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## Meshing Piecewise Linear Complexes by Constrained Delaunay Tetrahedralizations (2005)

Venue: | In Proceedings of the 14th International Meshing Roundtable |

Citations: | 65 - 3 self |

### Citations

304 | Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms.
- Edelsbrunner, Mucke
- 1990
(Show Context)
Citation Context ...The constant C usually is no larger than 4. 6 Removing Local Degeneracies In order to fulfill the condition of Theorem 2, all local degeneracies have to be removed from D1. Techniques of perturbation =-=[6]-=- are effective to remove degeneracies. However, they must be carefully applied in CDT algorithms. We say a vertex of a PLC is perturbable if there exists an arbitrarily small perturbation on it which ... |

241 | A Delaunay Refinement Algorithm for Quality 2-Dimensional
- Ruppert
- 1995
(Show Context)
Citation Context ...s algorithm can be proved by showing that the length of every segment created is bounded by the local feature size divided by constant depending only on the input. For a PLC X, the local feature size =-=[9]-=- lfs(v) of any point v in X is the radius of the smallest ball centered ej psMeshing PLCs by CDTs 9 at v that intersects two segments or vertices in X that do not intersect each other. The lfs() defin... |

179 | Incremental topological flipping works for regular triangulations
- Edelsbrunner, Shah
- 1996
(Show Context)
Citation Context ...the newly inserted points respectively. (4) Recover the subfaces of X2 in D2 by a cavity retetrahedralization method. In step (1), D0 can be efficiently constructed by any standard algorithm, such as =-=[7]-=-. D0 probably does not respect the segments and subfaces of X0. After step (2) is done, D1 is a Delaunay tetrahedralization of vertices of X1 which contains all segments of X1. However, D1 may not res... |

133 | Tetrahedral mesh generation by Delaunay refinement
- Shewchuk
- 1998
(Show Context)
Citation Context ... properties inherited from Delaunay tetrahedralizations. Many applications can be envisaged after getting a CDT. For instance, it is a good initial mesh for getting a quality conforming Delaunay mesh =-=[13, 23, 24]-=- which is suitable for numerical methods. A key question for constructing a CDT is to decide its existence, i.e., whether a given PLC has a CDT without adding points. So far, Shewchuk [12] has proved ... |

122 |
TetGen: A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangulator - User’s Manual,"
- Si
- 2006
(Show Context)
Citation Context ...special type of PLCs - each facet is a triangle. Hence PLCs are able to approximate arbitrary complicated and curved shapes. In addition, many popular polygonal file formats (e.g., STL, OFF, PLY, and =-=[26]-=-) can be directly used or slightly modified to describe PLCs. Given a PLC X, Shewchuk [15] defined a CDT of X as follows: Let V be the set of vertices of X. σ is any simplex (tetrahedron, triangle, ed... |

80 |
Efficient threedimensional Delaunay triangulation with automatic point creation and imposed boundary constraints. Inter Algorithm for 3D Mesh Generation national Journal for Numerical Methods in
- Weatherill, Hassan
- 1994
(Show Context)
Citation Context ...tional points than other methods which do edge protect provably [17, 18, 19] too. When the existence of a CDT is known, another key issue is to recover facets of the PLC. Generally non-CDT algorithms =-=[17, 18, 19, 10, 20, 22]-=- will continuously insert points on the missing facets or the inside of the PLC. While CDT algorithms [15, 16, 25] recover the missing facets without introducing additional points. This again reduces ... |

79 | Convex partitions of polyhedra: a lower bound and worstcase optimal algorithm,
- Chazelle
- 1984
(Show Context)
Citation Context ...ly more difficult for arbitrarily shaped three dimensional domains. Such domains can be described by piecewise linear complexes (PLCs) [11], which are objects more general than polyhedra. It is known =-=[1, 2, 3, 21]-=- even a simple polyhedron may not be tetrahedralizable without adding new vertices. The simplest example is the so-called Schönhardt polyhedron [1], which is a non-convex twisted triangular prism. Mor... |

56 |
On the Difficulty of Triangulating ThreeDimensional Non-convex Polyhedra
- Ruppert, Seidel
- 1992
(Show Context)
Citation Context ...est example is the so-called Schönhardt polyhedron [1], which is a non-convex twisted triangular prism. Moreover, the problem of deciding whether a simple polyhedron can be tetrahedralized is NP-hard =-=[8]-=-. PLCs are usually much more complicated than simple polyhedra. To guarantee an arbitrary PLC can always be meshed, methods must resort to adding Steiner points. However, a number of difficult issues ... |

52 |
Uber die Zerlegung von Dreieckspolyedern in
- Schonhardt
- 1928
(Show Context)
Citation Context ...ly more difficult for arbitrarily shaped three dimensional domains. Such domains can be described by piecewise linear complexes (PLCs) [11], which are objects more general than polyhedra. It is known =-=[1, 2, 3, 21]-=- even a simple polyhedron may not be tetrahedralizable without adding new vertices. The simplest example is the so-called Schönhardt polyhedron [1], which is a non-convex twisted triangular prism. Mor... |

49 |
Constrained delaunay triangulations. Algorithmica,
- Chew
- 1989
(Show Context)
Citation Context ...wn that every polygonal domain can be triangulated into triangles without adding new vertices (the Steiner points). Almost optimal algorithms (with a linear complexity in practice) have been proposed =-=[4, 5]-=-. The problem is significantly more difficult for arbitrarily shaped three dimensional domains. Such domains can be described by piecewise linear complexes (PLCs) [11], which are objects more general ... |

44 | Control Volume Meshes using Sphere Packing- generation, refinement, and coarsening.
- Millar, Talmor, et al.
- 1996
(Show Context)
Citation Context ...n practice) have been proposed [4, 5]. The problem is significantly more difficult for arbitrarily shaped three dimensional domains. Such domains can be described by piecewise linear complexes (PLCs) =-=[11]-=-, which are objects more general than polyhedra. It is known [1, 2, 3, 21] even a simple polyhedron may not be tetrahedralizable without adding new vertices. The simplest example is the so-called Schö... |

44 | Constrained Delaunay Tetrahedralizations and Provably Good Boundary Recovery
- Shewchuk
- 2002
(Show Context)
Citation Context ...d K. Gärtner to be resolved, such as the placement of the Steiner points, the minimum bound on such points, and so on. The Constrained Delaunay tetrahedralization (CDT), proposed recently by Shewchuk =-=[15]-=- is a Delaunay-like tetrahedralization that is constrained to respect the shape of a PLC. CDTs are obviously suitable structures for resolving the above problem. Not only they respect the boundary but... |

37 | A condition guaranteeing the existence of higher-dimensional constrained Delaunay triangulations.
- Shewchuk
- 1998
(Show Context)
Citation Context ...esh [13, 23, 24] which is suitable for numerical methods. A key question for constructing a CDT is to decide its existence, i.e., whether a given PLC has a CDT without adding points. So far, Shewchuk =-=[12]-=- has proved the condition: if all segments of the PLC are strongly Delaunay, then the CDT exists. The hint gained from this condition is: additional points can be inserted only on the segments of the ... |

26 | Sweep algorithms for constructing higher-dimensional constrained Delaunay triangulations. In:
- Shewchuk
- 2000
(Show Context)
Citation Context ...[15, 16, 25] recover the missing facets without introducing additional points. This again reduces the number of Steiner points in a CDT. By now Shewchuk has provided several facet recovery algorithms =-=[14, 15, 16]-=-. The incremental facet insertion algorithm [15] recovers facets one after one and the CDT is updated accordingly. For each facet, a gift-wrapping algorithm is used for retetrahedralizing the two cavi... |

13 | Updating and constructing constrained delaunay and constrained regular triangulations by flips - Shewchuk - 2003 |

8 |
On indecomposable polyhedra
- Bagemihl
- 1948
(Show Context)
Citation Context ...ly more difficult for arbitrarily shaped three dimensional domains. Such domains can be described by piecewise linear complexes (PLCs) [11], which are objects more general than polyhedra. It is known =-=[1, 2, 3, 21]-=- even a simple polyhedron may not be tetrahedralizable without adding new vertices. The simplest example is the so-called Schönhardt polyhedron [1], which is a non-convex twisted triangular prism. Mor... |

7 | Constrained boundary recovery for three dimensional Delaunay triangulations
- Du, Wang
- 2004
(Show Context)
Citation Context ...tional points than other methods which do edge protect provably [17, 18, 19] too. When the existence of a CDT is known, another key issue is to recover facets of the PLC. Generally non-CDT algorithms =-=[17, 18, 19, 10, 20, 22]-=- will continuously insert points on the missing facets or the inside of the PLC. While CDT algorithms [15, 16, 25] recover the missing facets without introducing additional points. This again reduces ... |

6 | A Priori Delaunay-Conformity
- Pébay
- 1998
(Show Context)
Citation Context ...e sizes. Moreover sharp corners are implicitly handled during the creation of the Steiner points. Both algorithms tend to use fewer additional points than other methods which do edge protect provably =-=[17, 18, 19]-=- too. When the existence of a CDT is known, another key issue is to recover facets of the PLC. Generally non-CDT algorithms [17, 18, 19, 10, 20, 22] will continuously insert points on the missing face... |

5 | On a generalization of Schönhardt’s polyhedron
- Rambau
- 2005
(Show Context)
Citation Context |

4 |
A Robust 3D Delaunay Refinement Algorithm
- Pav, Walkington
- 2004
(Show Context)
Citation Context ... properties inherited from Delaunay tetrahedralizations. Many applications can be envisaged after getting a CDT. For instance, it is a good initial mesh for getting a quality conforming Delaunay mesh =-=[13, 23, 24]-=- which is suitable for numerical methods. A key question for constructing a CDT is to decide its existence, i.e., whether a given PLC has a CDT without adding points. So far, Shewchuk [12] has proved ... |

2 |
Gable (2000) A PointPlacement Strategy for Conforming Delaunay Tetrahedralization
- Murphy, Mount, et al.
(Show Context)
Citation Context ...e sizes. Moreover sharp corners are implicitly handled during the creation of the Steiner points. Both algorithms tend to use fewer additional points than other methods which do edge protect provably =-=[17, 18, 19]-=- too. When the existence of a CDT is known, another key issue is to recover facets of the PLC. Generally non-CDT algorithms [17, 18, 19, 10, 20, 22] will continuously insert points on the missing face... |

2 |
Borouchaki H, Saltel E (2003) ’Ultimate’ Robustness in Meshing an Arbitrary Polyhedron
- George
(Show Context)
Citation Context ...tional points than other methods which do edge protect provably [17, 18, 19] too. When the existence of a CDT is known, another key issue is to recover facets of the PLC. Generally non-CDT algorithms =-=[17, 18, 19, 10, 20, 22]-=- will continuously insert points on the missing facets or the inside of the PLC. While CDT algorithms [15, 16, 25] recover the missing facets without introducing additional points. This again reduces ... |

1 |
de Verdière E, Yvinec M (2002) Conforming Delaunay Triangulations in 3D
- Cohen-Steiner, Colin
(Show Context)
Citation Context ...e sizes. Moreover sharp corners are implicitly handled during the creation of the Steiner points. Both algorithms tend to use fewer additional points than other methods which do edge protect provably =-=[17, 18, 19]-=- too. When the existence of a CDT is known, another key issue is to recover facets of the PLC. Generally non-CDT algorithms [17, 18, 19, 10, 20, 22] will continuously insert points on the missing face... |

1 |
E A, Ray T (2004) Quality Meshing for Ployhedra with Small Angels
- Cheng, Dey, et al.
(Show Context)
Citation Context ... properties inherited from Delaunay tetrahedralizations. Many applications can be envisaged after getting a CDT. For instance, it is a good initial mesh for getting a quality conforming Delaunay mesh =-=[13, 23, 24]-=- which is suitable for numerical methods. A key question for constructing a CDT is to decide its existence, i.e., whether a given PLC has a CDT without adding points. So far, Shewchuk [12] has proved ... |

1 |
Gärtner K (2004) An Algorithm for Three-Dimensional Constrained Delaunay Triangulations
- Si
(Show Context)
Citation Context ...elds a provably good bound on edge lengths. However, it requires to compute the local feature size explicitly for protecting sharp corners (vertices with angles less than 90 ◦ formed by segments). In =-=[25]-=- we proposed a new segment recovery strategy which exploits the available local geometric information to efficiently construct Steiner points on segments. It needs not to compute the local feature siz... |