All Questions
Tagged with discrete-geometry triangulations
25 questions
9
votes
4
answers
475
views
Minimum number of common edges of triangulations
Let $S$ and $T$ be two triangulations.
We define
$c(S,T)$ as the number of edges shared by $S$ and $T$.
With this, we can define
$f(n) = \min_{P} \min_{S,T} c(S,T)$.
Here the first minimum goes over ...
- 479
8
votes
2
answers
850
views
Three-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?
A side question:
If one represents this ...
- 183
8
votes
3
answers
921
views
Alternating Sum Involving Catalan Numbers
I was wondering if anyone knew how to obtain a simpler closed form of the following sum(or had any other insights regarding it):
$$\sum_{k=0}^n (-1)^k{n \choose k} C_{2n-2-k} $$
Here $C_n = \frac{1}{n+...
- 509
8
votes
0
answers
170
views
Is there a combinatorial representation of general topological manifolds similar to triangulations?
Piece-wise linear manifolds are combinatorially represented by simplicial complexes modulo Pachner moves. However, for dimensions greater than $3$, the notions of piece-wise linear and topological ...
- 3,001
7
votes
1
answer
137
views
Dropping altitudes to achieve nonobtuse planar triangulations: finite or infinite?
Given a planar triangulation of (say) a convex region,
imagine the following process to convert it to a triangulation with
no obtuse angles:
Pick an arbitrary obtuse angle at vertex $a$ of $\triangle ...
- 151k
7
votes
1
answer
186
views
How many maximal triangulations of a rectangle?
I posted the following question on MathStackExchange, but I didn't any answer. So please let me post it on MathOverflow.
Let $L_{m,n}\subset\mathbb R^2$ be a rectangle given by $[0,m]×[0,n]$ with $m,...
- 71
6
votes
1
answer
153
views
Combinatorial curvature of real projective plane
There is a notion of combinatorial curvature due to Forman, see here (published paper) or here (preprint). I checked for a couple of small triangulations of $\mathbb{RP}^2$ (6-vertex, 7-vertex, 9-...
- 17.4k
5
votes
1
answer
261
views
Do random triangulation edge-flips maintain randomness?
Let $S$ be a fixed set of $n$ points in the plane in general position.
Let $T$ be a triangulation
of $S$, (somehow) selected
uniformly at random from all triangulations of $S$.
(There are an ...
- 151k
4
votes
1
answer
304
views
Do combinatorially equivalent polytopes have the same triangulations?
A triangulation of a convex polytope $P\subset\Bbb R^n$ is a partition of $P$ into $n$-simplices $\{\Delta_1,...,\Delta_m\}$ each of which has all its vertices among the vertices of $P$. A polytope ...
- 13.6k
4
votes
1
answer
323
views
What properties does generalized Delaunay triangulation have?
Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ...
- 19.1k
4
votes
0
answers
286
views
Mean number of $n$-simplices per $(n-2)$-simplex in a triangulated $n$-manifold
Work by Tamura (extending results by Luo and Stong) shows the following.
Theorem: For any closed 3-manifold $M$ and any rational number $4.5 < r < 6$ there is a triangulation $T$ of $M$ for ...
- 175
3
votes
2
answers
323
views
Minimum weight triangulation of lattice points in a circle
Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points
$S$ inside or on the circle $C$ of radius $r$ centered on the origin.
Let $P$ be the convex hull of $S$; so $P$ is inscribed ...
- 151k
3
votes
1
answer
142
views
Uniqueness constraints for Delaunay triangulation
Commonly the assumption that is made on point sets that shall be Delaunay-triangulated is that no three are collinear and no four are cocircular.
Those assumptions are however too restrictive: if ...
- 13.2k
3
votes
0
answers
93
views
Minimal set of geometric moves in various equivalence classes of triangulated geometries
I would like to get to know what is the minimal set of geometric changes "aka. moves" (topology preserving modifications / Pachner moves / bistellar moves) that can transform any 3-...
- 183
2
votes
1
answer
92
views
Calculating a relaxed Delaunay Triangulation
The triangles of a planar Delaunay Triangulations are essentially characterized by the property that no triangle's corner is inside another triangle's circumcircle; Delaunay Triangulations can be ...
- 13.2k
2
votes
1
answer
115
views
mean length of the non-crossing graphs on n points
My original question is rather vague so I'll start with a precise example and then indicate possible generalisations.
Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...
- 1,352
2
votes
0
answers
75
views
Triangulations with discrete metrics and conformal equivalence
A discrete metric for a triangulation of a 2-dimensional manifold is a map associating $\mathbb{R}_+$-valued lengths to all edges, such that the triangle inequality holds on every triangle. In many ...
- 3,001
2
votes
0
answers
23
views
What is known about $\operatorname{card}_E(\mathrm{MST}\cap\mathrm{MWT})$?
It is a wellknown fact of computational geometry that the edges of Minimum-weight Spanning Tree are also found in the Delaunay Triangulation of a planar pointset $\mathcal{P}$, i.e. $\operatorname{...
- 13.2k
2
votes
0
answers
87
views
Existence of a "generic enough" lattice point interior to a lattice triangle
Let $T$ be a lattice triangle in $\Bbb R^2$ (i.e. the convex hull of three noncolinear points in $\Bbb Z^2$), and assume it has at least one interior lattice point. Is it always possible to find a ...
- 3,079
1
vote
1
answer
97
views
Is every triangulation the projection of a convex hull
Question:
given the triangulation $T$ of a set $P$ of $n$ points $p_1,\dots,p_n$ in the euclidean plane whose convex hull is a triangle, can we always find a set $Q$ of $n+1$ points $q_0,q_1,\dots,q_n$...
- 13.2k
1
vote
1
answer
73
views
Partitioning polygons into obtuse isosceles triangles
Ref:
Partitioning polygons into acute isosceles triangles
Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles
https://math.stackexchange.com/questions/1052063/...
- 5,979
1
vote
1
answer
179
views
Relation of MSTs in the Euclidean plane to Delaunay triangulations
It is known that the Minimum Spanning Tree (MST) of a finite set of points in the Euclidean plane is contained in the point set's Delaunay triangulation, but is that all that can be said about their ...
- 13.2k
1
vote
0
answers
62
views
What are some other methods for partitioning an n-dimensional space based on a set of points in that space?
So this is a very general question, but I'm curious if there are any other methods for partitioning an n-dimensional space based on the location of a set of points, either randomly chosen or specified,...
- 11
1
vote
0
answers
179
views
Regular triangulation of hypercube
I have started studying regular subdivisions of the $n$-cube, and came across the following post: Regularity of Delaunay triangulation of a hypercube.
My question is whether the "standard ...
- 11
0
votes
0
answers
98
views
Number of tetrahedra inside a sphere with boundary A
I understand, that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many ...
- 183