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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Laplace Challenge in One Examples, Is there any help?

this question is taken from 2014 exam on CE Entrance Exam, Question $32$ on the end of page $6$. Consider the Laplace equation of following polar coordination, $$\frac{1}{r}\frac{\partial}{\partial ...
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35 views

Shortest possible distance to locate an unknown road

You are stranded in the middle of a large desert and the only way home is a through a straight road, which unfortunately you do not know the location of. If the perpendicular distance from you to ...
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Points with shortest distance always form an edge of a delaunay triangulation.

So I have to proof the following: Points with shortest distance to each other always form an edge of a delaunay triangulation/minimal spanning tree. Since minimal spanning tree is a subtree of a ...
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Find $x$ such that $\sin(\arccos(\tan(\arcsin x))))=x$

Find all real numbers $x$ such that $$\sin(\arccos(\tan(\arcsin x))))=x.$$ Using Wolfram|Alpha we get $$\sin(\arccos(\tan(\arcsin x))))=\sqrt{\dfrac{2x^2-1}{x^2-1}}$$ Why? Because some case such $\...
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How many triangulations are at least possible for a set of points in 2d?

I'm a little confused, because I thought, there would be C(n-2) triangulations, where C(n) is the n-th catalan number and n the amount of points in the set. But it turns out, that there seems to be ...
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reference for simplicial complexes and triangulation

I am trying to understand simplicial complexes and triangulations, am following Basic Algebraic topology by Prof. Anant R Shastri and not very comfortable with the way it has been explained. Suggest ...
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1answer
57 views

How to prove this inequality $x\sin^2{A}+y\sin^2{B}\ge xy\sin^2{C}$

In $\Delta ABC$,if $x,y>0$ and $x+y=1$.show that $$x\sin^2{A}+y\sin^2{B}\ge xy\sin^2{C}$$ I have looked at the simpler methods,? Here is one solution $$\dfrac{\sin^2{A}}{y}+\dfrac{\sin^2{B}}{...
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split a rectangle with triangles into polygons as uniformly as possible

Given a rectangle $A$ and $n$ triangles $\{B_1,B_2,...,B_n\}$, I put the triangles inside $A$, at least one vertex of each triangle is not outside $A$ (inside $A$ or on the edge of $A$). So that A is ...
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39 views

Triangulation with two camera setup - Result in world or camera coordinate system?

I have some problems to understand how I can triangulate a 3D-point using a two camera setup. Let's assume I'm using a right handed coordinate system and the camera is looking in positive z-...
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Is this triangulation consistent with Sperner's Lemma?

Since any two triangles which intersect have an edge or a vertex in common, the triangulation is simplicial. However, I am concerned about triangle $A$. Is every sub-triangle supposed to have a ...
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Clarification of Sperner's Lemma

From Graph Theory by Bondy, Murty Image from wikipedia I don't see how the picture holds according to the definition from the Graph Theory book. Specifically, the definition says to assign ...
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19 views

Using GPS coordinates in trillateration

for a project we need to find a certain position. The info we have : 3 surrounding positions and the distance between those positions and the point we are looking for. We've got a setup like this ...
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1answer
34 views

Voronoi edges example

I have 4 line segments: 0 0 2 0 // 1st line segment 2 0 2 1 // 2nd line segment 2 1 0 1 0 1 0 0 and I wrote some CGAL code to print the Voronoi edges. However, <...
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46 views

Fundamental group of cylinder

I calculated the fundamental group of the cylinder, $C$, using the following method: triangulate $C$ find max contractable subspace realise generators on remaining 1-simplices I found the ...
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1answer
20 views

orientation independent of triangulation

I have $2$-dimensional compact connected orientable manifolds... It is known that those can be triangulated in such a way that everything fits nicely: number of triangles is finite, each edge is edge ...
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1answer
63 views

What is the number of interior faces adjacent to an interior vertex in a triangulation in $\mathbb{R}^3$?

Let $\Omega$ be a polygonal domain in $\mathbb{R}^3$. Assume $\Omega$ is partitioned into tetrahedra using the most common admissible triangulation, that is, roughly speaking, two adjacent tetrahedra ...
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32 views

Triangulation of double-holed torus to calculate fundamental group

Show that the fundamental group of the double-holed torus is given by: $\pi_1=<a, b, c, d | aba^{-1}b^{-1}=cdc^{-1}d^{-1}>$ I have ended up with the following identification diagram ...
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1answer
48 views

The fundamental group of a point is $1$

Show that the fundamental group of the point space $p$ is given as $\pi(p, w_0)=1$ where $w_0$ is the base point This is probably somewhat trivial, but I am looking for a proof. I am familiar with ...
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8 views

Why describe basis multipliers as barycentric coordinates?

So a disclaimer up front: I'm from a EECS background as opposed to pure math, so if possible keep that in mind for your answers. I've been reading a paper on 2D-3D triangulation and came across the ...
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1answer
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Area of Convex hull

For every point set $A \subset R^2$, prove that in general the sum of the coordinates of $\phi(T)$ is independent of a triangulation T and is associated to the area of the Convexv_Hull(A). We define ...
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22 views

Triangulate this identification space

$Z$ is an identification space of the unit square $Q=\{(x, y) | 0 \leq x, y \leq 1\}$ with the following identifications: $(0, y)$~$(1, y)$ $(x, 0)$~$(x+ \frac{1}{2}, 0) $ $(x, 1) $~ $(x + \frac{1}{...
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Triangulation of torus - understanding why

Note: in relation to the answer of the duplicate question, I see that the second picture below refers to the triangulation when we consider simplicial complexes. I do not understand why the triangles ...
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How can I find the distance between two points within a triangle if I have the distance between each point and each vertex of the triangle?

Title says it all. It would be useful to extend the question to finding the distance if any of the points is outside of the triangle, but I'm trying to figure out the basic problem first.
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Catalan numbers and triangulation

Assume $C_n$ is the number of triangulations of a polygon with $n+2$ sides. Using a combinatorial proof, show that $(4n+2)C_n=(n+2)C_{n+1}$. I don't even know where to start with this one. I ...
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How to solve this geometry problem which involves triangles and triangulation

I need to solve this trig problem. Can you please help me? Based on this image: I need to calculate $PO$ based on the values of $\alpha$, $\beta$ and $AB$ ( Assume that I know the values of $\alpha$...
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23 views

Show that the intersection between a polygon and convex hull can be computed in the O(n+m)

I am trying to understand triangulation, explained in the book "Computational Geometry Algorithms and Applications, 3rd Ed - de Berg et al". Unfortunately, I don't know how to solve the following ...
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Working with Triangulations

I would like to prove that $\chi _(M_1\# M_2)=\chi(M_1)+\chi(M_2)-2$. However, the notes I'm using only mention following statement: Let $T_1,T_2$ be two finite triangulations of a compact surface....
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Find the Radius of Sphere using TDOA

My goal is to calculate Position of impact using Trilateration. I followed this guide on wikipedia : Trilateration Wikipedia I don't know how to find the Radius R1,R2,R3.(Normally it is PA,PB,PC,...
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30 views

Orientation of a triangulated compact surface, using orientations of triangles

The questions I am working on asks me to :"Give the definition of an orientation of a triangulated compact surface by using orientations of triangles" I know that a surface is orientable if the ...
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Sampling from Irwin-Hall distribution using triangular distribution

So I need to sample from the Irwin-Hall distribution using rejection sampling with the triangular distribution. I built 2 functions: The first is d_irwin which receives an $x\in supp(g)$ and the n we ...
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triangulation of a surface, adapted to curvature

This is about my printed models of mathematical objects. All of the designs that I've published so far consist of grids of bent ‘rods’, and in most of them the spacing of vertices depends on the rod'...
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66 views

Maximal planar graph

A maximal planar graph $G$ with at least 3 vertices is a simple finite planar graph for which we cannot add any new edge $e$ such that $G \cup e$ is still planar. Is there an easy and rigorous way to ...
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40 views

Triangulation of torus $S^1\times S^1$

I want to find a triangulation of torus To do this we need $14$ triangles How can is this true ? (I do not want a proof but triangulation) Define a complex $K$ : If $V=\{v_1,\cdots, v_n\}$ is vertex ...
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Is this a triangulation of a cylinder?

I am currently a beginner in Algebraic topology. I don't know whether triangulations of a thing are unique or not. So I thought to ask here whether the "triangulation" I've come up with is really a ...
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Existence of subdivision of PL manifold triangulation which is combinatorial manifold

Suppose $X$ is a PL manifold with triangulation $\psi:|\Delta| \to X$. Does there exist a subdivision of $\psi$ which is a combinatorial triangulation?
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convex polygon triangulation

Suppose we have given a convex polygon on $n$ vertices $P= \{ a_1, \cdots , a_n \}$ in the plane (arranged clockwise). How can we prove that there exist atleast two indices $i$ such that circle $a_{i-...
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Compact 3-manifold implies finite triangulability

I know that it's a theorem by Moise that every compact 3-manifold admits a finite triangulation but to me the astounding part of that statement is the existence part instead of the finite one. So I ...
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56 views

What trigonometric identity makes the method of triangulation work?

I've read the article on Wikipedia, but I don't get how to construct the relationships between sides and angles to reach a solution for the distance between two points. All the other sites I read ...
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Triangulations of combinatorially equivalent polytopes

I am wondering which relation(s) there are between triangulations of combinatorially equivalent polytopes that use no new points: Let $P,Q$ be a $n$-polytopes such that their face lattices are ...
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finding edges of a triangle graph from degrees of points

my theory: Given a list of points on a 2 dimensional plane, and the degree of each point, there should correspond only one way to arrange the edges between points so that the final graph is a mesh of ...
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Identification space of square. Net, triangulation and surface classification

Space Z is made as an identification space of unit square $Q=${$(x,y) | 0\leq x, y \leq 1$} by making the following identifications: $ (0,y)$~$(1,y) $ for all $0\leq y\leq 1 $, $ (x,0)...
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Triangulation of simplotope (Cartesian product of simplices)

Let $s,n\in\mathbb{N}$ and $\mathcal{S}_{s,n}=\operatorname{conv}\left(\{\mathbf{0}\}\cup\{\mathbf{e}_{s,i}\mid i\in\{1,\ldots,s\}\}\right)^n\subseteq\mathbb{R}^{sn}$ where $\mathbf{0}$ is the zero ...
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Can someone explain how I can triangulate using angles and one side of a Right-angled triangle?

I've been looking around, trying to find a simple way explaining why and how to calculate distance using the triangulation technique, but I'm still pretty confused, I've got some simple math notions, ...
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Triangulation of Torus in three different ways, but two of them are wrong.

In my geometry notes the writer states that the following two, are not triangulations for the torus: On the contrary this is a good triangulation: I tried to wrap a piece of paper in order the ...
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Can this be triangulated?

Given N people with their phones that can sense the signal strength of every other phone knowing what phone it is. Phones don't know their absolute location (underground). There is a formula that ...
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Trigonometry Calculate Distance and Angle of object in camera frame

I have an application where I am trying to build a handheld scanner that can draw a 2d profile of a 3d surface (using structured light scanning). The handheld device consists of a line laser and a ...
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How to triangulate from a Voronoï diagram?

I computed a Voronoï diagram from a set of point (with Boost.polygon). I try to find a Delaunay triangulation, connecting each cell center for each Voronoï edge, but I miss some edges. In the ...
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How to show $P^n$ (real projective space of dimension n)is triangulable?

How to show $P^n$ (real projective space of dimension n) is triangulable? That is, how to show there exists a triangulation of $P^n$? By triangulation, I mean a simplicial complex $K$ and a ...
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Barycentric subdivision of regular CW decomposition is a combinatorial manifold?

Suppose $X$ is a PL manifold (with boundary) and let $(X,X_{i})$ be a regular CW complex. Is the barycentric subdivision of $(X,X_{i})$ a combinatorial manifold? Answer given in comments. Definition: ...
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Lifting of triangulation

In "Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions" and many other books is described a lifting of triangulations for branched covers ...