Questions tagged [barycentric-coordinates]

Barycentric coordinates describe the place of certain points in a triangle.

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Given two cevians in a tetrahedron, how do I know if they intersect?

Given two cevians in the format of barycentric coordinates on the surface triangle of a n-scale tetrahedron. Next multiplying each cevian's barycentric coordinate by the lcm (value for two cevians' ...
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29 views

Barycentric subdivision and chain maps

In Rotman's An Introduction to Algebraic Topology, 4th printing, pp. 113, he provided a definition of barycentric subdivision in a convex set $E$: Let $E$ be a convex set. Then barycentric ...
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Formula for Area of a Triangle - nodal basis function

Let T be a triangle with corners $P_1, P_2, P_3$ and the nodal basis function $\lambda_1, \lambda_2, \lambda_3$ and $\alpha, \beta, \in \mathbb{N}_0$. I want to show that $$ \int_{T}^{} \lambda_1^\...
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Given area of three points on a plane and $P$ is a point inside a triangle, find $\vec{OP}$

I am working on my scholarship exam practice (high school/pre-university level) and stuck at question (2) below. Let $A, B, C$ be three points on a plane and $O$ be the origin point on this plane. ...
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Is it possible to recover a simplicial complex from its 1-skeleton via its barycentric subdivision?

As in the title, suppose I start with a simplicial complex, perform a barycentric subdivision once, compute its 1-skeleton, then hand it to you, and you are told about this procedure. Can you recover ...
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Clarification on Barycentric coordinates.

I know how to work on triangles a bit by assigning barycentric coordinates $(1,0,0),\ (0,1,0),$ and $(0,0,1)$ to vertices $A,\ B,$ and $C$ resepectively. Let's say that triangle $ABC$ is equilateral, ...
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I need a book recommendation on affine geometry, affine spaces, affine functions!

I am first year university student and I have four math, two informatics courses. I am learning Number Theory, Algebra, Analysis, and .. Geometry. First year Geometry was okay (2-d,3-d). My teacher ...
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Perpendicularity and Parallelism in Barycentric Coordinates

Given the barycentric equation of a line $l$ and barycentric coordinates of any point $P$ in the plane, how does one find the equation of the line passing through $P$ and (i) perpendicular to $l$? (ii)...
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163 views

Algorithms for projecting a point onto the convex hull spanned by a set of vectors

Given a set of vectors $V = \{ \mathbf{v}_1, \ldots, \mathbf{v}_n \} \subset \mathbb{R}^d$, I want to project a point $\mathbf{x}_0 \in \mathbb{R}^d$ onto the convex hull $\text{conv}(V)$ of the ...
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Calculating the barycentric coordinates for a point in a triangle?

How would I find the corresponding $\alpha$, $\beta$, and $\gamma$ for the point $p$ with respect to the triangle with vertices $a$, $b$, and $c$?
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Fastest way to find the abciss of the barycenter of a given trapezoid

I'm a student programmer and I need, given a trapeze of height 1 and given the abcisses of its vertices, to find the abcisse of its barycenter. How can I do this the fastest way ? Thank you
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93 views

The Foot of a Perpendicular from a Point to a Line

I have a triangle $ABC$ and a point $P$ with barycentric coordinates $(\alpha,\beta,\gamma)$ that I want to know the barycentric coordinates of the foot of a perpendicular from P to a line $l: ux + vy ...
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241 views

Prove triangle area formula for barycentric coordinates

Let $P_1, P_2, P_3$ be points with barycentric coordinates (with reference triangle $ABC$) $P_i = (u_i, v_i, w_i )$ for $i = 1, 2, 3$. Then the signed area of $\Delta P_1P_2P_3$ is given by the ...
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Multiple integral equation with Barycentric coordinate

$\Delta(i, j, k)$ is a triangle with vertex $i(x_i, y_i), j(x_j, y_j), k(x_k, y_k)$ in counterclockwise, $P(x, y)$ is a point in $\Delta(i, j, k)$, $S, S_i, S_j, S_k$ is the area of $\Delta(i, j, k), \...
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49 views

common point on the three lines…

Actually, I am dealing with a problem in barycentric coordinates I got the equations of three lines as I know that these three lines sharing a common point, I know if I prove their det is zero, they ...
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83 views

Reflection of point in barycentric coordinates

I have a triangle $ABC$ and a point $P$ with barycentric coordinates ($\alpha, \beta, \gamma)$ that I want to reflect about the sides $a,b$ and $c$. Calculating the general expression for a ...
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How to determine linear terms from the nonlinear dataset?

Let us take the parametric curve r($t$) = [$t^2$;$t$], $t$ = [0,1]. Using this equation, I generate 1000 points. Now my goal is to determine the value of $t$ for each point on the curve without using ...
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92 views

Volume and barycentric coordinates of $k$-simplex in $\Bbb{R}^{n}$

How can the volume and barycentric coordinates (aka area/triangular coordinates) of a $k$-simplex in $\Bbb{R}^{n}$ be calculated given the vertices? In general $k \le n$ but any special cases for $k=n$...
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76 views

how to use barycentric coordinates for polygons

suppose we want to deal with the problems with a sided polygons then how can we proceed in barycentric coordinates, that is how can we fame the coordinates of the hexagon in barycentric coordinates.
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81 views

Barycentric subdivision in convex sets: is Rotman's definition incorrect?

In Rotman's An Introduction to Algebraic Topology, 4th printing, pp. 113, he provided a definition of barycentric subdivision in a convex set $E$: Let $E$ be a convex set. Then barycentric ...
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0answers
46 views

Simultaneous movement toward barycenters - what can be guaranteed

Suppose a tiling is given in 2D (an embedding of a planar triangulated graph), with all faces convex. Now suppose one moves each point, one by one, to the barycenter of its neighbors. I think that ...
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170 views

Find the midpoint of two points in barycentric coordinates. [closed]

Though it is a very bad question for this site,I don't know the answer. So please help me. What is the midpoints of the points $(a:b:c)$ and $(x:y:z)$ in barycentric coordinates?
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55 views

Masspoints: The method of weights — Why does it work?

Mass point geometry involves systematically assigning 'weights' to points using ratios of lengths relating vertices, which can then be used to deduce other lengths, using the fact that the lengths ...
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192 views

How would one set up barycentric coordinates for a trapezoid?

Barycentric coordinates are great for triangles, but I'm interested in how to construct a barycentric coordinate system for an arbitrary trapezoid. I've seen this done for an arbitrary quadrilateral,...
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126 views

Intersection of the circles of Apollonius

Given triangle $\triangle ABC$. Consider an Apollonius circle $\omega_1$ of segment $AB$ with ratio $BC:CA$, i.e. for all points $X$ of this circle, we obtain: $\frac{XA}{XB}=\frac{BC}{CA}$. We define ...
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2answers
216 views

trilateration with dimensionless distances

I am trying to figure out how to calculate the coordinate of a point P using the coordinates of three nearby points (A, B and C). The only problem is that I don't know the actual distances to P, only ...
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1answer
220 views

Proof that the barycentric subdivision of a simplicial complex decreases the diameter of its simplexes.

I need help with a result about barycentric subdivision someone already asked about here. However, the part I've been struggling with is the one the original poster saw as trivial, statement (c). The ...
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68 views

How to find a triangle vertices given a barycenter and a normal

So, I'd need to retrieve the vertices of a triangle, given a point (let's assume being the barycenter) and a normal on that point. This is needed because then I have to find all the intersection ...
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1answer
150 views

How do you prove that an image preserving barycentric coordinates w.r.t two triangles is an affine transformation?

Given two triangles $ABC$ and $A'B'C'$ in $\mathbb A^2$. Define a map $F: \mathbb A^2 \to \mathbb A^2$ as follows: $F(P) = Q$ iff the barycentric coordinates of P w.r.t $A$, $B$ and $C$ are the same ...
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342 views

Clarificaiton on barycentric coordinates

This is related to ray tracing (which I learnt and then forgot). Given a triangle in 3D $\widehat{ABC}$, where $A,B,C$ are the points of the triangle And a parametric line described by $(O,\vec v)$ ...
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1answer
182 views

Decompose a cube into tetrahedra (more than one way)?

This must be a standard exercise but I have a shape $[0,1]^3$ and I must express it has the union of tetrahedra joined at the faces. The vertices and edges are clear: $V = \{ 0, 1 \} \times \{ 0, 1 \...
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209 views

Derive Barycentric coordinate distance formula

please pardon the poor formatting. (I'll work on learning it in time; I just started this account to see help with this question.) I've recently started learning about affine geometry and Barycentric ...
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46 views

Nine point center in barycentric coordinates

Is there a short way to compute the barycentric coordinates of the nine point center: $N=(a \cos (B-C): b \cos (C-A) : c \cos(B-A) )$ When I try I get lost in calculations..
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1answer
116 views

A problem of collinearity - is it appropriate to use barycentric coordinates?

Triangle $ABC$ is inscribed in circle $\omega$. Point $P$ lies on line $BC$ such that line $\overline{PA}$ is tangent to $\omega$. The bisector of $\angle APB$ meets segments $\overline{AB}$ and $\...
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21 views

Direction of position vectors

I've been trying to learn Barycentric coordinates and came across the following definition: I am confused if $\vec A, \vec B, \vec C$ are denoted as $\vec {PA}, \vec {PB}, \vec {PC}$ respectively or ...
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Build a positive Cartesian coordinate system in the space so that the $Oz$ axis is orthogonal to the unit triangle and goes through the centroid

Build a positive Cartesian coordinate system in the space so that the $Oz$ axis is orthogonal to the triangle defined by the vertices $(1,0,0),(0,1,0),(0,0,1)$ and goes through the centre of mass. ...
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399 views

Finding barycentric coordinates of a point $p$ in a triangle.

Suppose you have three points $a,b,c \in \mathbb{R}^3$, not collinear. Let $p$ belonging to the triangle formed by $a,b,c$. For $x,y \in \mathbb{R}^3$ we define: $$ \begin{array}{l} P_{y}(x) = \left\...
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169 views

formula for rational length cevians in an equilateral triangle

I wrote a computer program that iterates through progressively larger scale triangles. The program seeks whole numbers m & n that represent distances from the base vertices, and when a whole ...
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1answer
470 views

Is the integral of a linear function over a triangle always the average of the vertices?

Say I have a triangle with vertices$\ r_0$,$\ r_1$and$\ r_2$. At each vertex the value of some function$\ f(r)$ is known. I know that I can use barycenter coordinates to describe any point$\ t$ in ...
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205 views

Barycentric Coordinates: How do I keep track of Ratios

I am trying to write a java Program wich essentially lets a Point inside a given triangle move only in a straight line to one of the three corners (the specific corner being chosen randomly at every ...
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114 views

Is there a generalized equation for a line in barycentric coordinates?

I've read that in the plane the equation for points $\lambda=(\lambda_1,\lambda_2,\lambda_3)$ on the line passing through $\mu=(\mu_1,\mu_2,\mu_3)$ and $\nu=(\nu_1,\nu_2,\nu_3)$ is $$ \left|\begin{...
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418 views

How to make sense of the Barycentric coordinate system from Cartesian coordinate system?

Given a triangle (defined by three points: $p_1$, $p_2$, $p_3$) and a point ($t$) inside a triangle in 3D space. Then, I compute the Barycentric coordinate of point $t$ programmatically like this: <...
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Integrate a function over a triangle in barycentric coordinate

Let $A$ be the triangle with vertices $(0,0)$, $(2,0)$ and $(1,1)$ in $R^2$. I want to find $$I=\int\limits_A (3x^3y-1)\mathrm{d}A.$$ My problem is that I would like to write the triangle in ...
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256 views

Tetrahedron circumcenter in quadriplanar coordinates

The triangle circumcenter is conveniently expressed in trilinear coordinates as $$ \cos\alpha_1 : \cos\alpha_2 : \cos\alpha_3, $$ where the $\alpha_i$ are the angles opposite of the respective edge in ...
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Relation between barycentric coordinates for triangles of order n and n + 1.

I am taking course in Finite Element Method, and we are looking at barycentric coordinates for triangles. It appears that there is a relation between the coordinates for point in triangle of order n ...
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1answer
52 views

Is there a way I can diagnose errors in my FEM formulation from principles?

I'm working through an FEM calculation by hand to verify I understand the algorithm before I write a bunch of code based on incorrect assumptions. However, when I iterate through the calculations, the ...
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1answer
202 views

The Canonical Triangle of a Tetrahedron

Consider these 4 points as vertices of a tetrahedron. $$((0,-15,0),(0,9,-12),(12,9,0),(0,9,12))$$ Consider these 3 points as vertices of a triangle. $$(( \frac{1}{4} \left(49+31 \sqrt{3}\right) , \...
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269 views

Barycentric coordinate of the excenter proof?

I'm trying to show that the barycentric coordinate of excenter of triangle ABC, where BC=a, AC=b, and AB=c, and excenter opposite vertex A is Ia, is Ia=(-a:b:c). I've gotten to the point where after a ...
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543 views

Barycentric Coordinates of Incenter

I'm trying to prove the fact that the incenter has the barycentric coordinates $(a,b,c)$. My reasoning goes like this; consider the triangle $\Delta ABC$ with barycentric coordinates $(1,0,0)$, $(0,1,...
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43 views

The difference between barycentric and parallel coordinates

I am studying for an exam tomorrow and there are some topics which I do not understand from the slides of the professor. If anyone could give me a clear explanation, I would be eternally grateful. ...