# Chapter 31
2D Triangulations

Mariette Yvinec

This chapter describes the two dimensional triangulations of CGAL. Section recalls the main definitions about triangulations. Sections discusses the way two-dimensional triangulations are represented in CGAL . Section 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 ), Delaunay triangulations (Section ), regular triangulations (Section ), constrained triangulations (Section ), and constrained Delaunay triangulations (Section ). Section describes a class which implements a constrained or constrained Delaunay triangulation with an additionnal data structure to describe how the constraints are refined by the edges of the triangulations. Section describes a hierarchical data structure for fast point location queries. At last, Section explains how the user can benefit from the flexibility of CGAL triangulations using customized classes for faces and vertices.

## 31.1 Definitions

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 UT 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 subface.

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 subface. 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 UT 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 UT 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 stucture 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 tiangulation 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 , and ).

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.

## 31.2 Representation

#### The set of faces

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: Infinite vertex and infinite faces

#### A representation based on faces and vertices

Because a triangulation is a set of triangular faces with constant-size complexity, triangulations are not implemented as a layer on top of a planar map. CGAL uses a proper internal representation of triangulations based on faces and vertices rather than on edges. Such a representation saves storage space and results in faster algorithms [BDTY00].

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 neighbor 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 , 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 implicitely 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: Vertices and neighbors.

## 31.3 Software Design

The triangulations classes of CGAL provide high-level geometric functionalities such as location of a point in the triangulation, insertion or removal 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 of Chapter . The triangulation data structure defines the types used to represent the faces and vertices of the triangulation, as well as additionnal 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 base class and a face base class from which the vertex and face classes are respectively derived. 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 additionnal properties attached to the vertices or faces of a triangulation. See section for more details on the way to make use of this flexibility.

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

Figure: 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 summarizes the derivation dependancies 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 classes, 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: The derivation tree of 2D triangulations.

## 31.4 Basic Triangulations

### 31.4.1 Description

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 wether 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 circimcircles have the counterclockwise orientation.

The triangulation can be modified by several functions : insertion of a point, removal 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 body (see Figure ).

Figure: Flip.

#### Implementation

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(sqrt(n)) on average if the vertices are distributed uniformly at random. The class Triangulation_hierarchy_2<Traits,Tds>, described in section , 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 retriangulating the hole. Removal takes a time at most proportionnal to d^2, where d is the degree of the removed vertex, which is O(1) for a random vertex.

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.

#### Geometric Traits

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 Triangulation_euclidean_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 usefull 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 for an example. CGAL provides also the geometric traits classes Triangulation_euclidean_traits_yz_3<R> and Triangulation_euclidean_traits_zx_3<R> to deal with projections on the xz plane and yz-plane, respectively.

### 31.4.2 Example of a Basic Triangulation

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.C

    #include <fstream>

    #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
    #include <CGAL/Triangulation_2.h>

    struct K : CGAL::Exact_predicates_inexact_constructions_kernel {};

    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;
    }

## 31.5 Delaunay Triangulations

### 31.5.1 Description

The class Delaunay_triangulation_2<Traits,Tds> is designed to represent the Delaunay triangulation of a set of data points in the plane. A Delaunay triangulation fulfills the following empty circle property (also called Delaunay property): the circumscribing circle of any facet of the triangulation contains no data point in its interior. For a point set with no subset of four cocircular points the Delaunay triangulation is unique, it is dual to the Voronoi diagram of the set of points.

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>. Additionnal 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 a new point in the triangulation or remove a vertex from it 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.

### Geometric traits

The geometric traits has to be a model of the concept DelaunayTriangulationTraits_2 which refines the concept TriangulationTraits_2. In particular this concept provides the side_of_oriented_circle predicate which, given four points p,q,r,s decides the position of the point s with respect to the circle passing through p, q and r. The side_of_oriented_circle predicate actually defines the Delaunay triangulation. Changing this predicate allows to build variant of Delaunay triangulations for different metrics such that L1 or L metric or any metric defined by a convex object. However, the user of an exotic metric must be careful that the constructed triangulation has to be a triangulation of the convex hull which means that convex hull edges have to be Delaunay edges. This is granted for any smooth convex metric (like L2) and can be ensured for other metrics (like L ) by the addition to the point set of well chosen sentinel points.

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, Triangulation_euclidean_traits_xy_3<R>,
Triangulation_euclidean_traits_yz_3<R>, and
Triangulation_euclidean_traits_zx_3<R>
are also models of DelaunayTriangulationTraits_2 excapt that they do not fulfills the requirements for the duality functions and nearest vertex queries.

### Implementation

The insertion of a new point in the Delaunay triangulation is performed using first the insertion member function of the basic triangulation and second performing a sequence of flips to restore the Delaunay property. The number of flips that have to be performed is O(d) if the new vertex has degree d in the updated Delaunay triangulation. For points distributed uniformly at random, each insertion takes time O(1) on average, once the point has been located in the triangulation.

Removal calls the removal in the triangulation and then retriangulates 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.

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.

### 31.5.2 Example : a Delaunay Terrain

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.

    // file: examples/Triangulation_2/terrain.C

    #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
    #include <CGAL/Triangulation_euclidean_traits_xy_3.h>
    #include <CGAL/Delaunay_triangulation_2.h>

    #include <fstream>

    struct K : CGAL::Exact_predicates_inexact_constructions_kernel {};

    typedef CGAL::Triangulation_euclidean_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;
    }

### 31.5.3 Example : Voronoi Diagram

The following code computes the edges of Voronoi diagram of a set of data points and counts the number of finite edges and the number of rays of this diagram

    // file: examples/Triangulation_2/Voronoi.C

    #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
    #include <CGAL/Delaunay_triangulation_2.h>

    #include <fstream>

    struct K : CGAL::Exact_predicates_inexact_constructions_kernel {};

    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);
        K::Segment_2 s;
        K::Ray_2     r;
        if (CGAL::assign(s,o)) {++ns;}
        if (CGAL::assign(r,o)) {++nr;}
      }
      std::cout << "The voronoi diagram as " << ns << "finite edges "
    	    << " and " << nr << " rays" << std::endl;
      return 0;
    }

## 31.6 Regular Triangulations

### 31.6.1 Description

Let PW = {(pi, wi), i = 1, ..., n } be a set of weighted points where each pi is a point and each wi is a scalar called the weight of point pi. Alternatively, each weighted point (pi, wi) can be regarded as a sphere (or a circle, depending on the dimensionality of pi) with center pi and radius ri=sqrt(wi).

The power diagram of the set PW is a space partition in which each cell corresponds to a sphere (pi, wi) of PW and is the locus of points p whose power with respect to (pi, wi) 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= { pi, i = 1, ..., n } of center points and whose vertices form a subset of P. Such a triangulation is called a regular triangulation. Three points pi, pj and pk 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 (pi, wi), (pj, wj) and (pk, wk) 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 (pi, wi) and (pj, wj) as:

(pi, wi,pj, wj) = pipj 2 - wi - wj .

(pi, wi,pj, 0) is simply the power of point pj with respect to the sphere (pi, wi), and two weighted points are said to be orthogonal if their power product is null. The power circle of three weighted points (pi, wi), (pj, wj) and (pk, wk) is defined as the unique circle (, ) orthogonal to (pi, wi), (pj, wj) and (pk, wk).

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 pipjpk is a face of the regular triangulation of PW iff the power product of any weighted point (pl, wl) of PW with the power circle of (pi, wi), (pj, wj) and (pk, wk) is positive or null. We call power test of (pi, wi), (pj, wj), (pk, wk), and (pl, wl), the predicates which amount to compute the sign of the power product of (pl, wl) with respect to the power circle of (pi, wi), (pj, wj) and (pk, wk). This predicate amounts to computing the sign of the following determinant

|

1 | xi | yi | xi 2 + yi 2 - wi
1 | xj | yj | xj 2 + yj 2 - wj
1 | xk | yk | xk 2 + yk 2 - wk
1 | xl | yl | xl 2 + yl 2 - wl

|

A pair of neighboring faces pipjpk and pipjpl is said to be locally regular (with respect to the weights in PW) if the power test of (pi, wi), (pj, wj), (pk, wk), and (pl, wl) 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'= { (pi,pi 2 - wi ), 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. The vertices of the regular triangulation of a set of weighted points PW correspond only to a subset of PW. Some of the input weigthed 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.

#### The geometric traits

The geometric traits of a regular triangulation must provide a weighted point type and a power test on these weighted points. The concept RegularTriangulationTraits_2, is a refinement of the concept TriangulationTraits_2. CGAL provides the class Regular_triangulation_euclidean_traits_2<Rep,Weight> which is a model for the traits concept RegularTriangulationTraits_2. The class Regular_triangulation_euclidean_traits_2<Rep,Weight> derives from the class Triangulation_euclidean_traits_2<Rep> and uses a Weighted_point type derived from the type Point_2 of Triangulation_euclidean_traits_2<Rep>. There is also a class Regular_triangulation_filtered_traits_2<FK> which provides filtered predicates (efficient and exact).

#### The Vertex type and Face Type of a Regular Triangulation

The base vertex type of a regular triangulation includes a boolean 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.

### 31.6.2 Example : a Regular Triangulation

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

    // file: examples/Triangulation_2/regular.C

    #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
    #include <CGAL/Regular_triangulation_euclidean_traits_2.h>
    #include <CGAL/Regular_triangulation_2.h>

    #include <fstream>

    struct K : CGAL::Exact_predicates_inexact_constructions_kernel {};

    typedef double W;
    typedef CGAL::Regular_triangulation_euclidean_traits_2<K,W>  Gt;
    typedef CGAL::Regular_triangulation_2<Gt> 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;
    }

## 31.7 Constrained Triangulations

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 approximative.
  - 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 .

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 approximative 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 additionnal 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.

#### The Geometric Traits

The geometric traits of a constraint triangulation has to be a model of the concept TriangulationTraits_2. When intersections of input constraints are supported, the geometric traits class has to be a model of the concept ConstrainedTriangulationTraits_2 which refines the concept TriangulationTraits_2 providing additional function object types to compute the intersection of two segments.

#### The Base Face of a Constrained Triangulation

The information about constrained edges is stored in the faces of the triangulation. The base face of a Constrained Triangulation has to be a model for the concept ConstrainedTriangulationFaceBase_2 which refines the concept TriangulationFaceBase_2. The concept ConstrainedTriangulationFaceBase_2 requires member functions the get and set the constrained status of the edges.

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 booleans to store the status of its edges.

Figure: 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.

## 31.8 Constrained Delaunay Triangulations

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 approximative 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 .

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.

#### The Geometric Traits

The geometric traits of a constrained Delaunay triangulation is required to provide the side_of_oriented_circle predicate as the geometric traits of a Delaunay triangulation and has to a model of the concept DelaunayTriangulationTraits_2. When intersecting input constraints are supported the geometric traits is further required to provide function objects to compute constraints intersections. Then the geometric traits has to be at the same time a model of the concept ConstrainedTriangulationTraits_2.

#### The face base class

Information about the status (constrained or not) of the edges of the triangulation has to be stored in the face class and the base face class of a constrained Delaunay triangulation has to be a model of ConstrainedTriangulationFaceBase_2.

### 31.8.1 Example : a constrained Delaunay triangulation

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.C

    #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
    #include <CGAL/Constrained_Delaunay_triangulation_2.h>

    struct K : CGAL::Exact_predicates_inexact_constructions_kernel {};

    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;
    }

## 31.9 Constrained Triangulations Plus

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 subconstraints 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.

### 31.9.1 Example : Building a triangulated arrangement of segments

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.C

    #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>

    struct K : CGAL::Exact_predicates_exact_constructions_kernel {};

    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;
    }

## 31.10 The Triangulation Hierarchy

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 succedding level, the data structure stores a triangulation of a small random sample of the vertices of the triangulation at the preceeding 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 preceeding level. Because the number of vertices in each triangulation is only a small fraction of the number of vertices of the preceeding triangulation, the data structure remains small and achieves fast point location queries on real data. As proved in [Dev98], this structure has an optimal behaviour 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 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.

#### The Vertex of a Triangulation Hierarchy

The base vertex class of a triangulation hierarchy has to be a model of the concept TriangulationHierarchyVertexBase_2 which extends the concept TriangulationVertexBase_2. This extension adds access and setting member functions for two pointers to the corresponding vertices in the triangulations of the next and preceeding levels.

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 base class or a user customized vertex base with additionnal functionalities.

### 31.10.1 Examples of the Use of a Triangulation Hierarchy

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

    // file: examples/Triangulation_2/hierarchy.C

    #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>

    struct K : CGAL::Exact_predicates_inexact_constructions_kernel {};

    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 hierachy in conjonction with a constrained triangulation plus.

    // file: examples/Triangulation_2/constrained_hierarchy_plus.C

    #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>

    struct K : CGAL::Exact_predicates_inexact_constructions_kernel {};

    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;
    }

## 31.11 Flexibility: Using Customized Vertices and Faces

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 usefull flexibility however comes from the fact that the triangulation data structure itself has two template parameters to be instantiated by base classes for the vertices and faces of the triangulation. The vertex and face classes of the triangulation are derived from those base classes. Thus, using his own customized classes to instantiate these parameters, the user can easily build up a triangulation with additionnal informations or functionalities in the vertices and faces.

A cyclic dependancy 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 thenselves parameterized by the triangulation data structure, and there is a cyclic dependancy on template parameter.

Figure: 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 dependancy is based on a rebind mechanism similar to the mecanism used in the standard allocator class std::allocator. The rebind mecanism is described in Section of Chapter . 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 mecanism.

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

Adding colors 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 do 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 additionnal information. The following examples shows how to add a CGAL::Color in the triangulation faces.

    // file : examples/Triangulation_2/colored_face.C

    #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>

    struct K : CGAL::Exact_predicates_inexact_constructions_kernel {};

    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;
    }

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

    // file : examples/Triangulation_2/adding_handles.C

    #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
    #include <CGAL/Triangulation_2.h>
    #include <cassert>

    /* A facet 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_ ; }
    };

    struct K : CGAL::Exact_predicates_inexact_constructions_kernel {};

    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;
    }

## 31.12 Design and Implementation History

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.