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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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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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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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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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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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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Point distance to verices of triangle with given edges

I would like to find a formulation for the distances between a given point P(x,y), which is inside a general triangle with all edges values provided, to its vertices. Thanks,
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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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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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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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68 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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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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342 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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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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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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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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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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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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36 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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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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42 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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Is Delaunay triangulation allowed to add nodes?

I'm trying to compunte a triangulation in R with package RTriangle. The triangulations is based on 911 points, but after the process I have 1006 points. My question is: why there is an increase of ...
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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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25 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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340 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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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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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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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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125 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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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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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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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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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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59 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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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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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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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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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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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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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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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 ...
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How prove $\left(\sum\cos{\frac{2k-1}{p}\pi}\right)\cdot\left(\sum\cos{\frac{2k-1}{p}\pi}\right)$

Question:let $p$ be an odd prime number,let $A$ be the set of the (postive and less than $p$) quadratic residues modulo $p$,and $B$ be the set of the (positive and less than $p$ quadraric non-residues ...
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Narrowing down a location on a grid based on multiple data points

I'm working on a program for triangulating wireless device locations on a map. So far I've cooked up the triangulation algorithm, but the problem is that wireless signals can bounce around depending ...