# Questions tagged [delaunay-triangulation]

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### How many maximal planar graphs are there?

We denote by a triangulation a (simple) maximal planar graph. How many triangulations on $n$ vertices are there? How many triangulations are there if we cannot distinguish the vertices, i.e. ...
85 views

### Delaunay triangulation when some triangles have been pre-specified?

I have an implementation of the Bowyer-Watson algorithm for Delaunay triangulation which works well -- given a set of 2D points, it computes a set of triangles to fill the areas between the points. ...
78 views

### Example of Delaunay Triangulation where it does not minimize the maximum angle

I know that that the Delaunay triangulation maximizes the minimum angle of triangulation. And it does not minimize the maximum angle. If we consider the set of points in general position(no four ...
110 views

### Non-Midpoint Segment Splitting in Ruppert's Delaunay Triangulation Refinement Algorithm

Roughly speaking, in Ruppert's Delaunay Triangulation refinement algorithm, so called encroached edges are split until no more encroached edges remain. The algorithm specifies splitting the edges ...
29 views

### Extending Delaunay graphs in d-space

I am new to computational geometry so pardon me for the lack of formalism. I am currently experimenting with an algorithm of mine in which I need to extend recursively a Delaunay graph in $d$-space. ...
801 views

### Number of “3-edge” triangles in a planar triangulation

I'm working on a triangle partitioning problem, and I'm trying to find and prove some properties of specific triangulations. The triangulations I'm dealing with are constrained delaunay triangulations ...
118 views

### Topological properties of Delaunay triangulations

Do Delaunay triangulations satisfy any extra topological conditions over normal planar triangulations? In other words, if $T$ is a topological triangulation of the plane, when does there exist points ...
2k views

### Partitioning graphs while minimizing inter-partition edges

I'm working on trying to partition a triangulated graph into connected subgraphs with some guarantees on the number of inter-partition edges. Here's an example of a triangulated graph that has been ...
1k views

### Maximum number of triangles in a constrained delaunay triangulation

I'm looking for an upper bound for the number of triangles in a constrained planar delaunay triangulation. I know for d=2 delaunay triangulation, there are at most n+1 triangles where n is the number ...
108 views

### Worst-case optimal Delaunay algorithm based on spatial sort and walking?

Buchin [2] showed that under "reasonable" assumptions, (polynomial point spread) incremental construction of Delaunay triangulations using the biased randomized ordering of Amenta et al. [1] with a ...
62 views

### Partition planar graph of vertices with at most degree 3 into connected subgraphs

I'm currently working on my thesis which deals with pathfinding over a Delaunay triangulated graph. I want to be able to partition my Delaunay triangulation into disjoint (regarding vertices) ...
438 views

### Rectangular constraints in Delaunay Triangulation without edges within

I'm using a triangulation library to compute the Constrained Delaunay Triangulation of a set of rectangles within some large boundary. The algorithm returns all the edges, but also adds edges inside ...
335 views

### Properties about verticies in Delaunay Triangulations

I'm working on my thesis dealing with pathfinding over Delaunay triangulations. I have an algorithm that has running time in time proportional to the degree of a vertex. Are there any properties or ...
53 views

### Finding if an edge lies within a set of disjoint rectangles

I'm using a triangulation library to compute the Constrained Delaunay Triangulation of a set of rectangles within some large boundary. The algorithm returns all the edges, but also adds edges inside ...
187 views

### Delaunay Triangulation of Parallelepiped

Given $n$ linearly independent vectors $v_1, v_2, \ldots, v_n$ in $n$-dimensional space. Let $V$ be the set of $2^n$ points of the form $x_1 v_1 + x_2v_2 + \ldots + x_nv_n$, in which $x_i$ can be $0$ ...
57 views

### Voronoi diagrams applications where the input order has some known properties?

Are there applications of Voronoi diagrams or Delaunay triangulations where the order in which the points are generated (and given to the algorithm) have some known properties (e.g. concatenation of ...
370 views

### Strongly edge-guarding a 3d triangulation

Let $T$ be a planar triangulation. It is known that one can guard the faces of $T$ using at most $\lfloor n/3 \rfloor$ edge-guards (Worst-case-optimal algorithms for guarding planar graphs and ...
489 views

### How to partition 3d Voronoi graph into n-number of balanced cuts while minimizing the number of edges that go between the parts?

I have a 3d Delaunay triangulation and I construct a Voronoi diagram from it. I have a computation algorithm: for each node of the Voronoi diagram compute a value based on values that neighbouring ...
1k views

### What is the worst case of the randomized incremental delaunay triangulation algorithm?

I know that the expected worst-case runtime of the randomized incremental delaunay triangulation algorithm (as given in Computational Geometry) is $\mathcal O(n \log n)$. There is an exercise which ...
Given a point set $P$ and a triangulation $T$ of $P$ with $d$ triangles, let's define $$\alpha(T) = (\alpha_1, \alpha_2, \ldots, \alpha_{3d})$$ which denotes the series of interior angles of $T$, ...