\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 3D Triangulations

All Classes Namespaces Files Functions Variables Typedefs Enumerations Enumerator Groups Pages
User Manual

Authors
  Clément Jamin, Sylvain Pion and Monique Teillaud

The basic 3D-triangulation class of CGAL is primarily designed to represent the triangulations of a set of points \( A\) in \( \mathbb{R}^3\). It is a partition of the convex hull of \( A\) into tetrahedra whose vertices are the points of \( A\). Together with the unbounded cell having the convex hull boundary as its frontier, the triangulation forms a partition of \( \mathbb{R}^3\). Its cells ( \( 3\)-faces) are such that two cells either do not intersect or share a common facet ( \( 2\)-face), edge ( \( 1\)-face) or vertex ( \( 0\)-face).

# Representation

In order to deal only with tetrahedra, which is convenient for many applications, the unbounded cell can be subdivided into tetrahedra by considering that each convex hull facet is incident to an `infinite cell` having as fourth vertex an auxiliary vertex called the `infinite vertex`. In that way, each facet is incident to exactly two cells and special cases at the boundary of the convex hull are simple to deal with.

The class `Triangulation_3<TriangulationTraits_3,TriangulationDataStructure_3>` of CGAL implements this point of view and therefore considers the triangulation of the set of points as a set of finite and infinite tetrahedra. Notice that the infinite vertex has no significant coordinates and that no geometric predicate can be applied on it.

A triangulation is a collection of vertices and cells that are linked together through incidence and adjacency relations. Each cell gives access to its four incident vertices and to its four adjacent cells. Each vertex gives access to one of its incident cells.

The four vertices of a cell are indexed with 0, 1, 2 and 3 in positive orientation, the positive orientation being defined by the orientation of the underlying Euclidean space \( \mathbb{R}^3\) (see Figure 40.1). The neighbors of a cell are also indexed with 0, 1, 2, 3 in such a way that the neighbor indexed by \( i\) is opposite to the vertex with the same index.

Figure 40.1 Orientation of a cell (3-dimensional case).

As in the underlying combinatorial triangulation (see Chapter 3D Triangulation Data Structure), edges ( \( 1\)-faces) and facets ( \( 2\)-faces) are not explicitly represented: a facet is given by a cell and an index (the facet `i` of a cell `c` is the facet of `c` that is opposite to the vertex with index `i`) and an edge is given by a cell and two indices (the edge `(i,j)` of a cell `c` is the edge whose endpoints are the vertices of `c` with indices `i` and `j`). See Figure 41.1.

Degenerate Dimensions

The class `Triangulation_3` can also deal with triangulations whose dimension \( d\) is less than 3. A triangulation of a set of points in \( \mathbb{R}^d\) covers the whole space \( \mathbb{R}^d\) and consists of cells having \( d+1\) vertices: some of them are infinite, they are obtained by linking the additional infinite vertex to each facet of the convex hull of the points.

- dimension 2: when a triangulation only contains coplanar points (which is the case when there are only three points), it consists of triangular faces.
- dimension 1: the triangulation contains only collinear points (which is the case when there are only two points), it consists of edges.
- dimension 0: the triangulation contains only one finite point.
- dimension -1: this is a convention to handle the case when the only vertex of the triangulation is the infinite one.

The same cell class is used in all cases: triangular faces in 2D can be considered as degenerate cells, having only three vertices (resp. neighbors) numbered \( (0,1,2)\); edges in 1D have only two vertices (resp. neighbors) numbered \( 0\) and \( 1\).

The implicit representation of facets (resp. edges) still holds for degenerate dimensions (i.e. dimensions \( <3\)): in dimension 2, each cell has only one facet of index 3, and 3 edges \( (0,1)\), \( (1,2)\) and \( (2,0)\); in dimension 1, each cell has one edge \( (0,1)\).

Validity

A triangulation of \( \mathbb{R}^3\) is said to be locally valid iff

(a)-(b) Its underlying combinatorial graph, the triangulation data structure, is locally valid (see Section Introduction of Chapter 3D Triangulation Data Structure)

(c) Any cell has its vertices ordered according to positive orientation. See Figure 40.1.

When the triangulation is degenerated into a triangulation of dimension 2, the geometric validity reduces to:

(c-2D) For any two adjacent triangles \( (u,v,w_1)\) and \( (u,v,w_2)\) with common edge \( (u,v)\), \( w_1\) and \( w_2\) lie on opposite sides of \( (u,v)\) in the plane.

When all the points are collinear, this condition becomes:

(c-1D) For any two adjacent edges \( (u,v)\) and \( (v,w)\), \( u\) and \( w\) lie on opposite sides of the common vertex \( v\) on the line.

The method `Triangulation_3::is_valid()` checks the local validity of a given triangulation. This does not always ensure global validity [18], [10] but it is sufficient for practical cases.

# Delaunay Triangulation

The class `Delaunay_triangulation_3` represents a three-dimensional Delaunay triangulation.

Delaunay triangulations have the specific empty sphere property, that is, the circumscribing sphere of each cell of such a triangulation does not contain any other vertex of the triangulation in its interior. These triangulations are uniquely defined except in degenerate cases where five points are co-spherical. Note however that the CGAL implementation computes a unique triangulation even in these cases.

This implementation is fully dynamic: it supports insertions of points, vertex removals and displacements of points.

# Regular Triangulation

The class `Regular_triangulation_3` implements incremental regular triangulations, also known as weighted Delaunay triangulations.

Let \( {S}^{(w)}\) be a set of weighted points in \( \mathbb{R}^3\). Let \( {p}^{(w)}=(p,w_p), p\in\mathbb{R}^3, w_p\in\mathbb{R}\) and \( {z}^{(w)}=(z,w_z), z\in\mathbb{R}^3, w_z\in\mathbb{R}\) be two weighted points. A weighted point \( {p}^{(w)}=(p,w_p)\) can also be seen as a sphere of center \( p\) and radius \( \sqrt{w_p}\). The power product between \( {p}^{(w)}\) and \( {z}^{(w)}\) is defined as

\[ \Pi({p}^{(w)},{z}^{(w)}) = {\|{p-z}\|^2-w_p-w_z} \]

where \( \|{p-z}\|\) is the Euclidean distance between \( p\) and \( z\). \( {p}^{(w)}\) and \( {z}^{(w)}\) are said to be orthogonal iff \( \Pi{({p}^{(w)},{z}^{(w)})} = 0\) (see Figure 40.2).

Figure 40.2 Orthogonal weighted points (picture in 2D).

Four weighted points have a unique common orthogonal weighted point called the power sphere. The weighted point orthogonal to three weighted points in the plane defined by these three points is called the power circle. The power segment will denote the weighted point orthogonal to two weighted points on the line defined by these two points.

A sphere \( {z}^{(w)}\) is said to be regular if \( \forall {p}^{(w)}\in{S}^{(w)}, \Pi{({p}^{(w)},{z}^{(w)})}\geq 0\).

A triangulation of \( {S}^{(w)}\) is regular if the power spheres of all simplices are regular.

The regular triangulation of \( {S}^{(w)}\) is in fact the projection onto \( \mathbb{R}^3\) of the convex hull of the four-dimensional points \( (p,\|p-O\|^2-w_p),\) for \( {p}^{(w)}=(p,w_p)\in{S}^{(w)}\). Note that all points of \( {S}^{(w)}\) do not necessarily appear as vertices of the regular triangulation. To know more about regular triangulations, see for example [14].

When all weights are 0, power spheres are nothing more than circumscribing spheres, and the regular triangulation is exactly the Delaunay triangulation.

The implementation of 3D regular triangulation supports insertions of weighted points, and vertex removals. Displacements are not supported in the current implementation.

# Software Design

The main classes `Triangulation_3`, `Delaunay_triangulation_3` and `Regular_triangulation_3` are connected to each other by the derivation diagram shown in Figure 40.3. This diagram also shows another class: `Triangulation_utils_3`, which provides a set of tools operating on the indices of vertices in cells.

Figure 40.3 Derivation diagram of the 3D triangulation classes.

The three main classes (`Triangulation_3`, `Delaunay_triangulation_3` and `Regular_triangulation_3`) provide high-level geometric functionality such as location of a point in the triangulation [11], insertion and possibly removal of a point [8], and are responsible for the geometric validity. They are built as layers on top of a triangulation data structure, which stores their combinatorial structure. This separation between the geometry and the combinatorics is reflected in the software design by the fact that these three triangulation classes take the following template parameters :

- the geometric traits class, which provides the type of points to use as well as the elementary operations on them (predicates and constructions). The concepts for these parameters are described in more details in Section The Geometric Traits Parameter.
- the triangulation data structure class, which stores their combinatorial structure, described in Section Software Design of Chapter 3D Triangulation Data Structure.
- the location policy tag, which is supported only by the Delaunay triangulation class, described in Section The Location Policy Parameter.

Optionally, the main Delaunay and regular triangulations algorithms (insert, remove) support multi-core shared-memory architectures to take advantage of available parallelism. For this purpose, a model of the concept `SurjectiveLockDataStructure` can be given as fourth template parameter; it defaults to `Spatial_lock_grid_3`. This data structure allows to lock points with coordinates (x, y, z) in a 3D domain. When a thread owns the lock on a point, no other thread can lock this point. Locking a facet (resp. a cell) boils down to locking all its 3 (resp. 4) incident vertices. See Section Parallel Algorithms for more details.

## The Geometric Traits Parameter

The first template parameter of the triangulation class `Triangulation_3<TriangulationTraits_3, TriangulationDataStructure_3>` is the geometric traits class, described by the concept `TriangulationTraits_3`. It must define the types of the geometric objects (points, segments, triangles and tetrahedra) forming the triangulation together with a few geometric predicates on these objects: orientation in space, orientation in case of coplanar points, order of collinear points.

In addition to the requirements described before, the geometric traits class of `Delaunay_triangulation_3` must define predicates to test for the empty sphere property. It is described by the concept `DelaunayTriangulationTraits_3`, which refines `TriangulationTraits_3`.

The kernels provided by CGAL: `Cartesian`, `Homogeneous`, `Simple_cartesian`, `Simple_homogeneous` and `Filtered_kernel` can all be used as models for the geometric traits parameter. They supply the user with all the functionalities described for the concepts `TriangulationTraits_3` and `DelaunayTriangulationTraits_3`. In addition, the predefined kernels `Exact_predicates_inexact_constructions_kernel` and `Exact_predicates_exact_constructions_kernel` can also be used, the latter being recommended when the dual construction is used.

In order to be used as the traits class for `Regular_triangulation_3`, a class must provide functions to compute the power tests (see Section Regular Triangulation). `Regular_triangulation_euclidean_traits_3<K,Weight>` is a traits class designed to be used by the class `Regular_triangulation_3<RegularTriangulationTraits_3, TriangulationDataStructure_3>`. It provides `Weighted_point`, a class for weighted points needed by the regular triangulation, which derives from the three dimensional point class `K::Point_3`. It supplies the user with all the functionalities described for the concept `RegularTriangulationTraits_3`. It can be used as a traits class for `Regular_triangulation_3<RegularTriangulationTraits_3, TriangulationDataStructure_3>`.

Note that for regular triangulations, plugging a filtered kernel such as `Exact_predicates_inexact_constructions_kernel` or `Exact_predicates_exact_constructions_kernel` in `Regular_triangulation_euclidean_traits_3<K,Weight>` will provide exact and efficient filtered predicates.

## The Triangulation Data Structure Parameter

The second template parameter of the main classes (`Triangulation_3`, `Delaunay_triangulation_3` and `Regular_triangulation_3`) is a triangulation data structure class. This class can be seen as a container for the cells and vertices maintaining incidence and adjacency relations (see Chapter chapterTDS3). A model of this triangulation data structure is `Triangulation_data_structure_3`, and it is described by the `TriangulationDataStructure_3` concept . This model is itself parameterized by a vertex base and a cell base classes, which gives the possibility to customize the vertices and cells used by the triangulation data structure, and hence by the geometric triangulation using it. Depending on the kind of triangulation used, the requirements on the vertex and cell base classes vary, and are expressed by various concepts, following the refinement diagram shown in Figure 40.4.

Figure 40.4 Concepts refinement hierarchy for the vertex and cell base classes parameters.

A default value for the triangulation data structure parameter is provided in all the triangulation classes, so it need not be specified by the user unless he wants to use a different triangulation data structure or a different vertex or cell base class.

## The Location Policy Parameter

The Delaunay triangulation class supports an optional feature which maintains an additional data structure for fast point location queries. The fast location policy should be used when the user inserts points in a random order or needs to do many unrelated queries. If the user is able to give a good hint to help the point location of its queries (and its newly inserted points), then it should prefer the default policy. In such a case where good hints are provided, the default policy save some memory (few percents), and is faster. Notice that if points are not inserted one by one, but as a range, then a good hint is automatically computed using spatial sort.

Reading Section Complexity and Performance on complexity and performance can help making an informed choice for this parameter.

The point location strategy can be selected with the third template argument of `Delaunay_triangulation_3`, `LocationPolicy`, which enables a fast point location data structure when set to `Fast_location`. By default, it uses `Compact_location`.

Note that you can specify the `LocationPolicy` parameter without specifying the triangulation data structure, in case you are fine with the default there. In this case, the `LocationPolicy` appears as a second parameter after the geometric traits.The mechanism used behind the scenes to allow this syntactical convenience is called deduced parameters.

The `Fast_location` policy is implemented using a hierarchy of triangulations; it changes the behavior of functions `locate`, `insert`, `move`, and `remove`. As proved in [12], this structure has an optimal behavior when it is built for Delaunay triangulations.

In this setting, if you build a triangulation by iteratively inserting points, you should try to shuffle the points beforehand, as the time complexity is guaranteed only for a randomized order. For example, inserting points in lexicographic order is typically much slower. Note that this shuffling is performed internally by the constructor taking a range of points.

Prior to CGAL 3.6, this functionality was available through the `Triangulation_hierarchy_3` class, which is now deprecated.

## Flexibility of the Design

In order to satisfy as many uses as possible, a design has been selected that allows to exchange different parts to meet the users' needs, while still re-using a maximum of the provided functionalities. We have already seen that the main triangulation classes are parameterized by a geometric traits class and a triangulation data structure (TDS), so that each of them can be interchanged with alternate implementations.

The most useful flexibility is the ability given to the user to add his own data in the vertices and cells by providing his own vertex and cell base classes to `Triangulation_data_structure_3`. The Figure 40.5 shows in more detail the flexibility that is provided, and the place where the user can insert his own vertex and/or cell base classes.

Figure 40.5 Triangulation software design.

The design of the triangulation data structure gives the possibility to store any kind of data, including handles (an entity akin to pointers) directly in the vertex and cell base classes.

To do so, there are three possibilities. The simplest one is to use the class `Triangulation_vertex_base_with_info_3`, and this approach is illustrated in a following subsection Adding a Color. The most complicated one, and probably useless for almost all cases, is to write a vertex base class from scratch, following the documented requirements. This is mostly useless because most of the time it is enough to derive from the models that CGAL provides, and add the desired features. In this case, when the user needs to access some type that depends on the triangulation data structure (typically handles), then he should write something like:

...
template < class GT, class Vb = Triangulation_vertex_base<GT> >
class My_vertex
: public Vb
{
public:
typedef typename Vb::Point Point;
typedef typename Vb::Cell_handle Cell_handle;
template < class TDS2 >
struct Rebind_TDS {
typedef typename Vb::template Rebind_TDS<TDS2>::Other Vb2;
typedef My_vertex<GT, Vb2> Other;
};
My_vertex() {}
My_vertex(const Point&p) : Vb(p) {}
My_vertex(const Point&p, Cell_handle c) : Vb(p, c) {}
...
};
... // The rest has not changed

The situation is exactly similar for cell base classes. Section Software Design provides more detailed information.

## Parallel Algorithms

Parallel algorithms of `Delaunay_triangulation_3` and `Regular_triangulation_3` are enabled if the `TriangulationDataStructure_3::Concurrency_tag` type is `Parallel_tag` and a reference to a lock data structure instance is provided via the constructor or by using `Triangulation_3::set_lock_data_structure`. This data structure must be a model of the concept `SurjectiveLockDataStructure` and can be optionally given as a template parameter of the triangulation; it defaults to `Spatial_lock_grid_3`.

Note that the parallel Delaunay triangulation must use the default compact location policy (and not the fast one). If those conditions are fulfilled, the insertion/removal of a range of points will be performed in parallel, and the individual insert/remove operations will be optionally thread-safe.

Parallel algorithms require the program to be linked against the Intel TBB library. To control the number of threads used, the user may use the tbb::task_scheduler_init class. See the TBB documentation for more details.

# Examples

## Basic Example

This example shows the incremental construction of a 3D triangulation, the location of a point and how to perform elementary operations on indices in a cell. It uses the default parameter of the `Triangulation_3` class.

File Triangulation_3/simple_triangulation_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Triangulation_3.h>
#include <iostream>
#include <fstream>
#include <cassert>
#include <list>
#include <vector>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Triangulation_3<K> Triangulation;
typedef Triangulation::Cell_handle Cell_handle;
typedef Triangulation::Vertex_handle Vertex_handle;
typedef Triangulation::Locate_type Locate_type;
typedef Triangulation::Point Point;
int main()
{
// construction from a list of points :
std::list<Point> L;
L.push_front(Point(0,0,0));
L.push_front(Point(1,0,0));
L.push_front(Point(0,1,0));
Triangulation T(L.begin(), L.end());
Triangulation::size_type n = T.number_of_vertices();
// insertion from a vector :
std::vector<Point> V(3);
V[0] = Point(0,0,1);
V[1] = Point(1,1,1);
V[2] = Point(2,2,2);
n = n + T.insert(V.begin(), V.end());
assert( n == 6 ); // 6 points have been inserted
assert( T.is_valid() ); // checking validity of T
Locate_type lt;
int li, lj;
Point p(0,0,0);
Cell_handle c = T.locate(p, lt, li, lj);
// p is the vertex of c of index li :
assert( lt == Triangulation::VERTEX );
assert( c->vertex(li)->point() == p );
Vertex_handle v = c->vertex( (li+1)&3 );
// v is another vertex of c
Cell_handle nc = c->neighbor(li);
// nc = neighbor of c opposite to the vertex associated with p
// nc must have vertex v :
int nli;
assert( nc->has_vertex( v, nli ) );
// nli is the index of v in nc
std::ofstream oFileT("output",std::ios::out);
// writing file output;
oFileT << T;
Triangulation T1;
std::ifstream iFileT("output",std::ios::in);
// reading file output;
iFileT >> T1;
assert( T1.is_valid() );
assert( T1.number_of_vertices() == T.number_of_vertices() );
assert( T1.number_of_cells() == T.number_of_cells() );
return 0;
}

## Changing the Vertex Base

The following two examples show how the user can plug his own vertex base in a triangulation. Changing the cell base is similar.

## Adding a Color

When the user doesn't need to add a type in a vertex which depends on the `TriangulationDataStructure_3` (e.g. a `Vertex_handle` or `Cell_handle`), then he can use the `Triangulation_vertex_base_with_info_3` class to add his own information easily in the vertices. The example below shows how to add a `Color` this way.

File Triangulation_3/color.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/Triangulation_vertex_base_with_info_3.h>
#include <CGAL/IO/Color.h>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Triangulation_vertex_base_with_info_3<CGAL::Color, K> Vb;
typedef CGAL::Triangulation_data_structure_3<Vb> Tds;
typedef CGAL::Delaunay_triangulation_3<K, Tds> Delaunay;
typedef Delaunay::Point Point;
int main()
{
Delaunay T;
T.insert(Point(0,0,0));
T.insert(Point(1,0,0));
T.insert(Point(0,1,0));
T.insert(Point(0,0,1));
T.insert(Point(2,2,2));
T.insert(Point(-1,0,1));
// Set the color of finite vertices of degree 6 to red.
Delaunay::Finite_vertices_iterator vit;
for (vit = T.finite_vertices_begin(); vit != T.finite_vertices_end(); ++vit)
if (T.degree(vit) == 6)
vit->info() = CGAL::RED;
return 0;
}

## Adding Handles

When the user needs to add a type in a vertex which depends on the `TriangulationDataStructure_3` (e.g. a `Vertex_handle` or `Cell_handle`), then he has to derive his own vertex base class, as the following example shows.

File Triangulation_3/adding_handles_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/Triangulation_vertex_base_3.h>
template < class GT, class Vb = CGAL::Triangulation_vertex_base_3<GT> >
class My_vertex_base
: public Vb
{
public:
typedef typename Vb::Vertex_handle Vertex_handle;
typedef typename Vb::Cell_handle Cell_handle;
typedef typename Vb::Point Point;
template < class TDS2 >
struct Rebind_TDS {
typedef typename Vb::template Rebind_TDS<TDS2>::Other Vb2;
typedef My_vertex_base<GT, Vb2> Other;
};
My_vertex_base() {}
My_vertex_base(const Point& p)
: Vb(p) {}
My_vertex_base(const Point& p, Cell_handle c)
: Vb(p, c) {}
Vertex_handle vh;
Cell_handle ch;
};
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Triangulation_data_structure_3<My_vertex_base<K> > Tds;
typedef CGAL::Delaunay_triangulation_3<K, Tds> Delaunay;
typedef Delaunay::Vertex_handle Vertex_handle;
typedef Delaunay::Point Point;
int main()
{
Delaunay T;
Vertex_handle v0 = T.insert(Point(0,0,0));
Vertex_handle v1 = T.insert(Point(1,0,0));
Vertex_handle v2 = T.insert(Point(0,1,0));
Vertex_handle v3 = T.insert(Point(0,0,1));
Vertex_handle v4 = T.insert(Point(2,2,2));
Vertex_handle v5 = T.insert(Point(-1,0,1));
// Now we can link the vertices as we like.
v0->vh = v1;
v1->vh = v2;
v2->vh = v3;
v3->vh = v4;
v4->vh = v5;
v5->vh = v0;
return 0;
}

## Setting Information While Inserting a Range of Points

The most efficient method to insert (weighted) points in a Delaunay (or regular) triangulation is to provide an iterator range over (weighted) points to the insert function. However, an iterator range of (weighted) points does not allow to set different information to each vertex. To solve this problem, in the case the vertex type of the triangulation is a model of the concept `TriangulationVertexBaseWithInfo_3` (such as `Triangulation_vertex_base_with_info_3`), we provide three examples doing the same operation: set an unsigned integer as the information of each vertex. The value of this unsigned integer is the initial order of the corresponding point given in the range.

### Using an Iterator Over Pairs

Each point and its information are gathered into a pair. We provide the constructor of the triangulation (which is calling the `insert` function) with a range of such pairs.
File Triangulation_3/info_insert_with_pair_iterator.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/Triangulation_vertex_base_with_info_3.h>
#include <vector>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Triangulation_vertex_base_with_info_3<unsigned, K> Vb;
typedef CGAL::Triangulation_data_structure_3<Vb> Tds;
//Use the Fast_location tag. Default or Compact_location works too.
typedef CGAL::Delaunay_triangulation_3<K, Tds, CGAL::Fast_location> Delaunay;
typedef Delaunay::Point Point;
int main()
{
std::vector< std::pair<Point,unsigned> > points;
points.push_back( std::make_pair(Point(0,0,0),0) );
points.push_back( std::make_pair(Point(1,0,0),1) );
points.push_back( std::make_pair(Point(0,1,0),2) );
points.push_back( std::make_pair(Point(0,0,1),3) );
points.push_back( std::make_pair(Point(2,2,2),4) );
points.push_back( std::make_pair(Point(-1,0,1),5) );
Delaunay T( points.begin(),points.end() );
CGAL_assertion( T.number_of_vertices() == 6 );
// check that the info was correctly set.
Delaunay::Finite_vertices_iterator vit;
for (vit = T.finite_vertices_begin(); vit != T.finite_vertices_end(); ++vit)
if( points[ vit->info() ].first != vit->point() ){
std::cerr << "Error different info" << std::endl;
exit(EXIT_FAILURE);
}
std::cout << "OK" << std::endl;
return 0;
}

### Using the Boost Zip Iterator

Information and points are in separate containers. We use `boost::zip_iterator` to provide an iterator gathering them.

File Triangulation_3/info_insert_with_zip_iterator.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/Triangulation_vertex_base_with_info_3.h>
#include <boost/iterator/zip_iterator.hpp>
#include <vector>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Triangulation_vertex_base_with_info_3<unsigned, K> Vb;
typedef CGAL::Triangulation_data_structure_3<Vb> Tds;
typedef CGAL::Delaunay_triangulation_3<K, Tds> Delaunay;
typedef Delaunay::Point Point;
int main()
{
std::vector<unsigned> indices;
indices.push_back(0);
indices.push_back(1);
indices.push_back(2);
indices.push_back(3);
indices.push_back(4);
indices.push_back(5);
std::vector<Point> points;
points.push_back(Point(0,0,0));
points.push_back(Point(1,0,0));
points.push_back(Point(0,1,0));
points.push_back(Point(0,0,1));
points.push_back(Point(2,2,2));
points.push_back(Point(-1,0,1));
Delaunay T( boost::make_zip_iterator(boost::make_tuple( points.begin(),indices.begin() )),
boost::make_zip_iterator(boost::make_tuple( points.end(),indices.end() ) ) );
CGAL_assertion( T.number_of_vertices() == 6 );
// check that the info was correctly set.
Delaunay::Finite_vertices_iterator vit;
for (vit = T.finite_vertices_begin(); vit != T.finite_vertices_end(); ++vit)
if( points[ vit->info() ] != vit->point() ){
std::cerr << "Error different info" << std::endl;
exit(EXIT_FAILURE);
}
return 0;
}

### Using the Boost Transform Iterator

We define a functor `Auto_count` used together with `boost::transform_iterator` to set the order of each point in the range. Note that this is correct because the iterator is dereferenced only once per point during the insertion.
File Triangulation_3/info_insert_with_transform_iterator.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/Triangulation_vertex_base_with_info_3.h>
#include <boost/iterator/transform_iterator.hpp>
#include <vector>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Triangulation_vertex_base_with_info_3<unsigned, K> Vb;
typedef CGAL::Triangulation_data_structure_3<Vb> Tds;
typedef CGAL::Delaunay_triangulation_3<K, Tds> Delaunay;
typedef Delaunay::Point Point;
//a functor that returns a std::pair<Point,unsigned>.
//the unsigned integer is incremented at each call to
//operator()
struct Auto_count : public std::unary_function<const Point&,std::pair<Point,unsigned> >{
mutable unsigned i;
Auto_count() : i(0){}
std::pair<Point,unsigned> operator()(const Point& p) const {
return std::make_pair(p,i++);
}
};
int main()
{
std::vector<Point> points;
points.push_back(Point(0,0,0));
points.push_back(Point(1,0,0));
points.push_back(Point(0,1,0));
points.push_back(Point(0,0,1));
points.push_back(Point(2,2,2));
points.push_back(Point(-1,0,1));
Delaunay T( boost::make_transform_iterator(points.begin(),Auto_count()),
boost::make_transform_iterator(points.end(), Auto_count() ) );
CGAL_assertion( T.number_of_vertices() == 6 );
// check that the info was correctly set.
Delaunay::Finite_vertices_iterator vit;
for (vit = T.finite_vertices_begin(); vit != T.finite_vertices_end(); ++vit)
if( points[ vit->info() ] != vit->point() ){
std::cerr << "Error different info" << std::endl;
exit(EXIT_FAILURE);
}
std::cout << "OK" << std::endl;
return 0;
}

## The Simplex Class

The triangulation defines a `Triangulation_3::Simplex` class that represents a simplex (vertex, edge, facet or cell). This example demonstrates how simplices can be stored in a set.

File Triangulation_3/simplex.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Triangulation_3.h>
#include <iostream>
#include <fstream>
#include <list>
#include <set>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Triangulation_3<K> Triangulation;
typedef Triangulation::Finite_vertices_iterator Finite_vertices_iterator;
typedef Triangulation::Finite_edges_iterator Finite_edges_iterator;
typedef Triangulation::Finite_facets_iterator Finite_facets_iterator;
typedef Triangulation::Finite_cells_iterator Finite_cells_iterator;
typedef Triangulation::Simplex Simplex;
typedef Triangulation::Locate_type Locate_type;
typedef Triangulation::Point Point;
int main()
{
// construction from a list of points :
std::list<Point> L;
L.push_front(Point(0,0,0));
L.push_front(Point(1,0,0));
L.push_front(Point(0,1,0));
L.push_front(Point(0,1,1));
Triangulation T(L.begin(), L.end());
std::set<Simplex> simplices;
Finite_vertices_iterator vit = T.finite_vertices_begin();
simplices.insert(Simplex(vit));
Finite_cells_iterator cit = T.finite_cells_begin();
simplices.insert(Simplex(cit));
Finite_edges_iterator eit = T.finite_edges_begin();
simplices.insert(Simplex(*eit));
Finite_facets_iterator fit = T.finite_facets_begin();
simplices.insert(Simplex(*fit));
for (std::set<Simplex>::iterator it = simplices.begin();
it != simplices.end(); it++) {
std::cout << it->dimension() << std::endl;
}
return 0;
}

## Fast Point Location for Delaunay Triangulations

File Triangulation_3/fast_location_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/Random.h>
#include <vector>
#include <cassert>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Delaunay_triangulation_3<K, CGAL::Fast_location> Delaunay;
typedef Delaunay::Point Point;
int main()
{
// generating points on a grid.
std::vector<Point> P;
for (int z=0 ; z<20 ; z++)
for (int y=0 ; y<20 ; y++)
for (int x=0 ; x<20 ; x++)
P.push_back(Point(x,y,z));
// building their Delaunay triangulation.
Delaunay T(P.begin(), P.end());
assert( T.number_of_vertices() == 8000 );
// performing nearest vertex queries to a series of random points,
// which is a case where the Fast_location policy is beneficial.
for (int i=0; i<10000; ++i)
T.nearest_vertex(Point(CGAL::get_default_random().get_double(0, 20),
CGAL::get_default_random().get_double(0, 20),
CGAL::get_default_random().get_double(0, 20)));
return 0;
}

## Finding the Cells in Conflict with a Point in a Delaunay Triangulation

File Triangulation_3/find_conflicts_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/point_generators_3.h>
#include <vector>
#include <cassert>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Delaunay_triangulation_3<K> Delaunay;
typedef Delaunay::Point Point;
typedef Delaunay::Cell_handle Cell_handle;
typedef Delaunay::Facet Facet;
int main()
{
Delaunay T;
CGAL::Random_points_in_sphere_3<Point> rnd;
// First, make sure the triangulation is 3D.
T.insert(Point(0,0,0));
T.insert(Point(1,0,0));
T.insert(Point(0,1,0));
T.insert(Point(0,0,1));
assert(T.dimension() == 3);
// Inserts 100 random points if and only if their insertion
// in the Delaunay tetrahedralization conflicts with
// an even number of cells.
for (int i = 0; i != 100; ++i) {
Point p = *rnd++;
// Locate the point
Delaunay::Locate_type lt;
int li, lj;
Cell_handle c = T.locate(p, lt, li, lj);
if (lt == Delaunay::VERTEX)
continue; // Point already exists
// Get the cells that conflict with p in a vector V,
// and a facet on the boundary of this hole in f.
std::vector<Cell_handle> V;
Facet f;
T.find_conflicts(p, c,
CGAL::Oneset_iterator<Facet>(f), // Get one boundary facet
std::back_inserter(V)); // Conflict cells in V
if ((V.size() & 1) == 0) // Even number of conflict cells ?
T.insert_in_hole(p, V.begin(), V.end(), f.first, f.second);
}
std::cout << "Final triangulation has " << T.number_of_vertices()
<< " vertices." << std::endl;
return 0;
}

## Regular Triangulation

This example shows the building of a regular triangulation. In this triangulation, points have an associated weight, and some points can be hidden and do not result in vertices in the triangulation. Another difference is that a specific traits class has to be used (at least at the moment).

File Triangulation_3/regular_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Regular_triangulation_3.h>
#include <CGAL/Regular_triangulation_euclidean_traits_3.h>
#include <cassert>
#include <vector>
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Regular_triangulation_euclidean_traits_3<K> Traits;
typedef Traits::RT Weight;
typedef Traits::Bare_point Point;
typedef Traits::Weighted_point Weighted_point;
typedef CGAL::Regular_triangulation_3<Traits> Rt;
typedef Rt::Vertex_iterator Vertex_iterator;
typedef Rt::Vertex_handle Vertex_handle;
int main()
{
// generate points on a 3D grid
std::vector<Weighted_point> P;
int number_of_points = 0;
for (int z=0 ; z<5 ; z++)
for (int y=0 ; y<5 ; y++)
for (int x=0 ; x<5 ; x++) {
Point p(x, y, z);
Weight w = (x+y-z*y*x)*2.0; // let's say this is the weight.
P.push_back(Weighted_point(p, w));
++number_of_points;
}
Rt T;
// insert all points in a row (this is faster than one insert() at a time).
T.insert (P.begin(), P.end());
assert( T.is_valid() );
assert( T.dimension() == 3 );
std::cout << "Number of vertices : " << T.number_of_vertices() << std::endl;
// removal of all vertices
int count = 0;
while (T.number_of_vertices() > 0) {
T.remove (T.finite_vertices_begin());
++count;
}
assert( count == number_of_points );
return 0;
}

## Parallel insertion in Delaunay triangulation

This example shows the parallel building of a Delaunay triangulation.

File Triangulation_3/parallel_insertion_in_delaunay_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Delaunay_triangulation_3.h>
#include <CGAL/point_generators_3.h>
#include <iostream>
#include <fstream>
#include <vector>
int main()
{
#ifdef CGAL_LINKED_WITH_TBB
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
// Delaunay T3
typedef CGAL::Triangulation_data_structure_3<
CGAL::Triangulation_vertex_base_3<K>,
CGAL::Triangulation_cell_base_3<K>,
CGAL::Parallel_tag> Tds;
typedef CGAL::Delaunay_triangulation_3<K, Tds> Triangulation;
typedef Triangulation::Point Point;
const int NUM_INSERTED_POINTS = 5000;
CGAL::Random_points_in_cube_3<Point> rnd(1.);
// Construction from a vector of 1,000,000 points
std::vector<Point> V;
V.reserve(NUM_INSERTED_POINTS);
for (int i = 0; i != NUM_INSERTED_POINTS; ++i)
V.push_back(*rnd++);
// Construct the locking data-structure, using the bounding-box of the points
Triangulation::Lock_data_structure locking_ds(
CGAL::Bbox_3(-1., -1., -1., 1., 1., 1.), 50);
// Construct the triangulation in parallel
Triangulation T(V.begin(), V.end(), &locking_ds);
assert(T.is_valid());
#endif //CGAL_LINKED_WITH_TBB
return 0;
}

## Parallel insertion and removal in Regular triangulation

This example shows the parallel building of a Regular triangulation, followed by the parallel removal of the first 100,000 vertices.

File Triangulation_3/parallel_insertion_and_removal_in_regular_3.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/Regular_triangulation_3.h>
#include <CGAL/Regular_triangulation_euclidean_traits_3.h>
#include <CGAL/point_generators_3.h>
#include <iostream>
#include <fstream>
#include <vector>
int main()
{
#ifdef CGAL_LINKED_WITH_TBB
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Regular_triangulation_euclidean_traits_3<K> Traits;
typedef Traits::Bare_point Point;
// Regular T3
typedef CGAL::Triangulation_data_structure_3<
CGAL::Triangulation_vertex_base_3<Traits>,
CGAL::Regular_triangulation_cell_base_3<Traits>,
CGAL::Parallel_tag> Tds;
typedef CGAL::Regular_triangulation_3<Traits, Tds> Rt;
typedef Rt::Vertex_handle Vertex_handle;
const int NUM_INSERTED_POINTS = 5000;
CGAL::Random_points_in_cube_3<Point> rnd(1.);
// Construction from a vector of 1,000,000 points
std::vector<Point> V;
V.reserve(NUM_INSERTED_POINTS);
for (int i = 0; i != NUM_INSERTED_POINTS; ++i)
V.push_back(*rnd++);
// Construct the locking data-structure, using the bounding-box of the points
Rt::Lock_data_structure locking_ds(
CGAL::Bbox_3(-1., -1., -1., 1., 1., 1.), 50);
// Contruct the triangulation in parallel
std::cerr << "Construction and insertion" << std::endl;
Rt rtr(V.begin(), V.end(), &locking_ds);
assert(rtr.is_valid());
std::cerr << "Remove" << std::endl;
// Remove the first 1/10 vertices
std::vector<Vertex_handle> vertices_to_remove;
Rt::Finite_vertices_iterator vit = rtr.finite_vertices_begin();
for (int i = 0 ; i < NUM_INSERTED_POINTS/10 ; ++i)
vertices_to_remove.push_back(vit++);
// Parallel remove
rtr.remove(vertices_to_remove.begin(), vertices_to_remove.end());
assert(rtr.is_valid());
#endif //CGAL_LINKED_WITH_TBB
return 0;
}

# Complexity and Performance

In 3D, the worst case complexity of a triangulation is quadratic in the number of points. For Delaunay triangulations, this bound is reached in cases such as points equally distributed on two non-coplanar lines. However, the good news is that, in many cases, the complexity of a Delaunay triangulation is linear or close to linear in the number of points. Several articles [13], [15], [1], [2], [3] have proved such good complexity bounds for specific point distributions, such as points distributed on surfaces under some conditions.

## Running Time

There are several algorithms provided in this package. We will focus here on the following ones and give practical numbers on their efficiency :

- construction of a triangulation from a range of points,
- location of a point (using the `locate` function),
- removal of a vertex (using the `remove` function).

We will use the following types of triangulations, using `Exact_predicates_inexact_constructions_kernel` as geometric traits (combined with `Regular_triangulation_euclidean_traits_3` in the weighted case) :

- Delaunay : `Delaunay_triangulation_3`
- Delaunay - Fast location : `Delaunay_triangulation_3` with `Fast_location`
- Regular : `Regular_triangulation_3` (default setting : memorize hidden points)
- Regular - No hidden points : `Regular_triangulation_3` with hidden points discarded (using `Triangulation_cell_base_3` instead of `Regular_triangulation_cell_base_3`).

Figure 40.6 shows, for all these types of triangulations, the times in seconds taken to build a triangulation from a given number of points, then the average time to perform one point location in triangulations of various sizes, and the average time to perform one vertex removal (which is largely independent on the size of the triangulation).

The data sets used here are points randomly distributed in the unit cube (the coordinates are generated using the `drand48` C function). In the weighted case, the weights are all zero, which means that there are actually no hidden points during execution.

The measurements have been performed using CGAL 3.6, using the Gnu C++ compiler version 4.3.2, under Linux (Fedora 10 distribution), with the compilation options `-O3 -DCGAL_NDEBUG`. The computer used was equipped with a 64bit Intel Xeon 3GHz processor and 32GB of RAM (a recent desktop machine as of 2009).

| Delaunay | Delaunay | Regular | Regular
|  | Fast location |  | No hidden points
Construction from \( 10^2\) points | 0.00054 | 0.000576 | 0.000948 | 0.000955
Construction from \( 10^3\) points | 0.00724 | 0.00748 | 0.0114 | 0.0111
Construction from \( 10^4\) points | 0.0785 | 0.0838 | 0.122 | 0.117
Construction from \( 10^5\) points | 0.827 | 0.878 | 1.25 | 1.19
Construction from \( 10^6\) points | 8.5 | 9.07 | 12.6 | 12.2
Construction from \( 10^7\) points | 87.4 | 92.5 | 129 | 125
Point location in \( 10^2\) points | 9.93e-07 | 1.06e-06 | 7.19e-06 | 6.99e-06
Point location in \( 10^3\) points | 2.25e-06 | 1.93e-06 | 1.73e-05 | 1.76e-05
Point location in \( 10^4\) points | 4.79e-06 | 3.09e-06 | 3.96e-05 | 3.76e-05
Point location in \( 10^5\) points | 2.98e-05 | 6.12e-06 | 1.06e-04 | 1.06e-04
Point location in \( 10^6\) points | 1e-04 | 9.65e-06 | 2.7e-04 | 2.67e-04
Point location in \( 10^7\) points | 2.59e-04 | 1.33e-05 | 6.25e-04 | 6.25e-04
Vertex removal | 1e-04 | 1.03e-04 | 1.42e-04 | 1.38e-04

Figure 40.6 Running times in seconds for algorithms on 3D triangulations.

More benchmarks comparing CGAL to other software can be found in [17].

## Parallel Performance

Figure Figure 40.7 shows insertion and removal speed-ups obtained using the parallel version of the triangulation algorithms of CGAL 4.5. The machine used is a PC running Windows 7 64-bits with two 6-core Intel Xeon CPU X5660 clocked at 2.80 GHz with 32GB of RAM. The program has been compiled with Microsoft Visual C++ 2012 in Release mode.

Figure 40.7 Speed-up obtained for the insertion of 1M points randomly generated inside a cube (red), and the removal of 100K of them (blue), compared to the sequential version of the algorithm.

## Memory Usage

We give here some indication about the memory usage of the triangulations. Those structures being intensively based on pointers, the size almost doubles on 64bit platforms compared to 32bit.

The size also depends on the size of the point type which is copied in the vertices (hence on the kernel). Obviously, any user data added to vertices and cells also affect the memory used.

More specifically, the memory space used to store a triangulation is first a function of the size of its `Vertex` and `Cell` types times their numbers (and for volumic distribution, one sees about 6.7 times more cells than vertices). However, these are stored in memory using `Compact_container`, which allocates them in lists of blocks of growing size, and this requires some additional overhead for bookkeeping. Moreover, memory is only released to the system when clearing or destroying the triangulation. This can be important for algorithms like simplifications of data sets which will produce fragmented memory usage (doing fresh copies of the data structures are one way out in such cases). The asymptotic memory overhead of `Compact_container` for its internal bookkeeping is otherwise on the order of \( O(\sqrt{n})\).

Figure 40.8 shows the number of bytes used per points, as measured empirically using `Memory_sizer` for large triangulations ( \( 10^6\) random points).

| Delaunay | Delaunay | Regular | Regular
|  | Fast location |  | No hidden points
32bit | 274 | 291 | 336 | 282
64bit | 519 | 553 | 635 | 527

Figure 40.8 Memory usage in bytes per point for large data sets.

## Variability Depending on the Data Sets and the Kernel

Besides the complexity of the Delaunay triangulation that varies with the distribution of the points, another critical aspect affects the efficiency : the degeneracy of the data sets. These algorithms are quite sensitive to numerical accuracy and it is important to run them using exact predicates.

Using a kernel with no exact predicates will quickly lead to crashes or infinite loops once they are executed on non-random data sets. More precisely, problems appear with data sets which contain (nearly) degenerate cases for the `orientation` and `side_of_oriented_sphere` predicates, namely when there are (nearly) coplanar or (nearly) cospherical points. This unfortunately happens often in practice with data coming from various kinds of scanners or other automatic acquisition devices.

Using an inexact kernel such as `Simple_cartesian<double>` would lead to optimal performance, which is only about 30% better than `Exact_predicates_inexact_constructions_kernel`. The latter is strongly recommended since it takes care about potential robustness issues. The former can be used for benchmarking purposes mostly, or when you really know that your data sets won't exhibit any robustness issue.

Exact predicates take more time to compute when they hit (nearly) degenerate cases. Depending on the data set, this can have a visible impact on the overall performance of the algorithm or not.

Sometimes you need exact constructions as well, so `Exact_predicates_exact_constructions_kernel` is a must. This is the case for example when you need the `dual` functions to be exact, or when your input is stored in points of such a kernel for other reasons (because it is the output of another algorithm which has this requirement, for example). This will slow down the computations by a factor of 4 to 5 at least, and it can be much more.

Figure 40.9 gives more detailed timings about various kernels one the following data sets : random points in a cube, random points on the surface of an ellipsoid, points scanned on the surface of a Buddha statue, points on a molecular surface, and points scanned on a dryer handle. See Figure 40.10 for pictures of the last 3 objects, which respectively illustrate volumic data, surfacic data, and data with many degenerate cases. This last data set exhibits an infinite loop with an inexact kernel, and of course we are not sure whether what is computed for the other data sets with this inexact kernel is a Delaunay triangulation. General introductory information about these robustness issues can be found in [16]. More benchmarks around this issue can also be found in [7].

| Random | Ellipsoid | Buddha | Molecule | Dryer
|  |  |  |  | Handle
Number of points | 100000 | 100000 | 542548 | 525296 | 49787
`Simple_cartesian<double>` | 0.69 | 0.627 | 4.21 | 3.8 | \( \infty \)-loop
`Exact_predicates_inexact_constructions_kernel` | 0.824 | 0.749 | 4.99 | 4.64 | 1.68
`Exact_predicates_exact_constructions_kernel` | 4.59 | 3.85 | 30.1 | 26.4 | 4.57
`Simple_cartesian<Gmpq>` | 492 | 534 | 1120 | 1030 | 75.2

Figure 40.9 Running times (seconds) for various kernels and data sets.

Figure 40.10 Data sets used in the benchmark of Figure 40.9.

# Design and Implementation History

Monique Teillaud started to work on the 3D triangulation packages in 1997, following the design of the 2D triangulation packages. The notions of degenerate dimensions and infinite vertex were formalized [20] and induced changes in the 2D triangulation packages. The packages were first released in CGAL 2.1. They contained basic functionalities on triangulations, Delaunay triangulations, regular triangulations.

A first version of removal of a vertex from a Delaunay triangulation was released in CGAL 2.2. However, this removal became really robust only in CGAL 2.3, after some research that allowed to deal with degenerate cases quite easily [8]. Andreas Fabri implemented this revised version of the removal, and a faster removal algorithm for CGAL 3.0.

The latter algorithm was proposed by Mariette Yvinec, who contributed in several ways to the package, first since she was maintaining the close 2D triangulation package and participated in many discussions, she also wrote the traits classes for regular triangulations.

In 2000, Sylvain Pion started working on these packages. He improved the efficiency of triangulations in CGAL 2.3 and 2.4 in several ways [6] he implemented the Delaunay hierarchy [12] in 2.3, he improved the memory footprint in 2.4 and 3.0, he also performed work on arithmetic filters [7] (see `Filtered_kernel`) to improve the speed of triangulations. He changed the design in CGAL 3.0, allowing users to add handles in their own vertices and cells.

Olivier Devillers, co-author of preliminary versions of the CGAL 2d triangulations, participated in many discussions, in particular about the perturbations, and more concretely in the implementation of the Delaunay hierarchy.

In 2005, Christophe Delage implemented the vertex removal function for regular triangulations, using the symbolic perturbation proposed in [9], which allowed to release this functionality in CGAL 3.2.

In 2006, Nico Kruithof wrote the `Triangulation_simplex_3` class that can store simplices of any dimension and improved the internal organization of the code.

As of March 2007, Christophe Delage made the iterator range insert methods and constructors use `spatial_sort` to improve efficiency.

In 2008, Camille Wormser added a few more iterators in the package that were integrated in release 3.4.

In 2009, Sylvain Pion simplified the design of the Delaunay hierarchy so that it became the simple `Fast_location` policy in release 3.6.

In 2010, Pedro de Castro and Olivier Devillers added the point displacement in release 3.7.

In 2011, Pedro de Castro and Olivier Devillers implemented in release 3.8 the structural filtering method, improving the efficiency of point location.

A new demo of this package was introduced in CGAL 3.8, coded by Fei (Sophie) Che, who was co-mentored by Manuel Caroli and Monique Teillaud in the framework of the Google Summer of Code, 2010.

In 2013, Clément Jamin added parallel algorithms (insert, remove) to the Delaunay and regular triangulations.

The authors wish to thank Lutz Kettner for inspiring discussions about the design of CGAL. Jean-Daniel Boissonnat is also acknowledged [5].