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Questions tagged [barycentric-coordinates]

This tag is for questions relating to Barycentric coordinate systems which describe the place of certain points in a triangle. They do not use distances of points, but only ratios of segments. Thus, they belong to the geometry of the affine plane, which deals with parallels and ratios of collinear segments without to use a notion of distance between two points, as it does euclidean geometry. The system was introduced in 1827 by August Ferdinand Möbius.

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Points inside a triangle.

I am trying to solve a problem where we basically need the points inside a triangle . I found a formula ( not sure if this is barycentric coordinate one or not): $$(x_{1},x_{2})=(1-\sqrt{r_{1}})A+\...
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Intersection of lines in barycentric coordinates

Consider a barycentric coordinate system with points $v_0,v_1,v_2$ and $x=\alpha_1 v_1 + \alpha_2 v_2$ and $y=\beta_0 v_0 + \beta_2 v_2$. Find the intersection $s$ of the lines through $x$ and $y$ and ...
Magne Seier's user avatar
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Given a point $P$ in regular tetrahedron $T$, is there a triangle with vertices on the boundary of $T$ whose center of mass is $P$?

Here's a "simple'' problem in basic geometry. Given is a regular tetrahedron $T$. Consider the Barycentric coordinate system on $T$. Given is also a point $P\in T$ in the interior of $T$ such ...
Juergen's user avatar
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Geometric intuition: Barycentric subdivision homotopic to identity

My question (see below for the definitions used): Barycentric subdivision $B$ should not change the homology. In other words, we want to show that the chain map $B$ is a chain homotopy equivalence (...
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Concurrency point is centroid through barycentric coordinates

Consider a point $X$ inside $\triangle ABC$, and construct $AD, BE, CF$ passing through $X$ and ending on the opposite sides. Let these be vectors. Prove that if $\vec{AD}+\vec{BE}+\vec{CF}$ is equal ...
oiuio's user avatar
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Simplifying polynomials in barycentric coordinates

Polynomials defined within a simplex can be written in terms of barycentric coordinates. Such expressions are not unique because the coordinates add up to unity. I am deriving basis functions for high-...
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How to show that the 3 common definitions/ideas of descriptions of barycentric coordinates are equivalent?

Is there some paper or do you know how to show that the 3 typical definitions for barycentric coordinates are equivalent? I only found papers stating these definitions, not proofing their equivalence....
spectre42's user avatar
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Explanation of Barycentric coordinates

Let $ABC$ be a triangle in the plane $\mathbb{R}^2$ (in other words, the points $A,B,C$ are affinely independent). Let $AA_1, BB_1, CC_1$ be the medians of this triangle meeting at the point $M$. Find ...
End points's user avatar
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Prove every point on the plane is a unique affine combination of the vertices of any triangle

Given three non-colinear points on the plane, prove that any point on the plane can be uniquely represented as an affine combination of them (this is barycentric coordinates). My proof is below. ...
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Expressing barycentric coordinates as functions after series of geometric constructions

Given a triangle $ABC$ with side lengths $AB=c, AC=b, BC=a$. After some geometric constructions to create a point $X$, can we always calculate the barycentric coordinates of $X$ as a triplet of ...
RopuToran's user avatar
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Calculating the barycentric coordinates for a point based on edge lengths alone

I am trying to calculate the barycentric weights of a triangle from which I have no Cartesian coordinates- only a set of edge lengths describing the configuration of a flattened tetrahedron with ...
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Recovering the third internal barycentric distance from a 3-point triangle and two internal distances

I am trying to solve the following problem: Essentially, I have three points that represent a triangle embedded on a 2D plane, and a pair of scalar distances that describe the distance from two of ...
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Barycentric coordinates of the origin point of a quadrilateral [closed]

I have a quadrilateral formed by the points A, B, C, D. I want to get the barycentric coordinates u, v, w, x so that: Au + Bv + Cw + Dx = [0, 0] How do I find the values of u, v, w, x knowing the ...
newbye's user avatar
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Convert to-and-from a local coordinate system relative to a n-sided polygon

Question Let $P$ be a convex, regular $N$-sided 2D polygon; take for instance, a pentagon. Let $T$ be a point located within $P$, or alongside any of its sides. Is it possible to convert $T$'s ...
Jebyte's user avatar
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Barycentric coordinates on an affine space

In the book 'Geometry of Quantum States' the author states that in an $n$-dimensional affine space, select $n+1$ points $x_i$ so that an arbitrary point $x$ can be written as: $x=\mu_0x_0+\mu_1x_1+.......
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"Clamp" barycentric coordinates?

Let's begin with something we know how to do. Given a point in 2D, one can project it to a segment by projecting the point onto the line containing the segment. You do this by doing $s = (p - o)\cdot ...
Makogan's user avatar
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Find the Degree matrix $D$ using the Adjacency matrix $A$ and (half of) the coordinates of a graph.

Say, we have a graph $G$ with 6 points of which 3 are given. The coordinates of the other 3 points are unknown at this point. I got the (full) adjacency matrix and the 3 coordinates. How do I ...
Gerald's user avatar
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deriving an explicit formula for barycentric subdivision of simplices

I am trying to prove the explicit, non-recursive formula for the subdivision of an n-simplex, $\operatorname{Sd}_{n}:S_n(\triangle^{n}_{top}) \to S_n(\triangle^{n}_{top})$. I found the exercise in ...
Paul Joh's user avatar
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On the Elements of Coordinate Geometry

During my read through the Elements of Coordinate Geometry by S.L Lonely, I've stumbled upon his method of proving that one point is either on one side or the other of the graph of a straight line ...
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Given point/value pairs for the vertices of a tetrahedron, how do you get values for other points inside the tetrahedron?

We will look at the 3 dimensional space $\mathbb{R}^3$. Say we have a tetrahedron with the vertices $p_1$, $p_2$, $p_3$, and $p_4$, each with corresponding value $v_1$, $v_2$, $v_3$, and $v_4$. Given ...
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If two coordinate axes are scaled differently, do all the formulas still work?

If I scale the two coordinate axes differently (e.g: two units of length to represent one unit on the y-axis whule 5 units of length to represent one unit on the x-axis), would all the formulas ...
Camelot823's user avatar
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Can the coordinate axes be scaled differently in pure math?

My question is straightforward. Can the coordinate axes, while being measure by the same unit, be scaled differently (in pure math). Thanks in advance.
Camelot823's user avatar
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Test if point in 3D is above triangle using vertex normals

Please, my problem is very close to Find a point on triangle and interpolated triangle normal which points to specific point in 3D, but I do not need to find analytical/numerical solution. I rather ...
Michal Wirth's user avatar
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1 answer
144 views

Deriving barycentric coordinates of the isotomic conjugate

Consider a triangle $ABC$ and a point $P=(x:y:z)$(in Barycentric coordinates). If $P^t$ is the isotomic conjugate of $P$ prove that $$P^t=\left(\frac{1}{x}:\frac{1}{y}:\frac{1}{z}\right)$$ Note that $$...
PNT's user avatar
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Expressing a 'zero-sum' ratio as a point in space? ( Eg. $1:-9:8$ )

I have a collection of ratios (they are all the same degree) where the sum of their parts equate to $0$; and I need a way to represent these ratios as points in space (to perform k-means clustering on ...
Triangler's user avatar
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51 views

Barycentric coordinates with respect to non affinely independent points

Let $\mathbf p_1, \ldots,\mathbf p_n$ be points in $\Re^d$ and let $k$ be the maximum number of affinely independent points among them. Let $\{I_1,\ldots,I_m\}$ be the set of all sets $I_i=\{j_1,\...
Massimiliano Pavan's user avatar
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Reduce 4 barycentric coordinates to 3, outside and inside of tetrahedron

For any point inside of a tetrahedron, it is possible to describe it in 3 barycentric parameters (the 4th can be deduced from the other 3). Is this reduction from 4 to 3 coordinates also possible for ...
gandreadis's user avatar
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Points collinear with the orthocenter and incenter

Let $ABC$ be a triangle, $H$ be its orthocenter and $I$ its incenter. What properties does the line $HI$ have? What known points in the triangle belong to it? What is the barycentric equation of the ...
user986772's user avatar
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Complex number / bary geometric problem in a cyclic quadrilateral

Let $ABCD$ be a convex cyclic quadrilateral, such as triangles $BCD$ and $CDA$ are not equilateral. Prove that, if $A$-Simson line is perpendicular to Euler line in $BCD$, $B$-Simson line is ...
MathStackExchange's user avatar
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1 answer
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Help understanding the role of barycentric coordinates in affine mapping of reference element to actual element

I am slogging through FEM self study leaning heavily on the 1st Zienkiewicz text. I think the book is great but I cannot seem wrap my head around switching between the element of interest and the ...
TheCodeNovice's user avatar
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Generalized barycentric coordinates using only the H-representation (i.e., only facets, not vertices)

I have a polytope in $\mathbf{R}^n$ given by the inequality $\boldsymbol{A}\boldsymbol{x}\le \boldsymbol{b}$, where $\boldsymbol{x}\in\mathbf{R}^n$, and $\boldsymbol{A}$ is a rectangular matrix. I was ...
Lab's user avatar
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How to to find coordinates of the center of a triangle in a $3$-d environment if the $x, y, z$ vertexes are known?

Data provided: $x, y, z$ coordinates for 3 points in space (it's 3 stars in the solar system). I have the stars coordinates from the Galactic coordinates system and basically I want to find the $x,y,z$...
Odin Oji's user avatar
2 votes
2 answers
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How to calculate (a physical) ratio of colors to achieve a target color?

Sorry in advance about my way of expressing this - I am not really a math person to be fair. However, I have been pretty obsessed with an idea I got and that idea requires me to understand something ...
Karim Loberg's user avatar
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1 answer
415 views

uniqueness of barycentric coordinates of a simplex

Let $k,d\in\mathbb N$, $p_0,\ldots,p_k\in\mathbb R^d$ be affinely independent and $$\Delta:=\left\{\sum_{i=0}^k\lambda_ip_i:\lambda_0,\ldots,\lambda_k\ge0\text{ and }\sum_{i=0}^k\lambda_i=1\right\}.$$ ...
0xbadf00d's user avatar
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barycentric coordinates of the foot of an altitude in a tetrahedron

Given three sides $a=BC$, $b=CA$, $c=AB$ of a triangle $ABC$. The foot $H$ on the segment $BC$ of the altitude $AH$ can be computed via (its proof is easy, using the Law of Cosine, for example) $$H=\...
Black Mild's user avatar
2 votes
1 answer
178 views

Doubts about barycentric coordinates

I started studying barycentric coordinates and I have some questions. In my book states, word to word : Consider $\triangle{ABC}$ and a point $P\in\mathcal{P}$. For any point $M\in\mathcal{P}$ there ...
Neox's user avatar
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After finding barycentric coordinates in triangle - what's the actual check to know if point(x,y) is inside?

I read about barycentric coordinates and how to find out if a point is inside a triangle or not. point is P and the barycentric coordinates are u,v and triangle points are ABC: P = A + u(C−A) + v(B−A) ...
taraz's user avatar
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Barycentric coordinates.

Let a,b,c$\ne0$. Show that the set of all points whose barycentric coordinates λ,μ,ν that satisfy aλ+bμ+cν=0 is a line L is line in the plane iff L=p+[v] where v is direction vector If we have three ...
Abedalkareem Othman's user avatar
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121 views

Measurability of the Barycenter Map

Let $V$ be a Banach space considered with its Borel $\sigma$-algebra $\mathcal{B}(V).$ Let $V^*$the dual of $V.$ A probability measure $\mu$ on $(V, \mathcal{B}(V))$ is said to have a barycenter if ...
vekin pirna's user avatar
1 vote
0 answers
348 views

Converting from a generalized barycentric coordinate back to Cartesian coordinate

The barycentric coordinate system of a simplex satisfies the following equation: $$\Lambda \cdot V = \begin{bmatrix} \lambda_1 & \lambda_2 & \lambda_3 \end{bmatrix} \cdot \begin{bmatrix} u_1 &...
Edward Cui's user avatar
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Can you help me understand barycentric coordinates visually?

This is my first question here so I'm sorry if it is not a well written/formatted one. I love maths but I have always had a lot of trouble following long equations. It always helps me when visuals are ...
Tree3708's user avatar
1 vote
1 answer
54 views

in barycentric coordinates why does $[PBC] = x[ABC]$?

From the Euclidean Geometry in Mathematical Olympiads written by Even Chan, there is a chapter about barycentric coordinates. It is said in the chapter that Barycentric coordinates are also sometimes ...
Y.T.'s user avatar
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1 vote
1 answer
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Calculation of simplex coordinates that are bound by vectors and the simplex has to contain a target coordinate

Notation: $p_{i, j, k}$; the $k^{th}$ component of the $j^{th}$ boundary coordinate on the $i^{th}$ boundary Finding a weight in 2D I have 2 sets of 2 boundary coordinates in $\mathbb{R}^2$ space. ...
WillemRoos's user avatar
2 votes
1 answer
109 views

Parametrising $\mathbb{R^3}$ coordinates using a mesh with barycentric coordinates, and signed distance to the mesh

I need to find a way to parametrise $\mathbb{R^3}$ coordinates relative to a mesh, which is a deformed $UV$ sphere with triangular faces. Every vertex on the mesh has a $UV$ coordinate, and I want to ...
EBartrum's user avatar
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Property of triangle center when barycentric coordinates are given

Trying to learn.. what may be interesting geometric properties of associated center of a triangle (sides $a,b,c$) whose barycentric coordinates are: $$\left(a^2+b^2-c^2, b^2+c^2-a^2, c^2+a^2-b^2 \...
Narasimham's user avatar
3 votes
1 answer
784 views

point of intersection of two lines in barycentric coordinate system

I am looking for an efficient way to determine the intersection point of two lines which go through a triangle (face) of a 3D triangular surface mesh. For both lines I know the two points at which ...
Fab's user avatar
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Calculate barycentric coordinates of a point with respect to a 3D triangle using matrix

This paper shows an elegant way to calculate barycentric coordinates ($w_1$ : $w_2$ : $w_3$) of a point $x_4$ with respect to a 3D triangle formed by three points $x_1$, $x_2$ and $x_3$ (given $\vec{x}...
Nguyễn Đức Long's user avatar
1 vote
1 answer
212 views

How can I describe a triangle or tetrahedron with barycentric coordinates?

How would I, for example, describe a triangle given by the vertices $(2,1),(4,2),(2.5,2)$ with barycentric coordinates? My guess would be $(a:b:1-a-b)$ where $0\leq a,b\leq 0.75$ and $a+b\leq 0.75$ ...
Analysis's user avatar
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Integration over a triangular region using barycentric coordinantes?

Example problem: Compute $\int _A 3x^2+y \, \mathrm{d}A$, where $A$ is a triangle with vertices $(0,0),(2,0),(1,1)$. Can this be computed using barycentric coordinates? I found a Wikipedia article on ...
Analysis's user avatar
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Transform line equation with barycentric coordinates

From some testing I have done with Desmos, it appears that straight lines, when every point on that line is transformed using barycentric coordinates from one triangle to another, will stay straight. ...
LordQuaggan's user avatar