Sweep algorithms for constructing higher-dimensional constrained Delaunay triangulations

@inproceedings{Shewchuk2000SweepAF,
  title={Sweep algorithms for constructing higher-dimensional constrained Delaunay triangulations},
  author={Jonathan Richard Shewchuk},
  booktitle={SCG '00},
  year={2000}
}
  • J. Shewchuk
  • Published in SCG '00 1 May 2000
  • Computer Science
I discuss algorithms for constructing constrained Delaunay triangulations (CDTs) in dimensions higher than two. If the CDT of a set of vertices and constraining simplices exists, it can be constructed in time, where is the number of input vertices and is the number of output -simplices. In practice, the running time is likely to be in all but the most pathological cases. The CDT of a star-shaped polytope can be constructed in time, yielding an efficient way to delete a vertex from a CDT. 

The Employment of Regular Triangulation for Constrained Delaunay Triangulation

TLDR
A connection between a regular triangulation and a constrained Delaunay triangulating in 2D is demonstrated and an algorithm for edge enforcement in the constrainedDelaunay Triangulation is proposed based on the use of regularTriangulation.

Constrained Delaunay Triangulation using Plane Subdivision

TLDR
An algorithm for obtaining a constrained Delaunay triangulation from a given planar graph using an efficient Žalik’s algorithm, using a plane subdivison, is presented, which is fast and efficient and therefore appropriate for GIS applications.

The Strange Complexity of Constrained Delaunay Triangulation

The problem of determining whether a polyhedron has a con- strained Delaunay tetrahedralization is NP-complete. How- ever, if no five vertices of the polyhedron lie on a common sphere, the problem

Updating and constructing constrained delaunay and constrained regular triangulations by flips

I discuss algorithms based on bistellar flips for inserting and deleting constraining (d - 1)-facets in d-dimensional constrained Delaunay triangulations (CDTs) and weighted CDTs, also known as

Computing the 3D Voronoi Diagram Robustly: An Easy Explanation

  • H. Ledoux
  • Computer Science
    4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007)
  • 2007
TLDR
This paper describes a simple 3D Voronoi diagram (and Delaunay tetrahedralization) algorithm, and explains how to ensure that it outputs a correct structure, regardless of the spatial distribution of the points in the input.

Star splaying: an algorithm for repairing delaunay triangulations and convex hulls

TLDR
Star splaying can be a fast first step in repairing a high-quality finite element mesh that has lost the Delaunay property after its vertices have moved in response to simulated physical forces.

Incrementally constructing and updating constrained Delaunay tetrahedralizations with finite-precision coordinates

TLDR
This work proposes a new algorithm for vertex insertion, given a new vertex to be inserted into a CDT, that guarantees a new CDT including that vertex, and adds one or more Steiner points incrementally.

Incrementally Constructing and Updating Constrained Delaunay Tetrahedralizations with Finite Precision Coordinates

TLDR
This work proposes a new algorithm for vertex insertion, given a new vertex to be inserted into a CDT, that guarantees a new CDT including that vertex, and modify these algorithms to robustly succeed in practice for polygons whose vertices deviate from exact coplanarity.

Practical Ways to Accelerate Delaunay Triangulations

TLDR
This thesis proposes several new practical ways to speed-up some of the most important operations in a Delaunay triangulation, including a distribution-sensitive point location algorithm based on the classical Jump & Walk called Keep, Jump, & Walk and a filtering scheme based upon the concept of vertex tolerances.

General-Dimensional Constrained Delaunay and Constrained Regular Triangulations, I: Combinatorial Properties

TLDR
The main contributions are rigorous, theory-tested definitions of CDTs and piecewise linear complexes, a characterization of the combinatorial properties ofCDTs and weighted CDTs, the proof of several optimality properties of CDT when they are used for piecewiselinear interpolation, and a simple and useful condition that guarantees that a domain has a CDT.
...

References

SHOWING 1-10 OF 21 REFERENCES

On deletion in Delaunay triangulations

TLDR
This paper presents how the space of spheres and shelling may be used to delete a point from a d-dimensional triangu- lation efficiently, and Heller algorithm is false, as explained in this paper.

Constrained delaunay triangulations

TLDR
It is shown that the constrained Delaunay triangulation (CDT) can be built in optimalO(n logn) time using a divide-and-conquer technique and has a number of properties that make them useful for the finite-element method.

Generalized delaunay triangulation for planar graphs

TLDR
It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulated graph is maximum among all possible triangulations of the graph.

A condition guaranteeing the existence of higher-dimensional constrained Delaunay triangulations

TLDR
If the bounding segments of these facets are subdivided so that the subsegments are strongly Delaunay, then a constrained tetrahedralization exists and fewer vertices are needed than in the most common practice in the literature.

On the difficulty of triangulating three-dimensional Nonconvex Polyhedra

TLDR
The problem of deciding whether a given polyhedron can be triangulated is NP-complete, and hence likely to be computationally intractable.

Control Volume Meshes Using Sphere Packing

  • G. Miller
  • Computer Science, Physics
    IRREGULAR
  • 1998
We present a sphere-packing technique for Delaunay-based mesh generation, refinement and coarsening. We have previously established that a bounded radius of ratio of circumscribed sphere to smallest

Constructing higher-dimensional convex hulls at logarithmic cost per face

  • R. Seidel
  • Computer Science, Mathematics
    STOC '86
  • 1986
TLDR
The main tool in this new approach is the notion of a straight line shelling of a polytope in convex hull problems, which has best case time complexity O(m 2 -tFlogm), which is an improvement over the best previously achieved bounds for a large range of values of F.

A sweepline algorithm for Voronoi diagrams

TLDR
A geometric transformation is introduced that allows Voronoi diagrams to be computed using a sweepline technique and is used to obtain simple algorithms for computing the Vor onoi diagram of point sites, of line segment sites, and of weighted point sites.

A new algorithm for three-dimensional voronoi tessellation

Higher-dimensional voronoi diagrams in linear expected time

A general method is presented for determining the mathematical expectation of the combinatorial complexity and other properties of the Voronoi diagram ofn independent and identically distributed