Questions tagged [triangulations]
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124
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Estimating shortest paths in planar drawings of graphs
Consider a drawing (in $\mathbb{R}^2$) of a planar graph. (The drawing is given, contrarily to the common setup in graph theory where we are seeking to build a drawing with specific properties.)
For ...
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existence of triangulations of manifolds
Let $M$ be a smooth manifold.
Let $K$ be a simplicial complex.
Let ${\rm sd}(K)$ be the sub-division of $K$.
Suppose there exists a simplicial sub-complex $K_1$ of ${\rm sd}(K)$ such that $K_1$ ...
- 123
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What is an intuitive explanation for a manifold to have no triangulation?
It is known that some topological manifolds, even compact and simply-connected ones, do not have admit a triangulation. One example is the E8 manifold in a dimension as low as $4$.
I am trying to ...
- 4,926
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Triangulating piecewise-linear manifolds
Question 1: Is this the mainstream definition of a PL-manifold?
Definition. A PL-manifold is a manifold with an atlas $(\varphi_i)_{i\in I}$ in which all transition maps $\varphi_j\circ\varphi_i^{-1}$ ...
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Volume of all Voronoi cells in n-dimensional bounded space
How can one find the volume of all Voronoi cells (bounded and unbounded) in an $n$-dimensional bounded space? For instance, consider an $N$-dimensional space (hypercube) with bounds on each dimension ...
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Naming convention for different type of triangulations
When studying random geometries and related mathematical/physical stuff conflicting naming convention pops up regarding the naming of the different ensemble types of triangulations (in general ...
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Number of polyhedral covers of a triangulation of $S^2$
For a given triangulation (combinatorial Type I. or Type II.) of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?
Under polygonal cover, ...
- 75
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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 ...
- 469
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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-...
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References/applications/context for certain polytopes
First, let's consider an almost trivial notion. With any subspace $V\subset \mathbb R^n$ we associate a convex polytope $P(V)\subset V^*$ as follows. Each of the $n$ coordinates in $\mathbb R^n$ is a ...
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Sufficient condition for a Hamilton cycle $C$ in a planar triangulation $G$ s.t. every triangle in $G$ has an edge in $C$
Let $G$ be a $k$-connected planar triangulation ($k\geq 4$) and let $C$ be a Hamilton cycle of $G$. Then:
Which conditions would be sufficient to assure that every triangle of $G$ has at least one ...
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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+...
- 449
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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 ...
- 12.8k
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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 ...
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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 $q_0,q_1,\dots,q_n$...
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