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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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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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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30 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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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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13 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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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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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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47 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 ...
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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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64 views

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 ...
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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 + ...
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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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Get location / position of an object via 2 cameras

Best In the following image, you can view the setup of my problem. In general, i've 2 cameras which have a view-angle of 48° and 64°. Secondly, I know the position of my camera's (which means i can ...
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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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101 views

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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79 views

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 ...
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1answer
19 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 ...
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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 ...
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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 ...
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1answer
63 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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38 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 ...
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1answer
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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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48 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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1answer
46 views

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 $, $ ...
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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 ...
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Prove that for every two triangles, there exists a line that halves the areas of the triangles simultaneously.

The problem may sound somewhat funny, cause I haven't got it from a good source. Anyways, I think I somehow get what it wants and here's my way of looking at the solution: Two triangles of arbitrary ...
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How do I find a third side of a triangle with two sides and a bisecting line segment?

I am using a laser range finder to calculate the height of a second story wall. I have a fixed point and three separate lengths hitting the top, the bottom, and an indeterminate point on the wall. ...
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1answer
57 views

Triangulation for a 1-manifold

I'm taking a course on Algebraic Topology and I had a question while studying. I know the answer is affirmative but I don't know why. What I want to prove is There exist a triangulation for every ...
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Convex Hull given set of planes.

If I have some finite amount of planes, for example \begin{equation} z_1=2x, \\ z_2=2y, \\ z_3=3+x+y, \\ z_4= 2+x, \\ z_5=2+y \\ z_6 =3 \end{equation} And I wish to find the convex hull in order to ...
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RBF - triangle mesh interpolation - skinny triangles and incorrect results

I have triangle mesh, that I need to describe by RBF. I need to do this only locally on vertex neighborhood. All is working correctly if underlaying triangulation is reasonably regular. But if there ...
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Determine if parallel lines are aligned, by a measurement

Real World math question: given two poles of 3 m length, how do I easiest determine if they are parallel and aligned? By "aligned" I mean that you can draw a rectangular square by connecting the ...
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what is side of any oblique triangle if given length of other two sides and all three angles?

For any oblique triangle, which may be acute or obtuse, if I know length of two sides, and I know all three angles, how can I determine length of unknown side? All three sides may or may not be equal ...
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Every triangulation on a disk is orientable.

Since a disk doesn't contain a Möbius or any other non orientable surface, it is orientable. I want to prove it rigorously by showing every triangulation on a disk is orientable. For this, I was ...
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Can $S^2$ be homeomorphic to a simplicial complex with fewer than 3 two-simplices?

I think I can see why $S^2$ is homeomorphic to a simplicial complex with four 2-simplices (for example, it can be obtained from the tetrahedron). Can $S^2$ be homeomorphic to a simplicial complex with ...