TetGen

A Quality Tetrahedral Mesh Generator and

3D Delaunay Triangulator

Version 1.5
User’s Manual
Also available as WIAS Technical Report No. 13, 2013

Hang Si
Hang.Si@wias-berlin.de
http://www.tetgen.org
c 2002 – 2013
ii
 Abstract
TetGen is a software for tetrahedral mesh generation. Its goal is to
generate good quality tetrahedral meshes suitable for numerical meth-
ods and scientific computing. It can be used as either a standalone
program or a library component integrated in other software.
The purpose of this document is to give a brief explanation of
the kind of tetrahedralizations and meshing problems handled by Tet-
Gen and to give a fairly detailed documentation about the usage of the
program. Readers will learn how to create tetrahedral meshes using
input files from the command line. Furthermore, the programming
interface for calling TetGen from other programs is explained.

keywords: tetrahedral mesh generation, Delaunay tetrahedraliza-

tion, weighted Delaunay triangulation, constrained Delaunay tetra-
hedralization, mesh quality, mesh refinement, mesh adaption, mesh
coarsening

AMS Classification: 65M50, 65N50

iii
iv
Contents

Contents v

1 Introduction 1
1.1 Triangulations of Point Sets . . . . . . . . . . . . . . . . . . . 2
1.1.1 Delaunay Triangulations, Voronoi Diagrams . . . . . . 2
1.1.2 Weighted Delaunay Triangulations, Power Diagrams . . 3
1.1.3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Tetrahedral Meshes of 3d Spaces . . . . . . . . . . . . . . . . . 6
1.2.1 Piecewise Linear Complexes (PLCs) . . . . . . . . . . . 6
1.2.2 Steiner Points . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Boundary Conformity . . . . . . . . . . . . . . . . . . 9
1.2.4 Constrained Delaunay Tetrahedralizations . . . . . . . 10
1.2.5 Mesh Quality, Tetrahedron Shape Measures . . . . . . 11
1.2.6 Mesh Size, Mesh Adaptation . . . . . . . . . . . . . . . 13
1.2.7 Mesh Optimization . . . . . . . . . . . . . . . . . . . . 14
1.2.8 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Description of the Meshing Process . . . . . . . . . . . . . . . 16

2 General Information 18
2.1 Language, Platforms . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Memory requirement . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 CPU time estimation . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Getting Started 22
3.1 Compilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.1 Using make . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.2 Using cmake . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.3 Remarks on Using Shewchuk’s Robust Predicates . . . 24
3.1.4 Using CGAL’s Robust Predicates . . . . . . . . . . . . 25
3.2 A Short Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 TetView . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 Medit and Paraview . . . . . . . . . . . . . . . . . . . 30

v
4 Using TetGen 31
4.1 Command Line Syntax . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Command Line Switches . . . . . . . . . . . . . . . . . . . . . 31
4.2.1 Delaunay and weighted Delaunay tetrahedralizations . 33
4.2.2 Boundary conformity and recovery (-p, -Y) . . . . . . . 36
4.2.3 Quality mesh generation (-q) . . . . . . . . . . . . . . . 39
4.2.4 Adaptive mesh generation (-a, -m) . . . . . . . . . . . 41
4.2.5 Reconstructing a tetrahedral mesh (-r) . . . . . . . . . 43
4.2.6 Mesh optimization (-O) . . . . . . . . . . . . . . . . . 44
4.2.7 Mesh coarsening (-R) . . . . . . . . . . . . . . . . . . . 45
4.2.8 Inserting additional points (-i) . . . . . . . . . . . . . . 45
4.2.9 Assigning region attributes (-A) . . . . . . . . . . . . . 45
4.2.10 Mesh output switches (-f, -e, -n, -z, -o2) . . . . . . . . 46
4.2.11 Mesh statistics (-V) . . . . . . . . . . . . . . . . . . . . 47
4.2.12 Memory allocation (-x) . . . . . . . . . . . . . . . . . . 48
4.2.13 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . 49

5 File Formats 51
5.1 Useful Things to Know . . . . . . . . . . . . . . . . . . . . . . 51
5.1.1 A Boundary Description of PLCs . . . . . . . . . . . . 51
5.1.2 Boundary Markers . . . . . . . . . . . . . . . . . . . . 52
5.2 TetGen’s File Formats . . . . . . . . . . . . . . . . . . . . . . 52
5.2.1 .node files . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2.2 .poly files . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2.3 .smesh files . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2.4 .ele files . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2.5 .face files . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.6 .edge files . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.7 .vol files . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.8 .mtr files . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.9 .var files . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.10 .neigh files . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.11 .v.node, .v.edge, .v.face, .v.cell . . . . . . . . . . . . . . 66
5.3 Supported File Formats . . . . . . . . . . . . . . . . . . . . . 68
5.3.1 .off files . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.2 .ply files . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.3 .stl files . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.4 .mesh files . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4 File Format Examples . . . . . . . . . . . . . . . . . . . . . . 70
5.4.1 A PLC with Two Boundary Markers . . . . . . . . . . 70
5.4.2 A PLC with Two Sub-regions (Materials) . . . . . . . 73

vi
 5.4.3 A PLC with Two Sub-regions and Two Holes . . . . . 75

6 Calling TetGen from Another Program 77

6.1 The Header File . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 The Calling Convention . . . . . . . . . . . . . . . . . . . . . 77
6.3 The tetgenio Data Type . . . . . . . . . . . . . . . . . . . . 78
6.4 Description of Arrays . . . . . . . . . . . . . . . . . . . . . . . 79
6.4.1 Memory Management . . . . . . . . . . . . . . . . . . . 81
6.4.2 The facet Data Structure . . . . . . . . . . . . . . . . 82
6.5 A Complete Example . . . . . . . . . . . . . . . . . . . . . . . 83

A Basic Definitions 88
A.1 Simplices, Simplicial Complexes . . . . . . . . . . . . . . . . . 88
A.2 Polyhedra and Faces . . . . . . . . . . . . . . . . . . . . . . . 89
A.3 CSG and B-Rep Models of 3d Domains . . . . . . . . . . . . . 90

B List of Error Codes and Messages 90

References 91

Index 94

vii
 1

1 Introduction

TetGen is a robust, fast, and easy-to-use software for generating tetrahedral

meshes suitable in many applications.
For a set of 3d (weighted) points, TetGen generates the Delaunay and
weighted Delaunay tetrahedralization as well as their duals, Voronoi diagram
and power diagram, of it. For a 3d polyhedral domain, TetGen generates the
constrained Delaunay tetrahedralization and an isotropic adaptive tetrahe-
dral mesh of it. Domain boundaries (edges and faces) are respected and can
be preserved in the resulting mesh. The shapes of resulting tetrahedra can
be provably good for a large class of inputs. One of its main applications is to
simulate physical phenomena by numerical methods, such as finite element
and finite volume methods. A good quality mesh is essential to achieve high
accuracy and efficiency of the simulations.
The algorithms of TetGen are Delaunay-based. They can preserve arbi-
trary complex geometry and topology. It uses a constrained Delaunay refine-
ment algorithm which has theoretical guarantees on termination and mesh
quality. The robustness of TetGen is enhanced by using advanced technolo-
gies developed in computational geometry. A technical paper describing the
algorithms and technologies used in TetGen is available [24].
TetGen is written in C++. It uses only C standard library. It is easy to
compile and runs on all major 32-bit and 64-bit computer systems. The
source code of TetGen is freely available at http://www.tetgen.org.
TetGen is an outcome of a long-term research project supported by Weier-
strass Institute (WIAS). It is continuously developed and improved.
The remainder of this section is to give a brief description of the triangu-
lation and meshing problems considered in TetGen, and an overview of the
implemented algorithms. For basic usage of TetGen, most of the information
are not necessary to know, but Sections 1.2.1 and 1.2.5 contain some neces-
sary guidelines to create correct inputs and to generate quality tetrahedral
meshes.
2 1 INTRODUCTION

Figure 1: The Delaunay triangulation (left) and its dual Voronoi diagram
(right) of a 2d point set.

1.1 Triangulations of Point Sets

Triangulations are basic geometric structures. A triangulation of a set V of
points is a simplicial complex S whose vertex set is a subset of or equal to V ,
and the underlying space of S is the convex hull of V . Given a point set, there
are many triangulations of it. Among them, the Delaunay triangulation is
of the most interested one. Its dual is the Voronoi diagram of the point set.
Delaunay triangulations and Voronoi diagrams have many nice mathematical
properties [1, 12, 7]. They are extensively used in many applications.

1.1.1 Delaunay Triangulations, Voronoi Diagrams

Let V be a set of points in Rd , σ be a k-simplex (0 ≤ k ≤ d) whose vertices

are in V . The circumsphere of σ is a sphere that passes through all vertices of
σ. If k = d, σ has a unique circumsphere, otherwise, there are infinitely many
circumspheres of σ. We say that σ is Delaunay if there exists a circumsphere
of σ such that no vertex of V lies inside it.
A Delaunay triangulation D of V is a simplical complex such that all
simplices are Delaunay, and the underlying space of D is the convex hull of
V [6]. Figure 1 left illustrates a 2d Delaunay triangulation. A 3d Delaunay
triangulation is also called a Delaunay tetrahedralization.
A Delaunay triangulation of V is unique if V is in general position, i.e.,
no d + 2 points in V lie on a common sphere. Otherwise, we say that V
contains degeneracies, i.e., there are d+2 points in V lie on a common sphere.
Degeneracies can be removed by applying an arbitrary small perturbation
onto the coordinates of points in V .
The dual of the Delaunay triangulation is the Voronoi diagram defined on
the same vertex set (see Figure 1 right). For any vertex p ∈ V , the Voronoi
cell of p, is the set of points with distance to p not greater than to any other
1.1 Triangulations of Point Sets 3

p+

x/1.0e-01

Figure 2: The relation between Delaunay triangulation in Rd and convex

hull in Rd+1 (here d = 2). Left: Some 2d points and their corresponding 3d
lift points. Right: The Delaunay triangulation of a set of 2d points and the
lower convex hull of its 3d lifted points.

vertex of V , i.e. it is the set cell(p) = {x ∈ Rd ; kx−pk ≤ kx−qk, ∀q ∈ V },

where k · k stands for the Euclidean distance. The Voronoi diagram of V is a
subdivision of Rd into Voronoi cells (some of which may be unbounded) and
their faces [27]. It is a d-dimensional polyhedral complex. If the point set
V is in general position, there is a one-to-one correspondence between the
k-simplices of the Delaunay triangulation and the (d − k)-polyhedra of the
Voronoi diagram, where 0 ≤ k ≤ d. In R3 , the vertices of the Voronoi diagram
are the circumcenters of the tetrahedra of the Delaunay tetrahedralization.
There is a nice relation between a Delaunay triangulation in Rd and a
convex hull in Rd+1 . For any point p = (p0 , p1 , · · · , pd−1 ) ∈ Rd , define its
lifted point p+ = (p0 , p1 , · · · , pd−1 , pd ) ∈ Rd+1 , where pd = p20 + · · · + p2d−1 .
For any point set V ⊂ Rd , define V + = {p+ ; p ∈ V } ⊂ Rd+1 be the lifted
point set of V . All points in V + lie on a paraboloid in Rd+1 (see Figure 2
left). The convex hull of V + is a (d + 1)-dimensional convex polytope P .
A lower face of P is a face of P which is on the downside of P (visible by
points in V ). The Delaunay triangulation of V is the projection of the set of
lower faces of P onto d dimensions. Figure 2 right illustrates the relationship
when d = 2. A simplex σ is a Delaunay simplex if and only if there exists
a hyperplane in Rd+1 passing through the lifted vertices of σ such that no
other lifted vertices in V + lies below of it. Similarly, the Voronoi diagram of
V is the projection of the lower faces of a convex polytope Q ⊂ Rd+1 such
that P and Q are polar to each other [29].

1.1.2 Weighted Delaunay Triangulations, Power Diagrams

Weighted Delaunay triangulations are generalization of Delaunay triangula-
tions by replacing the Euclidean distance by “weighted distance”.
A weighted point, p0 = (p, p2 ) ∈ Rd × R, can be interpreted as a sphere
4 1 INTRODUCTION

Z
P
p z

Figure 3: Left: The weighted

Z distance (left) of two weighted points (p, p2 )
and (z, z 2p). Right: The
Figure orthosphere
2.13:
z
Orthospheres
p of three
are weighted
generalization
z points. (Figures are
of circumspheres.
taken from Damrong Guoy’s PhD thesis.)

centered at p with radius p. The weighted distance between p0 and z0 is

Figure 2.11: The weighted distance between two circles equals the length of the second
tangent line. The weighted distance from a point to a circle equals the length of the tangent
line. The weighted distance from one point to p
another is their Euclidean distance.
πp0 ,z0 = kp − zk2 − (p2 + z 2 ),

seeTwo
Figure 3 left
Euclidean pointsfor
havean example.
zero distance whenIn particular,
they points
are the same point. Two in Rd can be considered
weighted

weighted points
points p̂, ẑ have with distance
zero weighted zero weight.
when they are orthogonal, meaning that the two

Two weighted points p0z�

spheres intersect at right angle i.e. �p − , 2z=0 (P
are2
+ orthogonal if their
Z 2 ). We also define weighted
a point to be distance is
zero, i.e.,to a sphere when the sphere passes through the point, which means that the
orthogonal
weighted distance is also zero. Figure 2.12 kp − zk2 =orthogonality
demonstrates (p2 + z 2in).two dimensions.
The book by Pedoe [46] offers a comprehensive study on the system of orthogonal circles. For
We say that two weighted points are farther than orthogonal when their
completeness, we say two weighted
weighted distance is positive, points are furthercloser
than orthogonal when their weighted
Figure 2.14: Twoand weighted Delaunaythantriangulations
orthogonal ofwhen
a cube.the distance
becomes an imaginary number.
distance is positive, and closer than orthogonal when the distance becomes an imaginary

In general, d+1 points in Rd define a unique circumsphere passing through

number.

them. Similarly, d + 1 weighted points in Rd define a unique common or-

In general, four points in R3 define a unique circumsphere passing through them. Sim-
21
thosphere. When all points have zero weights, their orthosphere is just their
ilarly four weighted points p̂0 , p̂1 , p̂2 , p̂3 in three dimensions define a unique common ortho-
circumsphere. Figure 3 (right) gives an example of the orthosphere of three
sphere(orthogonal sphere) ẑ by the system of equations
weighted points in two dimensions.
Let V 0 ⊂ Rd �z ×−Rp �2be a finite set of weighted points. We say a sphere
− (Z 2 + Pi2 ) = 0, 0 ≤ i ≤ 3.
i
is empty if all weighted points in V 0 are farther than orthogonal of it. The
19
weighted Delaunay triangulation of V 0 is a simplicial complex D0 such that
every simplex has an orthosphere which is empty, and the underlying space
of D0 is the convex hull of V 0 . Obviously, if all the points have the same
weight, the weighted Delaunay triangulation is the same as the usual Delau-
nay triangulation. Note that, a weighted Delaunay triangulation does not
necessarily contains all points in V 0 .
The dual of a weighted Delaunay triangulation is a weighted Voronoi
diagram, also called the power diagram [1, 9] of the weighted point set V 0 .
Power diagrams can be similarly defined as the Voronoi diagram by using
the weighted distance instead of the Euclidean distance. If no d + 2 weighted
points of V 0 share a common orthesphere, i.e., it is in general position, then
1.1 Triangulations of Point Sets 5

the simplices of the weighted Delaunay triangulation and the cells of the
power diagram have a one-to-one correspondence. In R3 , the vertices of
the power diagram are the orthocenters of the tetrahedra of the weighted
Delaunay tetrahedralization.
A weighted Delaunay triangulation of V ⊂ Rd is also the projection
of the set of lower faces of a convex polytope P ⊂ Rd+1 . Any point in
p = {p0 , ..., pd−1 } ∈ V is lifted to a point p0 = {p0 , ..., pd−1 , pd } ∈ Rd+1 ,
where pd = p20 + · · · + p2d−1 − p2 (p is the weight of p). For p 6= 0, p0 does
not lie on a paraboloid in Rd+1 , but is moved vertically downward by p2 . A
simplex belongs to the weighted Delaunay triangulation of V (i.e., it has an
empty orthosphere) if and only if there exits a hyperplane passing through
the lifted weighted points of these simplex and no lifted weighted point of V
lie below the hyperplane.
Both weighted Delaunay triangulations and power diagrams are called
regular subdivisions of point sets [29]. Regular subdivisions have nice combi-
natorial structures. They are one of the important objects studied in higher-
dimensional convex polytopes [29, 5].

1.1.3 Algorithms
Algorithms for generating Delaunay (and weighted Delaunay) tetrahedraliza-
tions are well studied in computational geometry [7]. TetGen implemented
two algorithms, the Bowyer-Watson algorithm [3, 28] and the incremental
flip algorithm [9]. Both algorithms are incremental, i.e., insert points one
after one. Both have the worst case runtime O(n2 ). In most of the practical
applications, they are usually very fast. The expected running time of these
algorithm is O(n log n) if the points are uniformly distributed in [0, 1]3 .
The speed of incremental algorithms is very much affected by the cost of
point location. TetGen uses a spatial sorting scheme [2] to improve the point
location. The idea is to sort the points such that nearby points in space have
nearby indices. The points are first randomly sorted in different groups, then
points in each group are sorted along the Hilbert curve. The point location
can be done in nearly constant time by inserting points in this order
TetGen uses Shewchuk’s exact geometric predicates [15] for performing
the Orient3D, InSphere, and Orient4D tests. These suffice to guarantee
the numerical robustness of generating Delaunay and weighted Delaunay
tetrahedralizations. A simplified symbolic perturbation scheme [8] is used to
remove the degeneracies.
6 1 INTRODUCTION

A 3d PLC non-PLCs
Figure 4: Left: A 3d piecewise linear complex. The left shaded area shows a
facet, which is non-convex and has a hole in it. It has also edges and vertices
floating in it. The right shaded area shows an interior facet separating two
sub-domains. Right: Configurations which are not PLCs.

1.2 Tetrahedral Meshes of 3d Spaces

A tetrahedral mesh is a 3d simplicial complex that is a discrete representation
of a 3d continuous space (domain), both in its topology and geometry. Note
that a Delaunay tetrahedralization is a tetrahedral mesh of the convex hull
of its vertex set. In general, a geometric domain may not be convex and may
have arbitrarily complex boundaries.
The input domain of TetGen is modeled by a piecewise linear complex
(Section 1.2.1). The focuses of TetGen are the representation of the geometry
(the boundary) and the quality of the mesh. TetGen generates several types
of tetrahedral meshes to achieve these goals. They are explained in the
following subsections.

1.2.1 Piecewise Linear Complexes (PLCs)

At first we need a model to represent a 3d domain such that it can be easily
described and handled. A 3d piecewise linear complex (PLC) X is a set of
cells, that satisfies the following properties:

(1) The boundary of each cell in X is a union of cells in X .

(2) If two distinct cells f, g ∈ X intersect, their intersection is a union of

cells in X .

It is first introduced by Miller, et al. [11], see Figure 4 left or an example.

The boundary of a 3d PLC is the set of cells whose dimensions are less
than or equal to 2. A 0-dimensional cell is a vertex. In particular, we call
a 1-dimensional cell (an edge) a segment, and a 2-dimensional cell a facet.
Each facet of a PLC is a 2d PLC. It may contain holes, segments and vertices
in its interior, see Figure 4 left for an example.
1.2 Tetrahedral Meshes of 3d Spaces 7

Figure 5: Examples of surface meshes of PLCs.

PLCs are flexible in describing 3d geometric features. For instance, they

permit facets, segments and vertices to float in a domain, or segments and
vertices to float in the facet. One purpose of these floating cells is to constrain
how the PLC can be meshed, so that boundary conditions may be applied
at those cells.
The definition of a PLC disallows illegal intersections of its cells, see Fig-
ure 4 right for examples. Two segments only can intersect at a common
vertex that is also in X . Two facets of X may intersect only at a union of
vertices and segments which are also in X .
S
The underlying space of a PLC X , denoted |X |, is f ∈X f , which is the
domain to be triangulated. A tetrahedral mesh of X , is a 3d simplicial com-
plex T such that (1) X and T have the same vertices, (2) every cell in X is
a union of simplices in T , and (3) |T | = |X |.
Let T be a tetrahedral mesh of a 3d PLC X . The boundary of X is
respected by the elements of T , i.e., each segment of X is represented by a
union of edges in T , and each facet of X is represented a union of triangles
in T . To distinguish those edges and triangles of T which are on segments
and facets of X , we call them boundary edges and boundary faces.

TetGen uses a simple boundary representation (a surface mesh) to repre-

sent a 3d PLC. It is explained in Section 5.1.1 and in the file formats .poly
and .smesh of TetGen. Figure 5 shows two typical surface meshes of 3d
PLCs. The following points are useful to know.

• TetGen does not generate the surface mesh of the PLC. It must be
given by the users as the input of TetGen.

• TetGen is able to modify the surface mesh by further subdividing them.

This is necessary in order to conform to the constrained Delaunay prop-
8 1 INTRODUCTION

Figure 6: Polyhedra which can not be tetrahedralized without Steiner points.

Left: The Schönhardt polyhedron [14]. Right: The Chazelle’s polyhedron [4].

erty and improve the mesh quality. This is the default choice of the -p
switch.

• TetGen will preserve the surface mesh (do not subdivide them) when
the switch -Y is applied.

• If the input surface mesh contains self-intersections, TetGen will detect

them and stop the meshing process automatically.

• If the input surface mesh contains holes, i.e., it is not watertight, Tet-
Gen will finish the meshing process, but it returns an empty 3d tetra-
hedral mesh unless the -c switch (to keep the convex hull of the mesh)
is used.

Limitation of PLCs. A PLC only gives a piecewise linear approximation

of a 3d domain. It does not take the curvature of the surfaces into account.
When TetGen modifies the surface mesh, it only modifies the linear edges
and facets. This is unfortunately a limitation of using PLC.

1.2.2 Steiner Points

There are 3d polyhedra which may not be tetrahedralized with only its own
vertices, two typical examples are shown in Figure 6. Nevertheless, it is
always possible to tetrahedralize a polyhedron if Steiner points (which are
not vertices of the polyhedron) are allowed.
A Steiner tetrahedralization of a PLC X is a tetrahedralization of X ∪ S,
where S is a finite set of Steiner points (disjoint from the vertices of X ).
TetGen generates Steiner tetrahedralizations of PLCs. Two types of Steiner
points are used in TetGen:
1.2 Tetrahedral Meshes of 3d Spaces 9

• The first type of Steiner points are used in creating an initial tetra-
hedralization of PLC. These Steiner points are mandatory in order to
create a valid tetrahedralization.

• The second type of Steiner points are used in creating quality tetrahe-
dral meshes of PLCs. These Steiner points are optional. While they
may be necessary in order to improve the mesh quality or to conform
the mesh size.

In both cases, TetGen tries to generate the Steiner points efficiently and
limit the number of Steiner points as small as possible. The optimal locations
and the optimal number of Steiner points are still research topics.

1.2.3 Boundary Conformity

A fundamental problem in mesh generation is how to enforce a set of con-
straints, such as edges and triangles, to be preserved or represented by a
mesh. These constraints usually describe the special features in the domain
boundaries, such as the boundary complex of a PLC, and they are required
to be correctly represented in the generated meshes. It is generally referred
as the boundary conformity or boundary recovery problem.
Boundary conformity in 2d is very easy. One can enforce any edge (which
does not intersect any boundary) into a triangulation. Moreover, it does not
need any Steiner point. However, it is very difficult in 3d since it is not
always possible to enforce an edge or a triangle into a tetrahedralization
without using Steiner points.
TetGen always respects the boundary of the domain. TetGen can generate
different types of (Steiner) tetrahedralizations such that input segments and
facets of a PLC are respected.

• A conforming Delaunay tetrahedralization. It is a subcomplex of a De-

launay tetrahedralization, i.e., every tetrahedron is a Delaunay tetra-
hedron. It may contain Steiner points. Some Steiner points may lie on
the boundary of the PLC, i.e., the boundary triangulation of the PLC
may be a refinement of the original one.

• A constrained Delaunay tetrahedralization (CDT). Each tetrahedra of

it satisfies a constrained Delaunay criteria, and it has many properties
similar to those of a Delaunay tetrahedralization. It is further explained
in Section 1.2.4. A CDT may contain Steiner points. Moreover, most of
the Steiner points lie on the segments of the PLC. Hence, the boundary
triangulation of the PLC may be a refinement of the original one.
10 1 INTRODUCTION

p e

b c b
f
c
d
a
a

q
d

Figure 7: Left: The tetrahedron abcd is constrained Delaunay. Right: The

triangle abc is locally Delaunay.

• A constrained tetrahedralization. It is a tetrahedralization which pre-

serves the input surface mesh of the PLC. It may contain Steiner points,
but they must lie in the interior of the PLC. This type of tetrahedral-
ization may be neither Delaunay nor constrained Delaunay.

These different types of tetrahedralizations produced by TetGen may find

use in different situations. For instances, conforming DTs are desired for
applications which need the Delaunay property. While this type of tetra-
hedralization usually need a large number of Steiner points. CDTs require
much less Steiner points. They are an alternative choice of conforming DTs
when the applications can accept non-Delaunay elements. Constrained tetra-
hedralizations are useful in many engineering applications which the input
domain boundaries are needed to be preserved.

1.2.4 Constrained Delaunay Tetrahedralizations

A constrained Delaunay tetrahedralization (CDT) is a variation of a Delaunay
tetrahedralization that is constrained to respect the edges and facets of X .
CDTs in the plane were introduced by Lee and Lin [10]. Shewchuk [16, 21]
generalized them into three or higher dimensions.
In the following, we give two equivalent definitions of constrained Delau-
nay tetrahedralizations.
The visibility between two vertices p, q ∈ |X | is occluded if there is a facet
f ∈ X such that p and q lie on opposite sides of the plane that includes f ,
and the line segment pq intersects this facet (see Figure 7). A tetrahedron t
whose vertices are in X is constrained Delaunay if its circumsphere encloses
no vertex of X , which is visible from any point in the relative interior of t
(see Figure 7 Left).
A tetrahedralization T is a constrained Delaunay tetrahedralization of X
if it is a tetrahedralization of X and every tetrahedron of T is constrained
Delaunay.
1.2 Tetrahedral Meshes of 3d Spaces 11

Regular Needle Spindle Wedge Cap Sliver

Figure 8: Tetrahedra of different shapes.

Let s be a triangle in a tetrahedralization T of X . s is said to be locally

Delaunay if either it belongs to only one tetrahedron of T , or it is a face of
exactly two tetrahedra t1 and t2 and it has a circumsphere encloses no vertex
of t1 and t2 . Equivalently, the circumsphere of t1 encloses no vertex of t2 and
vice versa (see Figure 7 Right). A tetrahedralization T of P is a CDT of P
if every triangle in T not included in any facet of P is locally Delaunay.
The definitions of Delaunay tetrahedralization and constrained Delaunay
tetrahedralization are almost the same except that, for the CDT, we free
the requirement of being locally Delaunay for triangles in the facet. Hence
CDTs remain many nice properties of those of Delaunay tetrahedralizations,
see [21, 23]. Note that simplices (tetrahedra, triangles, and edges) in a CDT
are not always Delaunay.
A CDT of an arbitrary PLC X may not exist [21]. Steiner points are
necessary to ensure the existence of a CDT. A Steiner CDT of X is a CDT
of X ∪ S, where S ⊂ |X | is a set of Steiner points.
Compared to conforming Delaunay tetrahedralizations, (Steiner) CDTs
usually require much less Steiner points.

1.2.5 Mesh Quality, Tetrahedron Shape Measures

There is no unique definition of the term “mesh quality”. It depends on the
intended application and the numerical methods employed, see, e.g. [19].
As a general guideline, elements with very small and large angles (and
dihedral angels) should be avoided since they usually cause the negative
influence on the accuracy and performance of numerical methods.
Figure 8 shows six differently shaped tetrahedra. A tetrahedron shape
measure is a continuous function which evaluates the shape of a tetrahedron
by a real number. Various tetrahedron shape measures have been suggested,
some of them are equivalent.
The most general shape measure for a simplex is the aspect ratio. The
aspect ratio, η(τ ), of a tetrahedron τ is the ratio between the longest edge
length lmax and the shortest height hmin , i.e., η(τ ) = lmax /hmin . The aspect
ratio measures the “roundness” of a tetrahedron in terms of a value between
12 1 INTRODUCTION

R
L

L R R

Figure 9: The radius-edge ratio ( R

L
) of tetrahedra. Most of the badly shaped
tetrahedra will have a large radius-edge-ratio except slivers (Right).

√ √
2/ 3 and +∞. Low aspect ratio implies better shape. Other possible def-
initions of aspect ratio exist, such as the ratio between the circumradius and
inradius. These definitions are equivalent in the sense that if a tetrahedron
is bounded under one ratio implies a bound on the other.
The tetrahedron shape measures used in TetGen are the face an-
gles (angles between two edges) and dihedral angles (angles between two
faces) as the shape measures of tetrahedra. They work well with the Delau-
nay refinement algorithm used in TetGen. Also, the combination of them
achieve the same objective as the aspect ratio.
To bound the smallest face angle is equivalent to bound the radius-edge
ratio of the tetrahedron. The radius-edge ratio, ρ(τ ) of a tetrahedron τ is
the ratio between the radius r of its circumscribed ball and the length d of
its shortest edge, i.e.,
r 1
ρ(τ ) = ≥ ,
d 2 sin θmin
where θmin is the smallest
√ face angle of τ , see Figure 9. The radius-edge
ratio ρ(τ ) is at least 6/4 ≈ 0.612, achieved by the regular tetrahedron.
Most of the badly shaped tetrahedra will have a big radius-edge ratio (e.g.,
> 2.0) except the sliver, which is a type of very flat tetrahedra having no
small
√ edges, but nearly zero volume. A sliver can have a minimal value
2/2 ≈ 0.707, hence the radius-edge ratio is not equivalent to aspect ratio
(due to the slivers). Besides this, the radius-edge ratio is a useful shape
measure. It can be shown that if a tetrahedral mesh has a radius-edge ratio
bounded for all of its tetrahedra, then the point set of the mesh is well spaced,
and each node of the mesh has bounded degree [11, 23].
Each of the six edges of a tetrahedron τ is surrounded by two faces. At
a given edge, a dihedral angle between two faces is the angle between the
intersection of these faces and a plane perpendicular to the edge. A dihedral
angle in τ is between 0◦ and 180◦ . The minimum dihedral angle φmin (τ ) of
τ is a tetrahedron shape measure used by TetGen.
1.2 Tetrahedral Meshes of 3d Spaces 13

Figure 10: An adaptive tetrahedral mesh (left) and the calculated numerical
solution (right) of a heat conduction problem.

1.2.6 Mesh Size, Mesh Adaptation

Mesh size means the total number of mesh elements. It is determined by
the sizes (e.g., edge lengths) of the mesh elements. The goal of adaptive
mesh generation is to generate the mesh which achieves the desired mesh
quality with a small number of mesh elements. In numerical methods, such
mesh gives a good balance between the solution time and the accuracy of the
solution, see Figure 10 for an example.
There is no unique way to determine the mesh size in advance. It de-
pends on the actual applications and the numerical methods employed. As
a general guideline for adaptive mesh generation, we assume that a mesh
sizing function is available. Let X be a 3d PLC. A mesh sizing function
H : |X | → R, is a function that maps each point p ∈ |X | to a positive value
H(p) which specifies the desired edge lengths at the point location p.
H is isotropic if the edge length does not vary with respect to the direc-
tions at p, otherwise, it is anisotropic. The current version of TetGen only
supports isotropic mesh sizing functions. An ideal sizing function is C ∞ ,
∀p ∈ |X |. However, in most cases, H is approximated by a discrete function
specified at some points in |X |. The size on other points in |X | is obtained
by means of interpolation.
TetGen supports several ways of defining a sizing function.

• One can apply a bound of maximum volume (the -a switch) on every

tetrahedra of the mesh. It is the same as defining a global constant
sizing function. For a domain consists of multiple sub-domains (mate-
rials), the volume bound can vary in each sub-domain, see the .poly
and .smesh file formats.

• One can apply a bound of maximum area on each facet, or apply a

bound of maximum edge length on each boundary segment of the input
PLC, see the .var file format.
14 1 INTRODUCTION

• One can define a sizing function directly on the input PLC. In this way,
each node of the PLC is assigned a mesh size. It means the desired
mesh edge length at this node, see the .mtr file format.

• One can use a background mesh to define a sizing function. The back-
ground mesh can be any tetrahedral mesh. Its underlying space must
cover the input PLC. Each mesh node of the background tetrahedral
mesh contains the desired edge length at that location in the PLC.

All the above ways of defining sizing function can be used together. Tet-
Gen will automatically choose the smallest mesh size. In the last two ways,
i.e., defining mesh sizes on nodes, it is possible to set a size zero at a node.
In this case, the mesh size at this location is ignored.

1.2.7 Mesh Optimization

Mesh optimization is an essential way for further improving the mesh quality.
Typically, it improves one or several objective functions on mesh quality, such
as the aspect ratio, minimum or maximum dihedral angles, etc.
Mesh optimization is usually done by applying various local mesh opera-
tions which either change the node locations or change the mesh connections.
The most frequently used operations are: node smoothing, edge/face swap-
ping, edge contraction, and vertex insertion. These operations are combined
according to a schedule to iteratively improve the mesh quality. One can de-
cide to either optimize the whole mesh (global optimization) or only optimize
a part of the mesh (local optimization).
TetGen uses mesh optimization after the mesh generation. It locally opti-
mizes the tetrahedral mesh to restore the Delaunay property and to improve
the mesh quality. The current version uses the maximum dihedral angle as
the objective function. It provides options to choose local operations and the
maximal iterations, see the -O switch.

1.2.8 Algorithms
Algorithms for constructing 3d CDTs were first considered by Shewchuk.
In [16], a sufficient condition for the existence of a CDT of a PLC (or a poly-
hedron) is given. Based on this condition, several algorithms for constructing
Steiner CDTs are proposed [18, 20, 26, 25]. TetGen’s CDT algorithm is from
Si and Gärtner [26, 25].
The basic algorithm for generating quality tetrahedral meshes is the De-
launay refinement algorithm from Ruppert [13] and Shewchuk [17]. This
1.2 Tetrahedral Meshes of 3d Spaces 15

algorithm generates a quality mesh of Delaunay tetrahedra with no tetrahe-

dra has a radius-edge ratio greater than 2.0 (equivalently, no face angle less
than 14.5o ). The sizes of tetrahedra are graded from small to large over a
short distance. TetGen implemented this algorithm for improving the mesh
quality of a CDT of a PLC. In practice, the algorithm generates meshes
generally surpassing the theoretical bounds and is effectively in eliminates
tetrahedra with small or large dihedral angles.
There are two theoretical problems of the basic Delaunay refinement al-
gorithm. First, it does not remove slivers due to the use of radius-edge ratio
as the sole tetrahedral shape measure. Second, it may not terminate if the
input PLC X contains sharp features, i.e., there are two edges of X meet at
an acute angle, or two facets of X meet at an acute dihedral angle.
TetGen uses the minimal dihedral angle of tetrahedron as a second shape
measure for the Delaunay refinement algorithm, hence slivers are found by
this measure and are removed by the same Delaunay refinement iterations.
Since TetGen works with CDTs, it can detect all the sharp features in the
CDT in advance then it starts the Delaunay refinement process. Tetrahedra
which near or at the sharp features are never removed. The modified algo-
rithm in TetGen always terminates. While some badly-shaped tetrahedra
near the sharp features may survive.
TetGen accepts a user-defined mesh sizing function to control the mesh
size. If it is given, TetGen will generate an adaptive tetrahedral mesh ac-
cording to the input mesh sizing function. It uses a constrained Delaunay
refinement algorithm [22].
16 1 INTRODUCTION

Figure 11: The flowchart of the mesh generation process of TetGen.

1.3 Description of the Meshing Process

Figure 11 shows a graphic flowchart of the meshing process in TetGen.
Here is the general steps of TetGen to create a (quality) tetrahedral mesh.
Many of these steps can be skipped, depending on the command line switches.
1. Initialize constants and parse the command line.
2. Read the vertices from a (.node) file and either
- create the Delaunay tetrahedralization (DT) of them (no -r), or
- read an existing tetrahedral mesh from (.ele, .face, .edge) files
and reconstruct it (-r).
3. Read the boundary informations (segments and facets) from (.poly or
.smesh, .edge) files and triangulate them (-p).
1.3 Description of the Meshing Process 17

4. Read the background mesh from (.b.node, .b.ele, .b.mtr ...) files
(if it is provided) and interpolate the mesh size from the background
mesh to the current mesh (-m).

5. Insert the boundary segments and facets into the DT (-p) by either

– constructing a constrained Delaunay tetrahedralization (CDT) in

which the segments and facets may be split into smaller pieces (no
-Y), or
– recovering the segments and facets in a tetrahedralization which
contains them (-Y).

6. Read the holes (-p), regional attributes (-pA), and regional volume
constraints (-pa), and either

- remove the exterior tetrahedra (in the holes and concavities) (no
-c), or
- mark the exterior tetrahedra (-c),

and spread the regional attributes and volume constraints.

7. Coarsen the mesh (-R) by removing vertices which are either marked
or inconsistent according to a mesh sizing function (-m).

8. Read the list of additional vertices from a (.a.node) file (if it is pro-
vided) and insert into the current mesh (-i).

9. Enforce the constraints on minimum quality bound (-q) and maximum,

volume (-a), and a mesh size function (-m).

10. Optimize the mesh with respect to the optimization scheme (-O) spec-
ified quality measures (-o).

11. Write the output files and print the statistics.

12. Check the consistency of the mesh (-C).

18 2 GENERAL INFORMATION

2 General Information
This section gives some general information about TetGen. These are not
required in order to use TetGen.

2.1 Language, Platforms

TetGen is written entirely in C++, and it only uses the standard C (ANSI) li-
braries, hence the code is easily portable and should be compiled by a popular
C++ compilers, such as GNU’s gcc/g++, Intel’s and Mircosoft’s C++ compil-
ers. So far, TetGen has been successfully compiled on all major computer
architectures and major operating systems (Unix/Linux, Windows, Mac OS)
with both 32-bit and 64-bit versions.
The current version of TetGen contains about 25, 000 source lines and
7, 000 comment lines. These numbers do not include those of Shewchuk’s
robust predicates.

2.2 Memory requirement

TetGen dynamically allocates memory when it needs. There is no minimum
memory requirement to run TetGen. The maximum memory is only limited
by the physical memory available from your system. The more memory you
have, the larger mesh you can generate.
For an example, the 32-bit version of TetGen used about 694.5 Mega
bytes memory to generate the Delaunay tetrahedralization (DT) of a set of
2, 000, 000 (two million) points randomly distributed inside a unit cube. This
DT has 13, 504, 899 tetrahedra. This is approximately 364 bytes per vertex
or 54 bytes per tetrahedron. In other words, with 4 Giga byte memory,
the maximum Delaunay tetrahedralization may be generated by the 32-bit
version of TetGen has approximately 11, 799, 360 (ca. 11 million) vertices or
79, 536, 431 (ca. 79 million) tetrahedra.
More memory are needed in generating quality tetrahedral meshes. The
extra memory are used to store boundary information and working arrays
of algorithms. For an example, the 32-bit version of TetGen used about 770
Mega bytes memory to generate a quality tetrahedral mesh with 2, 000, 000
(two million) vertices and 12, 237, 300 tetrahedra.
So far, the largest quality tetrahedral meshes was generated by the 64-bit
version of TetGen. It contains 1, 007, 700, 944 (ca. 1 billion) vertices, and
6, 454, 556, 696 (ca. 6.45 billion) tetrahedra. TetGen used about 905.7 Giga
bytes memory on generating this mesh, see Table 2.
2.3 CPU time estimation 19

2.3 CPU time estimation

When generating Delaunay tetrahedraliations, the worst running time of Tet-
Gen is O(n2 ), where n is the input number of vertices. However, this only
happens for some very special point sets. For most point sets appear in
applications, TetGen runs almost in O(n log n) time, see Table 1.
The CPU time required to obtain the quality tetrahedral mesh is obvi-
ously related to the complexity of the geometry and topology of the inputs,
as well as the command line switches and parameters specified. Nevertheless,
the running time of TetGen increases almost linearly respect to the output
mesh size, see Table 2.

2.4 Performance
This section gives some practical information regarding the performance of
TetGen. In particular, Table 1 and Table 2 report the statistics of some test
runs of TetGen. The used version of TetGen was a 64-bit version compiled by
GNU gcc/g++ version 4.7.2 with the −O3 (optimization) switch. The used
computer was a computer of WIAS (erhard-03) with Intel(R) Xeon(R)
Ten-Core 2.40GHz CPU, 1, 024 Giga byte memory, and SuSE Linux. The
given CPU times exclude the file I/O time. Comparisons of TetGen with
other programs are available in paper [24].
Table 1 shows the statistics of TetGen on creating Delaunay tetrahedral-
izations. Three random point sets of different sizes are used in these tests.
They are generated by the tool rbox (command: rbox -D3 xxx) in the pro-
gram qhull (http://www.qhull.org). It is well known that the Delaunay
tetrahedralizations of random point sets have linear complexity. Both of the
memory usage and CPU time of TetGen are quasilinear.

# of points # of tets used memory CPU time

(input) (output) (Mega bytes) (seconds)
1, 000, 000 6, 748, 645 710 9.2
2, 000, 000 13, 504, 917 1, 420 19.0
20, 000, 000 135, 059, 867 14, 196 203.1

Table 1: Statistics of TetGen on generating Delaunay tetrahedralizations.

Table 2 shows the statistics of TetGen on creating quality tetrahedral

meshes. The input of these tests is a 3d unit cube with 8 vertices and 6
facets described in file cube.smesh. The resulting meshes were generated by
using the command: tetgen -pqa# -x1000000 cube.smesh, where “-a#”
20 2 GENERAL INFORMATION

specifies a very small tetrahedron volume constraint, and “ -x1000000” spec-

ifies a user-defined memory allocation size.

volume # of points # of tets used memory CPU time

constraint (output) (output) (Mega bytes) (seconds)
−7
1.0 × 10 3, 023, 518 18, 744, 549 2, 904 126
1.0 × 10−8 29, 412, 255 186, 090, 870 27, 599 1, 374
1.0 × 10−9 290, 323, 148 1, 854, 336, 524 268, 781 12, 124
−10
5.0 × 10 579, 301, 396 3, 706, 331, 655 534, 386 26, 872
2.87 × 10−10 1, 007, 700, 944 6, 454, 556, 696 927, 449 44, 261

Table 2: Statistics of TetGen on generating quality tetrahedral meshes.

2.5 Errors
When running TetGen, errors may happen. In this case, TetGen may fail to
generate a mesh. The typical reasons that cause failures of TetGen may be
one of the follows:

• The wrong use of command line switches and parameters.

• The input surface mesh contains self-intersections.

• Out of memory.

• A known bug of TetGen.

• An unknown bug of TetGen.

In most cases TetGen is able to detect the error. When running Tet-
Gen as a standalone program, when an error is detected, TetGen will report
a message that describes the error maybe together with a suggestion to fix
it, and terminate itself. Below is an example of such message.

Found two segments intersect each other.

1st: [13,7] 1.
2nd: [1114,1113] 1.
A self-intersection was detected. Program stopped.
Hint: use -d option to detect all self-intersections.
2.5 Errors 21

When running TetGen as a library, i.e., it is called inside other program,

the error message will not be seen, and TetGen will throw an error index
(integer), which can be catched by the standard C++ exception handler try
and catch. A list of error indices and messages are provided in Appendix.
When TetGen terminates on error, it automatically releases the used
memory before terminating itself. This avoids potential memory leak when
calling TetGen multiple times.
22 3 GETTING STARTED

3 Getting Started
TetGen is distributed in its source code (written in C++). The latest version
of TetGen is available at http://www.tetgen.org.
Section 3.1 briefly explains how to compile TetGen into an executable
program or a library.
Once TetGen is get compiled, and assume you have the executable file,
tetgen (or tetgen.exe in Windows), you can start testing TetGen with the
included example file by following the tutorial in Section 3.2.
TetGen does not have a graphic user interface (GUI). The TetView pro-
gram can be used to visualize the input and output of TetGen. Alternatively,
other popular mesh viewers are supported, see Section 3.3.

3.1 Compilation
The downloaded archive should include the following files:

README General information.

LICENSE Copyright notices.
tetgen.h C++ header file of TetGen.
tetgen.cxx C++ source file of TetGen.
predicates.cxx C++ source of Shewchuk’s predicates.
makefile make file for compiling TetGen.
CMakeLists.txt cmake file for compiling TetGen.
example.poly An example input file.

The file predicates.cxx is a modified C++ version of Shewchuk’s robust

geometric predicates http://www.cs.cmu.edu/~quake/robust.html.
To compile TetGen, use a C++ compiler for the system on which Tet-
Gen will be used, such as GNU’s g++, or Microsoft C++ on MS Windows
systems. TetGen may be compiled into an executable program, or a library
which can be embedded into another program.

3.1.1 Using make

The easiest way to compile TetGen is to edit and use the included makefile.
Before compiling, put all source files, tetgen.h, tetgen.cxx, and predicates.cxx
and the makefile in one directory (usually they are), read the makefile,
which describes your options, and edit it accordingly.
You should at least specify the C++ compiler and the level of optimiza-
tion. Default, the GNU C++ compiler (g++) is used, and there is no opti-
mization is used.
3.1 Compilation 23

Once you’ve done this, type make to compile TetGen into an executable
program or type make tetlib to compile TetGen into a library. The exe-
cutable file tetgen or the library libtet.a appears in the same directory as
the makefile.
Alternatively, the files are usually easy to compile directly on the com-
mand line. Assume you’re using g++, first compile the file predicates.cxx
to get an object file:

g++ -c predicates.cxx

To compile TetGen into an executable file, use the following command:

g++ -o tetgen tetgen.cxx predicates.o -lm

To compile TetGen into a library, the symbol TETLIBRARY is needed:

g++ -DTETLIBRARY -c tetgen.cxx

ar r libtet.a tetgen.o predicates.o

Some additional remarks to get an efficient executable version of TetGen.

• One should use the optimization options provided by the C++ com-
piler. Usually, an optimized version of TetGen may run double times
faster than an unoptimized one.

• There are some assertions inserted in the source code of TetGen. They
are used for catching program bugs. But they may slow down the
performance of TetGen. You can disable them by adding the symbol
-DNDEBUG in your commands of compilation.

Here is an example to get an optimized version of TetGen using GNU’s C++

compiler:

g++ -O3 -DNDEBUG -c predicates.cxx

g++ -O3 -DNDEBUG -o tetgen tetgen.cxx predicates.o -lm

3.1.2 Using cmake

As an alternative, one can compile TetGen using cmake (www.cmake.org).
It simplifies the compilation of TetGen with different compilers and architec-
tures (Linux, MacOSX, and MS Windows) at the same time.
Using the provided file CMakeLists.txt, the simplest sequence of com-
mands is:
24 3 GETTING STARTED

cd <tetgen-directory>
mkdir build
cd build
cmake ..
make

When the compilation is finished, you should get both of the executable
file tetgen and the library libtet.a under the directory build.
To get a debug version of TetGen, using the following commands

cd <tetgen-directory>
mkdir build
cd build
cmake -DCMAKE_BUILD_TYPE=Debug ..
make

The default is cmake -DCMAKE_BUILD_TYPE=Release ...

To specify a compiler by doing

CXX=icpc cmake -DCMAKE_BUILD_TYPE=Debug ..

3.1.3 Remarks on Using Shewchuk’s Robust Predicates

TetGen default uses Shewchuk’s robust geometric predicates which perform
exact floating-point arithmetic (predicates.cxx). The arithmetic are based
on the IEEE 754 floating-point standard. However, some processors may not
default use this standard for floating-point representations and arithmetic. If
so, a configure is needed to correctly execute the predicates. It is described on
its website http://www.cs.cmu.edu/~quake/robust.pc.html for details.
Below are my own experience in using Shewchuk’s predicates.

• If TetGen fails with “a segmentation fault” during the construction of

the Delaunay tetrahedralization, it is most likely that the predicates
was not correctly configured.

• I used gcc/g++ version 4.4 on a MacOSX system with an Intel Core 2

Duo, there is no need to configure the predicates. Just using the default
settings in the predicates. TetGen runs correctly both with debug and
with optimization options. The same when I used gcc/g++ version 4.6
on a Linux (Ubuntu) system with an Intel Pentium 2.8GHz.
3.1 Compilation 25

• I did encounter failures of TetGen when I used gcc/g++ version 4.2 on

a Linux system with an Intel Xeon CPU, and TetGen was compiled
with an optimization option -O3. However, TetGen ran correctly when
it was compiled with the debug option -g. The same issue happened
when TetGen was compiled using Intel’s C++ compiler (version 2012)
with the optimization -O3 option.

3.1.4 Using CGAL’s Robust Predicates

Alternatively, TetGen can use other robust predicates developed in com-
putational geometry, such as the filtered robust predicates in CGAL (http:
//www.cgal.org). TetGen includes the interface to the CGAL’s predicates in
the file predicates.cxx. It uses the following kernel,

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>

In this section, we show how to use CGAL’s predicates in TetGen. The

following steps are needed.

1. Download CGAL. The version I used was 4.1 (October 2012). Compile
and install CGAL by using the following commands:

cd CGAL-x.y # go to CGAL directory

cmake . # configure CGAL
make # build the CGAL libraries

If you have any problems in compiling CGAL, please see CGAL’s manual
about installation. Once CGAL is compiled, make sure that you get the
new file compiler config.h inside the directory CGAL-x.y/include/CGAL.

2. Set the following environment path variables:

BOOST=/opt/local/include
GMP_LIB=/opt/local/lib
CGAL_INC=$(HOME)/Programs/CGAL/CGAL-4.1/include

BOOST should point to the directory containing the boost library, ver-
sion 1.35 or later is required. GMP LIB should point to the directory
containing the library GMP, - the GNU Multiple Precision Arithmetic
Library. And CGAL INC should point to the C++ headers of CGAL’s
library.
26 3 GETTING STARTED

3. Compiling predicates.cxx by using the variable -DUSE CGAL PREDICATES,

i.e.,

g++ -I$(BOOST) -I$(CGAL_INC) -DUSE_CGAL_PREDICATES \

-O3 -c predicates.cxx

4. Compiling an executable version of TetGen by the following command:

g++ -O3 -o tetgen tetgen.cxx predicates.o -L$(GMP_LIB) -lgmp

3.2 A Short Tutorial

TetGen gives a short list of command line options if it is invoked without
arguments (that is, just type tetgen). A brief description of the usage is
printed by invoking TetGen with the -h switch:

tetgen -h

The enclosed example file, example.poly, is a simple 3d mesh domain (a

PLC), see Figure 24. Try out TetGen using:

tetgen -p example.poly

With the -p switch, TetGen will read the file, i.e., example.poly, and
generate its constrained Delaunay tetrahedralization. The resulting mesh is
saved in four files: example.1.node, example.1.ele, example.1.face, and
example.1.edge, which are a list of mesh nodes, tetrahedra, boundary faces,
and boundary edges, respectively. The file formats of TetGen are described
in Section 5. The screen output of the above command looks like this:

Opening example.poly.
Delaunizing vertices...
Delaunay seconds: 0.000864
Creating surface mesh ...
Surface mesh seconds: 0.000307
Constrained Delaunay...
Constrained Delaunay seconds: 0.000325
Removing exterior tetrahedra ...
Exterior tets removal seconds: 8.6e-05
Optimizing mesh...
Optimization seconds: 6.4e-05

Writing example.1.node.
Writing example.1.ele.
3.2 A Short Tutorial 27

Writing example.1.face.
Writing example.1.edge.

Output seconds: 0.000663

Total running seconds: 0.002451

Statistics:

Input points: 28
Input facets: 23
Input segments: 44
Input holes: 2
Input regions: 2

Mesh points: 28
Mesh tetrahedra: 68
Mesh faces: 160
Mesh faces on facets: 50
Mesh edges on segments: 44

The above mesh is pretty coarse, and contains many badly-shaped (e.g.,
long and skinny) tetrahedra. Now try:
tetgen -pq example.poly

The -q switch triggers the mesh refinement such that Steiner points are
added to remove badly-shaped tetrahedra. The resulting mesh is contained
in the same four files as before. While now it is a quality tetrahedral mesh
whose tetrahedra have no small face angle less than about 14o (the default
quality value). The screen output of the above command looks like this:
Opening example.poly.
Delaunizing vertices...
Delaunay seconds: 0.000384
Creating surface mesh ...
Surface mesh seconds: 0.000181
Constrained Delaunay...
Constrained Delaunay seconds: 0.000163
Removing exterior tetrahedra ...
Exterior tets removal seconds: 0.000368
Refining mesh...
Refinement seconds: 0.009291
Optimizing mesh...
Optimization seconds: 0.000517

Writing example.1.node.
Writing example.1.ele.
Writing example.1.face.
28 3 GETTING STARTED

Writing example.1.edge.

Output seconds: 0.004716

Total running seconds: 0.015802

Statistics:

Input points: 28
Input facets: 23
Input segments: 44
Input holes: 2
Input regions: 2

Mesh points: 216

Mesh tetrahedra: 689
Mesh faces: 1587
Mesh faces on facets: 430
Mesh edges on segments: 126
Steiner points inside domain: 1
Steiner points on facets: 105
Steiner points on segments: 82

Now try to run:

tetgen -pq1.2V example.poly

TetGen will again generate a quality mesh, which contains more points
than previous one, and all tetrahedra have radius-edge ratio bounded by
1.2, i.e., the element shapes are better than those in the previous mesh. In
addition, TetGen prints a mesh quality report (due to the -V switch) which
looks as below:

Mesh quality statistics:

Smallest volume: 0.00059866 | Largest volume: 0.09363

Shortest edge: 0.25 | Longest edge: 1.4142
Smallest asp.ratio: 1.3255 | Largest asp.ratio: 10.169
Smallest facangle: 24.898 | Largest facangle: 126.8698
Smallest dihedral: 8.6045 | Largest dihedral: 163.4980

Aspect ratio histogram:

< 1.5 : 23 | 6 - 10 : 26
1.5 - 2 : 364 | 10 - 15 : 1
2 - 2.5 : 480 | 15 - 25 : 0
2.5 - 3 : 248 | 25 - 50 : 0
3 - 4 : 125 | 50 - 100 : 0
4 - 6 : 56 | 100 - : 0
(A tetrahedron’s aspect ratio is its longest edge length divided by its
3.2 A Short Tutorial 29

smallest side height)

Face angle histogram:

0 - 10 degrees: 0 | 90 - 100 degrees: 465
10 - 20 degrees: 0 | 100 - 110 degrees: 132
20 - 30 degrees: 188 | 110 - 120 degrees: 54
30 - 40 degrees: 1059 | 120 - 130 degrees: 5
40 - 50 degrees: 1704 | 130 - 140 degrees: 0
50 - 60 degrees: 1865 | 140 - 150 degrees: 0
60 - 70 degrees: 1609 | 150 - 160 degrees: 0
70 - 80 degrees: 1105 | 160 - 170 degrees: 0
80 - 90 degrees: 736 | 170 - 180 degrees: 0

Dihedral angle histogram:

0 - 5 degrees: 0 | 80 - 110 degrees: 1831
5 - 10 degrees: 2 | 110 - 120 degrees: 274
10 - 20 degrees: 150 | 120 - 130 degrees: 193
20 - 30 degrees: 266 | 130 - 140 degrees: 105
30 - 40 degrees: 643 | 140 - 150 degrees: 62
40 - 50 degrees: 1056 | 150 - 160 degrees: 52
50 - 60 degrees: 1213 | 160 - 170 degrees: 24
60 - 70 degrees: 1121 | 170 - 175 degrees: 0
70 - 80 degrees: 946 | 175 - 180 degrees: 0

Instead of using the -q switch to get a finer mesh, one can use the -a
switch to impose a maximum volume constraint on the resulting tetrahedra.
By doing so, no tetrahedron in the resulting mesh has volume bigger than it.
Try to run the following command.
tetgen -pq1.2Va1 example.poly

Now the resulting mesh should contain much more vertices than the pre-
vious one. Besides of -q and -a switches, TetGen provides more switches to
control the mesh size and shape. They are described in Section 4.
To compute the Delaunay tetrahedralization and convex hull of the point
set of this PLC, try this:

cp example.poly example.node
tetgen example.node

The Delaunay tetrahedralization is saved in example.1.node and example.1.ele.

The convex hull is represented by a list of triangles in file example.1.face.
All these meshes and Delaunay tetrahedralizations can be visualized by
the programs introduced in the next section.
Detailed descriptions of the command line switches and file formats are
found in Section 4 and 5.
30 3 GETTING STARTED

3.3 Visualization
3.3.1 TetView
TetView is a graphic interface for viewing piecewise linear complexes and
simplical meshes. It can read the input and output files of TetGen and
display the objects. It also shows other information as well, such as boundary
types and materials. The interactive GUI allows the user to manipulate (i.e.,
rotate, translate, zoom in/out, cut, shrink, etc.) the viewing objects easily
through either mouse or keyboard. TetView can save the current window
contents into high quality encapsulated postscript files. Most of the figures
of this document were produced by TetView.
TetView is freely available from http://www.tetgen.org/tetview.html.
You will find a list of precompiled executable versions on different platforms.
Download the one corresponding to your system.
To show the PLC in example.poly, first copy the executable file (tetview)
to the directory where you have this file, it is loaded by running:
tetview example.poly
And the following command will display the mesh (in files example.1.node,
example.1.ele, and example.1.face):
tetview example.1.ele
The instruction for using TetView can be found in the above website.

3.3.2 Medit and Paraview

TetGen can export its tetrahedral mesh into the .mesh format. It can be
then visualized by the software Medit, which is freely available from http:
//www.ann.jussieu.fr/~frey/logiciels.
For viewing mesh under Medit, add a -g switch in the command line.
TetGen will additionally output a file named example.1.mesh, which can be
read and rendered directly by TetGen. Try running:
tetgen -pg example.poly
medit example.1
Alternatively, TetGen can also output its tetrahedral mesh into the .vtk
format by adding the switch -k, i.e.,
tetgen -pk example.poly
It will output a file named example.1.vtk. It can then be visualized by
the software Paraview: http://www.paraview.org.
 31

4 Using TetGen
This section describes the use of TetGen as a stand-alone program. It is in-
voked from the command line with a set of switches and an input file name.
Switches are used to control the behavior of TetGen and to specify the out-
put files. In correspondence to the different switches, TetGen will generate
the Delaunay tetrahedralization, or the constrained (Delaunay) tetrahedral-
ization, or the quality conforming (Delaunay) mesh, etc.

4.1 Command Line Syntax

tetgen [-pYrq_Aa_miO_S_T_XMwcdzfenvgkJBNEFICQVh] input_file

Underscores indicate that numbers may optionally follow certain switches.

Do not leave any space between a switch and its numeric parameter. These
switches are explained in Section 4.2.
input file can be different files depending on the switches you use. If
no command line switch is used, it must be a file with extension .node which
contains a list of 3d points and the Delaunay tetrahedralization of this point
set will be generated.
If the -p switch is used, input file must be one of the following ex-
tensions: .poly, .smesh, .off, .stl, .ply, and .mesh, which describes the
boundary (a surface mesh) of a 3d piecewise linear complex and the boundary
constrained (Delaunay) tetrahedralization of this object will be generated. If
the -q switch is used together, the quality conforming tetrahedral mesh will
be generated.
If the -r switch is used, an existing tetrahedral mesh will be read. you
must supply .node and .ele files which describe a tetrahedral mesh. Op-
tionally a .face and a .edge file can be supplied which contain the boundary
faces and edges of the mesh. input file can have no file extension.
Together with the -q switch, the mesh will be refined with respect to
the new quality measure and variant constraints. Optionally, and a .vol, a
.mtr, and a .var file can be supplied which contain the mesh size control
information.
File formats are described in Section 5.

4.2 Command Line Switches

An overview of all command line switches and a short description follows
each switch is shown by invoking TetGen without any switch and input file.
32 4 USING TETGEN

Detailed descriptions of these switches are given in the following subsections.

-p Tetrahedralizes a piecewise linear complex (PLC).

-Y Preserves the input surface mesh (does not modify it).
-r Reconstructs a previously generated mesh.
-q Refines mesh (to improve mesh quality).
-R Mesh coarsening (to reduce the mesh elements).
-A Assigns attributes to tetrahedra in different regions.
-a Applies a maximum tetrahedron volume constraint.
-m Applies a mesh sizing function.
-i Inserts a list of additional points.
-O Specifies the level of mesh optimization.
-S Specifies maximum number of added points.
-T Sets a tolerance for coplanar test (default 10−8 ).
-X Suppresses use of exact arithmetic.
-M No merge of coplanar facets or very close vertices.
-w Generates weighted Delaunay (regular) triangulation.
-c Retains the convex hull of the PLC.
-d Detects self-intersections of facets of the PLC.
-z Numbers all output items starting from zero.
-f Outputs all faces to .face file.
-e Outputs all edges to .edge file.
-n Outputs tetrahedra neighbors to .neigh file.
-v Outputs Voronoi diagram to files.
-g Outputs mesh to .mesh file for viewing by Medit.
-k Outputs mesh to .vtk file for viewing by Paraview.
-J No jettison of unused vertices from output .node file.
-B Suppresses output of boundary information.
-N Suppresses output of .node file.
-E Suppresses output of .ele file.
-F Suppresses output of .face and .edge file.
-I Suppresses mesh iteration numbers.
-C Checks the consistency of the final mesh.
-Q Quiet: No terminal output except errors.
-V Verbose: Detailed information, more terminal output.
-h Help: A brief instruction for using TetGen.
4.2 Command Line Switches 33

4.2.1 Delaunay and weighted Delaunay tetrahedralizations

Given a set of 3d points or weighted points, TetGen generates the Delaunay

tetrahedralization or the weighted Delaunay tetrahedralization of the point
set. It can also output the Voronoi diagram or the power diagram.
Save the set of points in a .node file, e.g., test.node. Run TetGen using
the command:

tetgen test.node

This command generates the Delaunay tetrahedralization (DT) of this

point set. Below is a screen output of TetGen:

Opening test.node.
Delaunizing vertices...
Delaunay seconds: 0.001695

Writing test.1.node.
Writing test.1.ele.
Writing test.1.face.

Output seconds: 0.001555

Total running seconds: 0.003615

Statistics:

Input points: 100

Mesh points: 100

Mesh tetrahedra: 514
Mesh faces: 1057
Mesh edges: 642
Convex hull faces: 58

Figure 12 shows an example of an input point set (100 vertices) and the
generated DT and its convex hull.
The default outputs of TetGen are three files listed in Table 3.
The set of all faces and edges of the DT can be obtained by adding the
output switches -f (output all faces) and -e (output all edges), respectively.
For example, by the following command
34 4 USING TETGEN

Figure 12: From left to right: a set of 100 randomly distributed points in a
unit cube, the Delaunay tetrahedralization, and the convex hull of the point
set, respectively.
test.1.node The list of vertices (same as input) of the DT.
test.1.ele The list of tetrahedra of the DT.
test.1.face The list of convex hull faces of the point set.

Table 3: The default output files of TetGen.

tetgen -fe test.node

TetGen will output four files listed in Table 4.

test.1.node The list of vertices (same as input) of the DT.

test.1.ele The list of tetrahedra of the DT.
test.1.face The list of all faces of the DT.
Convex hull faces have a face marker ‘1’.
Interior faces have a face marker ‘0’.
test.1.edge The list of all edges of the DT.
Convex hull edges have an edge marker ‘1’.
Interior edges have an edge marker ‘0’.

Table 4: The output files by the command: tetgen -fe test.node.

The adjacency graph of the list of tetrahedra of the DT can be obtained by

adding the -n switch in the command line. An additional file, test.1.neigh,
will be output by TetGen, see file format .neigh for details.

The -w switch creates a weighted Delaunay tetrahedralization from a set

of weighted points. Remember that a weighted point is defined as p0 =
{px , py , pz , p2x + p2y + p2z − w} ∈ R4 , where w is the weight (a real value) of the
point p = {px , py , pz } ∈ R3 [7].
4.2 Command Line Switches 35

Figure 13: Letf: a set of 164 randomly distributed points in a unit cube.
Right: the Delaunay tetrahedralization (shown in black edges) and Voronoi
diagram (shown in colored faces).

Save the set of weighted points in a .node file. The points in .node file
must have at least one attribute, and the first attribute of each point is its
weight. To generate a weighted DT of this point set, run TetGen with the
following command:

tetgen -w test.node

The weighted Delaunay tetrahedralization and its convex hull are saved in
the same files listed in Table 3. Note that some of the points in test.1.node
may not belong to any tetrahedron.

The Voronoi diagram or the power diagram of the point set is obtained
by taking the dual of the generated Delaunay or weighted Delaunay tetrahe-
dralization, see Figure 13 for an example.
By adding a-v switch in the command line, TetGen outputs the Voronoi
diagram or the power diagram in the four files shown in Table 5:

test.1.v.node The list of Voronoi vertices (or orthocenters).

test.1.v.edge The list of Voronoi edges.
test.1.v.face The list of Voronoi faces.
test.1.v.cell The list of Voronoi cells.

Table 5: The output files for Voronoi or power diagram.

The .v.node file has the same file format of .node file. The file formats
of .v.edge, .v.face, and .v.cell are described in the file format section.
Note that the switches -w and -v are only used for a point set.
36 4 USING TETGEN

4.2.2 Boundary conformity and recovery (-p, -Y)

The -p switch reads a boundary description (a surface mesh) of a 3d piece-
wise linear complex (PLC) stored in file .poly or .smesh and generates a
tetrahedral mesh of the PLC.
Default, TetGen generates a constrained Delaunay tetrahedralization (CDT)
of the PLC. Here is an example of creating a CDT of the PLC named
cami1a.poly (Figure 14 left). Run the following command:

tetgen -p cami1a.poly

This will produce the CDT of the PLC shown in Figure 14 middle. Below
is a screen output of TetGen:

Opening cami1a.poly.
Opening cami1a.node.
Delaunizing vertices...
Delaunay seconds: 0.019862
Creating surface mesh ...
Surface mesh seconds: 0.002374
Constrained Delaunay...
Constrained Delaunay seconds: 0.012435
Removing exterior tetrahedra ...
Exterior tets removal seconds: 0.000783
Optimizing mesh...
Optimization seconds: 0.000662

Writing cami1a.1.node.
Writing cami1a.1.ele.
Writing cami1a.1.face.
Writing cami1a.1.edge.

Output seconds: 0.003398

Total running seconds: 0.039744

Statistics:

Input points: 460

Input facets: 884
Input segments: 690
Input holes: 0
Input regions: 0

Mesh points: 542

4.2 Command Line Switches 37

Mesh tetrahedra: 1678

Mesh faces: 3904
Mesh faces on facets: 1118
Mesh edges on segments: 772
Steiner points on segments: 82

From the mesh statistics of the output (the last line), we can see that
TetGen added 82 Steiner points on the segments of the PLC.
If the -Y switch is used together, the input boundary edges and faces of
the PLC are preserved in the generated tetrahedral mesh. Steiner points (if
there exist) appear only in the interior space of the PLC. For example, run
the following command:

tetgen -pY cami1a.poly

This will produce a tetrahedral mesh of the PLC shown in Figure 14 right.
Below is a screen output of TetGen:

Opening cami1a.poly.
Opening cami1a.node.
Delaunizing vertices...
Delaunay seconds: 0.016072
Creating surface mesh ...
Surface mesh seconds: 0.001333
Recovering boundaries...
Boundary recovery seconds: 0.0432
Removing exterior tetrahedra ...
Exterior tets removal seconds: 0.001152
Suppressing Steiner points ...
Steiner suppression seconds: 0.001164
Recovering Delaunayness...
Delaunay recovery seconds: 0.016093
Optimizing mesh...
Optimization seconds: 0.004006
Jettisoning redundant points.

Writing cami1a.1.node.
Writing cami1a.1.ele.
Writing cami1a.1.face.
Writing cami1a.1.edge.
38 4 USING TETGEN

Figure 14: An input PLC (cami1a.poly, left), the generated Steiner CDT
(middle, -p switch) in which Steiner points are located on the boundary
edges of the PLC, and a constrained tetrahedralization (right, -pY switch)
in which Steiner points lie in the interior of the PLC.

Output seconds: 0.003188

Total running seconds: 0.086466

Statistics:

Input points: 460

Input facets: 884
Input segments: 1349
Input holes: 0
Input regions: 0

Mesh points: 461

Mesh tetrahedra: 1516
Mesh faces: 3498
Mesh faces on facets: 954
Mesh edges on segments: 1349
Steiner points inside domain: 1

From the mesh statistics of the output (the last line), we can see that
TetGen only added 1 Steiner point in the interior of the PLC. The input
facets and segments are preserved.
The default outputs of TetGen are four files listed in Table 6:

cami1a.1.node The list of vertices (including Steiner points) of the CDT.

cami1a.1.ele The list of tetrahedra of the CDT.
cami1a.1.face The list of boundary faces of the CDT.
cami1a.1.edge The list of boundary edges of the CDT.

Table 6: The output files by command: tetgen -p cami1a.poly

4.2 Command Line Switches 39

Other output switches are available by adding the switches: -f (output all
faces including interior faces), -e (output all edges including interior edges),
and -n (output the adjacency graph of tetrahedra).

4.2.3 Quality mesh generation (-q)

The -q switch adds new points to improve the mesh quality. It can be used
together with -p (to refine a CDT), or -r (to refine a previously generated
mesh), -a, or -m (to conform to a mesh sizing function).
TetGen enforces two quality constraints on tetrahedra: a maximum radius-
edge ratio bound and a minimum dihedral angle bound. Default, these two
constraints are 2.0 and 0 degrees, respectively. These quality constraints may
be specified after the -q. The two constraints are separated by a slash ‘/’
(or ‘,’):

• the first constraint is the maximum allowable radius-edge ratio, default

is 2.0; and

• the second constraint is the minimum allowable dihedral angle, default

is 0 (degree);

of any tetrahedron. For example, -q1.2 specifies a maximum radius-edge

ratio of 1.2; -q1.2/10 specifies the same plus a minimum dihedral angle of
10 degrees. -q/7 specifies the default radius-edge ratio bound of 2 and a
dihedral angle bound of 7 degrees.
For an example, the following command uses the default quality con-
straints. It is equivalent to -pq2.0/0.

tetgen -pq thepart.smesh

The screen output of the command line: -pq2.0/0 is shown below. Fig-
ure 15 illustrate three quality tetrahedral meshes of a PLC generated by
applying different radius-edge ratio bounds.

Opening thepart.smesh.
Opening thepart.node.
Delaunizing vertices...
Delaunay seconds: 0.03408
Creating surface mesh ...
Surface mesh seconds: 0.004497
40 4 USING TETGEN

Constrained Delaunay...
Constrained Delaunay seconds: 0.025309
Removing exterior tetrahedra ...
Exterior tets removal seconds: 0.001419
Refining mesh...
Refinement seconds: 0.489247
Optimizing mesh...
Optimization seconds: 0.014569

Writing thepart.1.node.
Writing thepart.1.ele.
Writing thepart.1.face.
Writing thepart.1.edge.

Output seconds: 0.048593

Total running seconds: 0.618028

Statistics:

Input points: 994

Input facets: 1995
Input segments: 1491
Input holes: 0
Input regions: 0

Mesh points: 8029

Mesh tetrahedra: 33773
Mesh faces: 73092
Mesh faces on facets: 11092
Mesh edges on segments: 5143
Steiner points inside domain: 2485
Steiner points on facets: 898
Steiner points on segments: 3652

Remarks

• The default output files (four files) are the same as in Table 6.

• Adding a -V switch in the command line, TetGen will print a mesh

quality report (aspect ratios, radius-edge ratios, dihedral angles) of the
generated tetrahedral mesh on the screen, see Section 4.2.11.

• If there is no sharp features in the input PLC, the Delaunay refinement

algorithm used in TetGen is guaranteed to terminate with a radius-edge
4.2 Command Line Switches 41

Figure 15: The quality tetrahedral meshes of a PLC (thepart.smesh) gen-

erated by the commands: -pq2/0, -pq1.4/0, and -pq1.1/0
.

ratio bound no smaller than 2.0, and there is no bound on the minimum
dihedral angle. In practice, the algorithm behaves much better, e.g., it
usually succeeds for a radius-edge ratio of 1.2 and a minimum dihedral
angle of 18 degrees.

• If there are sharp features in the PLC, TetGen will ensure the desired
quality constraints on most of the tetrahedra, but leave some bad-
quality tetrahedra in the final mesh. Usually, they are near the sharp
features.

4.2.4 Adaptive mesh generation (-a, -m)

TetGen supports several ways of generating adaptive tetrahedral meshes.
They are briefly described in Section 1.2.6.

Impose volume constraints (-a) The -a switch is used in mesh refine-

ment, i.e., together with -q. It imposes a maximum volume constraint on
all tetrahedra. If a number follows the -a, no tetrahedra is generated whose
volume is larger than that number. See Figure 18 for an example.

• One can impose both a fixed volume constraint and a varying volume
constraint for some sub-regions (defined in .poly or .smesh file) by
invoking the -a switch twice, once with and once without a number
following. Each volume specified may include a decimal point.

• If no number is specified and the -r switch is used, a .vol file is

expected, which contains a separate volume constraint for each tetra-
hedron. It is useful for refining a finite element or finite volume mesh
based on a posteriori error estimates.
42 4 USING TETGEN

Figure 16: Examples of applying facet and segment constraints (.var file).

Impose facet area and segment length constraints TetGen also sup-
ports other constraints such as the constraint of maximum face area and the
constraint of maximum edge length imposed on facets and segments of the
PLC, respectively.
Figure 16 shows two examples of the results of applying constraints on a
facet and a segment, respectively.
These constraints are imposed by using a .var file (Section 5.2.9).

Apply a mesh sizing function (-m) The -m switch is used in mesh

refinement, i.e., together with the -q switch. It applies a user-defined mesh
sizing function which specifies the desired edge lengths in the final mesh. It
aims to create an adaptive mesh whose edge lengths are conforming to this
function. At the moment, only isotropic mesh sizing function is supported.
TetGen assumes that the mesh sizing function is specified on a set of
discrete points whose convex hull covers the mesh domain (i.e., the underlying
space of the PLC). The mesh size at any point in the domain is automatically
computed by a linear interpolation from its adjacent points.
When the -m switch is used, TetGen will read a .mtr file, which stores the
nodal mesh size, i.e., the desired edge length at the location of the node in
the mesh domain. There are two possible ways to specify the sizing function.

• The mesh size is directly defined on the nodes of the input PLC (-p
switch) or the nodes of the input mesh (-r switch). In this case, its file
name is xxx.mtr, where xxx is the base file name of the input PLC or
the input mesh, see Figure 17 for an example.

• The mesh size is defined on the nodes of a background mesh. In this

case, there is a background mesh given by files xxx.b.node, xxx.b.ele,
and the mesh size file xxx.b.mtr.
4.2 Command Line Switches 43

Figure 17: The tetrahedral meshes of a PLC (L.smesh) generated by the

commands: -pqm. A sizing function (L.mtr) was applied on the nodes of the
PLC. Both input files are found in Section 5.2.8.
.

4.2.5 Reconstructing a tetrahedral mesh (-r)

The -r switch reconstructs an existing tetrahedral mesh. Usually, the pur-
pose of using this switch is to refine the mesh to improve its quality, i.e., to
use it together with the -q switch. Other usages of the -r switch are possible,
such as inserting additional points (-i switch), mesh adaptation (-m switch),
and linear function interpolation (-m switch plus a background mesh).

• The tetrahedral mesh is read from a .node and an .ele file. These two
files must be supplied.

• If a .face file exists, TetGen will read it and use it to find boundary
faces in the tetrahedral mesh. Note: only those faces with a non-
zero boundary marker are regarded as boundary faces. In either case,
TetGen will automatically identify the faces on the exterior of the mesh
domain and regard them as boundary faces. Interior boundary faces are
also identified by comparing the attributes of two adjacent tetrahedra.

• If an .edge file exists, TetGen will read it and use it to find boundary
edges in the mesh. Note: only those edges with a non-zero boundary
marker are regarded as boundary edges. TetGen will also automatically
identify boundary edges from the identified boundary faces.

• The reconstructed mesh is distinguished from its origin with a dif-

ferent iteration number. For example, tetgen -r xxx.1 reads the
mesh in files xxx.1.node, xxx.1.ele and possibly xxx.1.face and
xxx.1.edge if they exist; reconstructs the mesh; outputs it into three
44 4 USING TETGEN

files xxx.2.node, xxx.2.ele, xxx.2.face, and xxx.2.edge. Now,

xxx.2 can be used as input in the above command, the result is another
mesh saved files xxx.3.node, and so on. Mesh iteration numbers allow
you to create a sequence of successively finer meshes.
• -r should not be used with the -I together.

4.2.6 Mesh optimization (-O)

The -O switch specifies a mesh optimization level and chooses the operations.
Both are integers and are separated by a slash ‘/’.
The mesh optimization level is an integer ranged from 0 to 10, where 0
means no mesh optimization is executed. The larger the level is, more mesh
optimization iterations will be performed, and TetGen may run very slow.
Default the mesh optimization level is 2.
There are three local operations available in TetGen for optimizing the
mesh, which are:
• Edge/face flips.
• Vertex smoothing.
• Vertex insertion/Deletion.
The integer for choosing operations is ranged from 0 to 7. Where 0 means
no operation is chosen (hence no mesh optimization will be done). Each
operation is enabled/disabled by setting the corresponding bit in this integer.
• The 1st bit (the least significant bit) enables/disables edge/face flips.
• The 2nd bit enables/disables vertex smoothing.
• The 3rd bit enables/disables vertex insertion/deletion.
Default, it is 7, i.e., all of these three operations are enabled.
For examples, the switch -O2/7 specifies the optimization level 2 and uses
all optimizing operations. These are the default switches in TetGen. The
switch -O/1 chooses only the edge/face flip operation and uses the default
optimization level.

The following switch is temporarily (maybe change in the future)

The current objective function to be optimized by TetGen is to reduce the
maximum dihedral angle of the tetrahedra. Default, it is 165 degree. One
can set this value after the -o/. For example, -o/150 sets the maximum
dihedral angle to be 150 degree.
4.2 Command Line Switches 45

4.2.7 Mesh coarsening (-R)

The -R switch indicates that some vertices of an existing tetrahedral mesh
are to be removed. TetGen provides two ways to indicate those vertices to
be removed.

• Vertices whose boundary markers (see .node file format) are ‘-1’ are to
be removed.

• When a mesh sizing function is supplied (-m switch), vertices whose

mesh sizes are too large are to be removed.

The -R switch only removes vertices which can be removed, such as ver-
tices lie in the interior of the domain, or vertices lie in the interior of a facet
or a segment. Also, this switch does not guarantee all the marked vertices
are successfully removed.
Once the mesh has been coarsened, the mesh quality may decrease. You
may use -q switch together with the -R switch. It will trigger the mesh
improvement algorithm of TetGen to improve the mesh quality after the
mesh coarsening.

4.2.8 Inserting additional points (-i)

The -i switch indicates that a list of additional points is going to be inserted
into an existing tetrahedral mesh. The list of additional nodes is read from
files xxx.a.node, which xxx stands for the input file name (i.e., xxx.poly or
xxx.smesh, or xxx.ele, ...). This switch is useful for refining a finite element
or finite volume mesh using a list of user-defined points.

• Points which lie in the exterior of the mesh domain are simply dis-
carded.

• TetGen uses a relative tolerance (set by -T switch) to check whether a

point lies on the domain boundary or not, default it is 10−8 .

4.2.9 Assigning region attributes (-A)

The -A switch assigns an additional attribute (an integer number) to each
tetrahedron that identifies what facet-bounded region each tetrahedron be-
longs to. In the output mesh, all tetrahedra in the same region will get a
non-zero attribute.
Figure 18 shows an example of tetrahedral meshes of a PLC which con-
tains several sub-domains.
46 4 USING TETGEN

Figure 18: The tetrahedral meshes of a PLC (ts80305 nd32 cell834.off)

generated by the commands: -pqAa1e-12
.

• A region attribute is an integer which can be either positive or negative.

It must not be a zero. It is used for the exterior of the PLC.

• User-defined attributes are assigned to regions by the .poly or .smesh

file (in the region section). If a region is not explicitly marked by the
.poly file or .smesh file, tetrahedra in that region are automatically
assigned a non-zero attribute.

• Default, the region attributes are numbered from 1, 2, 3, · · ·. If there

are user-assigned region attributes (by the .poly or .smesh file), the
region attributes are shifted by a number i, i.e., i + 1, i + 2, i + 3, · · ·,
where i is either 0 or the maximum integer of the user-assigned region
attributes.

• The -A switch has an effect only when the -p switch is used and the
-r switch is not.

4.2.10 Mesh output switches (-f, -e, -n, -z, -o2)

TetGen provides various switches to output its mesh. They are summarized
below.

-f The -f switch outputs all triangular faces (including interior faces) of

the mesh into a .face file. Default, only the boundary faces or the convex
hull faces are output.
In the .face file, interior faces and boundary (or convex hull) faces are
distinguished from their boundary markers. Each interior face has a bound-
ary marker ‘0’. A non-zero boundary marker means a boundary face or
convex hull face.
4.2 Command Line Switches 47

-e The -e switch outputs all mesh edges (including interior edges) of the
mesh into a .edge file. Default, only the boundary edges are output.
In the .edge file, interior edges and boundary edges are distinguished
from their boundary markers. Each interior edge has a boundary marker ‘0’.
A non-zero boundary marker means a boundary edge.

-n The -n switch outputs the neighboring tetrahedra to a .neigh file. Each

tetrahedron has four neighbors. The first neighbor of this tetrahedron is
opposite the first corner of it, and so on. The neighbors are given by their
indices into the corresponding .ele file. A ‘-1’ indicates that there is no
neighbor at this face of the tetrahedron.
If the -nn switch is used, TetGen also outputs the neighboring tetrahedra
to each face of the mesh in the corresponding .face file.

-z The -z switch numbers all output items starting from zero. This switch
is useful in case of calling TetGen from another program.

-o2 With -o2 switch, TetGen will output the tetrahedral mesh with quadratic
elements which have 10 nodes per tetrahedron, 6 nodes per triangular face,
and 3 nodes per edge. The positions of these extra nodes in each element is
shown in Figure 20.

4.2.11 Mesh statistics (-V)

The -V switch gives detailed information about what TetGen is doing. More
‘V’s are increasing the amount of detail.
Specifically, -V gives information on algorithmic progress and more de-
tailed statistics including a rough mesh quality report. Below is a screen
output of the quality report.

Mesh quality statistics:

Smallest volume: 0.016741 | Largest volume: 125.77

Shortest edge: 0.30902 | Longest edge: 12.189
Smallest asp.ratio: 1.2927 | Largest asp.ratio: 16.964
Smallest facangle: 15.352 | Largest facangle: 141.8279
Smallest dihedral: 5.587 | Largest dihedral: 163.9413

Aspect ratio histogram:

< 1.5 : 5 | 6 - 10 : 33
1.5 - 2 : 105 | 10 - 15 : 4
2 - 2.5 : 228 | 15 - 25 : 1
2.5 - 3 : 215 | 25 - 50 : 0
48 4 USING TETGEN

3 - 4 : 321 | 50 - 100 : 0
4 - 6 : 150 | 100 - : 0
(A tetrahedron’s aspect ratio is its longest edge length divided by its
smallest side height)

Face angle histogram:

0 - 10 degrees: 0 | 90 - 100 degrees: 637
10 - 20 degrees: 122 | 100 - 110 degrees: 131
20 - 30 degrees: 556 | 110 - 120 degrees: 101
30 - 40 degrees: 700 | 120 - 130 degrees: 44
40 - 50 degrees: 1273 | 130 - 140 degrees: 5
50 - 60 degrees: 1085 | 140 - 150 degrees: 1
60 - 70 degrees: 1129 | 150 - 160 degrees: 0
70 - 80 degrees: 871 | 160 - 170 degrees: 0
80 - 90 degrees: 506 | 170 - 180 degrees: 0

Dihedral angle histogram:

0 - 5 degrees: 0 | 80 - 110 degrees: 1675
5 - 10 degrees: 10 | 110 - 120 degrees: 228
10 - 20 degrees: 141 | 120 - 130 degrees: 149
20 - 30 degrees: 362 | 130 - 140 degrees: 92
30 - 40 degrees: 487 | 140 - 150 degrees: 77
40 - 50 degrees: 762 | 150 - 160 degrees: 32
50 - 60 degrees: 770 | 160 - 170 degrees: 7
60 - 70 degrees: 812 | 170 - 175 degrees: 0
70 - 80 degrees: 768 | 175 - 180 degrees: 0

To get the statistics for an existing mesh, run TetGen on that mesh with
the -rNEF switches to read the mesh and print the statistics without writing
any files.
Moreover, -V also gives information on the memory usage of TetGen.
Below is a screen output of the memory usage report.

Memory usage statistics:

Maximum number of tetrahedra: 45309

Maximum number of tet blocks (blocksize = 8188): 6
Approximate memory for tetrahedral mesh (bytes): 8,920,640
Approximate memory for extra pointers (bytes): 1,775,824
Approximate memory for algorithms (bytes): 637,136
Approximate memory for working arrays (bytes): 2,092,580
Approximate total used memory (bytes): 13,426,180

-VV gives more details on the algorithms, and slow down the execution.
While -VVV is only useful for debugging.

4.2.12 Memory allocation (-x)

TetGen allocates memory in blocks. Each block is a chunk of memory al-
4.2 Command Line Switches 49

located once. It stores a number of mesh entities, i.e., vertices, tetrahedra,

boundary faces, and boundary edges. TetGen will dynamically allocate new
blocks when they are needed.
Default, each block consists of 8188 tetrahedra. This data size may be
too small for generating large meshes. This may slow down the performance
of TetGen. The -x switch allows users set the desired number of elements
allocated in one block.
If the -V switch is used, TetGen will report its memory usage, see Sec-
tion 4.2.11. A hint to enlarge the block size can be seen from the “Maximum
number of tet blocks” (the second line in this report). If this number is large
(for example 10000), it is reasonable to enlarge the block size.

4.2.13 Miscellaneous

-c The -c switch keeps the convex hull of the tetrahedral mesh. Default,
TetGen removes all tetrahedra which do not lie in the interior of the PLC
(the domain) which may have an arbitrary shape and topology, i.e., it may
be non-convex and may contain holes. If the -c switch is used, tetrahedra in
the exterior of the PLC are not removed. The union of the mesh elements is
a topological ball.
TetGen assigns all exterior tetrahedra a region attribute ‘-1’, so that they
can be distinguished from the interior tetrahedra.

-S The -S switch specifies a maximum number of Steiner points (points

that are not in the input) which are added by mesh refinement to improve
the mesh quality. The default is to allow an unlimited number of Steiner
points.

-C The -C switch indicates TetGen to check the consistency of the mesh on

finish. If it is specified twice, i.e., -CC, TetGen also checks constrained De-
launay (for the -p switch) or conforming Delaunay (for -q, -a, or -i) property
of the mesh.

-I The -I switch does not use the iteration numbers, it suppresses the
output of .node file, so your input file won’t be overwritten. It cannot be
used with the -r switch, because that would overwrite your input .ele file. It
shouldn’t be used with the -q switch if one is using a .node file for input,
because no .node file is written, so there is no record of any added Steiner
points.
50 4 USING TETGEN

-T The -T switch sets a user-defined tolerance used by many computations

of TetGen, default is 10−8 .
In principle, the vertices which are used to define a facet of a PLC should
be exactly coplanar. But this is very hard to achieve in practice due to the
inconsistent nature of the floating-point format used in computers.
TetGen accepts facets which vertices are not exactly but ”approximately
coplanar”. Four points a, b, c and d are assumed to be coplanar as long
as the ratio v/l3 is smaller than the given tolerance, where v and l are the
volume and the average edge length of the tetrahedron abcd, respectively.
To choose a proper tolerance according to the input point set will usually
reduce the number of adding points and improve the mesh quality.
 51

4 3
12 11
17
10
9
1 2

Figure 19: The facet (shown in pink) consists of four polygons and one hole.
The ordered vertex lists of the polygons are: (1, 2, 3, 4), (9, 10, 11, 12), (11, 3),
and (17). The last two polygons are degenerate.

5 File Formats
Files are used as input and output when using TetGen as a stand-alone
program. This section describes the input/output file formats of TetGen.
When using TetGen as a library, the data structure tetgenio (explained in
Section 6) is used to transfer the data stored in files.

5.1 Useful Things to Know

5.1.1 A Boundary Description of PLCs
In TetGen, a 3d PLC is described by a boundary discretization (e.g., a surface
mesh) of the PLC. This description can be viewed as a Boundary Representa-
tion (B-Rep) without topological information, i.e., there are no information
like incidences and orientations about edges and facets. This makes the
description simple and it can describe non-manifolds easily. TetGen will re-
cover and validate the topological information from this description during
its meshing process.
The boundary description of a PLC contains the set of vertices and facets
of the PLC. Recall that each facet of a PLC is a 2d PLC. It may contain
holes, segments and vertices in its interior, see Figure 19 for an example.
TetGen describes a facet by a list of polygons and a list of holes. Each
polygon of a facet is described by an ordered list of vertices. The order of
the vertices can be in either clockwise or counterclockwise order. A polygon
may be degenerate, i.e., it may contain only one or two vertices. A degenerate
polygon is used to represent an isolated vertex or segment in this facet, see
Fig. 19.
A hole in a facet is described by specifying an arbitrary point p ∈ R3 ,
such that the projection of p onto this facet lies strictly in the interior of this
52 5 FILE FORMATS

hole. Note that p is not a vertex of the PLC.

Remark. By the definition of a PLC, all vertices of a facet must lie in
the same affine subspace of that facet. However, this requirement is generally
impossible to be satisfied in practice due to the floating-point numbers used in
computer. TetGen only requires that all vertices of a facet be approximately
coplanar.
In addition, a list of holes and sub-regions of the PLC can be defined. A
hole of the PLC is described by specifying an arbitrary point p ∈ R3 that
lies strictly in the interior of the hole. Sub-regions are described exactly the
same way. Note that the points used to define holes and sub-regions are not
vertices of the PLC.
This description of a PLC is further explained in the input file formats
of TetGen, i.e., the .node, .poly, and .smesh file formats.

5.1.2 Boundary Markers

In TetGen, the mesh entities like vertices, edges, and faces, are assigned with
a boundary marker. Boundary markers are tags (integers) used mainly to
identify which entities are associated with which boundary element of the
input PLC, such as, a segment or a facet. A common use is to determine
where boundary conditions should be applied to a finite element mesh. You
can prevent boundary markers from being written into files produced by
TetGen by using the -B switch.
Mesh entities which are not on the boundary of PLC must have their
boundary markers ‘0’.
Mesh entities which are on the boundary will assign a boundary marker
that is the same as the boundary marker of that boundary of the PLC.
However, if a boundary of a PLC does not have a boundary marker or have a
marker ‘0’, TetGen will assign a ‘1’ to those entities belong to this boundary
in the output files. This way, TetGen is able to distinguish them from other
interior mesh entities.

5.2 TetGen’s File Formats

Table 7 lists all file formats that used by TetGen. All files are of ASCII form
and may contain comments prefixed by the character ’#’. Points, tetrahedra,
facets, edges, holes, and maximum volume constraints must be numbered
consecutively, starting from either 1 or 0. However, all input files must be
consistent. TetGen automatically detects your choice while reading the .node
(or .poly or .smesh) file. When calling TetGen from another program, use
the -z switch if you wish to number objects from zero.
5.2 TetGen’s File Formats 53

.node input/output a list of nodes.

.poly input a PLC.
.smesh input/output a PLC.
.ele input/output a list of tetrahedra.
.face input/output a list of triangular faces.
.edge input/output a list of edges.
.vol input a list of maximum volumes.
.mtr input/output a mesh sizing function.
.var input a list of variant constraints.
.neigh output a list of neighbors.

Table 7: Overview of TetGen’s file formats.

Remark: in the following description ‘#’ stands for ‘number’ – it should

not cause confusion with the comment prefix.

5.2.1 .node files

A .node file contains a list of 3d points.

First line: <# of points> <dimension (3)> <# of attributes>

<boundary markers (0 or 1)>
Remaining lines list # of points:
<point #> <x> <y> <z> [attributes] [boundary marker]
...

Each point has three coordinates (x, y and z), probably has one or several
attributes, and a boundary marker as well. The .node files used as both
input and output files to represent the point set of a PLC, or the point set
of a mesh, or a set of additional points (for the -i switch) which need to
be inserted into a mesh. The example below demonstrates the layout of the
.node file.

# Node count, 3 dim, no attribute, no boundary marker

8 3 0 0
# Node index, node coordinates
54 5 FILE FORMATS

1 0.0 0.0 0.0

2 1.0 0.0 0.0
3 1.0 1.0 0.0
4 0.0 1.0 0.0
5 0.0 0.0 1.0
6 1.0 0.0 1.0
7 1.0 1.0 1.0
8 0.0 1.0 1.0

The attributes, which are typically floating-point values of physical quan-

tities (such as mass or conductivity) associated with the points, are copied
unchanged to the output mesh.
If -p, -q, -a, or -i is selected, each Steiner point added to the mesh has
attributes zero.
If the -w (weighted Delaunay tetrahedralization) switch is specified, the
first attribute is regarded as the weight of that point.
If the <boundary marker> of the first line is 1, the last column of the
remainder of the file is assumed to contain boundary markers. Boundary
markers are used to identify boundary points (points resting on PLC facets).
The .node file produced by TetGen contains boundary markers in the last
column unless they are suppressed by the -B switch. The boundary marker
associated with each point in an output .node file is chosen as follows:

• If a point is assigned a nonzero boundary marker in the input file, then

it is assigned the same marker in the output .node file.

• Otherwise, if the node lies on a PLC facet with a nonzero boundary

marker, then the node is assigned the same marker that facet has. If the
node lies on several such facets, one of the markers is chosen arbitrarily.

• Otherwise, if the node occurs on a boundary of the mesh, then the node
is assigned the marker 1.

• Otherwise, the point is assigned the marker 0.

TetGen can determine which points are on the boundary, input with the
boundary marker zero (or use no markers at all) will result in output with
boundary marker 1 for all points on the boundary.
If -R (mesh coarsening) switch is used, points with boundary markers
equal to −1 will be removed.
5.2 TetGen’s File Formats 55

5.2.2 .poly files

A .poly file is a B-Rep description of a piecewise linear complex (PLC) as
well as some additional information. It consists of four parts.

• The first part is a a list of points.

• The second part is a list of facets.

• The next part is a list of hole points.

• The fourth part is a a list of region attributes.

The first three parts are mandatory, but the fourth part is optional. They
are respectively described below.

Part 1 - node list Part 1 lists all the points, and is identical to the format
of .node files. <# of points> may be set to zero to indicate that the points
are listed in a separate .node file.

First line: <# of points> <dimension (3)> <# of attributes>

<boundary markers (0 or 1)>
Remaining lines list # of points:
<point #> <x> <y> <z> [attributes] [boundary marker]
...

Part 2 - facet list The facet list is given by

One line: <# of facets> <boundary markers (0 or 1)>

Following lines list # of facets:
<facet #>
...

The format of a single facet is:

56 5 FILE FORMATS

One line: <# of polygons> [# of holes] [boundary marker]

Following lines list # of polygons:
<# of corners> <corner 1> <corner 2> ... <corner #>
...
Following lines list # of holes:
<hole #> <x> <y> <z>
...

Each facet is a polygonal region which may contain segments, single points
and holes. It consists of a list of polygons. Each polygon is specified by giving
the number of corners n, n ≥ 1, followed by the list of ordered indices of those
corners. It doesn’t matter which order (counterclockwise or clockwise) you
choose to list the indices. It can be degenerate, i.e., n = 1 or n = 2 indicates
a single point or a segment, respectively.
A hole in a facet is specified by identifying a point inside the hole. The
list of hole points (consecutively) follows the list of polygons.
Boundary markers of facets are tags (integers) used mainly to identify
which faces of the tetrahedralization are associated with which PLC facet,
hence identify which faces occur on a boundary of the tetrahedralization. A
common use is to determine where different boundary condition types should
be applied to a mesh.
If [boundary marker] is 1, each facet is assumed to have a boundary
marker (an integer). TetGen will produce an additional boundary marker
for each face in .face (output) file (in the last column of each record).
If [boundary marker] is 0, TetGen will automatically assign a 1 for all
boundary faces (which belong to facets of the PLC) in .face (output) file.
You can prevent boundary markers from being written into .face file by
using the -B switch.
Note that each line of indices should not be arbitrarily long because the
maximum characters per line read by TetGen is 1024. The list can be broken
into several lines.

Part 3 - hole list Holes in the volume are specified by identifying a point
inside each hole.

One line: <# of holes>

5.2 TetGen’s File Formats 57

Following lines list # of holes:

<hole #> <x> <y> <z>
...

After the constrained Delaunay tetrahedralization is formed, TetGen cre-

ates holes by removing tetrahedra. This exactly is the reason that TetGen re-
quires a closed boundary of the PLCs. In case of non-closed PLC facets the
whole tetrahedralization will be “eaten” away. If two tetrahedra abutting a
boundary face are removed, the boundary face itself is also vanished.
Hole points have to be placed inside a region, else the rounding error
determines which side of the facet is being transformed into the hole.

Part 4 - region attributes list The optional fourth section lists regional
attributes (to be assigned to all tetrahedra in a region) and regional con-
straints on the maximum tetrahedron volume. TetGen will read this section
only if the -A switch is used or the -a switch without a number is invoked.
Regional attributes and volume constraints are propagated in the same man-
ner as holes.

One line: <# of region>

Following lines list # of region attributes:
<region #> <x> <y> <z> <region number> <region attribute>
...

If two values are written on a line after the x, y and z coordinate, the
former is assumed to be a regional attribute (but will only be applied if
the -A switch is selected), and the latter is assumed to be a regional volume
constraint (but will only be applied if the -a switch is selected). It is possible
to specify just one value after the coordinates. It can serve as both an
attribute and a volume constraint, depending on the choice of switches. A
negative maximum volume constraint allows to use the -A and the -a switches
without imposing a volume constraint in this specific region.

An example In the following, a unit (1 × 1 × 1) cube is described by the

poly file format.
58 5 FILE FORMATS

# Part 1 - node list

# node count, 3 dim, no attribute, no boundary marker
8 3 0 0
# Node index, node coordinates
1 0.0 0.0 0.0
2 1.0 0.0 0.0
3 1.0 1.0 0.0
4 0.0 1.0 0.0
5 0.0 0.0 1.0
6 1.0 0.0 1.0
7 1.0 1.0 1.0
8 0.0 1.0 1.0

# Part 2 - facet list

# facet count, no boundary marker
6 0
# facets
1 # 1 polygon, no hole, no boundary marker
4 1 2 3 4 # front
1
4 5 6 7 8 # back
1
4 1 2 6 5 # bottom
1
4 2 3 7 6 # right
1
4 3 4 8 7 # top
1
4 4 1 5 8 # left

# Part 3 - hole list

0 # no hole

# Part 4 - region list

0 # no region

5.2.3 .smesh files

A .smesh file is also a B-Rep description of a PLC. It describes a simple
B-Rep model which each of its facet only has exactly one polygon, no holes,
no segment and no point inside. It is less flexible than the .poly file format
while it is much simpler and useful when the boundary of PLC consists of
only simple polygons, i.e., triangles, quads, etc.
The same as .poly file format, the .smesh file format consists of four
5.2 TetGen’s File Formats 59

parts, which are points, facets, holes and regions, respectively. Only the
second part which describes facets is different. It is described below.

Part 2 - facet list Each facet consists of exactly one polygon. The corner
list of each polygon can be distributed over a number of lines. The optional
boundary marker of each facet is given at the end of the corner list.

One line: <# of facets> <boundary markers (0 or 1)>

Following lines list # of facets:
<# of corners> <corner 1> ... <corner #> [boundary marker]
...

The following example demonstrates the layout of facet part of the unit
cube.

# Part 2 - facet list

# facet count, no boundary marker
6 0
# facets
4 1 2 3 4 # front
4 5 6 7 8 # back
4 1 2 6 5 # bottom
4 2 3 7 6 # right
4 3 4 8 7 # top
4 4 1 5 8 # left

5.2.4 .ele files

An .ele file contains a list of tetrahedra.

First line: <# of tetrahedra> <nodes per tet. (4 or 10)>

<region attribute (0 or 1)>
Remaining lines list # of tetrahedra:
<tetrahedron #> <node> <node> ... <node> [attribute]
...
60 5 FILE FORMATS

5
6

9
3
1 10
8
7

Figure 20: The local numbering of the vertices (corners) and the second-order
nodes of a tetrahedron.

Each tetrahedron has four corners (or ten corners if -o2 switch is used).
Nodes are indices into the corresponding .node file. The first four nodes
are the corner vertices. If -o2 switch is used, the remaining six nodes are
generated on the midpoints of the edges of the tetrahedron. Figure 20 shows
how these corners and the second-order nodes are locally numbered. Second
order nodes are output only. They are omitted by the mesh reconstruction
(the -r switch).
If the <region attribute> in the first line is 1, each tetrahedra has an
additional region attribute (an integer) in the last column. Region attributes
of tetrahedra are tags used mainly to identify which tetrahedra of the tetra-
hedralization are associated with which facet-bounded region (sub-domain)
of the PLC, set in the fourth part of a .poly or a .smesh file. Region at-
tributes do not diffuse across facets, all tetrahedra in the same region have
exactly the same region attribute. A common use of the region attribute is
to determine which material a tetrahedron has.
The .ele file is the default output file of TetGen. However, it can be
omitted by -E switch. If -r switch is used, TetGen reads a .ele file and
reconstructs a tetrahedral mesh from it.
The following example illustrates a layout of the .ele file.

154 4 0
1 4 107 3 50
5.2 TetGen’s File Formats 61

2 4 108 3 107
3 9 97 95 94
4 4 107 50 93
5 56 1 50 47
6 94 98 97 95
7 97 9 95 55
...

5.2.5 .face files

A .face file contains a list of triangular faces of the tetrahedralization.

First line: <# of faces> <boundary marker (0 or 1)>

Remaining lines list # of faces:
<face #> <node> <node> <node> ... [boundary marker] ...
...

In its basic form, each face has three corners and possibly has a boundary
marker. Nodes are indices into the corresponding .node file.
If the <boundary marker> in the first line is 1, each face has an additional
boundary marker (an integer) in the last column. Boundary markers of
facets are defined in the .poly or the .smesh files. The optional column of
Boundary markers can be suppressed by the -B switch.
If the -o2 switch is used, each face has three corners and three second-
order nodes generated on the midpoints of the edges of this face. Figure 21
shows the local numbering of the corners and the second-order nodes of a
face. They are listed after the corners of this face, and before its boundary
marker. Second order nodes are output only. They are omitted by the mesh
reconstruction (the -r switch).
If the -nn switch is used, each face contains two additional indices (after
the boundary marker) into the corresponding .ele file. They are tetrahedra
containing this face. A −1 indicates there is no adjacent tetrahedron at this
side, i.e., it is “outer space”.
TetGen default only outputs the boundary faces or the convex hull faces
into a .face file. If -f switch is used, TetGen outputs all faces (including
interior faces) of the tetrahedralization. In this case, each interior face will
always have a ‘0’ as its boundary marker. This file can be omitted by the -F
switch.
62 5 FILE FORMATS

5
3
1
4
6

Figure 21: The local numbering of the corners and the second-order nodes
of a face.

If -r switch is used, TetGen can also read the .face file for identifying
boundary faces in a reconstructed mesh.

5.2.6 .edge files

An .edge file contains a list of edges of the tetrahedralization.

First line: <# of edges> <boundary marker (0 or 1)>

Remaining lines list # of edges:
<edge #> <endpoint> <endpoint> ... [boundary marker] ...
...

In its basic form, each edge has two endpoints and possibly a boundary
marker (if the <boundary marker> in the first line is 1). Endpoints are
indices into the corresponding .node file.
If the -o2 switch is used, each edge has a second order node (its midpoint).
It is listed after its endpoints, and before its boundary marker. Second order
nodes are output only. They are omitted by the mesh reconstruction (the -r
switch).
If the -nn switch is used, each edge contains one additional index (af-
ter the [boundary marker]) into the corresponding .ele file. It is one of
tetrahedra which contain this edge. A −1 indicates there is no adjacent
tetrahedron at this edge.
If -r switch is used, TetGen can also read the .edge file for identifying
boundary edges in a reconstructed mesh.
It is the default output of TetGen when -p or -r switch is used. Default,
it only contains a list of boundary edges (segments) of the tetrahedralization.
5.2 TetGen’s File Formats 63

If the -e switch is used, it contains a list of all edges (including interior edges)
of the tetrahedralization. This file can be omitted by the -F switch.

5.2.7 .vol files

A .vol file associates with each tetrahedron a maximum volume that is used
for mesh refinement. It is read by TetGen in case the -r switch is used.

First line: <# of tetrahedra>

Remaining lines list # of maximum volumes:
<tetrahedron #> <maximum volume>
...

As with other file formats, every tetrahedron must be represented, and

they must be numbered consecutively. A tetrahedron may be left uncon-
strained by assigning it a zero or negative maximum volume.

5.2.8 .mtr files

The .mtr file assigns a sizing function defined on the nodes of an input PLC,
or an input tetrahedral mesh, or a background tetrahedral mesh. It is used
when the -m switch is applied.

First line: <# of nodes> <size of metric (always 1)>

Remaining lines list # of point metrics:
<value>
...

Each node is assigned a value that is interpreted by TetGen as the desired

length for all edges connecting to the node. TetGen use this size information
to generate an adaptive tetrahedral mesh.
Below is an example of an L-shaped PLC (L.smesh) (see Figure 17):
64 5 FILE FORMATS

# L.smesh
12 3 0 0
1 0 0 0
2 4 0 0
3 4 2 0
4 2 2 0
5 2 6 0
6 0 6 0
1 0 0 2
2 4 0 2
3 4 2 2
4 2 2 2
5 2 6 2
6 0 6 2
8 0
6 1 2 3 4 5 6
6 7 8 9 10 11 12
4 1 2 8 7
4 2 3 9 8
4 3 4 10 9
4 4 5 11 10
4 5 6 12 11
4 6 1 7 12
0
0

Below is an example file L.mtr to assign a sizing function on the nodes

of the PLC:

# L.mtr
12 1
0.25
0.25
0.25
0.025
0.25
0.25
0.25
0.25
0.25
0.025
0.25
0.25
5.2 TetGen’s File Formats 65

The following command generates the adaptive tetrahedral mesh shown

in Figure 17.

tetgen -pqm L.smesh

It is possible to set a size 0 to a node. In this case, TetGen ignores the

mesh size at this node. For example, it is possible to use a L.mtr file like
following;

# L.mtr
12 1
0
0
0
0.025
0
0
0
0
0
0.025
0
0

5.2.9 .var files

A .var file allows you to specify maximum area constraints on facets and
maximum length constraints on segments. They are used for mesh refine-
ment.

One line: <# of facet constraints>

Remaining lines list # of facet constraints:
<constraint #> <boundary marker> <maximum area>
...
One line: <# of segment constraints>
Remaining lines list # of segment constraints:
<constraint #> <point1> <point2> <maximum length>
...
66 5 FILE FORMATS

A constraint of maximum area on facet is set by specifying the boundary

marker of that facet, which is the integer assigned to that facet in the corre-
sponding .poly or .smesh file. A constraint of maximum length on segment
is set by specifying the indices of the two endpoints of that segment.
Figure 16 shows two examples of the results of applying constraints on a
facet and a segment, respectively.

5.2.10 .neigh files

A .neigh file associates with each tetrahedron its neighbors (adjacent tetra-
hedra), which are indices into the corresponding .ele file.

First line: <# of tetrahedra> 4

Following lines list # of neighbors:
<tetrahedra #> <neighbor> <neighbor> <neighbor> <neighbor>
...

An index of −1 indicates no neighbor (because the tetrahedron is on a

boundary of mesh domain). The first neighbor of tetrahedron i is opposite
the first corner of tetrahedron i, and so on. It is output by TetGen when -n
switch is used.

5.2.11 .v.node, .v.edge, .v.face, .v.cell

A .v.node file contains a list of vertices of the Voronoi diagram or the power
diagram ( -w switch). Each Voronoi vertex is the circumcenter (or the or-
thocenter) of a Delaunay (or weighted Delaunay) tetrahedron. The format
of .v.node is the same as the .node file.

A .v.edge file contains a list of edges of the Voronoi diagram or the power
diagram ( -w switch). Each edge corresponds to a face of the Delaunay (or
weighted Delaunay) tetrahedralization. Each Voronoi edge is either a line
segment connecting two Voronoi vertices or a ray starting from a Voronoi
vertex. The file format of a .v.edge file is

First line: <# of edges>

Following lines list # of edges:
5.2 TetGen’s File Formats 67

<edge #> <vertex 1> <vertex 2> <V_x> <V_y> <V_z>

...

<vertex 1> and <vertex 2> are two indices pointing to the list of Voronoi
vertices. <vertex 1> must be non-negative, while <vertex 2> may be −1
which means it is a ray, in this case, the unit vector of this ray is given by
<vertex V x>, <vertex V y>, and <vertex V z>.

A .v.face file contains a list of faces of the Voronoi diagram or the power
diagram ( -w switch). Each face corresponds to an edge of the Delaunay
(or weighted Delaunay) tetrahedralization. It is formed by a list of Voronoi
edges, it may not be closed. The file format of a .v.face file is

First line: <# of faces>

Following lines list # of faces:
<face #> <cell 1> <cell 2> <# of edges> <edge 1> <edge 2> ...
...

<cell 1> and <cell 2> are two indices pointing into the list of Voronoi
cells, i.e., the two cells share this face. <edge 1>, <edge 2> ..., are indices
pointing into the edge list, i.e., they are the edges of this face, there are total
<# of edges’>. If the face is not closed, the index of the last edge of this
face is −1.

A .v.cell file contains a list of cells of the Voronoi diagram or the power
diagram ( -w switch). Each cell corresponds to a vertex of the Delaunay (or
weighted Delaunay) tetrahedralization. A cell is formed by a list of Voronoi
faces, it may not be closed. The file format of a .v.cell file is

First line: <# of cells>

Following lines list # of cells:
<cell #> <# of faces> <face 1> <face 2> ...
...
68 5 FILE FORMATS

.off input Geomview’s polyhedral file format.

.ply input Polyhedral file format.
.stl input Stereolithography format.
.mesh input/output Medit’s mesh file format.

Table 8: Overview of supported file formats.

<face 1>, <face 2>, ... are indices pointing into Voronoi face list. There
are total <# of faces’>. If the cell is not closed, the index of the last face
in this cell is −1.

5.3 Supported File Formats

TetGen supports some polyhedral file formats as well. Table 8 lists the sup-
ported file formats. TetGen recognizes the file formats by the file extensions.

5.3.1 .off files

The .off is the one of the file formats of Geomview http://www.geomview.

org - an interactive 3d viewing program for Unix/Linux. It represents collec-
tions of planar polygons with possibly shared vertices, a convenient way to
describe polyhedra. The polygons may be concave, but there’s no provision
for polygons containing holes.
The description of .off file format can be found elsewhere on the internet.
Below is a simple description of this file format.

OFF numVertices numFaces numEdges

x y z
x y z
... numVertices like above
NVertices v1 v2 v3 ... vN
MVertices v1 v2 v3 ... vM
... numFaces like above

Note that vertices are numbered starting at 0 (not starting at 1), and
that numEdges will always be zero.
5.3 Supported File Formats 69

5.3.2 .ply files

The .ply file format is a simple object description that was designed as a
convenient format for researchers who work with polygonal models. Early
versions of this file format were used at Stanford University and at UNC
Chapel Hill.
A description as well as examples, codes of the PLY file format can be
found at http://paulbourke.net/dataformats/ply/.

5.3.3 .stl files

The .stl or stereolithography format is an ASCII or binary file used in
manufacturing. It is a list of the triangular surfaces that describe a computer
generated solid model. This is the standard input for most rapid prototyping
machines.
The description of .stl file format can be found elsewhere on the web. An
elaborated description can be found at http://en.wikipedia.org/wiki/
STL_(file_format). Below is an example.

solid
...

facet normal 0.00 0.00 1.00

outer loop
vertex 2.00 2.00 0.00
vertex -1.00 1.00 0.00
vertex 0.00 -1.00 0.00
endloop
endfacet
...
endsolid

5.3.4 .mesh files

The .mesh is the file formats of Medit - an interactive 3d mesh viewing pro-
gram http://www.ann.jussieu.fr/frey/software.html. This file format
is described in the documentation of Medit.
A repository of free 3d models in this file format are available at IN-
RIA’s Free 3d Meshes Download http://www-roc.inria.fr/gamma/gamma/
70 5 FILE FORMATS

3 7

4 −2
8
−1
2 6

1 5

Figure 22: A bar having two boundary markers (−1 and −2) defined.

Accueil/.

5.4 File Format Examples

This section provides three examples. They are designed to support inter-
active learning. The topics are describing PLCs using TetGen’s .poly file
format and constructing different quality meshes through the command line
switches.

5.4.1 A PLC with Two Boundary Markers

Figure 22 shows the geometry of a rectangular bar. This bar consists of
eight nodes and six facets (which are all rectangles). In addition, there
are two boundary markers (−1 and −2) associated to the leftmost facet
and the rightmost facet, respectively. This simple model has its physical
meaning. It can be seen as a typical heat transfer problem. The task is
to compute the temperature diffusion in the bar, in which the flow of heat
moves from hot side to cold side. The two boundary markers can represent
two different boundary conditions, one has high temperature and the other
has low temperature. Here is the input file bar.poly describing the bar:

# Part 1 - the node list.

# A bar with 8 nodes in 3d, no attributes, no boundary marker.
8 3 0 0
# The 4 leftmost nodes:
1 0 0 0
2 2 0 0
3 2 2 0
4 0 2 0
# The 4 rightmost nodes:
5 0 0 12
6 2 0 12
7 2 2 12
8 0 2 12
# Part 2 - the facet list.
5.4 File Format Examples 71

# Six facets with boundary markers.

6 1
# The leftmost facet.
1 0 -1 # 1 polygon, no hole, boundary marker (-1)
4 1 2 3 4
# The rightmost facet.
1 0 -2 # 1 polygon, no hole, boundary marker (-2)
4 5 6 7 8
# Other facets.
1
4 1 5 6 2 # bottom side
1
4 2 6 7 3 # back side
1
4 3 7 8 4 # top side
1
4 4 8 5 1 # front side
# Part 3 - the hole list.
# There is no hole in bar.
0
# Part 4 - the region list.
# There is no region defined.
0

The command line is chosen as follows: first mesh the PLC (-p), then
impose the quality constraint (-q). This will result a quality mesh in four
files: bar.1.node, bar.1.ele, bar.1.face, and bar.1.edge.

> tetgen -pq bar.poly

Here is the output file bar.1.node. It contains 47 points. The additional

points were added by TetGen automatically to meet the quality measure.

47 3 0 0
1 0 0 0
2 2 0 0
3 2 2 0
4 0 2 0
5 0 0 12
6 2 0 12
7 2 2 12
8 0 2 12
9 1.0000469999999999 0 0
10 0 0.999668 0
11 0 0.99944500000000003 12
12 1.000594 0 12
...
# Generated by tetgen -pq bar.poly
72 5 FILE FORMATS

Here is the output file bar.1.ele, which contains 83 tetrahedra.

83 4 0
1 18 33 20 34
2 9 2 3 25
3 17 18 20 34
4 43 32 18 37
5 19 20 30 33
6 14 41 13 42
7 12 26 7 6
8 10 28 1 9
9 28 33 18 34
10 35 41 38 45
11 10 9 25 28
12 3 25 19 30
...
# Generated by tetgen -pq bar.poly

Here is the output file bar.1.face with 90 boundary faces. Faces 1 and
2 are on the leftmost facet thus have markers −1; faces 3 and 4 have markers
−2 indicating they are on the rightmost facet. Other faces have the default
markers zero.
90 1
1 3 4 10 -1
2 10 9 3 -1
3 7 12 11 -2
4 7 11 8 -2
5 18 37 43 0
6 24 46 39 0
7 26 6 7 0
8 35 22 24 0
9 29 7 8 0
10 39 7 29 0
11 29 11 27 0
12 23 45 38 0
...
# Generated by tetgen -pq bar.poly

However, the mesh above may be too coarse for numerical simulation
using finite element method or finite volume method. Using either -q or -a
switch or both of them will results a more dense quality mesh:
> tetgen -pq1.414a0.1 bar.poly
TetGen generates a mesh with 330 points and 1092 tetrahedra. The added
points are due to both the -q and -a switches we’ve applied. To see a quality
report of the mesh, type:
> tetgen -rNEFV bar.1.ele
5.4 File Format Examples 73

3 11 7
12 −2
4 8
−1
2 10 6

1 5
9
−10 −20

Figure 23: A bar having two regions (with region attributes −10 and −20,
respectively) and two boundary markers (−1 and −2) defined.

5.4.2 A PLC with Two Sub-regions (Materials)

In this example, we add an internal facet into the bar (in Figure 22), so
that create two sub-regions (separated by the newly added facet) in the bar.
Figure 23 shows the modified geometry. This bar consists of twelve nodes
(which I numbered) and seven facets (Note, some of them are not a single
polygon any more). In addition, there are two regions defined, which have
region attributes −10 and −20, respectively. Physically, you can associate
two different materials to each of these two regions, and the two boundary
markers (−1 and −2) in last example still remain. Here is the input file
bar2.poly describing the modified bar:
# Part 1 - the node list.
# The model has 12 nodes in 3d, no attributes, no boundary marker.
12 3 0 0
# The 4 leftmost nodes:
1 0 0 0
2 2 0 0
3 2 2 0
4 0 2 0
# The 4 rightmost nodes:
5 0 0 12
6 2 0 12
7 2 2 12
8 0 2 12
# The 4 added nodes:
9 0 0 3
10 2 0 3
11 2 2 3
12 0 2 3
# Part 2 - the facet list.
# Seven facets with boundary markers.
7 1
# The leftmost facet.
1 0 -1 # 1 polygon, no hole, boundary marker (-1)
74 5 FILE FORMATS

4 1 2 3 4
# The rightmost facet.
1 0 -2 # 1 polygon, no hole, boundary marker (-2)
4 5 6 7 8
# Each of following facets has two polygons, which are
# one rectangle (6 corners) and one segment.
2
6 1 9 5 6 10 2 # bottom side
2 9 10
2
6 2 10 6 7 11 3 # back side
2 10 11
2
6 3 11 7 8 12 4 # top side
2 11 12
2
6 4 12 8 5 9 1 # front side
2 12 9
# The internal facet separates two regions.
1
4 9 10 11 12
# Part 3 - the hole list.
# There is no hole in bar.
0
# Part 4 - the region list.
# There are two regions (-10 and -20) defined.
2
1 1.0 1.0 1.5 -10 0.1
2 1.0 1.0 5.0 -20 -1

The command line tetgen -pqaA bar2.poly generates the file bar2.1.ele.
The first eight lines are listed next. It differs from bar.1.ele in that each
record has an additional region attribute.

431 4 1
1 32 57 50 60 -20
2 51 23 50 49 -20
3 88 138 116 149 -10
4 76 96 95 36 -20
5 29 55 56 52 -20
6 132 138 88 139 -10
7 65 138 132 139 -10
8 16 54 53 15 -20
...
# Generated by tetgen -pqaA bar2.poly

Visualization of the resulting meshes (by TetView or other tools) shows

the refinement in the region with attribute −10 is denser than the other.
5.4 File Format Examples 75

This is due to the volume constraints (0.1) defined in the file bar2.poly and
the -aA switches.

5.4.3 A PLC with Two Sub-regions and Two Holes

This is the example file (example.poly) coming together with the TetGen’s
source distribution. The geometry as well as some generated mesh are shown
in Figure 24.

# Part 1 - the node list.

28 3 0 1
1 0 0 0 1
2 2 0 0 1
3 2 2 0 1
4 0 2 0 1
5 0 0 4 9
6 2 0 4 9
7 2 2 3 9
8 0 2 3 9
9 0 0 5 2
10 2 0 5 2
11 2 2 5 2
12 0 2 5 2
13 0.25 0.25 0.5 4
14 1.75 0.25 0.5 4
15 1.75 1.5 0.5 4
16 0.25 1.5 0.5 4
17 0.25 0.25 1 4
18 1.75 0.25 1 4
19 1.75 1.5 1 4
20 0.25 1.5 1 4
21 0.25 0 2 4
22 1.75 0 2 4
23 1.75 1.5 2 4
24 0.25 1.5 2 4
25 0.25 0 2.5 4
26 1.75 0 2.5 4
27 1.75 1.5 2.5 4
28 0.25 1.5 2.5 4
# Part 2 - the facet list
23 1
1 0 1 # 1
4 1 2 3 4
1 0 9 # 2
4 5 6 7 8
2 1 3 # 3
4 1 2 6 5
4 21 22 26 25
76 5 FILE FORMATS

1 1 0 2.25
1 0 3 # 4
4 2 3 7 6
1 0 3 # 5
4 3 4 8 7
1 0 3 # 6
4 4 1 5 8
1 0 2 # 7
4 9 10 11 12
1 0 3 # 8
4 9 10 6 5
1 0 3 # 9
4 10 11 7 6
1 0 3 # 10
4 11 12 8 7
1 0 3 # 11
4 12 9 5 8
1 0 4 # 12
4 13 14 15 16
1 0 4 # 13
4 17 18 19 20
1 0 4 # 14
4 13 14 18 17
1 0 4 # 15
4 14 15 19 18
1 0 4 # 16
4 15 16 20 19
1 0 4 # 17
4 16 13 17 20
1 0 4 # 18
4 21 22 23 24
1 0 4 # 19
4 25 26 27 28
1 0 4 # 20
4 21 22 26 25
1 0 4 # 21
4 22 23 27 26
1 0 4 # 22
4 23 24 28 27
1 0 4 # 23
4 24 21 25 28
# Part 3 - the hole list
2
1 1 0.4 2.25
2 1 0.4 0.75
# Part 4 - the region list
2
1 1 0.25 0.1 10 0.001
2 1 0.5 4 20 0.01
 77

Figure 24: Left: A PLC with two sub-regions (materials) and two holes.
Middle: the CDT generated by the command line: -pA. Right: The quality
tetrahedral mesh generated by the command line: -pqAa.

6 Calling TetGen from Another Program

One can use TetGen as a library so that it can be called directly from an-
other program. This section gives the necessary instructions for using the
TetGen library. Users are supposed to be able to use TetGen, i.e., know its
command line switches and the input and output file formats. Please refer
to Section 3 for the instructions of how to compile TetGen into a library.

6.1 The Header File

Programs calling TetGen must include the header file tetgen.h.

#include "tetgen.h"

It includes all data types and function declarations of the TetGen library.
It defines the function tetrahedralize() and the data type tetgenio,
which are provided for users to call TetGen with all its functionality. They
are described in Section 6.2 and Section 6.3, respectively.

6.2 The Calling Convention

The function tetrahedralize() is declared as follows:
78 6 CALLING TETGEN FROM ANOTHER PROGRAM

void tetrahedralize(char *switches, tetgenio *in, tetgenio *out,

tetgenio *addin = NULL, tetgenio *bgmin = NULL);

The parameter switches is a string containing the command line switches

for this call. In this string, no initial dash ’-’ is required. The Q (quiet) switch
is recommended in the final code. Some file output switches, like I and g are
ignored.
The parameters in and out, which are two pointers pointing to objects
of tetgenio, describing the input and the output. in and out may never be
NULL.
Two additional parameters addin and bgmin may be supplied. When the
switch -i is used, addin contains a list of additional vertices to be inserted.
When the switch -m is used, bgmin contains a background mesh with is used
to provide a mesh sizing function.

6.3 The tetgenio Data Type

The tetgenio structure is used to pass data into and out of the tetrahe-
dralize() procedure. It replaces the input and output files of TetGen by
a collection of arrays, which are used to store points, tetrahedra, boundary
markers, and so forth. It is a C++ class including data fields and functions.
The data fields of tetgenio:

int firstnumber; // 0 or 1, default 0.

int mesh_dim; // must be 3.

REAL *pointlist;
REAL *pointattributelist;
REAL *pointmtrlist;
int *pointmarkerlist;
int numberofpoints;
int numberofpointattributes;
int numberofpointmtrs;

int *tetrahedronlist;
REAL *tetrahedronattributelist;
REAL *tetrahedronvolumelist;
int *neighborlist;
int numberoftetrahedra;
int numberofcorners;
int numberoftetrahedronattributes;

facet *facetlist;
int *facetmarkerlist;
int numberoffacets;
6.4 Description of Arrays 79

REAL *holelist;
int numberofholes;

REAL *regionlist;
int numberofregions;

REAL *facetconstraintlist;
int numberoffacetconstraints;

REAL *segmentconstraintlist;
int numberofsegmentconstraints;

int *trifacelist;
int *trifacemarkerlist;
int numberoftrifaces;

int *edgelist;
int *edgemarkerlist;
int numberofedges;

6.4 Description of Arrays

In all cases, the first item in any array is stored starting at index [0]. However,
that item is item number firstnumber (0 or 1) unless the z switch is used,
in which case it is item number ‘0’. Following is the description of arrays.

pointlist An array of point coordinates. The first point’s x coordinate is

at index [0], its y coordinate at index [1], and its z coordinate at index [2],
followed by the coordinates of the remaining points. Each point occupies
three REALs.

pointattributelist An array of point attributes. Each point’s attributes

occupy numberofpointattributes REALs.

pointmarkerlist An array of point markers; one int per point.

pointmtrlist An array of metric tensors at points. Each point’s tensor

occupies numberofpointmtrs REALs.

tetrahedronlist An array of tetrahedron corners. The first tetrahedron’s

first corner is at index [0], followed by its other three corners, followed
80 6 CALLING TETGEN FROM ANOTHER PROGRAM

by any other nodes if the ’-o2’ switch is used. Each tetrahedron occupies
numberofcorners (4 or 10) ints.

tetrahedronattributelist An array of tetrahedron attributes. Each tetra-

hedron’s attributes occupy numberoftetrahedronattributes REALs.

tetrahedronvolumelist An array of tetrahedron volume constraints; one

REAL per tetrahedron. Input only.

neighborlist An array of tetrahedron neighbors; four ints per tetrahedron.

Output only.

facetlist An array of PLC facets. Each facet is an object of type facet

(see Section 6.4.2).

facetmarkerlist An array of facet markers; one int per facet.

holelist An array of holes. The first hole’s x, y and z coordinates are at

indices [0], [1] and [2], followed by the remaining holes. Three REALs per
hole.

regionlist An array of regional attributes and volume constraints. The

first constraints’ x, y and z coordinates are at indices [0], [1] and [2], followed
by the regional attribute at index [3], followed by the maximum volume at
index [4], followed by the remaining volume constraints. Five REALs per
volume constraint. Each regional attribute is used only if the A switch is
used, and each volume constraint is used only if the a switch (with no number
following) is used, but omitting one of these switches does not change the
memory layout.

facetconstraintlist An array of facet maximum area constraints. Two

REALs per constraint. The first one is the facet marker (cast the type to
integer), the second is its maximum area bound. Note the ’facetconstraintlist’
is used only for the ’q’ switch.

segmentconstraintlist An array of segment length constraints. Two RE-

ALs per constraint. The first one is the index (pointing into pointlist) of
the node, the second is its maximum length bound. Note the ’segmentcon-
straintlist’ is used only for the ’q’ switch.
6.4 Description of Arrays 81

trifacelist An array of triangular faces. The first face’s corners are at

indices [0], [1] and [2], followed by the remaining faces. Three ints per face.

trifacemarkerlist An array of face markers; one int per face.

edgelist An array of segment endpoints. The first segment’s endpoints are

at indices [0] and [1], followed by the remaining segments. Two ints per
segment.

edgemarkerlist An array of segment markers; one int per segment.

6.4.1 Memory Management

Two routines defined in tetgenio are used for memory initialization and clean-
ing. They are:
void initialize();
void deinitialize();

initialize() initializes all fields, that is, all pointers to arrays are initial-
ized to NULL, and other variables are initialized to zero except the variable
’numberofcorners’, which is 4 (a tetrahedron has 4 nodes). Initialization is
implicitly called by the constructor of tetgenio. For an example, the following
line creates an object of tetgenio named io, all fields of io are initialized:
tetgenio io;

The next step is to allocate memory for each array which will be used.
In C++ the memory allocation and deletion can be done by the new and
delete operators. Another pair of functions (preferred by C programmers)
are malloc() and free(). Whatever you use, you must stick with one of
these two pairs, e.g., ’new’/’delete’ and ’malloc’/’free’ cannot be mixed. For
example, the following line allocates memory for io.pointlist:
io.pointlist = new REAL[io.numberofpoints * 3];

deinitialize() frees the memory allocated in objects of tetgenio by us-

ing ’delete’. It is automatically called on deletion of the tetgenio objects. If
the memory was allocated by using the function malloc(), the user is re-
sponsible to free it. After having freed all memory, one call of initialize()
disables the automatic memory deletion.
To reuse an object is possible: first call deinitialize(), then call initialize()
before the next use.
82 6 CALLING TETGEN FROM ANOTHER PROGRAM

6.4.2 The facet Data Structure

The facet data structure defined in tetgenio can be used to represent any
facet of a PLC. The structure of facet shown below consists of a list of
polygons and a list of hole points.

typedef struct {
polygon *polygonlist;
int numberofpolygons;
REAL *holelist;
int numberofholes;
} facet;

A polygon is again an object of type polygon. It consists of a list of

corner points (vertexlist). The structure is shown below.

typedef struct {
int *vertexlist;
int numberofvertices;
} polygon;

The structure of a facet corresponds to the facet description in a .poly

file format, described in Section 5.2.2. The front facet of Figure 23 serves an
example for setting a PLC facet into an object of facet. It has two polygons,
one has six vertices, and the other is a segment, no holes, the ASCII data is:

2
6 4 12 8 5 9 1 # front side
2 12 9

The following C++ code does the translation. Assume the object of
tetgenio is io and has already be created.

tetgenio::facet *f; // Define a pointer of facet.

tetgenio::polygon *p; // Define a pointer of polygon.

// All indices start from 1.

io.firstnumber = 1;

...

// Use ’f’ to point to a facet of ’facetlist’.

f = &io.facetlist[i];
// Initialize the fields of this facet.
// There are two polygons, no holes.
f->numberofpolygons = 2;
// Allocate memory for polygons.
6.5 A Complete Example 83

f->polygonlist = new tetgenio::polygon[2];

f->numberofholes = 0;
f->holelist = NULL;

// Set the data of the first polygon into facet.

p = &f->polygonlist[0];
p->numberofvertices = 6;
// Allocate memory for vertices.
p->vertexlist = new int[6];
p->vertexlist[0] = 4;
p->vertexlist[1] = 12;
p->vertexlist[2] = 8;
p->vertexlist[3] = 5;
p->vertexlist[4] = 9;
p->vertexlist[5] = 1;

// Set the data of the second polygon into facet.

p = &f->polygonlist[1];
p->numberofvertices = 2;
p->vertexlist = new int[2]; // Alloc. memory for vertices.
p->vertexlist[0] = 12;
p->vertexlist[1] = 9;

6.5 A Complete Example

This section gives an example of how to call TetGen from another program
by using the tetgenio data structure and the function tetrahedralize().
The input PLC in Section 5.4.1 (Figure 22) is used again.
The complete C++ source code is given below. It is also available in
TetGen’s website:http://www.tetgen.org/files/tetcall.cxx. The code
illustrates the following basic steps:

• at first creates an input object in of tetgenio containing the data of

the bar;

• then it calls function tetrahedralize() to create a quality mesh of

the bar with output in out.

In addition, it outputs the PLC in the object in into two files (barin.node
and barin.poly), and outputs the mesh in the object out into three files
(barout.node, barout.ele, and barout.face).
This example can be compiled into an executable program.

• Compile TetGen into a library named libtet.a (see Section 3.1 for
compiling);
84 6 CALLING TETGEN FROM ANOTHER PROGRAM

• Save the file tetcall.cxx into the same directory in which you have
the files tetgen.h and libtet.a;

• Compile it using the following command:

g++ -o test tetcall.cxx -L./ -ltet

which will result an executable file test.

The complete source codes are given below:

#include "tetgen.h" // Defined tetgenio, tetrahedralize().

int main(int argc, char *argv[])

{
tetgenio in, out;
tetgenio::facet *f;
tetgenio::polygon *p;
int i;

// All indices start from 1.

in.firstnumber = 1;

in.numberofpoints = 8;
in.pointlist = new REAL[in.numberofpoints * 3];
in.pointlist[0] = 0; // node 1.
in.pointlist[1] = 0;
in.pointlist[2] = 0;
in.pointlist[3] = 2; // node 2.
in.pointlist[4] = 0;
in.pointlist[5] = 0;
in.pointlist[6] = 2; // node 3.
in.pointlist[7] = 2;
in.pointlist[8] = 0;
in.pointlist[9] = 0; // node 4.
in.pointlist[10] = 2;
in.pointlist[11] = 0;
// Set node 5, 6, 7, 8.
for (i = 4; i < 8; i++) {
in.pointlist[i * 3] = in.pointlist[(i - 4) * 3];
in.pointlist[i * 3 + 1] = in.pointlist[(i - 4) * 3 + 1];
in.pointlist[i * 3 + 2] = 12;
}

in.numberoffacets = 6;
in.facetlist = new tetgenio::facet[in.numberoffacets];
in.facetmarkerlist = new int[in.numberoffacets];
6.5 A Complete Example 85

// Facet 1. The leftmost facet.

f = &in.facetlist[0];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 1;
p->vertexlist[1] = 2;
p->vertexlist[2] = 3;
p->vertexlist[3] = 4;

// Facet 2. The rightmost facet.

f = &in.facetlist[1];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 5;
p->vertexlist[1] = 6;
p->vertexlist[2] = 7;
p->vertexlist[3] = 8;

// Facet 3. The bottom facet.

f = &in.facetlist[2];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 1;
p->vertexlist[1] = 5;
p->vertexlist[2] = 6;
p->vertexlist[3] = 2;

// Facet 4. The back facet.

f = &in.facetlist[3];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
86 6 CALLING TETGEN FROM ANOTHER PROGRAM

p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 2;
p->vertexlist[1] = 6;
p->vertexlist[2] = 7;
p->vertexlist[3] = 3;

// Facet 5. The top facet.

f = &in.facetlist[4];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 3;
p->vertexlist[1] = 7;
p->vertexlist[2] = 8;
p->vertexlist[3] = 4;

// Facet 6. The front facet.

f = &in.facetlist[5];
f->numberofpolygons = 1;
f->polygonlist = new tetgenio::polygon[f->numberofpolygons];
f->numberofholes = 0;
f->holelist = NULL;
p = &f->polygonlist[0];
p->numberofvertices = 4;
p->vertexlist = new int[p->numberofvertices];
p->vertexlist[0] = 4;
p->vertexlist[1] = 8;
p->vertexlist[2] = 5;
p->vertexlist[3] = 1;

// Set ’in.facetmarkerlist’

in.facetmarkerlist[0] = -1;
in.facetmarkerlist[1] = -2;
in.facetmarkerlist[2] = 0;
in.facetmarkerlist[3] = 0;
in.facetmarkerlist[4] = 0;
in.facetmarkerlist[5] = 0;

// Output the PLC to files ’barin.node’ and ’barin.poly’.

in.save_nodes("barin");
in.save_poly("barin");
6.5 A Complete Example 87

// Tetrahedralize the PLC. Switches are chosen to read a PLC (p),

// do quality mesh generation (q) with a specified quality bound
// (1.414), and apply a maximum volume constraint (a0.1).

tetrahedralize("pq1.414a0.1", &in, &out);

// Output mesh to files ’barout.node’, ’barout.ele’ and ’barout.face’.

out.save_nodes("barout");
out.save_elements("barout");
out.save_faces("barout");

return 0;
}
88 A BASIC DEFINITIONS

(a) (b) (c)

Figure 25: (a) A two-dimensional simplical complex K consists of 2 trian-

gles(which are shaded), 7 edges, and 5 vertices. (b) The underlying space.
(c) A subcomplex, which is a 1-dimensional simplical complex, consists of 4
edges, and 4 vertices.

A Basic Definitions
This section gives simplified explanations of some basic notions of combina-
torial topology as a quick reference.

A.1 Simplices, Simplicial Complexes

Convex Hull A point set V ⊂ Rd is convex if it contains every line segment
whose end points are in this set. There are infinitely many convex sets of
V . The smallest convex set of V is called the convex hull of V , denoted as
convV . The dimension of the convex hull of V is the dimension of the affine
space of V .

Simplex A simplex σ is the convex hull of an affinely independent set S

of points. The dimension of σ is one less than the number of points of S.
Specifically, in R3 the maximum number of affinely independent points is 4,
so we have non-empty simplices of dimensions 0, 1, 2 and 3 referred to as
vertices, edges, triangles, and tetrahedra, respectively. For any subset T ⊆ S,
the simplex τ = convT is a face of σ and we write τ ≤ σ. τ is a proper face
of σ if T is a proper subset of S.

Simplical Complex A simplical complex K is a finite set of simplices such

that (i) any face of a simplex in K is also in K, and (ii) the intersection of
any two simplices in K is a face of both. Condition (ii) allows for the case
in which two simplices are disjoint because the empty set is the unique (-1)-
dimensional simplex, which is a face of any simplex. Figure 25 (a) illustrates
a two-dimensional simplical complex.
A.2 Polyhedra and Faces 89

Figure 26: A three-dimensional non-convex polyhedron. In left, the vertices

(0-faces) and edges (1-faces) of the polyhedron are shown. All its facets
(2-faces) are shown in right.

Underlying Space The underlyingS space of a set of simplices L, denoted

|L|, is the union of interiors, σ∈L intσ (see an illustration in Figure 25 (b)).
|L| is a topologically closed set if and only if L is a simplical complex.

Subcomplex A subcomplex of K is a subset of simplices of K that is also

a simplical complex. For an example see Figure 25 (c).

A.2 Polyhedra and Faces

Convex polyhedra and their faces are well defined objects. It is a central
topic in discrete geometry to study their structures and properties, see [29].
However, general polyhedra which are not necessarily convex are much com-
plex objects. There exist various definitions in the literature. We adopt the
definitions given by Edelsbrunner [7].
A polyhedron P is the union of convex polyhedra and the space of P is
connected. It is not necessarily convex, see Figure 26 left for an example. The
dimension of P is the dimension of the smallest affine space that contains P .
The interior of P , denoted as int(P ) is the point set such that every point
p ∈ int(P ) has a neighborhood (e.g., an open ball centered at this point)
which is a subset of P . The boundary of P is the point set bd(P ) = P −int(P ).
A face F of P is the closure of the connected set of points in the boundary
of P and all points in F belong to the same minimal dimensional affine space
that contains F . It is also a polyhedron whose dimension is the dimension
of the affine space that determines F . 0-, 1-, and 2-dimensional faces of P
are called vertices, edges, and facets of P . Each facet is a polygonal region
which may not be convex and may contain holes in it, as illustrated in the
right of Figure 26.
90 B LIST OF ERROR CODES AND MESSAGES

A.3 CSG and B-Rep Models of 3d Domains

In geometric and solid modeling, constructive solid geometry (CSG) and
boundary representation (B-Rep) are two popular representations for three-
dimensional objects.
A CSG model implicitly describes the domain as a combination of simple
primitives or other solids in a series of Boolean operations. It can describe
rather complicated shapes simply. However, the domain boundary must be
calculated numerically in order to find the intersecting points, curves, and
patches. This involves solving many non-linear equations in three variables.
Obtaining a PLC from a CSG model is generally not a simple task.
A B-Rep model explicitly describes the domain boundary by a set of
non-overlapping facets (may be curved surfaces) together with topological
information (such as incidence and adjacency) between the facets. The vol-
ume of the domain is implicitly bounded by them. However, it is not trivial
to correctly define such a model for a complex object. Nevertheless, The
B-Rep model is popularly used in describing 3d geometries.

B List of Error Codes and Messages

The list of error codes and messages can also be found in the function
terminatetetgen() defined in the file tetgen.h.

code 1
message Error: Out of memory

code 2
message Please report this bug to Hang.Si@wias-berlin.de. Include
the message above, your input data set, and the exact
command line you used to run this program, thank you
description This failure was caused by a known bug of TetGen.

code 3
message A self-intersection was detected. Program stopped.
Hint: use -d option to detect all self-intersections.
description This failure was caused by an input error.
REFERENCES 91

code 4
message A very small input feature size was detected. Program stopped.
Hint: use -T option to set a smaller tolerance.
description This failure was caused by a possible input error.
For example, there are two segments nearly intersect each.
other. If you want to ignore this possible error, set a smaller
tolerance by the -T switch, default is 10−8 .

code 5
message Two very close input facets were detected. Program stopped.
Hint: use -Y option to avoid adding Steiner points in boundary.
description This failure was caused by a possible input error.
For example, there are two facets nearly overlap each
other. Once Steiner points are added into one of the facets,
it will cause a self-intersection.

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[3] A. Bowyer. Computing Dirichlet tessellations. Comp. Journal,

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case optimal algorithm. SIAM Journal on Computing, 13(3):488–507,
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[5] J. A. de Loera, J. Rambau, and F. Santos. Triangulations, Structures for

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[6] B. N. Delaunay. Sur la sphère vide. Izvestia Akademii Nauk SSSR,

Otdelenie Matematicheskikh i Estestvennykh Nauk, 7:793–800, 1934.

[7] H. Edelsbrunner. Geometry and topology for mesh generation. Cam-

bridge University Press, England, 2001.
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code 10
message An input error was detected. Program stopped.
description This failure was caused by an input error.

[8] H. Edelsbrunner and M.P. Mücke. Simulation of simplicity: A technique

to cope with degenerate cases in geometric algorithm. ACM Transac-
tions on Graphics, 9(1):66–104, 1990.

[9] H. Edelsbrunner and N. R. Shah. Incremental topological flipping works

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[10] D. T. Lee and A. K. Lin. Generalized Delaunay triangulations for planar

graphs. Discrete and Computational Geometry, 1:201–217, 1986.

[11] G. L. Miller, D. Talmor, S.-H. Teng, N. J. Walkington, and H. Wang.

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[12] V. T. Rajan. Optimality of the Delaunay triangulation in Rd . Discrete

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[13] J. Ruppert. A Delaunay refinement algorithm for quality 2-dimensional

mesh generation. Journal of Algorithms, 18(3):548–585, 1995.

[14] E. Schönhardt. Über die zerlegung von dreieckspolyedern in tetraeder.

Mathematische Annalen, 98:309–312, 1928.

[15] J. R. Shewchuk. Adaptive precision floating-point arithmetic and fast

robust geometric predicates. Discrete and Computational Geometry,
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[16] J. R. Shewchuk. A condition guaranteeing the existence of higher-

dimensional constrained Delaunay triangulations. In Proc. 14th Ann.
Symp. on Comput. Geom., pages 76–85, 1998.

[17] J. R. Shewchuk. Tetrahedral mesh generation by Delaunay refinement.

In Proc. 14th Ann. Symp. on Comput. Geom., pages 86–95, 1998.

[18] J. R. Shewchuk. Constrained Delaunay tetrahedralizations and prov-

ably good boundary recovery. In Proc. 11th International Meshing
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[19] J. R. Shewchuk. What is a good linear element? interpolation, con-

ditioning, and quality measures. In Proc. 11th International Meshing
Roundtable, pages 115–126, Ithaca, New York, September 2002. Sandia
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[20] J. R. Shewchuk. Updating and constructing constrained Delaunay and

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[21] J. R. Shewchuk. General-dimensional constrained Delaunay and con-

strained regular triangulations, i: combinatorial properties. Discrete
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[22] H. Si. Adaptive tetrahedral mesh generation by constrained delaunay re-

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[23] H. Si. Three dimensional boundary conforming Delaunay mesh gen-

eration. PhD thesis, Institut für Mathematik, Technische Universität
Berlin, Strasse des 17. Juni 136, D-10623, Berlin, Germany, August
2008. Available online: http://opus.kobv.de/tuberlin/volltexte/
2008/1966/.

[24] H. Si. TetGen, towards a quality tetrahedral mesh generator. WIAS

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Mathematics. Springer-Verlag, New York, second edition edition, 1997.
Index
anisotropic, 13 -k, 30
aspect ratio, 11 -m, 39, 42, 43, 63
attribute -n, 34, 39, 47
of point, 35, 54 -nn, 61, 62
of region, 45, 57, 60 -o2, 47, 60–62
-p, 31, 36, 39
background mesh, 14 -q, 31, 39, 41, 45
boundary
-r, 31, 39, 41, 43
of PLC, 6
-v, 35
boundary conformity, 9
-w, 34
boundary edges, 7
-x, 49
boundary faces, 7
-z, 47
boundary marker, 52, 56
command line syntax, 31
of face, 61
compile
of point, 54
cmake, 23
boundary recovery, see boundary con-
CGAL’s predicates, 25
formity
CMakeLists.txt, 23
CDT, see constrained Delaunay tetra- make, 22
hedralization makefile, 22
circumsphere, 2 Shewchuk’s predicates, 5, 24
command line switches, 31 constrained Delaunay tetrahedraliza-
-A, 45 tion, 10
-B, 52, 54, 56, 61 constrained Delaunay tetrahedron, 10
-C, 49 constrained tetrahedralization, 10
-F, 61, 63 constraint
-I, 44, 49 area, 42
-N, -E, -F, 48 length, 42
-O, 44 volume, 41
-R, 45 convex, 88
-S, 49 convex hull, 88
-T, 45, 50 convex polyhedra, 89
-V, 40, 47 convex polytope, 89
-Y, 37
-a, 39, 41 degeneracy, 2
-c, 49 degenerate polygon, 51
-e, 33, 39, 47, 63 Delaunay refinement, 14
-f, 33, 39, 46, 61 Delaunay simplex, 2
-g, 30 Delaunay tetrahedralization, 2
-i, 43, 45 conforming, 9

94
INDEX 95

constrained, 9, 10 memory allocation, 48

generation, 33 memory usage report, 48
Delaunay triangulation, 2 mesh adaption, 13, 41
dihedral angle, 12 mesh coarsening, 45
dimension, 88 mesh element size, 13
mesh iteration number, 44
facet, 6, 51, 56 mesh optimization, 14, 44
data structure, 80, 82 level, 44
planarity, 52 local operations, 44
polygon, 51 mesh process, 16
file formats, 52 mesh quality, 11
.edge, 62 report, 28, 47
.ele, 59 mesh reconstruction, 43
.face, 61 mesh refinement, 39
.mesh, 69 mesh size, 13
.mtr, 42, 63 mesh sizing function, 13, 42
.neigh, 66 mesh statistics, 47
.node, 52, 53 mesh validation, 49
.off, 68
.ply, 69 orthogonal, 4
.poly, 52, 55 orthosphere, 4
.smesh, 52, 58
.stl, 69 piecewise linear complex, 6
.v.cell, 67 PLC, seepiecewise linear complex6
.v.edge, 66 polar, see lifting map
.v.face, 67 polygon, see polygon of facet, 56
.v.node, 66 data structure, 82
.var, 42, 65 degenerate, 56
.vol, 41, 63 polyhedron, 89
boundary, 89
general position, 2 face, 89
interior, 89
hole power diagram, 4
of facet, 52, 56
of PLC, 52, 56 radius edge ratio, 12
region attribute, see attribute of re-
isotropic, 13 gion
regular subdivision, 5
lifted point, see lifting map regular tetrahedron, 12
lifting map, 3
locally Delaunay, 11 segment, 6
lower face, see lifting map sharp feature, 15
96 INDEX

sharp features, 40
simplex, 88
face, proper face, 88
simplical complex, 88
subcomplex, 89
sliver, 12
solid modeling
B-Rep, 90
CSG, 90
Steiner point, 8, 49
Steiner tetrahedralization, 8
sub-region, 52

terminatetetgen() function, 90
tetgen.h C++ source, 77
tetgenio structure, 51, 78
tetrahedral mesh, 6
of a 3d PLC, 7
of a 3d point set, 6
tetrahedralize() function, 77
tetrahedron shape measure, 11
triangulation, 2

underlying space, 7, 89

vertex, 6
visible, 10
visualization
Medit, 30, 69
Paraview, 30
TetView, 30
Voronoi cell, 2
Voronoi diagram, 3

weight, see attribute of point

weighted Delaunay triangulation, 4
weighted distance, 4
weighted point, 3
weighted Voronoi diagram, see power
diagram