Questions tagged [delaunay-triangulation]

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9
votes
1answer
660 views

Do Delaunay triangulations on the sphere maximize the minimum angle?

Delaunay triangulations in the plane maximize the minimum angle in a triangle. Does the same hold true for the Delaunay triangulation of points on the sphere ? (here the "angle" is the local angle in ...
9
votes
1answer
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 ...
8
votes
4answers
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 ...
6
votes
3answers
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 ...
5
votes
1answer
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 ...
5
votes
0answers
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 ...
4
votes
2answers
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 ...
4
votes
1answer
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 ...
4
votes
0answers
196 views

Triangulation with maximum greatest area

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 $T$, ...
3
votes
0answers
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 ...
2
votes
0answers
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 ...
2
votes
0answers
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 ...
1
vote
1answer
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 $0$ ...
1
vote
1answer
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. ...
1
vote
0answers
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. ...
1
vote
0answers
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) ...
1
vote
0answers
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 ...
0
votes
1answer
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 ...
0
votes
0answers
87 views

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. ...
0
votes
0answers
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 ...
-1
votes
1answer
85 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 ...