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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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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74 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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66 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
18 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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15 views

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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22 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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8 views

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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11 views

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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34 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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20 views

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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1answer
23 views

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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53 views

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
28 views

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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45 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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22 views

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
44 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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12 views

Topological nets and triangulations

How does one construct a net and triangulation for a space? For example the identification space of the unit square with these identifications $(0,y)$~$(1, 1-y)$ for all $0 \leq y \leq 1$ ...
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75 views

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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15 views

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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31 views

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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1answer
87 views

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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2answers
72 views

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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74 views

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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37 views

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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52 views

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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44 views

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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33 views

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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1answer
45 views

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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2answers
49 views

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
53 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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36 views

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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29 views

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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1answer
42 views

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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29 views

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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69 views

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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1answer
68 views

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 ...
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1answer
37 views

Is a compact triangulated surface obtained by edge-pairing on a polygon?

I guess that one can make the compact surface by edge-pairing. I'm trying to rearrange the triangles so that they could be considered as parts of a disk. Let $S=\bigcup_{i=1}^n T_i$. Then we are able ...
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2answers
94 views

Polygons with a Unique Triangulation

For each n > 3, find a polygon with n vertices that has a unique triangulation. I want to say that you can somehow build these polygons by continuously adding triangles somehow, but I'm not sure.
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82 views

What is the Wrong in this Triangulation of the Torus

On pg 133 of Roman's Introduction to Algebraic Topology it is stated that one requires at least 14 triangles in any triangulation of the torus. Admittedly, I do not have a very good understanding of ...
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77 views

Two cevians divide a triangle into 4 parts. Calculate the area of the 4th part, given the other 3.

Good day Here is the question: Connecting $AF$ and setting areas $\triangle ADF = x$ and $\triangle AFE = y$: $\frac {9+x}{12} =\frac y{15}$ $\frac{15+y}{12} =\frac x9$ from the ratios of the ...
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1answer
185 views

Trouble With a Triangulation of the Torus

On pg. 133 of Rotman's Introduction to Algebraic Topology, we have a figure which claims to be a triangulation of the torus. Now a triangulation of a topological space is defined as Definition. ...
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45 views

Is it known whether or not the 'Hauptvermutung' is true for finite simplicial complexes in $\mathbb{R}^4$?

If I have two finite simplicial 4-complexes embedded linearly in $\mathbb{R}^4$ (as in all the lines and faces are straight and flat and there are only a finite number of 4-simplices) do they have a ...
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Gluing 3 dimensional tetrahedra with orientation reversing edge

I am not sure how to proceed on exercise 3.2.3 in Thurston's book "Three Dimensional Geometry and Topology". The wording is as follows: "In a gluing of three dimensional simplices, each edge enters ...
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Adjusting density distribution with smallest changes possible for vertexes

I am doing research on my master thesis where I am going to calculate time dependance of surface movement for liquid drop. Fortunately the problem for me is simplified to the boundary integral ...
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102 views

Multiple objects triangulation in 3D, intersecting the right vectors (rays)

I am working on a project in which I should be able to triangulate the position of multiple objects when they are seen by (at least) two cameras. Single object Currently I am able to triangulate a ...
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26 views

Polyhedral surface with infinitely many triangulations with same combinatorics

Is there an example of a polyhedral surface that has infinitely many triangulations with the same combinatorics?
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57 views

find point at distance $d_1$,$d_2$,$d_3$ from $p_1$,$p_2$,$p_3$ in 3d

There are three points in 3d space: $p_1$, $p_2$, $p_3$ (or more). These points form a triangle, so you can assume are not collinear. There exist an additional unknown point $p_\star$ for which I ...
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2answers
223 views

How do I properly read a clinometer?

If the weight hangs down at roughly 42 degrees, would the angle be 90 degrees - 42 degrees = 48 degrees?