3$.

The right equation can be found here (note that the equation has error in calculating $\gamma$ although it can be corrected easily.)

What the equation should've been is like this:

$\lambda _{1}={\frac {(x_{1}-x_{3})(y-y_{3})-(x-x_{3})(y_{1}-y_{3})}{(x_{2}-x_{1})(y_{3}-y_{1})-(x_{3}-x_{1})(y_{2}-y_{1})}}\,,$

$\lambda _{2}={\frac {(x_{2}-x_{1})(y-y_{1})-(x-x_{1})(y_{2}-y_{1})}{(x_{2}-x_{1})(y_{3}-y_{1})-(x_{3}-x_{1})(y_{2}-y_{1})}}\,,$

$\lambda _{3}=1-\lambda _{1}-\lambda _{2}\,.$

(I cannot verify the reliability of the above equation, but my initial tests have shown no issues (I'll cite a verifiable implementation for test somewhere))

AvinashANBhat (talk)

The pdf link you attached is no longer working. ScientistBuilder (talk) 19:46, 15 June 2022 (UTC)

Trilinear coordinates

"Trilinear coordinates" redirects to "barycentric coordinates", but they're not the same thing. Essentially, if I put three masses (m,p,q) at the vertices of a given triangle, the point described by those barycentric coordinates is the center of mass of that system. On the other hand, trilinear coordinates describe the relative distances from the three sides of a given triangle. See for example, the Mathworld page on trilinear coordinates.

Another example: the centroid in barycentric coords is (1,1,1). On the other hand, the incenter in trilinear coordinates is (1,1,1). Lunch 23:22, 1 September 2006 (UTC)

Errr, nevermind. I changed the redirect myself. But there's still a stub there if anyone wants to embellish it... Lunch 23:48, 1 September 2006 (UTC)

Affine coordinates

"Affine Coordinates" redirects to this page but do not appear in the article. --91.23.217.47 11:58, 2 November 2007 (UTC)

reliability

I'm interested in the reliability of barycentric coordinates, both the two variable and three variable forms and alternatives when applied to texture mapping for example. Does the accuracy fall for extremely narrow triangles? 68.144.80.168 (talk) 05:14, 10 July 2008 (UTC)

In accurate, theoretic math no, it doesn't. In the real world of approximated floating-point math, obviously yes, as FP can only work on a fixed footprint to store values. As the triangle approaches degeneration, that is det(A) appoaches 0 this algo will be less and less tolerant to error. 82.88.245.35 (talk) 10:12, 29 June 2010 (UTC)

Generalised Barycentric Coordinates

This section states:

More abstractly, generalized barycentric coordinates express a polytope with n vertices, regardless of dimension, as the image of the standard $(n-1)$-simplex, which has n vertices – the map is onto: $\Delta ^{n-1}\twoheadrightarrow P.$

However, it seems to me that this map is onto the convex hull of the polytope. Even the example given of the quadrilateral fails for the non-convex case. Zteve (talk) 11:34, 17 November 2010 (UTC)

suggested move

Does the article title really need a disambiguator? —Tamfang (talk) 04:12, 28 June 2011 (UTC)

Hi! I just wanted to say that in section 'Barycentric coordinates on triangles' points ABC and P are extensively referenced to explain something, but that these points are not drawn on any schematic. This leads to understanding the text to be a piece of guesswork at best... Thanks! 145.53.101.116 (talk) 09:36, 17 October 2013 (UTC)

Criticism

Like so many math articles on Wikipedia written by putative experts, this one is heavy on notation and extraordinarily light on conceptual explanation. I submit that the author, while handy with the formalisms, possesses no actual UNDERSTANDING of the material such as can be conveyed to another intelligent but as-yet unaware individual. The whole thing just seems so overwhelmingly self-serving, festooned with largely useless (as Professor Paul Bailyn at The Cooper Union would derisively have termed it) "alphabet soup." And that's a crying shame.

73.49.1.133 (talk) 17:48, 3 September 2014 (UTC)

Please comment on content, not on contributors. If you see something that you think could be improved, you can either just change it yourself, or suggest the changes here. It might help to review Wikipedia_talk:WikiProject_Mathematics/FAQ where points like this are covered. Deltahedron (talk) 18:12, 3 September 2014 (UTC)

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More images would clear up derivation

Specifically, in the section describing the coordinates for triangles, it's difficult to tell what subtriangles and vectors and such the author is describing, especially considering they're described with elements the author just introduced.

104.7.12.222 (talk) 17:34, 18 January 2017 (UTC)

Lead

@D.Lazard: It is a good idea that You moved the general definition on the top. But I regret that You removed my remarks on the geometric meanings. They were essential findings of Möbius and are essential properties for its usage in triangle geometry. I think the new comment on CAD should be replaced/added. Barycentric coordinates are used for the generation of a special kind of Bezier surfaces, triangular BS, which are rather different from the "usual" ones (tensor product BS). So: triangular should appear. I think,from the present lead a non-mathematician gets no idea of the advantages of barycentric coordinates.--Ag2gaeh (talk) 16:06, 7 September 2020 (UTC)

AS you can read, my edit summary contains "rm ... details that clearly do not belong to the lead". This does not mean that the removed text does not belong to the article. On the opposite, the removed text about triangle geometry must be added to the section "On a plane", when the mess of this section will be clarified. A section on "triangular BS" deserves to be added to the "Applications" section. However, for mentioning "triangular Bézier surfaces" in the lead, would require to have a wikilink for them, which is not the case. Nevertheless, I agree that my formulation is not correct, and I'll replace "understanding and manipulating Bézier surfaces" by "defining some sorts of Bézier surfaces". D.Lazard (talk) 17:12, 7 September 2020 (UTC)

I am astonished reading Your rating of my efforts improving the article a "mess". Collaboration looks in a different way.--Ag2gaeh (talk) 20:40, 7 September 2020 (UTC)

My contributions

@D.Lazard: OK, You are writing for pure mathematicians, I wrote my contribution along Möbius' ideas, which may not fit into Your idea of the article. By the way, the physics needed for understanding of my contributions is highschool level. In order to clear the way for a version You have in mind, I shall remove my contributions. Go ahead ! --Ag2gaeh (talk) 12:15, 13 September 2020 (UTC)

The Wikipedia articles must be written for the widest possible audience. This includes people who have had US highschool courses, but also people who have forgot these courses and people from other countries that have never learnt the law of the lever (note that it is referred to in the article, but not linked). I am not against developping the relationship with physics, but this must not be a prerequesty. This must appear after the mathematical computations, as an explanation of their meaning. This would allows things to remain understandable for people ignoring physics. For everybody, it is useful to distinguish what is physics, and what is mathematics. In particular, using physics terminology in what is presented as a mathematical proof is definitively confusing.

This being said, I have tagged these sections because I think that they deserve to be improved, but I will not edit them soon, because I am working on articles that have a much larger audience. Go ahead yourself ! D.Lazard (talk) 13:32, 13 September 2020 (UTC)

Cramers Rule

I think the sentence

The area interpretation of the barycentric coordinates can be recovered by applying Cramer's rule to this linear system.

in the section on converting from cartesian coordinates could be expanded to explain how the area interpretation of barycentric coordinates can be recovered. ScientistBuilder (talk) 18:32, 15 June 2022 (UTC)

Typo in definition

The first sentence of the definition currently goes

Let $A_{0},\ldots ,A_{n}$ be n + 1 points in a Euclidean space, a flat or an affine space $\mathbf {A}$ of dimension n that are affinely independent; this means that there is no affine subspace of dimension n that contains all the points, or, equivalently that the points define a simplex.

The first remark on the meaning of affine independence, "this means that there is no affine subspace of dimension n that contains all the points", is false.

First, it's impossible for any set of points to satisfy this condition. $\mathbf {A}$, an n-dimensional affine space, has one affine subspace of dimension n, namely itself. This subspace contains any set of points contained in $\mathbf {A}$.

Here's a source cited in the affine space wiki which gives a correct definition of affine independence:

• Reventós Tarrida, Agustí (2011), "Affine spaces", Affine Maps, Euclidean Motions and Quadrics, Springer, ISBN 978-0-85729-709-9

It can be found in section 1.8 (Linear Varieties Generated by Points). It says that a set of n + 1 points are affinely independent when the dimension of the smallest affine subspace which contains them is n.

To bring the false remark in line with this definition, we can just subtract one from the n which appears there: "this means that there is no affine subspace of dimension n - 1 that contains all the points".

Seems like someone just made a typo at some point. I tried to fix this but it was undone...

- FionaLovesCats (talk) 01:26, 27 April 2023 (UTC)

Thanks, I was mistaken the first time. It might help to add that reference there. This page needs more inline citations in general. Apocheir (talk) 02:49, 27 April 2023 (UTC)

Good idea. I'm a little hesitant to add a new source to the page before I check out the ones that are already here some more. If it was a typo then it might have been copied from one of them.

It might be good to change the phrasing to more closely match that in the wiki for affine spaces. Also, it's tempting to shorten "affine subspace of dimension n - 1" to hyperplane. FionaLovesCats (talk) 03:18, 27 April 2023 (UTC)

Special points in some special spaces

I've just put through an edit to the Examples of special points section, which is a part of the Barycentric coordinates on triangles section. I'd like to note that:

- I changed the names $A$, $B$, $C$ for the angles of the triangle $ABC$ to $\alpha$, $\beta$, $\gamma$ in order to avoid confusion with the points of the triangle. This is in line with the notation on a law of sines diagram, which might be a useful graphic to borrow.

- I separated the special points which don't depend on length/angle (vertices and centroid) from those which do (the rest of them). This is to be clear that in a space which has no notion of length/angle, the coordinates of the former points are still valid (mostly, see next note).

- The centroid of three points in an affine space over a field of characteristic three (e.g. of three points in the Hesse configuration) does not exist. It's one of those points at infinity like is mentioned in the Definition section. I didn't add anything about this since I was mostly trying to simplify the section, but I think it's worth carefully mentioning for readers who care. FionaLovesCats (talk) 03:32, 5 May 2023 (UTC)

Here's a citation for my third claim:

"Given $m$ points $a_{1},\ldots ,a_{m}$ of $E$, whose number $m$ is not a multiple of the characteristic of $K$ (V, §1), the point $g=\sum _{i=1}^{m}{\frac {1}{m}}a_{i}$ is called (by an abuse of language) the barycentre of the points $a_{i}$ $(1\leq i\leq m)$ ..."

Bourbaki, Nicolas (1974). Algebra. Paris: Hermann, Publishers in arts and science. p. 326. ISBN 2705656758.

Here $E$ is an affine space over a field $K$. This is a special case of what the text calls the "barycentre of the points $x_{i}$ given the masses $\lambda _{i}$". FionaLovesCats (talk) 19:06, 5 May 2023 (UTC)

Triangle Circumcenter Errata:

This is an excellent and helpful article, but there is an error in the Barycentric Coordinates given for triangle circumcenters:

        Sin2α     :     sin2β      :     sin2γ

=      1-cos β cos γ            :        1-cos γ cos α            :        1-cos α cos β

=     a²(-a² + b² + c²)        :       b²(a² - b² + c²)          :      c²(a² + b² - c²)

The first and third expressions are indisputable, but the middle expression is clearly incorrect. For example, the triangle {a, b, c} = {4, 3, 2} yields the following data:

       cos {α, β, γ} = {-1/4, 11/16, 7/8}

       1 – cos {β, γ, α} cos {γ, α, β} = {51/128, 39/32, 75/64} --> (17 : 52 : 50)

       sin 2{α, β, γ} = {-√15/8, 33√15/128, 7√15/32} --> (-16 : 33 : 28)

       {a2(-a2 + b2 + c2), …} = {-48, 99, 84} --> (-16 : 33 : 28)

The (-16 : 33 : 28) point can be verified as circumcenter by any number of methods, so the (17 : 52 : 50) point can’t possibly be the circumcenter. I find the 1 – cos {β, γ, α} cos {γ, α, β} point does not pass any of the tests for circumcenter. It is not on the Euler line, not on any of the triangle edge perpendicular bisectors, and not a distance of R from any triangle vertex (R = circumradius = a/2sin α, typically). I don’t know what the errant point represents.

As a cosine equivalent to the sine expression for circumcenter, I would recommend:

               (a cos α : b cos β : c cos γ).

This is based on “sin 2α = 2sin α cos α” and “sin {α, β, γ} = {a, b, c}/2R”.

I don’t know the original author’s intent here. The errant expression seems to be a typographical misplacement of some sort. It can be simply deleted or replaced by some other helpful expression. JFSather (talk) 19:15, 2 April 2024 (UTC)