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Constrained Delaunay triangulation
About: Constrained Delaunay triangulation is a research topic. Over the lifetime, 1418 publications have been published within this topic receiving 44070 citations.
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27 May 1996
TL;DR: Triangle as discussed by the authors is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunayer refinement algorithm for quality mesh generation, and it is shown that the problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is impossible for some PSLGs.
Abstract: Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.
2,268 citations
1,517 citations
TL;DR: This paper provides a unified discussion of the Delaunay triangulation and two algorithms are presented for constructing the triangulations over a planar set ofN points.
Abstract: This paper provides a unified discussion of the Delaunay triangulation. Its geometric properties are reviewed and several applications are discussed. Two algorithms are presented for constructing the triangulation over a planar set ofN points. The first algorithm uses a divide-and-conquer approach. It runs inO(N logN) time, which is asymptotically optimal. The second algorithm is iterative and requiresO(N
2) time in the worst case. However, its average case performance is comparable to that of the first algorithm.
1,460 citations
TL;DR: The essential algorithms and techniques used to develop TetGen are presented, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations, and filtered exact geometric predicates, which can robustly handle arbitrary complex 3D geometries and is fast in practice.
Abstract: TetGen is a Cpp program for generating good quality tetrahedral meshes aimed to support numerical methods and scientific computing. The problem of quality tetrahedral mesh generation is challenged by many theoretical and practical issues. TetGen uses Delaunay-based algorithms which have theoretical guarantee of correctness. It can robustly handle arbitrary complex 3D geometries and is fast in practice. The source code of TetGen is freely available.This article presents the essential algorithms and techniques used to develop TetGen. The intended audience are researchers or developers in mesh generation or other related areas. It describes the key software components of TetGen, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations (combination of flips and edge removal), and filtered exact geometric predicates. The essential algorithms include incremental Delaunay algorithms for inserting vertices, constrained Delaunay algorithms for inserting constraints (edges and triangles), a new edge recovery algorithm for recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation. Experimental examples as well as comparisons with other softwares are presented.
1,290 citations
TL;DR: An intuitive framework for analyzing Delaunay refinement algorithms is presented that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and helps to solve the difficult problem of meshing nonmanifold domains with small angles.
Abstract: Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles Although small angles inherent in the input geometry cannot be removed, one would like to triangulate a domain without creating any new small angles Unfortunately, this problem is not always soluble A compromise is necessary A Delaunay refinement algorithm is presented that can create a mesh in which most angles are 30^o or greater and no angle is smaller than arcsin[(3/2)sin(@f/2)]~(3/4)@f, where @f=<60^ois the smallest angle separating two segments of the input domain New angles smaller than 30^o appear only near input angles smaller than 60^o In practice, the algorithm's performance is better than these bounds suggest Another new result is that Ruppert's analysis technique can be used to reanalyze one of Chew's algorithms Chew proved that his algorithm produces no angle smaller than 30^o (barring small input angles), but without any guarantees on grading or number of triangles He conjectures that his algorithm offers such guarantees His conjecture is conditionally confirmed here: if the angle bound is relaxed to less than 265^o, Chew's algorithm produces meshes (of domains without small input angles) that are nicely graded and size-optimal
1,156 citations