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...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 ...

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...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...

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...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...

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... 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 ...

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...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...

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...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 ...

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...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...

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...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 ...

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...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...

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...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 ...

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...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...

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...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...

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...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 ...

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...[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...

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...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...

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...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 ...

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...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...

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...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...

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... 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 ...

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...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...

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...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 ...

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...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...

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... 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 ...

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...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...