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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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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36 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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48 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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25 views

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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120 views

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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81 views

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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66 views

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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71 views

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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45 views

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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178 views

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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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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292 views

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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43 views

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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66 views

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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39 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 ...
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homology of face links of a triangulated manifold

In a triangulation of a general topological space, we can define a face link (for any face in the triangulation). Intuitively, this is a kind of "$\epsilon-$sphere" in metric space. In chapter 3.8 ...
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What is the equivalent of Delaunay tringulations in high dimensions?

For 2D manifolds, Delaunay triangulation is a very useful tool for coarse graining. It has the nice property that in the flat/euclidian manifold case, it reduces to a 2D simplicial tesselation of the ...
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Delaunay Triangulation Possible Triangles

I have a set of points and I want to find all possible triangles which have empty circumcircle. I want to use Delaunay Triangulation. I have read some papers on the subject but I am not sure whether ...
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82 views

What is a power Delaunay triangulation, and how is it computed?

What is a power Delaunay triangulation? and how would I compute it in n-Dimension?
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79 views

Circumsphere of a tetrahedron undefined?

I am trying to find 3D alpha shapes from my data-set. In doing so, I am keeping only those tetrahedra that have circumradius below a certain threshold. However, while finding the circumradius of the ...
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Triangulation in periodically repeated 3D box

I have a set of points in a 3D box (rectangular parallelepiped), which is periodically repeated out of two its opposite sides. How do I find the Delaunay triangulation for this set of points? Here is ...
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How prove this $S_{n}\neq 0$

let $n$ is postive integer numbers,and let $$S_{n}=\sin{1}-\sin{4}+\cdots+(-1)^{n-1}\sin{(3n-2)}$$ show that $$S_{n}\neq 0, \forall n\in N^{+}$$ My try: maybe this problem use ...
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Triangulation of a 3-sphere

If one wants to generate a Simplicial complex of the topology of the 3-sphere, one can just take the boundary of a 5-cell, 16-cell or 600-cell. The curvature is concentrated on the edges meeting the ...
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Trangulation method to find the location

I am trying to finding the location of the user using mobile tower signal strengths. Scenario is given in the picture Here, I know the coordinates (latitude and longitude, like x,y) of Tower 1, 2 ...
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60 views

Software to produce graphics of triangulated surfaces

I would like to find a software that lets me create graphics of a surface with a triangulation on it. It doesn't need to be very fancy; I just need to explain to a bunch of high schoolers what a ...
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60 views

Finding a 3D co-ordinate using triangluation

I'm trying to find a real world coordinate where 3 spheres collide and interact. At the moment I have been able to set up my triangulation equations so that I can work out the 2D position of where my ...
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153 views

3 Intersecting lines point, triangulation.

I currently have 3 circles that intersect each other. At these points I create a line, although I am stumped at how I can find the point of which all 3 lines intersect? I know the coordinate center ...
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328 views

How can I prove a point is within a triangle, given three other points?

Could someone please explain the formula behind this, and then provide an example of how to do this? Basically I have 4 points, each with a longitude and latitude number. (They make a polygon quad, so ...
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Cycles and Closed Paths

$\newcommand{\Im}{\operatorname{Im}}$ Show that if $\Delta$ is a triangulation, then $[a_1, b_1]+[a_2, b_2]+\cdots+[a_n,b_n]$ is a $1$-cycle precisely when the indicated oriented edges $1$-faces ...
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triangulation n-dimensional cube into exactly n! simplices

This question is similar to Find the smallest triangulation of the n-dimensional but easier : How to show the n-dimensional cube can be triangulated into exactly n! simplices?