# Questions tagged [triangulation]

For questons about triangulation, that is a) the subdivision of the plane or other topological spaces into triangles (or, more generally, simplices) or b) the methods used in surveying for locating points by measuring angles and accessible lengths of triangles

318 questions
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### Systematic approach to triangulation closed combinatorial surfaces

I was wondering whether there is a systematic approach to the triangulation of closed combinatorial surfaces, which we know can be shown to be homeomorphic to polygons with complete set of side ...
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### Triangulation with 3 known points and time is involved?

I am completely stuck on how to visualize this problem, let alone code it. Understanding the mathematics behind it could help me out a lot. Thanks! Let's suppose that the unknown point is actually a ...
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### Riemann Mapping theorem in triangulations

I am reading the paper 'Rotation Distance, Triangulations, and Hyperbolic Geometry' by Thurston et al. The authors are constructing a sequence of triangulation from a regular icosahedron. Each face of ...
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### intersection in a simplex

In a triangulation $\Gamma$ of a (oriented) 2-manifold, consider a 2-simplex labeled by ($123$), where $1,2,3$ denote the order of vertices. Consider the dual $\Gamma^*$ of $\Gamma$, and then denote ...
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### Coloring triangles in a Delaunay triangulation on the surface of a 3d sphere.

Suppose a delaunay triangulation over the surface of a 3d sphere (or generally some 3d surface of something topologically equivalent to the sphere). How many colors do I need to color its triangles so ...
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### Estimate coordinates of vertices

Let $\hat{t}$ be the reference triangle with the vertices $\hat{A_1} = (0,0)$, $\hat{A_2} = (1,0)$, $\hat{A_3} = (0,1)$ and let $t$ be the triangle with the vertices $A_1 = (0,0)$, $A_2 = (h_1, 0)$ ...
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### How can I create an evenly distributed mesh from a shape?

I'm trying to convert some 2D shapes (without holes) into meshes with evenly distributed vertices. Before the conversion the shapes are edge loops with no internal vertices. After the conversion I ...
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### Estimation of relation between vertices of a triangle

Let $\hat{t}$ be the reference triangle with the vertices $\hat{A_1} = (0,0)$, $\hat{A_2} = (1,0)$, $\hat{A_3} = (0,1)$ and let $t$ be the triangle with the vertices $A_1 = (0,0)$, $A_2 = (h_1, 0)$ ...
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### Is there an easy way to find the sign of this determinant without calculating it directly?

There exist real numbers $A_x, A_y, B_x, B_y, C_x, C_y, D_x$ and $D_y$. Is there an easy way to find the sign of following determinant without calculating it directly? BTW, the determinant appears ...
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### Trilateration when only combinations of distance are available

My problem setup is as shown below: I know the location (x,y) of fixed points p1+, p1-, <...
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### Calculate C point of triangle given A, B, angle a, angle b

One pic is worth thousand words... I know angle a, angle b, pointA, ...
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### Biholomorphic functions and delaunay triangulation

Lets have a look at the two simply connected domains $D,G \subset \mathbb{C}$ and a biholomorphic function $f:D \rightarrow G$ which maps $D$ conformal onto $G$. For some $n \in \mathbb{N}$ there ...
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### Numerical integration in Finite Element Method (and implementation in Matlab)?

i'm trying to solve the p-Laplace Equation: \begin{align} \begin{cases} \text{div} (\sigma |\nabla u|^{p-2} \nabla u) = f &\quad \text{in } \Omega\\ u = g &\quad \text{in } \partial\Omega \...
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### Lower bound on the number of faces incident to a set of vertices in a planar triangulation

Suppose that $G$ is a planar triangulation (also the outer face has to be triangle) on $n \geq 4$ vertices. Let X be a subset of vertices of $G$ such that $|X| \leq n-3$. Let $F(X)$ be the set of all ...
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### Blindly removing inessential diagonals from a triangulation can lead to a bad convex partitioning

Assume that a simple polygon $P$ and a triangulation of it only using the diagonals is given. We say a diagonal $d$ is essential for vertex $v$ if removing $d$ creates a piece that is nonconvex at $v$....
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### Is this a triangulation for the 2-torus?

I am not quite sure I understand simplicial comlexes/triangulations. For instance, I think that the below image represents a triangulation for the 2-torus. Am I correct?
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### Induction Problem with Polygon Triangulation rules

Problem: Let P be a convex polygon with consecutive vertices v1,v2,...,vn. Use some form of induction to show that when P is triangulated into n−2 triangles, the n−2 triangles can be numbered 1,2,...,...
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### triangulation of a circle and the way to solve a problem

Consider the circle $S^1$ with multiplication given by the complex numbers. Prove that the map $f(x) = x ^n$ , $n$ a positive integer, has degree $n$. What is the degree of the map $g(x) = 1/x$. ...
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### Why does skinny triangle is avoided in triangulation algorithm?

I recently learned about Delaunay triangulation algorithm.. One property of this algorithm is to prevent the generation of skinny triangles.. However, I haven't really seen any good explanation of why ...
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### Use of the integrals in the graph theory

I hope to know some good references about the use of integrals to study the graph theory: For example, it seems that $$\int^{\infty}_{-\infty} dx \exp(-x^2/2+\lambda x^3/3!)$$ whose coefficients in ...
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### Catalan numbers and triangulations

The number of ways to parenthesize an $n$ fold product is a Catalan number in the list $1,1,2,5,14,\cdots$ where these are in order of the number of terms in the product. The $n$th such number is also ...
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### 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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### Why does “angle defect” work as a global measure of flatness or lack thereof?

In "Non-Euclidean Geometry and Curvature" by James W. Cannon, the author states that the "angle defect" def(D) of a polyhedral disk D works "as a global measure of the degree to which D fails to be ...
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### Number of triangles in any triangulation of a 2-d figure

We are given a figure like this in the plane. Does any triangulation without addition of new vertices of such a figure have the same number of triangles? For a polygon, I know any triangulation gives ...
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### Probability that a delaunay triangle contains the center of its circumcircle

A Delaunay triangulation for a given set P of discrete points in a plane is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). https://en.wikipedia.org/...
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### Euler Characteristic of figure(piecewise linear complex - 2d)

I want to calculate the Euler characteristic of this $2$-dimensional piecewise linear complex. A piecewise linear complex is a finite set of linear cells - vertices, edges and polygons( not ...
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### What separates a cyclic polytope from a projective polytope?

I am having trouble understanding the difference between a cyclic polytope and a convex projective polytope as positive geometries. The link https://arxiv.org/pdf/1703.04541.pdf is the source of ...
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### Triangulation and Linear Systems

I'd like to ask your help to solve a linear system related to a triangulation problem involving two rays (vectors). Let $a\textbf{p}_{l}$ ($a \in \mathbb{R}$) be the ray $l$ through $O_{l}$ ($a = 0$) ...
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### Find a point related to triangle.

I have two triangles that are not similar. The Only thing that I know is that AB and C points from triangle 1 are related to $A^1 B^1$ and $C^1$ points from triangle 2. Based on these inputs I want ...
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### Interpolation via triangulation between a set of points on two parallel lines

I'm trying to develop a fast algorithm to perform 2D interpolation between two parallel lines. Along the way, I found an interesting problem and have been wrestling with it for longer than I should. ...
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### Can't find an equation for calculating when 4 moving points have a circle passing through them

First thing - please forgive me if my way of explaining my problem is not formal or not accurate to standards, I am an amature mathematician and I have much to learn, I welcome you to let me know ...
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### Determine relative coordinates of a point inside a triangle with only distances known

Say I have a triangle with points A, B, and C, and I know the lengths of AC, AB, and BC. The triangle may or may not be a right triangle. Example: click here, I can't embed the image since I don't ...