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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Triangulations of manifolds are non-branching
Let $X$ be an $n$-manifold and let $K$ be a triangulation of that manifold. I am looking for a proof of the fact that $K$ is non-branching, which means:
There is no simplex $S \in K$ of dimension $n-1$...
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Delaunay Triangulation but in 3D
I guess this is the right place to ask this question. Let me tell you why did I ask this question, so I have a pointcloud data that I want to calculate it's volume, I know that pointcloud lib has ...
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Existence of smooth triangulation for Riemannian 2-manifold
Most proofs that I can find of the Gauss-Bonnet Theorem for a compact Riemannian $2$-manifold $M$ always start with the assumption that $M$ has a smooth triangulation, i.e. a triangulation where the ...
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A certain proof method for Ky Fan's sphere covering theorem
Ky Fan's theorem (1952) for sphere covering states the following:
Let $A_1, A_2,\dots,A_m$ be an antipodal-free (which means $A_i\cap (-A_i)=\varnothing$) closed (can be open, let's go with closed ...
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Finding all empty triangles of a plane
I have a set of $N$ points ${(x_i,y_i)}_{i=1,...,N}$. I am looking for an efficient algorithm to find the set of all empty triangles (i.e., that do not contain any points).
The brute-force method that ...
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How to triagulate multiple sound locations
Other people posed the question of how to triangulate sound from multiple locations.
Approximate (but as accurate as it can) location of sound
Sound Triangulation
My question is how to seperate ...
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Is there a straightforward way to triangulate this tetrahedrally-symmetric convex surface according to these criteria?
I have a tetrahedrally-symmetric surface of constant width defined in spherical coordinates by the support function
$$
h(θ, φ) = \frac{S}{16} ⋅ \left(\sin(θ)^3 ⋅ \cos(3 ⋅ φ) + \frac{5 ⋅ \cos(θ)^3 - 3 ⋅...
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Real-life interpretations of a Miklos Schweitzer problem
This is P8 from the 2002 Miklós Schweitzer competition:
Given $n$ points in general position. Show that one can color these points using at most $c\log n$ colors for some constant $c$, so that any ...
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Proof of the fact that every 4-dimensional triangulation is PL
I've been trying to wrap my head around the fact that every triangulation of a 4-manifold must be PL. I have found the following answer:
Equivalence of triangulations and piecewise-linear ...
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Please help me to understand triangulation
Yesterday, my teacher gave us an example of triangulation of torus($18$ triangles) without gave us the exactly definition of triangulation and told us if you want to know more, just read book about ...
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What are the possible surfaces that one can construct from a finite set ot triangles?
I am looking for references in discrete differential geometry for a concept I've been interested in.
It is very common to approximate smooth surfaces using discrete triangulations. I am interested in ...
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Visualising the interior gluings of a 3D shape in 2D
I have a small triangulation of a 3-ball that I'm trying to form a nice 'visualisation' of for a paper/talk.
The best I've got so far is a few rough sketches like the one below, where I've tried to ...
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Distribution of the lengths of edges of the Delaunay triangulation?
Consider the unit square $I^2 = [0,1]^2$ and suppose we have choose $n$ points at random from $I^2$ where the points are taken from the uniform distribution on $I^2$. Call this space $X_{n}$. Can ...
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A triangle is cut into several triangles, one isosceles (not equilateral) and the rest equilateral. Determine the angles of the original triangle.
This question has been taken from III GEOMETRICAL OLYMPIAD IN HONOUR OF I.F.SHARYGIN:
A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all ...
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Minimum number of points to have a point inside every triangle formed by $n$ points
Place $n$ points in a general position on the plane. Call a set $S$ of any points stabbing if every triangle formed by the $n$ chosen points contains at least one point from $S$ in its interior. For ...
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