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 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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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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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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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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37 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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25 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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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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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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66 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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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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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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81 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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48 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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69 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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24 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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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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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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24 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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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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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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52 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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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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36 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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32 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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55 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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107 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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39 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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629 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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625 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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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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55 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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39 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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51 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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81 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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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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606 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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31 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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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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215 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 ...