# Tagged Questions

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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### split a rectangle with triangles into polygons as uniformly as possible

Given a rectangle $A$ and $n$ triangles $\{B_1,B_2,...,B_n\}$, I put the triangles inside $A$, at least one vertex of each triangle is not outside $A$ (inside $A$ or on the edge of $A$). So that A is ...
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### Triangulation with two camera setup - Result in world or camera coordinate system?

I have some problems to understand how I can triangulate a 3D-point using a two camera setup. Let's assume I'm using a right handed coordinate system and the camera is looking in positive z-...
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### Is this triangulation consistent with Sperner's Lemma?

Since any two triangles which intersect have an edge or a vertex in common, the triangulation is simplicial. However, I am concerned about triangle $A$. Is every sub-triangle supposed to have a ...
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### Clarification of Sperner's Lemma

From Graph Theory by Bondy, Murty Image from wikipedia I don't see how the picture holds according to the definition from the Graph Theory book. Specifically, the definition says to assign ...
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### Using GPS coordinates in trillateration

for a project we need to find a certain position. The info we have : 3 surrounding positions and the distance between those positions and the point we are looking for. We've got a setup like this ...
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### Voronoi edges example

I have 4 line segments: 0 0 2 0 // 1st line segment 2 0 2 1 // 2nd line segment 2 1 0 1 0 1 0 0 and I wrote some CGAL code to print the Voronoi edges. However, <...
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### Fundamental group of cylinder

I calculated the fundamental group of the cylinder, $C$, using the following method: triangulate $C$ find max contractable subspace realise generators on remaining 1-simplices I found the ...
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### orientation independent of triangulation

I have $2$-dimensional compact connected orientable manifolds... It is known that those can be triangulated in such a way that everything fits nicely: number of triangles is finite, each edge is edge ...
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### What is the number of interior faces adjacent to an interior vertex in a triangulation in $\mathbb{R}^3$?

Let $\Omega$ be a polygonal domain in $\mathbb{R}^3$. Assume $\Omega$ is partitioned into tetrahedra using the most common admissible triangulation, that is, roughly speaking, two adjacent tetrahedra ...
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### Triangulation of double-holed torus to calculate fundamental group

Show that the fundamental group of the double-holed torus is given by: $\pi_1=<a, b, c, d | aba^{-1}b^{-1}=cdc^{-1}d^{-1}>$ I have ended up with the following identification diagram ...
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### The fundamental group of a point is $1$

Show that the fundamental group of the point space $p$ is given as $\pi(p, w_0)=1$ where $w_0$ is the base point This is probably somewhat trivial, but I am looking for a proof. I am familiar with ...
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### Why describe basis multipliers as barycentric coordinates?

So a disclaimer up front: I'm from a EECS background as opposed to pure math, so if possible keep that in mind for your answers. I've been reading a paper on 2D-3D triangulation and came across the ...
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### Area of Convex hull

For every point set $A \subset R^2$, prove that in general the sum of the coordinates of $\phi(T)$ is independent of a triangulation T and is associated to the area of the Convexv_Hull(A). We define ...
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### Compact 3-manifold implies finite triangulability

I know that it's a theorem by Moise that every compact 3-manifold admits a finite triangulation but to me the astounding part of that statement is the existence part instead of the finite one. So I ...
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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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### 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 ...