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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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
38 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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67 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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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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2answers
27 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
53 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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65 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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49 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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1answer
29 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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41 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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35 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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25 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
43 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
41 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
51 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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35 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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21 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
36 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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2answers
26 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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53 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
64 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
36 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
81 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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65 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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1answer
58 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
124 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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43 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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47 views

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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74 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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25 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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56 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
134 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?
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41 views

What is the unknown angle?

So first off I started with the pythagorean theorem to find the missing leg of the triangle. \begin{align*} 5^2 + b^2 ={}& 8^2 \\ 25 + b^2 ={}& 64 \\ 64 - 25 ={}& 39 \\ \text{missing ...
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30 views

Polygons - necessity of checking for collinearity with edge incident to diagonal's vertices?

I'm reading a book on Computational Geometry ('CG in C' by Joseph O'Rourke). It is quite enlightening but there is one thing I feel like I have to ask about when it comes to triangulation of a ...
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3answers
294 views

Finding an Unknown Location with known distances from location

Lets say that I have a map and an unknown location. If I have multiple locations in which I know the distance away from the unknown location, can I pinpoint the unknown location? I am aware of ...
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1answer
51 views

triangulation of the cube of whose vertices are in the set $\lbrace (\pm 1 , \pm 1 , \dots , \pm 1)\rbrace$

Take the cube centered at the origin whose vertices are $\lbrace (1 ,1 , 1) , (-1 ,1 , 1) , (1 ,-1 , 1) , (1 ,1 , -1) , (1 ,-1 , -1) , (-1 ,1 , -1) , (-1 ,-1 , 1) , (-1 ,-1 , -1) \rbrace$. We can ...
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1answer
133 views

Find my coordinates from distance with unknown coordinates

I am trying to work out if there is a way to calculate some coordinates relative to each other simply by knowing $3$ or more distances from some unknown points. I do not have a distance matrix, I ...
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61 views

Quadrilateral Based on Delaunay Triangulation

I have read about Delaunay triangulation. Is there such thing as Delaunay Quadrangulation or quadrilateral based on Delaunay triangulation? This is the only thing ...
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2answers
126 views

Triangulation of the projective plane

I just worked a little bit with triangulations of surfaces. I think the following "triangulation" of the real projective plane is false: The red (blue) edges are identified in an inverse way. Sorry ...
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1answer
40 views

Decomposition of hyper-rectangles into congruent simplices

Let $(a_1, \ldots, a_d) \in \mathbb{N}_+^d$ be positive integers and the semi-axes of the $d$-dimensional $\ell_1$-ellipse $$ E_{\bf a} := \{{\bf x} \in \mathbb{R}_{\geq 0}^d: \sum_{j=1}^d ...
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1answer
30 views

Finding Both Missing Co-ordinates in distance formula

Hi I am using this to find location of a device in a 2d plane based on the distance formula. The co-ordinates of reference points and the distance of the device from the device is known. How can we ...
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1answer
25 views

Calculate a jetplane's distance from my location

So i was sitting outside my workplace and saw this jet flying. I was really curious if there is a way to calculate the jet's distance between the jet and my location. (I have very little knowledge ...
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Why do we care about triangle density and triangle freeness in large graphs?

There seems to be a lot of research done about determining whether large graphs are triangle free or counting the number of triangles. Aside from coloring, why is this important?
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405 views

Triangulation of matrices

Suppose that $A$ is some triangularizable matrix in $M_n(\mathbb R)$. The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their ...
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1answer
57 views

Are PL-homeomorphic manifolds diffeomorphic?

Take two smooth manifolds. Since they are smooth, they both possess triangulations. Now assume that the triangulations are related by Pachner moves, that is, the triangulated manifolds are ...
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1answer
31 views

Triangulations of surface.

Let $R$ be e regolar region of a surface $\Sigma$ such that $R$ is the closure `of an open set whose bourdary $\partial R$ is the union of simple closed regular curves. Let $T$ be a trangulation of ...
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1answer
111 views

Proving the continuation of the Cayley-Hamilton theorem from Schur's triangularization theorem

The Cayley-Hamilton theorem says that every square matrix can satisfy its own characteristic equation, $p(\lambda) = 0$, or $p(\mathbf{A}) = \mathbf{0}$. The question is to show how the ...