Mesh Generation and Optimal Triangulation

Bern, Marshall; Eppstein, David

doi:10.1142/9789812831699_0003

Cited by 278 publications

(186 citation statements)

References 18 publications

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“…In x2, we require a formula for the minimal number of triangles required to triangulate the interior of a fes. The interior of a fes comprises the interiors of its separate connected subdomains, S, with boundary curves C(F k ) for k = 0 to m, as described in (2). If the boundary curves of S are all simple closed curves, then there is a standard formula for the number of triangles in a mesh on the interior of S. I f w e let T be the number of triangles, E b the number of edges on the boundary, a n d V b and V i be the number of vertices on the boundary and in the interior of S respectively, t h e n E b = V b and T = E b + 2 V i + 2 ( m ; 1)…”

Section: Resultsmentioning

confidence: 99%

A framework for advancing front techniques of finite element mesh generation

Farestam

Simpson

1995

Bit Numer Math

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Advancing front t e c hniques are a family of methods for nite element mesh generation that are particularly e ective in dealing with complicated boundary geometries. In the rst part of this paper, conditions are presented which ensure that any planar aft algorithm that meets these conditions terminates in a nite number of steps with a valid triangulation of the input domain. These conditions are described by specifying a framework of subtasks that can accommodate many aft methods and by prescribing the minimal requirements on each subtask that ensure correctness of an algorithm that conforms to the framework.An important e ciency factor in implementing an aft is the data structure used to represent the unmeshed regions during the execution of the algorithm. In the second part of the paper, we discuss the use of the constrained Delaunay triangulation as an e cient abstract data structure for the unmeshed regions. We indicate how the correctness conditions of the rst part of the paper can be met using this representation. In this case, we also discuss the additional requirements on the framework which ensure that the generated mesh is a constrained Delaunay triangulation for the original boundary.

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Section: Resultsmentioning

confidence: 99%

A framework for advancing front techniques of finite element mesh generation

Farestam

Simpson

1995

Bit Numer Math

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“…Note that, e.g., for computing the minimum weight triangulation of a simple polygon, the currently known worst case appears when there are no reflex vertices [6].…”

Section: Related Workmentioning

confidence: 99%

Geodesic-Preserving Polygon Simplification

Aichholzer

Hackl

Korman

et al. 2013

Algorithms and Computation

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Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon P by a polygon P such that (1) P contains P, (2) P has its reflex vertices at the same positions as P, and (3) the number of vertices of P is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both P and P , our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of P. We describe several of these applications (including linear-time post-processing steps that might be necessary).

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“…Among all triangulations of a fixed two-dimensional vertex set, the Delaunay triangulation is optimal by a variety of criteria-maximizing the smallest angle in the triangulation [28], minimizing the largest circumcircle among the triangles [4], and minimizing a property called the roughness of the triangulation [35,37]. A twodimensional CDT shares these same optimality properties, if it is compared with every other constrained triangulation of the same PSLG [4,29].…”

Section: Interpolation Criteria Optimized By Cdtsmentioning

confidence: 99%

“…Algorithms by Murphy et al [33], Cohen-Steiner et al [12], Cheng and Poon [8], and Pav and Walkington [34] can construct a conforming Delaunay tetrahedralization of any three-dimensional polyhedron by inserting carefully chosen vertices on the boundary of the polyhedron. (Their algorithms work not only on polyhedra, but also on a more general input called a piecewise linear complex, defined 4 Schönhardt's untetrahedralizable polyhedron (b) is formed by rotating one end of a triangular prism (a), thereby creating three diagonal reflex edges. Every tetrahedron defined on the vertices of Schön-hardt's polyhedron sticks out (c) below.)…”

Section: Introductionmentioning

confidence: 99%

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

Shewchuk

2007

Discrete Comput Geom

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Two-dimensional constrained Delaunay triangulations are geometric structures that are popular for interpolation and mesh generation because they respect the shapes of planar domains, they have "nicely shaped" triangles that optimize several criteria, and they are easy to construct and update. The present work generalizes constrained Delaunay triangulations (CDTs) to higher dimensions and describes constrained variants of regular triangulations, here christened weighted CDTs and constrained regular triangulations. CDTs and weighted CDTs are powerful and practical models of geometric domains, especially in two and three dimensions.The main contributions are rigorous, theory-tested definitions of CDTs and piecewise linear complexes (geometric domains that incorporate nonconvex faces with "internal" boundaries), a characterization of the combinatorial properties of CDTs and weighted CDTs (including a generalization of the Delaunay Lemma), the proof of several optimality properties of CDTs when they are used for piecewise linear interpolation, and a simple and useful condition that guarantees that a domain has a CDT. These results provide foundations for reasoning about CDTs and proving the correctness of algorithms. Later articles in this series discuss algorithms for constructing and updating CDTs.

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Mesh Generation and Optimal Triangulation

Cited by 278 publications

References 18 publications

A framework for advancing front techniques of finite element mesh generation

A framework for advancing front techniques of finite element mesh generation

Geodesic-Preserving Polygon Simplification

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

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