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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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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 ...
Danendra's user avatar
2 votes
1 answer
34 views

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 ...
Tob Ernack's user avatar
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2 votes
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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 ...
HackR's user avatar
  • 1,802
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47 views

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 ...
Quentin PLOUSSARD's user avatar
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30 views

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 ...
James Hall's user avatar
1 vote
1 answer
19 views

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 ⋅...
Lawton's user avatar
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40 views

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 ...
mathlover's user avatar
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27 views

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 ...
homologic's user avatar
2 votes
0 answers
69 views

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 ...
MGIO's user avatar
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2 votes
1 answer
54 views

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 ...
Einav Brin's user avatar
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18 views

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 ...
Finn T's user avatar
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0 answers
12 views

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 ...
Bazza's user avatar
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3 votes
0 answers
173 views

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 ...
curious's user avatar
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71 views

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 ...
Kangaroo976's user avatar
4 votes
1 answer
64 views

Triangulation in Monsky's Theorem

In the proof of Monsky's Theorem, which states that it is not possible to dissect a square into an odd number of triangles of equal area, it is common to use a triangulation of a unit square. The ...
Cleto Pereira's user avatar
4 votes
3 answers
178 views

Prove that every simple polygon has a ear without resorting to triangulation

We can establish the existence of a triangulation for each simple polygon relying on the fact that every simple polygon has at least one ear, utilizing induction. Conversely, we can establish that ...
log2's user avatar
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2 answers
58 views

Does geometric realization of a Δ-set collapse connected simplices to a single point?

According to Wikipedia, the geometric realization of a Δ-set is defined as the following quotient space: Each Δ-set has a corresponding geometric realization, defined as $|S|=\left(\coprod _{{n=0}}^{{...
GolDDranks's user avatar
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47 views

Finding a tetrahedron with known sides of the base and the angles opposite to these sides of the side faces.

Given a tetrahedron ABCD with known sides $a, b, c - AB, AC, BC$ of the base and the angles $\alpha, \beta, \gamma - \angle{ADB}, \angle{ADC}, \angle{BDC}$ of the side faces opposite to these sides. ...
Cyril's user avatar
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1 answer
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The formal definition of a Δ-set doesn't guarantee orientation, and its implications for gluing?

It seems to be that the formal definition of a Δ-set doesn't forbid identifying different faces of a simplex, that is, face maps $d_i$ and $d_j$, $i \ne j$ may map element $a : S_n$ to a same element $...
GolDDranks's user avatar
3 votes
3 answers
348 views

Definition of a Δ-set?

According to Wikipedia, Formally, a Δ-set is a sequence of sets $\{S_{n}\}_{{n=0}}^{{\infty}}$ together with maps ${\displaystyle d_{i}\colon S_{n+1}\rightarrow S_{n}}$ with ${\displaystyle i=0,1,\...
GolDDranks's user avatar
0 votes
0 answers
15 views

Surface Discretization with Equi-Sized Squares

I am currently working with a triangulated surface, represented by a list of vertices (x, y, z coordinates) and triangles defined by three vertices each. Notably, the triangles vary in size, and their ...
roymustang's user avatar
1 vote
1 answer
64 views

is every triangulation of regular n-gon has same minimum angle?

Although my knowledge maybe not correct, but I will state it nonetheless; "Delaunay triangulation maximizes the minimum angle among all triangulation" "Every triangulation of regular n-...
K.R.Park's user avatar
  • 113
1 vote
1 answer
56 views

Triangulation trigonometry

I’ve been looking at triangulation calculations, and I’ve become a bit stumped as to how the authors of the attached document have come to the following calculations. In the page it explains how one ...
Richard's user avatar
  • 11
5 votes
1 answer
567 views

Can one compute the location of the unseen point?

My question is quite simple. I have two images, on the first one I know the location of points $P1, P2, P3$, and $P4$. In the second image, I know the location of $P2'$, $P3'$, $P4'$, and point $Q'$. ...
apraglez's user avatar
0 votes
0 answers
59 views

Ordered Delaunay triangulations

I would like to show that, given n points in the plane $q_1 ... q_n$ such that the distance between $q_0$ and $q_i$ is smaller than or equal to the distance between $q_0$ and $q_j$ for every $i < j$...
Baksel's user avatar
  • 81
1 vote
1 answer
88 views

Do two triangulations of a smooth manifold have a common subdivision?

The Hauptvermutung (ie. the question in the title) is known to be false for PL manifolds and topological manifolds, but I can't find a result for smooth manifolds (with boundary), though I recall ...
JLA's user avatar
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1 vote
0 answers
58 views

Why are there only finitely many simplicial maps from one polyhedron to another?

I don't understand why for two polyhedra $|X|$ and $|Y|$, there are finitely many choices of simplicial maps $$s: |X^m| \rightarrow |Y|$$ for some large enough $m \in \mathbb{N}$. Multiple sources say ...
Carson Newman's user avatar
1 vote
1 answer
62 views

Bicolored triangulations of $S^2$ with certain conditions on degrees of vertices

Finite sets $B,W\subset\mathbb N^2$ are given. Suppose $G=(V,E)$ is a triangulation of a two-dimensional sphere so that $V=V_b\sqcup V_w$. We say that $G$ is a $(B,W)$-triangulation if $(\deg_bv,\...
te4's user avatar
  • 235
2 votes
1 answer
417 views

Triangulation vs. Trilateration

Are triangulation and trilateration different words for the exact same procedure, or is there a small technical difference between the two? If there is a small difference, which is the correct term ...
Nate's user avatar
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1 vote
0 answers
42 views

Does every smooth manifold have a Hamiltonian triangulation?

Call a triangulation of a smooth manifold Hamiltonian if its 1-skeleton has a Hamiltonian cycle. I have several questions about these that I haven't been able to find answers to. First, every smooth ...
JLA's user avatar
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0 votes
0 answers
34 views

Are these maps homotopic relative to a finite set of points?

Let $M,N$ be (compact) smooth manifolds and consider a triangulation $\mathcal{T}$ of $M.$ Consider two maps $f,g:M\to N$ which agree on the vertices of the triangulation. Is it true that $f$ and $g$ ...
JLA's user avatar
  • 6,534
1 vote
1 answer
77 views

Why does direct linear transform (DLT) yield a unique solution?

I try to triangulate point correspondences from 2 images in order to reconstruct the 3D positions of those points. I found the DLT method as an easy way to achieve that. The system which needs to be ...
NMO's user avatar
  • 151
6 votes
0 answers
206 views

Counting problem about polygon triangulations

I have the following question about triangulations (by non-intersecting diagonals, and edges) of regular polygons. What is the number of triangulations of a regular n-gon, up to all symmetry (i.e. the ...
Andrea B.'s user avatar
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0 votes
0 answers
79 views

3-dimensional triangulations with fixed number of vertices

My question is the following: Are there triangulations of $S^3$ which (a) are non-degenerate, (b) have four vertices, and (c) have no edges of degree two? Here are the definitions. We use labeled ...
Kregnach's user avatar
1 vote
1 answer
51 views

Package to enumerate all regular triangulations for point configuration

I am trying to enumerate the regular triangulation of some point configurations. The sage-math can enumerate all triangulations, but cannot check if each one is regular or not. It seems TOPCOM can ...
Jay's user avatar
  • 301
4 votes
1 answer
195 views

Does every surface admit a quadrangulation?

Every surface (2-manifold) admits a triangulation, and I wonder if the same can be said for quadrangulation. My intuition is that every orientable surface can be quadrangulated, but I'm not sure about ...
chaohuang's user avatar
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0 votes
1 answer
143 views

Why isn't this a correct triangulation of the torus/projective plane?

This is to be proven an incorrect triangulation of the torus: And this is to be proven an incorrect triangulation of the projective plane: I would like to know what part of the definition of ...
David Huélamo Longás's user avatar
2 votes
1 answer
135 views

Triangulation of a surface

In their 2015 paper on random simplicial complexes, Costa and Farber casually mention without a proof that in triangulation of a closed surface, there is a formula $$\frac{\text{number of $1$-faces}}{\...
the_dude's user avatar
  • 596
1 vote
1 answer
110 views

Non-delaunay triangulation of a set of points and its convex hull boundary

I have a set of n random points on a 2D plane, and its convex hull boundary. Some points lie interior to the convex hull boundary. I want to triangulate this in a non-delaunay random fashion, such ...
mark_52's user avatar
  • 63
0 votes
0 answers
60 views

Rotating matrices to display data from radar onto a HUD

I'm trying to write a code (in-game scripts) that takes data from radar (distance, azimuth, elevation) and displays points on a HUD in front of the player. Yesterday, I've managed to visualize radar ...
N3ttX's user avatar
  • 1
0 votes
1 answer
39 views

Second closest red-blue theorem

I want to prove/disprove that for a set $S = R \cup B$ of $n \geq 3$ colored (red and blue) points in general position (no 3 are collinear, no 4 are cocircular, and the distances between the $n \...
mark_52's user avatar
  • 63
3 votes
0 answers
75 views

Identifying the delaunay simplex the point resides in without computing all the possible simplices

My question is whether it is possible to identify the delaunay simplex the point $\mathbf{x}_{\textrm{query}}$ resides in without pre-calculating all the possible simplices by triangulation? Can we ...
Karbo Lei's user avatar
1 vote
0 answers
78 views

Show that length of longest edge in Delaunay triangulation goes to zero

Suppose that we have a set of vertices $V$ in a certain space $\Omega$ and that we iteratively add a vertex to $V$ and build a Delaunay triangulation on these vertices. Now, if $\mathcal{T}^1$, $\...
user675763's user avatar
2 votes
1 answer
212 views

Lifting triangulation by branched covering map

Let $X, Y$ be compact connected manifolds of the same dimension, and let $f : X \to Y$ be a branched covering with finitely many branch points. (For example, $f$ could be a holomorphic function ...
Frank's user avatar
  • 2,625
4 votes
1 answer
77 views

A single-color path in a two-color triangulated square

Take any triangulation of a square (= a partition into triangles such that any two triangles are either disjoint or intersect in a common face). Suppose that there are no triangulation vertices on the ...
Erel Segal-Halevi's user avatar
1 vote
0 answers
29 views

Triangulation of the complement VS Complement of the triangulation

Consider a triangulable space $X$ in the $n$-sphere $S^n$ whose complement $X^c=S^n\backslash X$ is also triangulable: there exists finite simplicial complexes $T_X$ and $T_{X^c}$ whose geometric ...
OneC2's user avatar
  • 13
0 votes
0 answers
98 views

How do you find an unknown point via the known location of other points and the angle of the unknown point when facing the known points?

Based on this previous question, but slightly different: Find a point using multilateration or triangulation Suppose we have 3 points in a 3d coordinate system with the following locations. A=(100,0,0)...
Christian Blevens's user avatar
1 vote
0 answers
28 views

Isolate downward-facing faces of a convex hull

long time fan / first time poster here! My question: is there an efficient algorithm for isolating just the downward-facing faces of a 3D convex hull? In two dimensions, if one draws a line from the ...
thermo_charlie's user avatar
2 votes
0 answers
67 views

Theorem 5, Section 6.4 of Hoffman’s Linear Algebra

Let $V$ be a finite-dimensional vector space over the field $F$ and let $T$ be a linear operator on $V$. Then $T$ is triangulable if and only if the minimal polynomial for $T$ is a product of linear ...
user264745's user avatar
  • 4,227
2 votes
1 answer
138 views

Ratio of largest to smallest distance in a set of six points is $\ge \sqrt 3$?

Here is problem A1 of 25th Putnam 1964. Let $A_1, A_2, A_3, A_4, A_5, A_6$ be distinct points in the plane. Let $D$ be the longest distance between any pair, and $d$ the shortest distance. Show that $...
Oliver G's user avatar
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