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

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Classification of Triangulated Surface

this is for a homework problem, although not the problem itself, and I'm looking for a little guidance. In the problem, I am given a very long list of triangles, approximately 40, and asked to ...
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calculating position of a point knowing two reference lengths

Hi, I would like to know if there is a way to calculate a unique position for Point A knowing the lengths l1 and l2 which are variable string lengths. Point A can move within the range shown below. ...
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Pachner moves for graph of 4-valent nodes

For 3-simplices (i.e. tetrahedra), I understand the basic idea behind the Pachner moves 1 $\leftrightarrow$ 4, which takes one tetrahedron and replaces it with four (or vice versa), and 2 ...
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Is Delaunay triangulation allowed to add nodes?

I'm trying to compunte a triangulation in R with package RTriangle. The triangulations is based on 911 points, but after the process I have 1006 points. My question is: why there is an increase of ...
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A plane triangulation is 3-connected: Proof

I want to prove: "A plane triangulation $G$ with at least 4 vertices is 3-connected" I have found this proof. I don't like it but I took some ideas out of it: ...
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For planar triangulation, equivalence between 4-connectedness and non existence of separating triangle.

I want to prove the following equivalence: "A planar triangulation is 4-connected if and only if it has no separating triangle." My attempts so far: $\Rightarrow$: If there is a separating ...
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naming n-dimensional triangulation

I wonder why a triangulation of an n-dimensional point set is called triangulation and not something like "simplicication". Formally, the name of "triangles" is only used for 3-simplices and actually ...
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Triangulation of Torus

I was asked to find out the simplicial homology groups of the torus $T=S^1\times{}S^1$ embedded in $R^3$. I triangulated the torus like this : Here the $0$-simplices are $\{v_0\}$. $1$-simplices ...
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Triangulations of the concave polygon

It is known that the amount of possible triangulations of the convex polygon by disjoint diagonals is the Catalan number. But can we somehow know possible amount of the triangulations of the concave ...
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How prove this triangulation with indentity

let $x,y,z\in (0,\pi)$, prove or disprove $$\sin{(x+y)}\cdot\sin{(y+z)}\cdot\sin{(x+z)}\cdot\sin{(x+y+z)} ...
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How solve this equation $\sin x\cdot \sin20=2\sin(110-x) (\sin10)^2$

let $0<x<90$, and such $$\sin x\cdot \sin20=2\sin{(110-x)}(\sin10)^2$$ find the $x$ my idea: since $$\sin x\cdot 2\sin10\cos10=2\sin(70+x)(\sin10)^2$$ so $$\cot10=\dfrac{\sin(70+x)}{\sin ...
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Triangulation Definition Via Cell Partitions

There are two ways of defining a CW-complex. The first is to "inductively build" CW-complexes: you start with 0-cells as your 0-skeleton, attach 1-cells to that to get your 1-skeleton,... and so on. ...
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Calculate 3D-coordinates of a cube's points from the points on the projections

I have a following optical system: 3 cams (left and top, which is orthogonal to the left, and right, which is parallel to the left and orthogonal to the top) and the 2 cubes in the 3D-space with ...
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74 views

Distance between two barycentric coordinates

I am developing a system, and generally in this system we examine the effect of a number of factors on our data. We choose to use Barycentric coordinates to help us to achieve that. I have no problem ...
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33 views

Smallest triangle in a convex polygon triangulation

I have been working on this problem for quite a while and it seems necessary to prove or disprove this particular problem. Suppose $T$ is the set of all possible triangles made from the vertices of a ...
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Is it possible to reconstruct a triangulation from its $1$-skeleton?

Let's restrict to triangulations $T$ of compact and closed smooth manifolds $M$ with $\dim M=2,3$. Such a triangulation is a PL manifold homeomorphic to $M$ which geometric realization is a simplicial ...
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Finding a 3rd coordinate of the rectangle points in 3d

I have a 4 3-D-points, each of them has only 2 of 3 known coordinates, as follow (? is unknown here): P5 (P5x, P5y?, P5z) P6 (P6x, P6y?, P6z) P3 (P3x, P3y, P3z?) P4 (P4x, P4y, P4z?) They build ...
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Data structure issues with incremental Delaunay triangulation

I am implementing the incremental algorithm of Delaunay triangulation with a data structure based on Faces (triangles): 3 vertex indices and 3 Neighbor indices. The issue I have is that the structure ...
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Understand and an algorythm to Maximize number of triangles from a set of points on XY plane

Given: Set of points (x, y) Looking to: Maximize count of triangles that can be formed. Each triangle which is enclosed within another (with/without shared edge) will be counted again. Specifics on ...
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49 views

Proof of existence of Delaunay triangulation in 2D

I want to know references(papers/books/online articles) to the proof of existence of Delaunay triangulation of arbitrary set of vertices(in general position) on 2D euclidean plane. I do find a ...
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Create a configuration - graph theory

I've encountered this (startling) difficult, to me, question: Create a configuration in the plane with a ring size 4, so that every internal vertex is of degree 5. Now, I assume I may not use ...
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54 views

Triangulate the triangle with edges identified

Consider the closed triangle (simplex) $ \Delta = [(0,1), (0,0) , (1,0)] \subset \mathbb{R}^2 $, ie, $\Delta$ is the convex hull of the points $v_1 = (0,1)$, $v_0 = (0,0)$ and $v_2 = (1,0)$ in ...
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Triangulation of clusters of points

I am trying to solve a triangulation problem, but I am not really sure what is the best way to tackle it. I have a series of points $P$ in an $n$-dimensional space. These points are clustered in $k$ ...
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38 views

About the Degree of a Map

I am reading Elements of Homotopy Theory by George W. Whitehead. In the section about maps of the $n$-sphere into itself, in the second last paragraph of the text quoted below, he says that "Then an ...
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What is “Triangulable Triad”?

I am reading George W. Whitehead's Homotopy Theory; Corollary 1.0.2 mentioned the term "Triangulable triad" without definition. May I know how it is defined?
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Graph theoretic view on manifold triangulations

To make the question (hopefully) clearer, I reformulated it: Some triangulation $T$ of a smooth manifold $M$ is a piecewise linear manifold, because smooth manifolds are topological manifolds. Such a ...
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Why is this not a triangulation of the torus?

I refer to example 4, fig.3.6, p.17 of Munkres' Algebraic Topology. He says the given triangulation scheme "does more than paste opposite edges together". Not clear to me. For those who don't have the ...
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What is the point of triangulating topological spaces?

In a general sense, what is the purpose to triangulating, for example, a 3-dimensional topological space? What advantages does it give if we can triangulate a Seifert-Weber space into 23 tetrahedra? ...
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Drawing a triangle with 2 known corners and all side lengths

Assume that there are three points $A$, $B$ and $C$. All the pairwise distances are known $(|AB|, |AC|, |BC|)$. But none of the coordinates are known. I want to draw a triangle using those points. ...
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How find the equation $\cot x=\frac{\sin 20^\circ - \sin 80^\circ \cos 20^\circ}{\sin 80^\circ \sin 20^\circ}$

let $x\in R$, and such $$\cot x =\frac{\sin 20^\circ -\sin 80^\circ \cos 20^\circ}{\sin 80^\circ \sin 20^\circ}$$ Find $x$ my idea: $$\cot x=\csc 80^\circ - \cot 20^\circ$$ then I can't
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Proof of “Japanese Theorem” — Triangulation of Cyclic Polygon

On Mathoverflow, I saw this great result on the "Japanese Theorem". “Japanese Theorem” on cyclic polygons: Higher-dimensional generalizations? Given triangulation of a cyclic polygon, the sum of ...
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Lifting triangulations to universal covers

I thought I would have been able to find more information about this by simply googling than I have been; suppose I have the information that $X$ is constructed by taking a finite disjoint collection ...
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How prove $\left(\sum\cos{\frac{2k-1}{p}\pi}\right)\cdot\left(\sum\cos{\frac{2k-1}{p}\pi}\right)$

Question:let $p$ be an odd prime number,let $A$ be the set of the (postive and less than $p$) quadratic residues modulo $p$,and $B$ be the set of the (positive and less than $p$ quadraric non-residues ...
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Narrowing down a location on a grid based on multiple data points

I'm working on a program for triangulating wireless device locations on a map. So far I've cooked up the triangulation algorithm, but the problem is that wireless signals can bounce around depending ...
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Fitting a surface to scalar functions given on the edges of a triangulation

Given a triangle mesh $\mathcal{T}$ with vertices $V=\{\mathbf{v}_i\}_{i=1}^n$ in $\mathbb{R}^3$ and triangles $T_{ijk}=[\mathbf{v}_i, \mathbf{v}_j, \mathbf{v}_k]$. For each vertex $\mathbf{v}_i$, I ...
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Fundamental group: Attaching a disc to a path connected triangulable space

What's the effect on the fundamental group of a path connected triangulable space X if you attach a disc? So far I've figured out that a disc has trivial fundamental group. So the free product of ...
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How to prove the integral function with cosine is increasing

Prove that the following function is an increasing function on $x\in (0,1)$ when $n\ge2$. $f(x) = \int_0^{\pi} \frac{1}{(1-2x\cos\theta+x^2)^n} d\theta$
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How prove this $ x\neq \lambda\pi,\lambda\in Q$,if $\tan{x}=\frac{4}{3}$

if $0<x<\dfrac{\pi}{2}$,and such $$\tan{x}=\dfrac{4}{3}$$ show that $$x\neq\lambda\pi,\lambda\in Q$$
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Hijacked Malaysian plane position geometry

Sorry to get geeky in the midst of a tragedy and likely horrible crime, but does anyone know how they got this diagram showing the possible last known positions of the possibly hijacked Malaysian ...
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Ellipsoid Triangulation

I'd like to ask if there are three different types how to triangulate ellipsoid (which are different). The requirement is that every point of triangle should lie on the ellipsoid. Thank you, Adam
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Triangulation Question - Comb Space

Let $X = \{(x,y) \in \mathbb{R}^2 : 0 \le y \le 1 ,x=0 \text{ or }1/n \text{ for some } n \in \mathbb{N} \} \cup ([0,1] \times \{0\} ) $ I want to show $X$ cannot be triangulated. I have that given a ...
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67 views

Understanding McMullen's Upper Bound Theorem

I'm a computer science student working on a paper regarding constrained delaunay triangulations. I have been searching for a proof regarding the upper bound for the number of triangles in a ...
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Over Constrained Triangulation

I have a set of points, which I would like to know the locations of. What I have is noisy distance measurements between many of the points, but not necessarily all of them. Additionally, the points ...
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Simplicial Complexes, Triangulation general question.

I am taking a first course in topology, and I am struggling with simplicial complexes. Specifically the triangulation of subspaces of $ \mathbb{R}^n $ confuses me. If you could help me on the ...
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Ways to color an octagon's vertices with three colors?

In how many ways can we color the 8 vertices of an octagon each red, green, or blue, so that no two adjacent vertices are the same color? I think there should be something to do with Catalan numbers ...
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find set of points for lots of triangulations

I should find a set of $n$ points $Q$ in a plane, so that $t(Q)$ (the number of possible triangulations) the following equation holds: $$t(Q) \ge 2^{n-2\sqrt{n}}$$ I solved the problem using the ...
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How prove this $\left|\prod_{k=1}^{n}\sin{(\theta-a_{k})}\right|\ge\dfrac{1}{2^n}$

let $a_{1},a_{2},a_{3},\cdots,a_{n}\in(0,\pi)$, show that there exsit $\theta\in(0,\pi)$ such $$\left|\prod_{k=1}^{n}\sin{(\theta-a_{k})}\right|\ge\dfrac{1}{2^n}$$ My try: $$\Longleftrightarrow ...
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43 views

find a point in 3D space

Suppose we have $3$ fixed points $P_1, P_2, P_3$ in $3$-D space, their coordinates are $(x_i, y_i, z_i)$ for $i=1,2,3$. The problem is to find a point $P$ so that the distances from $P$ to ...
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Maximal size of triangulation in 17-gon

Given convex 17-gon. What is the maximal count of triangles we can divide it if we draw all it's diagonals? (for 4-gon,answer is 4, for 5-gon answer is 11)
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How find this sum $\sin^2{x}+\sin^2{(2x)}+\sin^2{(3x)}+\cdots+\sin^2{(nx)}$

Question: Find the value $$f^{(2)}_{n}(x)=\sin^2{x}+\sin^2{(2x)}+\sin^2{(3x)}+\cdots+\sin^2{(nx)}$$ My solution: since $$\sin^2{x}=\dfrac{1}{2}(1-\cos{(2x)})$$ so ...