Questions tagged [triangulation]

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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Upper triangulation of a matrix versus diagonalization

I am trying to google this question, but could not find any hints. This is important to me because of I am dealing with 3-4D matrices. It's true that an upper triangulation (Gauss elimination) of a ...
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Trilateration solution for N points with $D_{NxN}$ distances

Problem: Let there be $N $ points $P_i(x_i,y_i,z_i)$ and a $\mathbf M$ the $NxN$ matrix of distances $d_{ij}$ between each point. Lastly, let's consider that the 3 of the N points $P_1$, $P_2$ and $...
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Fast algorithm to embed a triangulation into plane

Let $G = (V, E)$ be a planar graph such that $|E| = 3|V| - 6$ (so $G$ must be a triangulation without Kuratowski subgraphs). Given the adjacent matrix $A$ of $G$, please design an algorithm to embed $...
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A Matrix is diagonalizable over Q if not all eigenvalues are in Q?

I'm not too sure about this, just trying to wrap my head around this. I'm asked to prove/disprove this statement: A matrix $A \in M_3(\mathbb{Q})$ with two eigenvalues $\lambda_1,\lambda_2 \in \mathbb{...
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Triangulations of the Hyperbolic Plane

I'm studying triangulations of the hyperbolic plane and have come across the following theorem: If we are given a triangle $\Delta_0$ with angles $\pi$/l,$\pi$/m,$\pi$/n, where the integers l, m, n ...
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Find triangle that envelops all points in point cloud

One of the steps of Delaney Triangulation using Bowyer-Watson algorithm in R2 space is to create triangle large enough to contain all points in point cloud (so not necessarily the triangle with ...
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Convex Hull with Buffer Radius

I am using a convex hull (via a Delauney Triangulation) around a point cloud to define a given region on a manifold. The problem I encountered was that the triangulation will never accurately describe ...
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What does "bottom side of the convex hull" mean as mentioned in the Delaunay wikipedia page and how to efficiently find it?

As show here in the quote and screenshot. https://en.wikipedia.org/wiki/Delaunay_triangulation The problem of finding the Delaunay triangulation of a set of points in d-dimensional Euclidean space ...
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Triangulation of 3-sphere and Hopf Fibration

I am currently reading the paper A Minimal Triangulation of the Hopf Map and its Application. In the paper, the authors are trying to describe a triangulation of the 3-sphere into a (abstract) ...
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Is there a name for the triangulation which minimizes the sum of the longest edge for each triangle?

Consider a point set $P\subset \mathbb{R}^2$. Let $T$ be a triangulation of $P$. For $t\in T$ a triangle, define $l(t)$ as the length of the longest side of the triangle. I want to find a ...
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Elevated Delaunay triangulation

This is what I call an elevated Delaunay triangulation: That is, this is a Delaunay triangulation of a surface. It is implemented in the R packages deldir and RCGAL. I used these packages to evaluate ...
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How do I triangulate a Tetrahedron?

I have a question in triangulating a Tetrahedron. Is it correct that it works as follows (the green arrows should point in the direction of the orientation): And if yes for what can I use this now? ...
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How can I compute the fundamental group of a surface (Fulton Chapter 17)?

I'm reading the chapter 17 of Fultons Book "Algebraic topology a first course" and I somehow have some problems in understanding the triangulation of a surface and what it has to do with ...
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Constructing geodesic polygons in uniformly normal balls

I am studying the book "Riemannian Manifolds: An Introduction to Curvature" by John Lee. In the chapter on Gauss-Bonnet Theorem, there is an exercise problem that outlines the proof of the ...
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3 votes
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Equivalence of triangulations and piecewise-linear triangulations in dimension $d\leq 4$

There are two different notions of triangulations for a manifold $\mathcal{M}$: A (simplicial) triangulation is an abstract simplicial complex $\Delta$ such that $\vert\Delta\vert\cong \mathcal{M}$. ...
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Triangulable manifolds vs manifolds admitting a handle decomposition

In differential topology we have a chain of strict inclusions $$\mathsf{Diff} \subsetneq \mathsf{PL} \subsetneq \mathsf{Triang} \subsetneq \mathsf{Top}$$ among the classes of smooth, PL, triangulable ...
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Expressing $\phi$ and $\theta$ in terms of time difference of arrival

I have an experimental setup consisting of three receivers with known locations $\langle x_i, y_i, z_i \rangle$, and a transmitter with unknown location $\langle x,y,z \rangle$ emitting a signal at ...
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Triangulations of the Torus (Example 4 from Munkres' Chapter 1.3)

Below it is possible to find an extract from Chapter 1.3 of Munkres' "Elements of Algebraic Topology", which concerns the triangulation of the torus. I have the following question regarding ...
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Recursivley count triangulations of a convex polygon

I am trying to find a recursive number of different triangulations of a convex polygon with $n$ vertices. After some searching I found that the number can be expressed using catalan numbers, this ...
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A combinatorical approach to classical Riemann-Roch

I am reading "Riemann-Roch and Abel-Jacobi Theory on a Finite Graph" by Baker and Norine (2007, arXiv 0608360). In this paper, the authors formulate abstract criterions for a set X, its set ...
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How does a triangulation of the punctured plan look like?

I know by abstract results that $\mathbb R^2-\{\mathrm{pt}.\}$ has a triangulation. By how can I visualise one?
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Triangulate point

I am reading linear algebra and there is a problem involving hyperbolic curves that struggle with. Three points $P_0=(0,0), P_1=(0,\frac{21}{4}), P_2=(0,\frac{25}{3})$. A point is located $\frac{5}{3}$...
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Are Steiner points always the vertex in Delaunay triangulation?

I'm reading up about Steiner point, and it's quite unclear whether I can use Steiner points to form the vertexes in the Delaunay triangulation. The wiki entry above doesn't seem to be clear on this. ...
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Triangulation of $S^{1}\times S^{2}$

I do not know if there are any answered question to it. To construct the triangulation of $S^{3}$ is possible to use the fact that by taking two 3-balls and identifying their boundaries $S^{2}$, is ...
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Topology and Geometry, $1$-simplex is $2$-boundary

consider page 172-173 in Topology and Geometry by Glen Bredon. In particular, Lemma 3.1. The result is: If $f,g$ are paths in $X$ such that $f(1)=g(0)$ then the $1$-chain $f\cdot g - f - g$ is a ...
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2 votes
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Gluings of tetrahedra

I am looking for all the possible ways of gluing tetrahedra to form simply connected manifolds with the topology of a sphere. I am interested in a function $f(n) = N$ that tells me, that $n$ number of ...
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Non-triangulation of torus proof

Consider $I\times I$, the unit square. (Triangulation is defined in terms of finite simplicial complexes). Then, insert a diagonal in $I\times I$ between endpoints $b$ and $d$. Then, this does not ...
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Triangulate position in 3d space from angles between known points

Suppose the observer is in 3d space at some unknown position and orientation, but there are known points scattered in 3d space. Unlike trilateration, I would only have cameras so I only know the the ...
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Longest-newest-edge bisection convergence results

I'm writing a thesis on triangulations and I'm treating the longest-edge bisection refinement technique. A bisection of a simplex $\Delta=\mathrm{conv}\,(v_0,\dots,v_n)$ is a partition of $\Delta$ ...
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euclidean distance in Voronoi polygons

Let be $E$ the set of $n \choose 2$ straight line segments ( edges ). The region $V(i) = \{ x \in E^2 \vert d(x, v_i) \leq d(x, v_j), j = 1,\dots,N\}$ is called Voronoi polygon associated with the ...
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Estimating location based on many distances and no known points

I am trying to estimate the relative positions of many points based on the distances between them, but no known points. This is for a project where I have 5-10 or more devices which are using ...
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Triangulate the polar coordinates to a 3D point by measuring the distances from three local points

Example: There is a person and a point of interest. The position of the POI is unknown. The position of the person is also unknown. But the person can take 3 distance measures from local points ...
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Triangulations of $3$-dimensional Manifold: Which Topology?

When I have some (abstract) $3$-dimensional simplicial complex, is it possible to "read of" which topology it describes. A little bit more precisely: Suppose I have a simplicial compelx and ...
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Finding a triangle that contains a given point

Let S be a set of points in $\mathbb{R}^2$. Given a point $p \in \mathbb{R}^2$, how can we find a set of $3$ points $(a,b,c)$ in $S$ so that the triangle $abc$ contains $p$ and the circumscribed ...
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A question on Polygon triangulation

The following is an extract from Dennis Zill's A First Course in Complex Analysis with Applications. The concern is in the highlighted sentence which is not even true for the Figure (7 sides, 5 ...
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Counting number of vertices in a simplicial complex

In some computation I have to do, I have to deal with the following situation: I have an arbitrary number of tetrahedra (=3 simplexes) and I am allowed to identify pairs of faces to each other. There ...
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4 votes
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How to derive an orientation from a triangulation?

Suppose some connected, closed 2-manifold $M$, and suppose I have a triangulation $t:M\rightarrow S$ where $S$ is a homeomorphic simplicial complex such that for each individual 2-cell I can specify a ...
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Are the compact three-dimensional submanifolds $\mathbb{R}^3$ unions of cubes of the unit Cartesian lattice?

Is it true that the following two classes of subsets $\mathbb{R} ^ 3$ coincide: $\{f (D_ε (A)) \,\mid\, f \, \text{is an ambient isotopy, A is the union of a finite collection of closed cubes of the ...
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2 votes
1 answer
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Find coordinates of right angle vertex in a right triangle.

I have a right triangle like shown in the image: Right triangle I know the coordinates of $\mathbf V_1$ and $\mathbf V_3$, as well as the lengths of all sides $(A, B, C)$ and angles of the vertices $(...
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How does one calculate the tracks like the ones used in tracks vehicle?

Please bear with me my English isn't good. I'm trying to make a simple Wall-E mechanism for some game. Please take a look at the simple drawing I made -> wall-e drawing Now, how can I find out the &...
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Simple triangulations of solid Cylinders

For some explicit calculation I have to do, I have to choose a triangulation of a solid cylinder as a starting point (viewed as a $3$-dimensional manifold with boundary). Can anyone tell me if the ...
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How to calculate the 3D cartesian coordinate of the Apex of a non equilateral triangle-based pyramid? [closed]

I was hoping someone could help provide a formula that can solve the (X,Y,Z) dimensions/coordinate for the Apex of the given pyramid? How can this be done using triangulation? I have modelled it in ...
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How to test if a partial triangle mesh is inside out?

I want to categorize partial trimeshes (actually regions on trimeshes) based on whether they are inside out or outside out. (We can assume consistent orientation.) Above is an example of what I am ...
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On the quality of triangles in triangulations

By indicating respectively with $\theta_{\min}$ and $\theta_{\max}$ the minimum angle and the maximum angle of a triangle, we have: $$ \text{quality}(\theta_{\min},\,\theta_{\max}) := \max\left(\frac{\...
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1 vote
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Triangulation with 4 points

I have a square room and in each corner there is a microphone that pics up noises. I need to find the position in space of this sound by using the time the sound needs to reach each microphone. I know ...
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1 vote
1 answer
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How to choose the points to triangulate a domain

Choosing Mathematica to expose the problem, defined the points: ...
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1 vote
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What are all possibilities for how many triangles when a square is triangulated?

A square with 3 points on each side used as vertices for triangulation (in addition to the vertices of the square) and 20 points inside the square is triangulated. What are all the possibilities for ...
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Is 9th grade geometry unique in the pythagorean theorem?

I've been taking Geometry and the year is ending. I have come to realize that it is centered around the Pythagorean theorem. Is it unique in this feature or will many classes be centered around this ...
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Finiteness in Prime Decomposition Theorem for 3 manifolds

In Allen Hatcher's text on 3 manifolds, he proves the Prime Decomposition Theorem by showing that a collection S of 2 spheres embedded in a smooth and compact 3-manifold M satisfying the condition ...
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3 votes
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Find the perimeter of a four-sided polygon that is formed between two triangles

Problem Statement An acute triangle intersects with another triangle to form a four-sided polygon. Given all angles, and given the distance of two sides of the polygon, find the other two sides of ...
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