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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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
21 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
20 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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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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34 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
27 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
92 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 ...
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68 views

Why can't this triangulate $\mathbb{RP}^2$?

I understand that an actual minimal triangulation of $\mathbb{RP}^2$ has at least 10 2-simplices, but I don't understand why. Without appealing to the computation of the homology groups of ...
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32 views

Rotational matrices

I apologize ahead of time that math isn't my strong suit, I understand most the basic concepts but lots of gaps. So forgive me if i miss use a concept. So I am working in a 3d engine integrating a ...
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1answer
27 views

Finding the coordinates of the third point in triangle

How would you find $x$ and $y$ coordinates of the third point in triangle($A$, $B$, $C$), if you know coordinates of $A$ and $B$, and angles at $A$ and $B$?
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33 views

Triangulation - third coordinate of triangle

I would like to ask : I have coordinates of two towers on the beach : A[x,y] B[x,y] . I know distane between them. My task is now to find out coordinate of the ship on sea.I also know both angles that ...
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82 views

How to compute QR decomposition of a product of matrices

Suppose I have $A=A_nA_{n-1}\cdots A_2A_1$ How can I compute the $QR$ factorization of $A$ without explicitly multiplying $A_1, A_2, \ldots, A_n$ together? The suggestion I got is that, suppose ...
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31 views

Distance between 2 Delaunay triangulations

I am making a Delaunay triangulation from a set of nodes. We will call it triangulation1. From the same set of nodes with acquisition problems (some nodes missing or maybe more nodes detected) i'm ...
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1answer
436 views

Calculate 3rd point of a triangle, given 2 points and all angles in 2D

I have stumbled upon an interesting problem. I tried to find an answer here but there are just too many similar threads which did not really help me, so I was trying to figure it out by myself. The ...
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1answer
571 views

Triangulation of the Klein Bottle

Why is this no triangulation of the Klein Bottle? Is it because the top and the bottom triangle share 3 vertices but have different edges? How do I find a triangulation?
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1answer
28 views

Calculating if an object is blocked from sight by another object

Is there an equation to determine if an object at altitude A can be seen at altitude B if there is an object between them at altitude C? Something to do with triangles I think... I know it has ...
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1answer
64 views

Interior angles of a polygon.

I was solving problems from Paul Zeitz's book "The Art and Craft of Problem Solving." There is a problem which states 3.2.11 Fix the proof in Example 2.3.5 on page 45. Show that even a concave ...
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1answer
29 views

Number of triangles in a triangulation

Wikipedia Delaunay Triangulation On this page, I read (with $n$ the number of edges): "In the plane (d = 2), if there are b vertices on the convex hull, then any triangulation of the points has at ...
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find diagonals of quadrilateral

I have 4 points and need to determine which pairs of these points represent the diagonals. In other words, I am trying to triangulate a quadrilateral. I realize that triangulation of polygons is a ...
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52 views

Put a set of triangles into proper mathematical equations / objects

I have a set of $n$ points $\{A_1,A_2,...,A_n\}$ of the plane. Three points $A$ should never form a line (so we can still draw a proper triangle). I draw every triangle formed with $3$ points $A$. I ...
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1answer
37 views

Classification of Triangulated Surface

this is for a homework problem, although not the problem itself, and I'm looking for a little guidance. In the problem, I am given a very long list of triangles, approximately 40, and asked to ...
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16 views

calculating position of a point knowing two reference lengths

Hi, I would like to know if there is a way to calculate a unique position for Point A knowing the lengths l1 and l2 which are variable string lengths. Point A can move within the range shown below. ...
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1answer
45 views

Pachner moves for graph of 4-valent nodes

For 3-simplices (i.e. tetrahedra), I understand the basic idea behind the Pachner moves 1 $\leftrightarrow$ 4, which takes one tetrahedron and replaces it with four (or vice versa), and 2 ...
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1answer
61 views

For planar triangulation, equivalence between 4-connectedness and non existence of separating triangle.

I want to prove the following equivalence: "A planar triangulation is 4-connected if and only if it has no separating triangle." My attempts so far: $\Rightarrow$: If there is a separating ...
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26 views

naming n-dimensional triangulation

I wonder why a triangulation of an n-dimensional point set is called triangulation and not something like "simplicication". Formally, the name of "triangles" is only used for 3-simplices and actually ...
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470 views

Triangulation of Torus

I was asked to find out the simplicial homology groups of the torus $T=S^1\times{}S^1$ embedded in $R^3$. I triangulated the torus like this : Here the $0$-simplices are $\{v_0\}$. $1$-simplices ...
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Triangulations of the concave polygon

It is known that the amount of possible triangulations of the convex polygon by disjoint diagonals is the Catalan number. But can we somehow know possible amount of the triangulations of the concave ...
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1answer
30 views

How prove this triangulation with indentity

let $x,y,z\in (0,\pi)$, prove or disprove $$\sin{(x+y)}\cdot\sin{(y+z)}\cdot\sin{(x+z)}\cdot\sin{(x+y+z)} ...
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3answers
77 views

How solve this equation $\sin x\cdot \sin20=2\sin(110-x) (\sin10)^2$

let $0<x<90$, and such $$\sin x\cdot \sin20=2\sin{(110-x)}(\sin10)^2$$ find the $x$ my idea: since $$\sin x\cdot 2\sin10\cos10=2\sin(70+x)(\sin10)^2$$ so $$\cot10=\dfrac{\sin(70+x)}{\sin ...
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1answer
61 views

Triangulation Definition Via Cell Partitions

There are two ways of defining a CW-complex. The first is to "inductively build" CW-complexes: you start with 0-cells as your 0-skeleton, attach 1-cells to that to get your 1-skeleton,... and so on. ...
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Calculate 3D-coordinates of a cube's points from the points on the projections

I have a following optical system: 3 cams (left and top, which is orthogonal to the left, and right, which is parallel to the left and orthogonal to the top) and the 2 cubes in the 3D-space with ...
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156 views

Distance between two barycentric coordinates

I am developing a system, and generally in this system we examine the effect of a number of factors on our data. We choose to use Barycentric coordinates to help us to achieve that. I have no problem ...
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1answer
39 views

Smallest triangle in a convex polygon triangulation

I have been working on this problem for quite a while and it seems necessary to prove or disprove this particular problem. Suppose $T$ is the set of all possible triangles made from the vertices of a ...
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28 views

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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45 views

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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1answer
50 views

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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1answer
67 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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1answer
33 views

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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1answer
65 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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1answer
39 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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66 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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1answer
76 views

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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2answers
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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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257 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 ...