2D Triangulations

Mariette Yvinec

This chapter describes the two dimensional triangulations of Cgal. Section 36.1 recalls the main definitions about triangulations. Sections 36.2 discusses the way two-dimensional triangulations are represented in Cgal. Section 36.3 presents the overall software design of the 2D triangulations package. The next sections present the different two dimensional triangulations classes available in Cgal : basic triangulations (section 36.4), Delaunay triangulations (Section 36.5), regular triangulations (Section 36.6), constrained triangulations (Section 36.7), and constrained Delaunay triangulations (Section 36.8). Section 36.9 describes a class which implements a constrained or constrained Delaunay triangulation with an additional data structure to describe how the constraints are refined by the edges of the triangulations. Section 36.10 describes a hierarchical data structure for fast point location queries. At last, Section 36.11 explains how the user can benefit from the flexibility of Cgal triangulations using customized classes for faces and vertices.

A two dimensional triangulation can be roughly described as a set T
of triangular facets such that :

- two facets either are disjoint or share a lower dimensional
face (edge or vertex).

- the set of facets in T is connected for the adjacency relation.

- the domain U_{T} which is the union
of facets in T has no singularity.

More precisely, a triangulation can be described as a simplicial complex. Let us first record a few definitions.

A simplicial complex is a set T of simplices such that

- any face of a simplex in T is a simplex in T

- two simplices in T either are disjoint or share
a common sub-face.

The dimension d of a simplicial complex is the
maximal dimension of its simplices.

A simplicial complex T is pure if any simplex of T
is included in a simplex of T with maximal dimension.

Two simplexes in T with maximal dimension d are said to be
adjacent if they share a d-1 dimensional sub-face.
A simplicial complex is connected if the adjacency relation
defines a connected graph
over the set of simplices of T with maximal dimension.

The union U_{T} of all simplices in T is called the domain of T.
A point p in the domain of T is said to singular
if its surrounding in U_{T}
is neither a topological ball nor a topological disc.

Then, a two dimensional triangulation can be described as a two dimensional simplicial complex that is pure, connected and without singularity.

Each facet of a triangulation can be given an orientation which in turn induces an orientation on the edges incident to that facet. The orientation of two adjacent facets are said to be consistent if they induce opposite orientations on their common incident edge. A triangulation is said to be orientable if the orientation of each facet can be chosen in such a way that all pairs of incident facets have consistent orientations.

The data structure underlying Cgal triangulations allows to represent the combinatorics of any orientable two dimensional triangulations without boundaries. On top of this data structure, the 2D triangulations classes take care of the geometric embedding of the triangulation and are designed to handle planar triangulations. The plane of the triangulation may be embedded in a higher dimensional space.

The triangulations of Cgal are complete triangulations which means that their domain is the convex hull of their vertices. Because any planar triangulation can be completed, this is not a real restriction. For instance, a triangulation of a polygonal region can be constructed and represented as a subset of a constrained triangulation in which the region boundary edges have been input as constrained edges (see Section 36.7, 36.8 and 36.9).

Strictly speaking, the term *face* should be used
to design a face of any dimension,
and the two-dimensional faces of a triangulation
should be properly called *facets*.
However, following a common usage, we hereafter often call *faces*, the facets
of a two dimensional triangulation.

A 2D triangulation of Cgal can be viewed as a planar partition
whose bounded faces are triangular and cover
the convex hull of the set of vertices.
The single unbounded face of this partition
is the complementary of the convex hull.
In many applications, such as Kirkpatrick's hierarchy
or incremental Delaunay construction, it is convenient to
deal with only triangular faces. Therefore,
a fictitious vertex, called the *infinite vertex*
is added to the triangulation as well as
*infinite edges* and *infinite faces* incident to it..
Each infinite edge
is incident to the infinite vertex and to a vertex of the convex hull.
Each infinite face is incident to the infinite vertex
and to a convex hull edge.

Therefore, each edge of the triangulation is incident to exactly two faces and the set of faces of a triangulation is topologically equivalent to a two-dimensional sphere. This extends to lower dimensional triangulations arising in degenerate cases or when the triangulations as less than three vertices. Including the infinite faces, a one dimensional triangulation is a ring of edges and vertices topologically equivalent to a 1-sphere. A zero dimensional triangulation, whose domain is reduced to a single point, is represented by two vertices that is topologically equivalent to a 0-sphere.

Note that
the *infinite vertex* has no significant
coordinates and that no geometric predicate can be applied on it
nor on an infinite face.

**Figure 36.1: **Infinite vertex and infinite faces

The basic elements of the representation are vertices and faces. Each triangular face gives access to its three incident vertices and to its three adjacent faces. Each vertex gives access to one of its incident faces and through that face to the circular list of its incident faces.

The three vertices of a face are indexed with 0, 1 and 2
in counterclockwise order. The neighbors of a face are also
indexed with 0,1,2 in such a way that the neighbor indexed by *i*
is opposite to the vertex with the same index.
See Figure 36.2,
the functions *ccw(i)*
and *cw(i)* shown on this figure
compute respectively i+1 and i-1 modulo 3.

The edges are not explicitly represented, they are only implicitly
represented through the adjacency relations of two faces.
Each edge has two implicit representations: the edge
of a face *f* which is opposed to the vertex indexed *i*,
can be represented as well as an edge of the *neighbor(i)* of
*f*.

**Figure 36.2: **Vertices and neighbors.

The triangulations classes of Cgal provide high-level geometric functionalities such as location of a point in the triangulation, insertion, removal, or displacement of a point. They are build as a layer on top of a data structure called the triangulation data structure. The triangulation data structure can be thought of as a container for the faces and vertices of the triangulation. This data structure also takes care of all the combinatorial aspects of the triangulation.

This separation between the geometric aspect and the combinatorial part is reflected in the software design by the fact that the triangulation classes have two template parameters:

- the first parameter stands for a
**geometric traits**class providing the geometric primitives (points, segments and triangles) of the triangulation and the elementary operations (predicate or constructions) on those objects. - the second parameter stands for a
**triangulation data structure**class. The concept of triangulation data structure is described in Section 37.2 of Chapter 37. The triangulation data structure defines the types used to represent the faces and vertices of the triangulation, as well as additional types (handles, iterators and circulators) to access and visit the faces and vertices.Cgal provides the class

*Triangulation_data_structure_2<Vb,Fb>*as a default model of triangulation data structure. The class*Triangulation_data_structure_2<Vb,Fb>*has two template parameters standing for a vertex class and a face class. Cgal defines concepts for these template parameters and provide default models for these concepts. The vertex and base classes are templated by the geometric traits which allows them to have some knowledge of the geometric primitives of the triangulation. Those default vertex and face base classes can be replaced by user customized base classes in order, for example, to deal with additional properties attached to the vertices or faces of a triangulation. See section 36.11 for more details on the way to make use of this flexibility.

The Figure 36.3 summarizes the design of the triangulation package, showing the three layers (base classes, triangulation data structure and triangulation) forming this design.

**Figure 36.3: **The triangulations software design.

The top triangulation level, responsible for the geometric
embedding of the triangulation comes in different flavors
according to the different kind of triangulations:
basic, Delaunay, regular, constrained or constrained Delaunay.
Each kind of triangulations correspond to a different
class.
Figure 36.4 summarizes the derivation dependencies
of Cgal 2D triangulations classes.
Any 2D triangulation class is parametrized by
a geometric traits and a triangulation data structure.
While a unique concept *TriangulationDataStructure_2*
describes the triangulation data structure requirements
for any triangulation class,
the concept of geometric traits actually depends
on the triangulation class.
In general, the requirements for the vertex and face base classes
are described by the basic concepts *TriangulationVertexBase_2*
and *TriangulationFaceBase_2*. However, some triangulation
classes requires base classes implementing
refinements
of the basic concepts.

**Figure 36.4: **The derivation tree of 2D triangulations.

The class *Triangulation_2<Traits,Tds>*
serves as a base class for the other
2D triangulations classes
and
implements the user
interface to a triangulation.

The vertices and faces of the triangulations are accessed through
*handles*,
*iterators* and *circulators*.
A handle is a model of the concept *Handle* which basically
offers the two dereference operators *** and *->* .
A circulator is a type devoted to visit circular sequences.
Handles are used whenever the accessed element
is not part of a sequence.
Iterators and circulators are used
to visit all or parts of the triangulation.

The iterators and circulators are all bidirectional and non mutable. The circulators and iterators are convertible to the handles with the same value type, so that when calling a member function, any handle type argument can be replaced by an iterator or a circulator with the same value type.

The triangulation class allows to visit the vertices and neighbors of a face in clockwise or counterclockwise order.

There are circulators to visit the edges or faces incident to a given vertex or the vertices adjacent to it. Another circulator type allows to visit all the faces traversed by a given line. Circulators step through infinite features as well as through finite ones.

The triangulation class offers some iterators to visit all the faces, edges or vertices and also iterators to visit selectively the finite faces, edges or vertices.

The triangulation class provides methods to test the infinite character of any feature, and also methods to test the presence in the triangulation of a particular feature (edge or face) given its vertices.

The triangulation class provides a method to locate a given point with respect to a triangulation. In particular, this method reports whether the point coincides with a vertex of the triangulation, lies on an edge, in a face or outside of the convex hull. In case of a degenerate lower dimensional triangulation, the query point may also lie outside the triangulation affine hull.

The triangulation class also provides methods to locate a point with respect to a given finite face of the triangulation or with respect to its circumcircle. The faces of the triangulation and their circumcircles have the counterclockwise orientation.

The triangulation can be modified by several functions: insertion of a point, removal of a vertex, displacement of a vertex, flipping of an edge. The flipping of an edge is possible when the union of the two incident faces forms a convex quadrilateral (see Figure 36.5).

Locate is implemented by a line walk. The walk
begins at a vertex of the face which
is given
as an optional argument or at an arbitrary vertex of the triangulation
if no optional argument is given. It takes
time O(n) in the worst case, but only O(√n)
on average if the vertices are distributed uniformly at random.
The class *Triangulation_hierarchy_2<Traits,Tds>*,
described in section 36.10,
implements a data structure designed to
offer an alternate more efficient point location algorithm.

Insertion of a point is done by locating a face that contains the point, and splitting this face into three new faces. If the point falls outside the convex hull, the triangulation is restored by flips. Apart from the location, insertion takes a time O(1). This bound is only an amortized bound for points located outside the convex hull.

Removal of a vertex is done by removing all adjacent triangles, and
re-triangulating the hole. Removal takes a time at most proportional to
d^{2}, where
d is the degree of the removed vertex,
which is O(1) for a random vertex.

Displacement of a vertex is done by: first, verifying if the triangulation embedding remains planar after the displacement; if yes the vertex is directly placed at the new location; otherwise, a point is inserted at the new location and the vertex at the obsolete location is removed.

The face, edge, and vertex iterators on finite features are derived from their counterparts visiting all (finite and infinite) features which are themselves derived from the corresponding iterators of the triangulation data structure.

The geometric traits of a triangulation
is required to provide
the geometric objects (points, segments and triangles)
building up the triangulation
together with the geometric predicates on those objects.
The required predicates are:

- comparison of the *x* or *y* coordinates of two points.

- the orientation test which computes
the order type of three given point.

The concept
*TriangulationTraits_2* describes the requirements for the
geometric traits class of a triangulation.
The Cgal kernel classes
are models for this concept.
The Cgal library also provides dedicated models
of *TriangulationTraits_2*
using the kernel geometric objects and predicates.
These classes are themselves templated with a Cgal kernel
and extract the required types and predicates from the kernel.
The traits class *Triangulation_euclidean_traits_2<R>*
is designed to deal with ordinary two dimensional points.
The class *Projection_traits_xy_3<R>*
is a geometric traits class to build the triangulation
of a terrain. Such a triangulation is a two-dimensional
triangulation embedded in three dimensional space.
The data points are three-dimensional points.
The triangulation is
build according to the projections of those points
on the xy plane and then lifted up to the original
three-dimensional data points.
This is especially useful
to deal with GIS terrains.
Instead of really projecting the three-dimensional points and
maintaining a mapping between each point and its projection
(which costs space and is error prone),
the traits class supplies geometric predicates that ignore the
*z*-coordinates of the points.
See Section 36.5 for an example.
Cgal provides also the geometric traits classes
*Projection_traits_yz_3<R>* and
*Projection_traits_xz_3<R>* to
deal with projections on the
*yz* plane and *xz*-plane,
respectively.

The following program creates a triangulation of 2D points
using the default kernel
*Exact_predicate_inexact_constructions_kernel*
as geometric traits and the default triangulation data structure.
The input points are read from a file
and inserted in the triangulation.
Finally points on the convex hull are written to `cout`.

File:examples/Triangulation_2/triangulation_prog1.cpp

#include <fstream> #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Triangulation_2.h> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Triangulation_2<K> Triangulation; typedef Triangulation::Vertex_circulator Vertex_circulator; typedef Triangulation::Point Point; int main() { std::ifstream in("data/triangulation_prog1.cin"); std::istream_iterator<Point> begin(in); std::istream_iterator<Point> end; Triangulation t; t.insert(begin, end); Vertex_circulator vc = t.incident_vertices(t.infinite_vertex()), done(vc); if (vc != 0) { do { std::cout << vc->point() << std::endl; }while(++vc != done); } return 0; }

The class *Delaunay_triangulation_2<Traits,Tds>* derives
from the class *Triangulation_2<Traits,Tds>*.

The class *Delaunay_triangulation_2<Traits,Tds>*
inherits the types defined by the
basic class *Triangulation_2<Traits,Tds>*.
Additional types, provided by the traits class,
are defined to represent the dual Voronoi diagram.

The class *Delaunay_triangulation_2<Traits,Tds>*
overwrites the member functions that insert, move, or remove a point
in the triangulation to maintain the Delaunay property.
It also has a member function (*Vertex_handle nearest_vertex(const Point& p)*)
to answer nearest neighbor queries
and member functions to construct the elements (vertices and edges)
of the dual Voronoi diagram.

The Cgal kernel classes
and the class *Triangulation_euclidean_traits_2<R>*
are models of the concept *DelaunayTriangulationTraits_2*
for the euclidean metric.
The traits class for terrains,
*Projection_traits_xy_3<R>*,

*Projection_traits_yz_3<R>*, and

*Projection_traits_xz_3<R>*

are also models of *DelaunayTriangulationTraits_2*
except that they do not fulfill
the requirements for the duality functions and nearest vertex
queries.

Removal calls the removal in the triangulation and then re-triangulates
the hole created in such a way that the Delaunay criterion is
satisfied. Removal of a
vertex of degree d takes time O(d^{2}).
The degree d is O(1) for a random
vertex in the triangulation.
When the degree of the removed vertex is small ( ≤ 7) a special
procedure is used that allows to increase global removal time by a factor of 2
for random points [Dev09].

The displacement of a vertex v at a point p to a new location p', first checks whether the triangulation embedding remains planar or not after moving v to p'. If yes, it moves v to p' and simply performs a sequence of flips to restore the Delaunay property, which is O(d) where d is the degree of the vertex after the displacement. Otherwise, the displacement is done by inserting a vertex at the new location, and removing the obsolete vertex. The complexity is O(n) in the worst case, but only O(1 + δ√n) for evenly distributed vertices in the unit square, where δ is the Euclidean distance between the new and old locations.

After having performed a point location, the nearest neighbor of a point is found in time O(n) in the worst case, but in time O(1) for vertices distributed uniformly at random and any query point.

The following code creates a Delaunay triangulation with the usual Euclidean metric for the vertical projection of a terrain model. The points have elevation, that is they are 3D points, but the predicates used to build the Delaunay triangulation are computed using only the x and y coordinates of these points.

The class *Projection_traits_xy_3<K>* is part of the 2D and 3D Linear Geometric Kernel,
and replaces the class *Triangulation_euclidean_traits_xy_3<K>* which is deprecated.

File:examples/Triangulation_2/terrain.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Projection_traits_xy_3.h> #include <CGAL/Delaunay_triangulation_2.h> #include <fstream> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Projection_traits_xy_3<K> Gt; typedef CGAL::Delaunay_triangulation_2<Gt> Delaunay; typedef K::Point_3 Point; int main() { std::ifstream in("data/terrain.cin"); std::istream_iterator<Point> begin(in); std::istream_iterator<Point> end; Delaunay dt; dt.insert(begin, end); std::cout << dt.number_of_vertices() << std::endl; return 0; }

File:examples/Triangulation_2/voronoi.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Delaunay_triangulation_2.h> #include <fstream> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Delaunay_triangulation_2<K> Triangulation; typedef Triangulation::Edge_iterator Edge_iterator; typedef Triangulation::Point Point; int main( ) { std::ifstream in("data/voronoi.cin"); std::istream_iterator<Point> begin(in); std::istream_iterator<Point> end; Triangulation T; T.insert(begin, end); int ns = 0; int nr = 0; Edge_iterator eit =T.edges_begin(); for ( ; eit !=T.edges_end(); ++eit) { CGAL::Object o = T.dual(eit); if (CGAL::object_cast<K::Segment_2>(&o)) {++ns;} else if (CGAL::object_cast<K::Ray_2>(&o)) {++nr;} } std::cout << "The Voronoi diagram has " << ns << " finite edges " << " and " << nr << " rays" << std::endl; return 0; }

The power diagram of the set PW is a space partition in which
each cell corresponds to a sphere (p_{i}, w_{i}) of PW
and is the locus of points p whose power with respect to (p_{i}, w_{i})
is less than its power with respect to any other sphere
in PW. In the two-dimensional space,
the dual of this diagram is a triangulation
whose domain covers the convex hull of the set
P= { p_{i} | i = 1, … , n } of center points
and whose vertices form a subset of P.
Such a triangulation is called a regular triangulation.
Three points p_{i}, p_{j} and p_{k} of P
form a triangle in the regular triangulation of PW
iff there is a point p of the plane with equal
powers with respect to (p_{i}, w_{i}), (p_{j}, w_{j})
and (p_{k}, w_{k}) and such that this power
is less than the power of p
with respect to any other sphere in PW.

Let us defined the power product of two weighted points
(p_{i}, w_{i}) and (p_{j}, w_{j}) as:

Π(p_{i}, w_{i}, p_{j}, w_{j}) = p_{i}p_{j} ^{2} - w_{i} - w_{j} . |

The regular triangulation of the sets PW
satisfies the following *regular property* (which just reduces to the
Delaunay property when all the weights are null):
a triangle p_{i}p_{j}p_{k} is a face of the regular triangulation
of PW iff the power product of any weighted point
(p_{l}, w_{l}) of PW with the power circle of
(p_{i}, w_{i}), (p_{j}, w_{j}) and (p_{k}, w_{k}) is positive or null.
We call power test of (p_{i}, w_{i}), (p_{j}, w_{j}), (p_{k}, w_{k}),
and (p_{l}, w_{l}), the predicates which amount to compute
the sign of
the power product of (p_{l}, w_{l}) with respect to
the power circle of
(p_{i}, w_{i}), (p_{j}, w_{j}) and (p_{k}, w_{k}).
This predicate amounts to computing the sign of
the following
determinant

| |
| | |

A pair of neighboring faces p_{i}p_{j}p_{k}
and p_{i}p_{j}p_{l} is said to be locally regular
(with respect to the weights in PW)
if the power test of (p_{i}, w_{i}), (p_{j}, w_{j}), (p_{k}, w_{k}),
and (p_{l}, w_{l}) is positive.
A classical result of computational geometry
establishes that a triangulation of the convex hull of P
such that any pair of neighboring faces is regular with respect
to PW, is a
regular triangulation of PW.

Alternatively, the regular triangulation
of the weighted points set PW
can be obtained as the projection
on the two dimensional plane of the convex hull of the set of three
dimensional points
P'= { (p_{i},p_{i} ^{2} - w_{i} ) | i = 1, … , n }.

The class *Regular_triangulation_2<Traits, Tds>*
is designed to maintain the
regular triangulation of a set of 2d weighted points.
It derives from the class *Triangulation_2<Traits, Tds>*.
The functions *insert* and
*remove* are overwritten to handle weighted points
and maintain the regular
property.
In current version, function *move* is not
overwritten and thus does not preserve the regular property.
The vertices of the regular triangulation
of a set of weighted points PW correspond only to a subset
of PW.
Some of the input
weighted points have no cell in the dual power diagrams
and therefore do not correspond to a vertex of the regular
triangulation.
Such a point is called a hidden point.
Because hidden points can reappear later on as vertices
when some other point is removed,
they have to be stored somewhere.
The regular triangulation store those points in special vertices, called
hidden vertices.
A hidden point can reappear as vertex of the triangulation
only when the two dimensional face that hides it
is removed from the triangulation. To deal with this feature,
each face of a regular triangulation stores a list of hidden vertices.
The points in those vertices
are reinserted in the triangulation when the face
is removed.

Regular triangulation have member functions to construct the vertices and edges of the dual power diagrams.

**Note that, since the type Weighted_point is not defined
in Cgal kernels, plugging a filtered kernel such as
Exact_predicates_exact_constructions_kernel in
Regular_triangulation_euclidean_traits_2<K,Weight> will in fact
not provide exact predicates on weighted points.**
To solve this, there is also another model of the traits concept,

The base vertex type of a regular triangulation
includes a Boolean data member to mark the hidden state of the vertex.
Therefore Cgal defines the concept
*RegularTriangulationVertexBase_2* which refine
the concept *TriangulationVertexBase_2*
and provides a default model
for this concept.

The base face type of a regular triangulation
is required to provide a list of hidden vertices,
designed to store the points hidden by the face. It has to be a model
of the concept *RegularTriangulationFaceBase_2*.
Cgal provides the templated class
*Regular_triangulation_face_base_2<Traits>*
as a default base class for faces of regular triangulations.

The following code creates a regular triangulation of a set of weighted points and output the number of vertices and the number of hidden vertices.

File:examples/Triangulation_2/regular.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Regular_triangulation_euclidean_traits_2.h> #include <CGAL/Regular_triangulation_filtered_traits_2.h> #include <CGAL/Regular_triangulation_2.h> #include <fstream> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Regular_triangulation_filtered_traits_2<K> Traits; typedef CGAL::Regular_triangulation_2<Traits> Regular_triangulation; int main() { Regular_triangulation rt; std::ifstream in("data/regular.cin"); Regular_triangulation::Weighted_point wp; int count = 0; while(in >> wp){ count++; rt.insert(wp); } rt.is_valid(); std::cout << "number of inserted points : " << count << std::endl; std::cout << "number of vertices : " ; std::cout << rt.number_of_vertices() << std::endl; std::cout << "number of hidden vertices : " ; std::cout << rt.number_of_hidden_vertices() << std::endl; return 0; }

A constrained triangulation is a triangulation of a set of points
that has to include among its edges
a given set of segments joining the points. The corresponding
edges are called *constrained edges*.

The endpoints of constrained edges are of course vertices of the triangulation. However the triangulation may include include other vertices as well. There are three versions of constrained triangulations.

- In the basic version, the constrained triangulation does not handle intersecting constraints, and the set of input constraints is required to be a set of segments that do not intersect except possibly at their endpoints. Any number of constrained edges are allowed to share the same endpoint. Vertical constrained edges or constrained edges with null length are allowed.
- The two other versions support intersecting input constraints.
In those versions, input constraints are allowed to be
intersecting, overlapping or partially
overlapping segments.
The triangulation introduces additional vertices at each point that
is the proper intersection points of two
constraints. A single constraint intersecting other
constraints will then appear as several edges in the triangulation.
The two versions dealing with intersecting constraints differ
in the way intersecting constraints are dealt with.
- One of them is designed to be robust when predicates are evaluated exactly but constructions (i. e. intersection computations) are approximate.
- The other one is designed to be used
with exact arithmetic (meaning exact
evaluation of predicates and exact computation of intersections.)
This last version finds its full efficiency when used in conjunction
with a constraint hierarchy data structure (which allows one to avoid the
cascading of intersection computations)
as provided in the class
*Constrained_triangulation_plus_2*. See section 36.9.

A constrained triangulation is represented in the Cgal library as an
object of the class *Constrained_triangulation_2<Traits,Tds,Itag>*.
The third parameter *Itag* is the intersection tag
which serves to choose how intersecting constraints
are dealt with. This parameter has to be instantiated
by one of the following classes :
*CGAL::No_intersection_tag* when input constraints do not
intersect

*CGAL::Exact_predicates_tag* if the geometric traits provides
exact predicates but approximate constructions

*CGAL::Exact_intersections_tag* when an exact predicates
and exact constructions are provided.

The class *Constrained_triangulation_2<Traits,Tds, Itag>*
inherits from *Triangulation_2<Traits,Tds>*.
It defines an additional type *Constraint*
to represent the constraints. A
constraint is represented as a pair of points.

A constrained triangulation can be created
from a
list of constrained edges.
The class *Constrained_triangulation_2<Traits,Tds,Itag>*
overrides the insertion and removal of a point to take care of the
information about constrained edges. The class also allows inline
insertion of a new constraint, given by its two endpoints
or the removal of a constraint.
In current version, function *move* is not
overwritten and thus does not take care of the constraints.

Cgal provides a default base face class
for constrained triangulations. This class, named
*Constrained_triangulation_face_base_2<Traits>*,
derives from the class
*Triangulation_face_base_2<Traits>*
and adds three Boolean data members to store the status of its edges.

**Figure 36.6: **Constrained and Constrained Delaunay triangulation:
the constraining edges are the green edges, a constrained
triangulation is shown on the left, the constrained Delaunay
triangulation with two examples of circumcircles is shown on the right.

A constrained Delaunay triangulation is a triangulation with
constrained edges which tries to be as much Delaunay as possible.
As constrained edges are not necessarily Delaunay edges,
the triangles of a constrained Delaunay triangulation do not
necessarily fulfill the empty circle property
but they fulfill a weaker *constrained empty circle property*.
To state this property,
it is convenient to think of constrained
edges as blocking the view. Then, a triangulation is
constrained Delaunay iff
the circumscribing circle
of any facet encloses
no vertex visible
from the interior of the facet.
As in the case of constrained triangulations, three different versions
of Delaunay constrained triangulations are provided. The first version
handle set of constraints which do not intersect except possibly
at the endpoints. The two other versions
handle intersecting input constraints. One of them
is designed to be designed to be robust
when used in conjunction with a geometric traits
providing exact predicates and approximate constructions
(such as a *CGAL::Filtered_Kernel* or any kernel providing
filtered exact predicates). The third version is designed to be used
with an exact arithmetic number type.

The Cgal class
*Constrained_Delaunay_triangulation_2<Traits,Tds,Itag>*
is designed to represent
constrained Delaunay triangulations.

As in the case of constraints triangulation, the third parameter
*Itag* is the intersection tag
and serves to choose how intersecting constraints
are dealt with. It can be instantiated with one of the following
class : *CGAL::No_intersection_tag*,
*CGAL::Exact_predicates_tag*,
*CGAL::Exact_intersections_tag*
(see Section 36.7.

A constrained Delaunay triangulation is not a Delaunay
triangulation but it is a constrained triangulation.
Therefore the class
*Constrained_Delaunay_triangulation_2<Traits,Tds,Itag>*
derives from
the class *Constrained_triangulation_2<Traits,Tds,Itag>*.

The constrained Delaunay triangulation has member functions to override the insertion and removal of a point or of a constraint. Each of those member function takes care to restore the constrained empty circle property.

File:examples/Triangulation_2/constrained.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Constrained_Delaunay_triangulation_2.h> #include <cassert> #include <iostream> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Triangulation_vertex_base_2<K> Vb; typedef CGAL::Constrained_triangulation_face_base_2<K> Fb; typedef CGAL::Triangulation_data_structure_2<Vb,Fb> TDS; typedef CGAL::Exact_predicates_tag Itag; typedef CGAL::Constrained_Delaunay_triangulation_2<K, TDS, Itag> CDT; typedef CDT::Point Point; int main( ) { CDT cdt; std::cout << "Inserting a grid of 5x5 constraints " << std::endl; for (int i = 1; i < 6; ++i) cdt.insert_constraint( Point(0,i), Point(6,i)); for (int j = 1; j < 6; ++j) cdt.insert_constraint( Point(j,0), Point(j,6)); assert(cdt.is_valid()); int count = 0; for (CDT::Finite_edges_iterator eit = cdt.finite_edges_begin(); eit != cdt.finite_edges_end(); ++eit) if (cdt.is_constrained(*eit)) ++count; std::cout << "The number of resulting constrained edges is "; std::cout << count << std::endl; return 0; }

The class *Constrained_triangulation_plus_2<Tr>*
provides a constrained triangulation with an additional data
structure called the constraint hierarchy
that keeps track of the input constraints and of their refinement
in the triangulation.
The class *Constrained_triangulation_plus_2<Tr>*
inherits from its template parameter Tr, which has to be instantiated
by a constrained or constrained Delaunay triangulation.
According to its intersection tag, the base class
will support intersecting input constraints or not.
When intersections of input constraints are supported,
the base class constructs a triangulation of the arrangement
of the constraints,
introducing new vertices at each proper intersection
points and refining the input constraints into sub-constraints
which appear as edges (more precisely as constrained edges) of the
triangulation. The data structure maintains for each
input constraint
the sequence of intersection vertices added on this constraint.
The constraint hierarchy also allows the user to retrieve the set
of constrained edges of the triangulation, and for each
constrained edge, the set of input constraints that overlap it.

The class *Constrained_triangulation_plus_2<Tr>*
is especially useful when the base constrained triangulation class
handles intersections of constraints and uses an exact number type,
i.e. when its intersection tag is *CGAL::Exact_intersections_tag*.
Indeed in this case, the *Constrained_triangulation_plus_2<Tr>*
is specially designed to avoid cascading in the computations of
intersection points.

The following code inserts a set of intersecting constraint segments into a triangulation and counts the number of constrained edges of the resulting triangulation.

File:examples/Triangulation_2/constrained_plus.cpp

#include <CGAL/Exact_predicates_exact_constructions_kernel.h> #include <CGAL/intersections.h> #include <CGAL/Constrained_Delaunay_triangulation_2.h> #include <CGAL/Constrained_triangulation_plus_2.h> #include <cassert> #include <iostream> typedef CGAL::Exact_predicates_exact_constructions_kernel K; typedef CGAL::Triangulation_vertex_base_2<K> Vb; typedef CGAL::Constrained_triangulation_face_base_2<K> Fb; typedef CGAL::Triangulation_data_structure_2<Vb,Fb> TDS; typedef CGAL::Exact_intersections_tag Itag; typedef CGAL::Constrained_Delaunay_triangulation_2<K,TDS,Itag> CDT; typedef CGAL::Constrained_triangulation_plus_2<CDT> CDTplus; typedef CDTplus::Point Point; int main( ) { CDTplus cdt; std::cout << "Inserting a grid 5 x 5 of constraints " << std::endl; for (int i = 1; i < 6; ++i) cdt.insert_constraint( Point(0,i), Point(6,i)); for (int j = 1; j < 6; ++j) cdt.insert_constraint( Point(j,0), Point(j,6)); assert(cdt.is_valid()); int count = 0; for (CDTplus::Subconstraint_iterator scit = cdt.subconstraints_begin(); scit != cdt.subconstraints_end(); ++scit) ++count; std::cout << "The number of resulting constrained edges is " << count << std::endl; return 0; }

The class *Triangulation_hierarchy_2<Tr>*
implements a triangulation augmented with
a data structure to answer efficiently point location queries.
The data structure is a hierarchy
of triangulations. The triangulation at the lowest level is
the original triangulation where operations and point location are to
be performed.
Then at each succeeding level, the data structure
stores a triangulation of a small random sample of the vertices
of the triangulation at the preceding level. Point location
is done through a top down nearest neighbor query.
The nearest neighbor query is first
performed naively in the top level triangulation.
Then, at each following level, the nearest neighbor at that level
is found through a linear walk performed from
the nearest neighbor found at the preceding level.
Because the number of vertices in each triangulation is only a small
fraction of the number of vertices of the preceding triangulation,
the data structure remains small and achieves fast point location
queries on real
data. As proved in [Dev98], this structure has an optimal behavior
when it is built for Delaunay triangulations.
However it can be used as well for other triangulations
and the class *Triangulation_hierarchy_2<Tr>* is templated by a parameter
which is to be instantiated by one of the Cgal triangulation
classes.

The class *Triangulation_hierarchy_2<Tr>* inherits from the
triangulation type passed as template parameter *Tr*.
The *insert*, *move*, and *remove* member functions
are overwritten to update the data structure at each operation.
The locate queries are also overwritten to take advantage of the data
structure for a fast processing.

Cgal provides the class *Triangulation_hierarchy_vertex_base_2<Vb>*
which is a model for the concept
*TriangulationHierarchyVertexBase_2*.
This class is templated by a parameter *Vb*
which is to be instantiated by a model of the concept
*TriangulationVertexBase_2*.
The class *Triangulation_hierarchy_vertex_base_2<Vb>* inherits
from its template parameter *Vb*.
This design allows to use for *Vb*
either the default
vertex class or a user customized
vertex class with additional functionalities.

The following program is example for the standard use of a triangulation hierarchy to enhance the efficiency of a Delaunay triangulation. The program outputs the number of vertices at the different levels of the hierarchy.

File:examples/Triangulation_2/hierarchy.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Delaunay_triangulation_2.h> #include <CGAL/Triangulation_hierarchy_2.h> #include <CGAL/point_generators_2.h> #include <CGAL/algorithm.h> #include <cassert> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Triangulation_vertex_base_2<K> Vbb; typedef CGAL::Triangulation_hierarchy_vertex_base_2<Vbb> Vb; typedef CGAL::Triangulation_face_base_2<K> Fb; typedef CGAL::Triangulation_data_structure_2<Vb,Fb> Tds; typedef CGAL::Delaunay_triangulation_2<K,Tds> Dt; typedef CGAL::Triangulation_hierarchy_2<Dt> Triangulation; typedef Triangulation::Point Point; typedef CGAL::Creator_uniform_2<double,Point> Creator; int main( ) { std::cout << "insertion of 1000 random points" << std::endl; Triangulation t; CGAL::Random_points_in_square_2<Point,Creator> g(1.); CGAL::copy_n( g, 1000, std::back_inserter(t)); //verbose mode of is_valid ; shows the number of vertices at each level std::cout << "The number of vertices at successive levels" << std::endl; assert(t.is_valid(true)); return 0; }

The following program shows how to use a triangulation hierarchy in conjunction with a constrained triangulation plus.

File:examples/Triangulation_2/constrained_hierarchy_plus.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Constrained_Delaunay_triangulation_2.h> #include <CGAL/Triangulation_hierarchy_2.h> #include <CGAL/Constrained_triangulation_plus_2.h> #include <cassert> #include <iostream> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Triangulation_vertex_base_2<K> Vbb; typedef CGAL::Triangulation_hierarchy_vertex_base_2<Vbb> Vb; typedef CGAL::Constrained_triangulation_face_base_2<K> Fb; typedef CGAL::Triangulation_data_structure_2<Vb,Fb> TDS; typedef CGAL::Exact_predicates_tag Itag; typedef CGAL::Constrained_Delaunay_triangulation_2<K,TDS,Itag> CDT; typedef CGAL::Triangulation_hierarchy_2<CDT> CDTH; typedef CGAL::Constrained_triangulation_plus_2<CDTH> Triangulation; typedef Triangulation::Point Point; int main( ) { Triangulation cdt; std::cout << "Inserting a grid 5 x 5 of constraints " << std::endl; for (int i = 1; i < 6; ++i) cdt.insert_constraint( Point(0,i), Point(6,i)); for (int j = 1; j < 6; ++j) cdt.insert_constraint( Point(j,0), Point(j,6)); int count = 0; for (Triangulation::Subconstraint_iterator scit = cdt.subconstraints_begin(); scit != cdt.subconstraints_end(); ++scit) ++count; std::cout << "The number of resulting constrained edges is "; std::cout << count << std::endl; //verbose mode of is_valid ; shows the number of vertices at each level std::cout << "The number of vertices at successive levels" << std::endl; assert(cdt.is_valid(true)); return 0; }

To be able to adapt to various needs, a highly flexible design has been selected for 2D triangulations. We have already seen that the triangulation classes have two parameters: a geometric traits class and a triangulation data structure class which the user can instantiate with his own customized classes.

The most useful flexibility however comes from the fact that the triangulation data structure itself has two template parameters to be instantiated by classes for the vertices and faces of the triangulation. Using his own customized classes to instantiate these parameters, the user can easily build up a triangulation with additional information or functionality in the vertices and faces.

To insure flexibility, the triangulation data structure is templated by the vertex and face base classes. Also since incidence and adjacency relations are stored in vertices and faces, the base classes have to know the types of handles on vertices and faces provided by the triangulation data structure. Thus the vertex and base classes have to be themselves parameterized by the triangulation data structure, and there is a cyclic dependency on template parameter.

**Figure 36.7: **The cyclic dependency in triangulations software design.

Previously, this cyclic dependency was avoided by
using only *void** pointers in the interface of base classes.
These *void** were converted to appropriate types at the
triangulation data structure levels. This solution had some drawbacks
: mainly the user could not add in the vertices or faces of the
triangulation a functionality related to types defined by
the triangulation data structure, for instance a handle to a vertex,
and he was lead to use himself
*void** pointers).
The new solution to resolve the template dependency
is based on a rebind mechanism similar to the mechanism used in the
standard allocator class std::allocator. The rebind mechanism
is described in Section 37.3
of Chapter 37.
For now, we will just notice that the design
requires
the existence in the vertex and face base classes
of a nested template class
*Rebind_TDS* defining a type *Other* used by
the rebinding mechanism.

The two following examples show how the user can put in use the flexibility offered by the base classes parameters.

The first example corresponds to a case where the user wishes to add in
the vertices or faces of the triangulation an additional information
that does not depend on types provided
by the triangulation data structure.
In that case, predefined classes
*Triangulation_vertex_base_with_info_2<Info,Traits,Vb>*
or *Triangulation_face_base_with_info_2<Info,Traits,Vb>*
can be used. Those classes have
a template parameter *Info* devoted to
handle additional information.
The following examples shows how to add a
*CGAL::Color* in the triangulation faces.

File:examples/Triangulation_2/colored_face.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/IO/Color.h> #include <CGAL/Triangulation_2.h> #include <CGAL/Triangulation_face_base_with_info_2.h> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Triangulation_vertex_base_2<K> Vb; typedef CGAL::Triangulation_face_base_with_info_2<CGAL::Color,K> Fb; typedef CGAL::Triangulation_data_structure_2<Vb,Fb> Tds; typedef CGAL::Triangulation_2<K,Tds> Triangulation; typedef Triangulation::Face_handle Face_handle; typedef Triangulation::Finite_faces_iterator Finite_faces_iterator; typedef Triangulation::Point Point; int main() { Triangulation t; t.insert(Point(0,1)); t.insert(Point(0,0)); t.insert(Point(2,0)); t.insert(Point(2,2)); Finite_faces_iterator fc = t.finite_faces_begin(); for( ; fc != t.finite_faces_end(); ++fc) fc->info() = CGAL::BLUE; Point p(0.5,0.5); Face_handle fh = t.locate(p); fh->info() = CGAL::RED; return 0; }

The second example shows how the user can still derive and plug in his own vertex or face class when he would like to have additional functionalities depending on types provided by the triangulation data structure.

File:examples/Triangulation_2/adding_handles.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Triangulation_2.h> #include <cassert> /* A vertex class with an additionnal handle */ template < class Gt, class Vb = CGAL::Triangulation_vertex_base_2<Gt> > class My_vertex_base : public Vb { typedef Vb Base; public: typedef typename Vb::Vertex_handle Vertex_handle; typedef typename Vb::Face_handle Face_handle; typedef typename Vb::Point Point; template < typename TDS2 > struct Rebind_TDS { typedef typename Vb::template Rebind_TDS<TDS2>::Other Vb2; typedef My_vertex_base<Gt,Vb2> Other; }; private: Vertex_handle va_; public: My_vertex_base() : Base() {} My_vertex_base(const Point & p) : Base(p) {} My_vertex_base(const Point & p, Face_handle f) : Base(f,p) {} My_vertex_base(Face_handle f) : Base(f) {} void set_associated_vertex(Vertex_handle va) { va_ = va;} Vertex_handle get_associated_vertex() {return va_ ; } }; typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef My_vertex_base<K> Vb; typedef CGAL::Triangulation_data_structure_2<Vb> Tds; typedef CGAL::Triangulation_2<K,Tds> Triangulation; typedef Triangulation::Vertex_handle Vertex_handle; typedef Triangulation::Finite_faces_iterator Finite_faces_iterator; typedef Triangulation::Point Point; int main() { Triangulation t; Vertex_handle v0 = t.insert(Point(0,1)); Vertex_handle v1 = t.insert(Point(0,0)); Vertex_handle v2 = t.insert(Point(2,0)); Vertex_handle v3 = t.insert(Point(2,2)); // associate vertices as you like v0->set_associated_vertex(v1); v1->set_associated_vertex(v2); v2->set_associated_vertex(v3); v3->set_associated_vertex(v0); assert( v0->get_associated_vertex() == v1); return 0; }

File:examples/Triangulation_2/info_insert_with_pair_iterator_2.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Delaunay_triangulation_2.h> #include <CGAL/Triangulation_vertex_base_with_info_2.h> #include <vector> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Triangulation_vertex_base_with_info_2<unsigned, K> Vb; typedef CGAL::Triangulation_data_structure_2<Vb> Tds; typedef CGAL::Delaunay_triangulation_2<K, Tds> Delaunay; typedef Delaunay::Point Point; int main() { std::vector< std::pair<Point,unsigned> > points; points.push_back( std::make_pair(Point(0,0),0) ); points.push_back( std::make_pair(Point(1,0),1) ); points.push_back( std::make_pair(Point(0,1),2) ); points.push_back( std::make_pair(Point(14,4),3) ); points.push_back( std::make_pair(Point(2,2),4) ); points.push_back( std::make_pair(Point(-4,0),5) ); Delaunay T; T.insert( 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; }

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

File:examples/Triangulation_2/info_insert_with_zip_iterator_2.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Delaunay_triangulation_2.h> #include <CGAL/Triangulation_vertex_base_with_info_2.h> #include <boost/iterator/zip_iterator.hpp> #include <vector> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Triangulation_vertex_base_with_info_2<unsigned, K> Vb; typedef CGAL::Triangulation_data_structure_2<Vb> Tds; typedef CGAL::Delaunay_triangulation_2<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)); points.push_back(Point(1,0)); points.push_back(Point(0,1)); points.push_back(Point(1,47)); points.push_back(Point(2,2)); points.push_back(Point(-1,0)); Delaunay T; T.insert( 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; }

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:examples/Triangulation_2/info_insert_with_transform_iterator_2.cpp

#include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Delaunay_triangulation_2.h> #include <CGAL/Triangulation_vertex_base_with_info_2.h> #include <boost/iterator/transform_iterator.hpp> #include <vector> typedef CGAL::Exact_predicates_inexact_constructions_kernel K; typedef CGAL::Triangulation_vertex_base_with_info_2<unsigned, K> Vb; typedef CGAL::Triangulation_data_structure_2<Vb> Tds; typedef CGAL::Delaunay_triangulation_2<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)); points.push_back(Point(1,0)); points.push_back(Point(0,1)); points.push_back(Point(4,10)); points.push_back(Point(2,2)); points.push_back(Point(-1,0)); Delaunay T; T.insert( 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 code of this package is the result of a long development process. Here follows a tentative list of people who added their stone to this package : Jean-Daniel Boissonnat, Hervé Brönnimann, Olivier Devillers, Andreas Fabri, Frédéric Fichel, Julia Flötotto, Monique Teillaud and Mariette Yvinec.

Next: Reference Manual

CGAL Open Source Project.
Release 3.9.
26 September 2011.