 CGAL 5.4 - 2D Triangulations
CGAL::Constrained_triangulation_2< Traits, Tds, Itag > Class Template Reference

#include <CGAL/Constrained_triangulation_2.h>

## Definition

A constrained triangulation is a triangulation of a set of points which has to include among its edges a given set of polylines joining the points.

The given polylines are called constraints and the corresponding edges in the triangulation are called constrained edges.

The endpoints of constrained edges are of course vertices of the triangulation. However the triangulation may 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 polylines that do not intersect except possibly at their endpoints. An exception of type Intersection_of_constraints_exception is thrown upon insertion of a constraint intersecting in its interior an already inserted constraint. Any number of constrained edges may share the same endpoint. Constrained edges may be vertical or have zero length.
• The two other versions support intersecting input constraints. In those versions, input constraints may consist of intersecting, overlapping or partially overlapping segments. The triangulation introduces additional vertices at each point which is a proper intersection point of two constraints. A single constraint intersecting other constraints will then appear as the union of several constrained edges of the triangulation. There are two ways to deal with intersecting constraints.
• The first one is robust when predicates are evaluated exactly but constructions (i. e. intersection computations) are approximate.
• The second one should be used with exact arithmetic (meaning exact evaluation of predicates and exact computation of intersections.)

In order to retrieve the constrained edges of a constraint, or the constraints overlapping with a constrained edge, we provide the class Constrained_triangulation_plus_2. This class maintains a constraint hierarchy data structure. See Section Constrained Triangulations with a Bidirectional Mapping between Constraints and Subconstraints for details. This class should also be used when doing exact intersection computations as it avoids the cascading of intersection computations. Template Parameters
 Traits is a geometric traits class and must be a model of the concept TriangulationTraits_2. When intersection of input constraints are supported, the geometric traits class is required to provide additional function object types to compute the intersection of two segments. It has then to be a model of the concept ConstrainedTriangulationTraits_2. Tds must be a model of the concept TriangulationDataStructure_2 or Default. The information about constrained edges is stored in the faces of the triangulation. Thus the nested Face type of a constrained triangulation offers additional functionalities to deal with this information. These additional functionalities induce additional requirements on the face base class plugged into the triangulation data structure of a constrained Delaunay triangulation. The face base of a constrained Delaunay triangulation has to be a model of the concept ConstrainedTriangulationFaceBase_2. Itag is the intersection tag which serves to choose between the different strategies to deal with constraints intersections. CGAL provides three valid types for this parameter: No_constraint_intersection_tag disallows intersections of input constraints except for the case of a single common extremity; No_constraint_intersection_requiring_constructions_tag (default value) disallows intersections of input constraints except for configurations where the intersection can be represented without requiring the construction of a new point such as overlapping constraints; Exact_predicates_tag is to be used when the traits class provides exact predicates but approximate constructions of the intersection points; Exact_intersections_tag is to be used in conjunction with an exact arithmetic type.

CGAL provides default instantiations for the template parameters Tds and Itag. If Gt is the geometric traits class parameter, the default triangulation data structure is the class is the class Triangulation_data_structure_2<Triangulation_vertex_base_2<Gt>, Constrained_triangulation_face_base_2<Gt> >. The default intersection tag is No_constraint_intersection_requiring_constructions_tag.

CGAL::Triangulation_2<Traits,Tds>
TriangulationDataStructure_2
TriangulationTraits_2
ConstrainedTriangulationTraits_2
ConstrainedTriangulationFaceBase_2

Implementation

The insertion of a constrained edge runs in time proportional to the number of triangles intersected by this edge.

## Related Functions

(Note that these are not member functions.)

template<typename Traits , typename Tds , typename Itag >
std::ostream & operator<< (std::ostream &os, const Constrained_triangulation_2< Traits, Tds, Itag > &ct)
writes the triangulation as for Triangulation_2<Traits,Tds> and, for each face f, and integers i=0,1,2, writes "C" or "N" depending whether edge (f,i) is constrained or not.

template<typename Traits , typename Tds , typename Itag >
std::istream & operator>> (std::istream &is, Constrained_triangulation_2< Traits, Tds, Itag > Ct &ct)
reads a triangulation from stream is and assigns it to ct. More...

## Types

typedef std::pair< Point, PointConstraint

typedef unspecified_type Constrained_edges_iterator
A bidirectional iterator to visit all the edges e of the triangulation which are constrained. More...

typedef Iterator_range< Constrained_edges_iteratorConstrained_edges
A range type to iterate over the constrained edges.

typedef Itag Intersection_tag
The intersection tag which decides how intersections between input constraints are dealt with.

## Creation

Constrained_triangulation_2 ()
Default constructor.

Constrained_triangulation_2 (const Constrained_triangulation_2 &ct1)
Copy constructor: All faces and vertices are duplicated and the constrained status of edges is copied.

## Queries

bool is_constrained (Edge e) const
returns true if edge e is a constrained edge.

bool are_there_incident_constraints (Vertex_handle v) const
returns true if at least one of the edges incident to vertex v is constrained.

template<class OutputItEdges >
OutputItEdges incident_constraints (Vertex_handle v, OutputItEdges out) const
outputs the constrained edges incident to v into the output iterator out and returns the resulting output iterator. More...

Constrained_edges_iterator constrained_edges_begin () const
returns an iterator that enumerates the constrained edges.

Constrained_edges_iterator constrained_edges_end () const
returns the past-the-end iterator.

Constrained_edges constrained_edges () const
returns a range of constrained edges.

## Insertion and Removal

Vertex_handle insert (Point p, Face_handle f=Face_handle())
inserts point p and restores the status (constrained or not) of all the touched edges. More...

Vertex_handle insert (const Point &p, Locate_type &lt, Face_handle loc, int li)
inserts point p in the triangulation at the location given by (lt,loc,i). More...

Vertex_handle push_back (const Point &p)
Equivalent to insert(p).

void insert_constraint (Point a, Point b)
inserts points a and b in this order, and inserts the line segment ab as a constraint. More...

void push_back (const std::pair< Point, Point > &c)
Equivalent to insert(c.first, c.second).

void insert_constraint (const Vertex_handle &va, const Vertex_handle &vb)
inserts the line segment s whose endpoints are the vertices va and vb as a constraint. More...

template<class PointIterator >
void insert_constraint (PointIterator first, PointIterator last, bool close=false)
inserts a polyline defined by the points in the range [first,last). More...

void remove (Vertex_handle v)
removes a vertex v. More...

void remove_incident_constraints (Vertex_handle v)
makes the edges incident to vertex v unconstrained edges.

void remove_constrained_edge (Face_handle f, int i)
makes edge (f,i) unconstrained.

bool is_valid (bool verbose=false, int level=0) const
checks the validity of the triangulation and the consistency of the constrained marks in edges. Public Types inherited from CGAL::Triangulation_2< Traits, Tds >
enum  Locate_type {
VERTEX =0, EDGE, FACE, OUTSIDE_CONVEX_HULL,
OUTSIDE_AFFINE_HULL
}
specifies which case occurs when locating a point in the triangulation. More...

typedef Tds::Vertex_handle Vertex_handle
handle to a vertex.

typedef Tds::Face_handle Face_handle
handle to a face.

typedef Tds::Face_iterator All_faces_iterator
iterator over all faces.

typedef Tds::Edge_iterator All_edges_iterator
iterator over all edges.

typedef Tds::Vertex_iterator All_vertices_iterator
iterator over all vertices.

typedef unspecified_type Finite_faces_iterator
iterator over finite faces.

typedef unspecified_type Finite_edges_iterator
iterator over finite edges.

typedef unspecified_type Finite_vertices_iterator
iterator over finite vertices.

typedef unspecified_type Point_iterator
iterator over the points corresponding to the finite vertices of the triangulation.

typedef Iterator_range< unspecified_typeAll_face_handles
range type for iterating over all faces (including infinite faces), with a nested type iterator that has as value type Face_handle.

typedef Iterator_range< All_edges_iteratorAll_edges
range type for iterating over all edges (including infinite ones).

typedef Iterator_range< unspecified_typeAll_vertex_handles
range type for iterating over all vertices (including the infinite vertex), with a nested type iterator that has as value type Vertex_handle.

typedef Iterator_range< unspecified_typeFinite_face_handles
range type for iterating over finite faces, with a nested type iterator that has as value type Face_handle.

typedef Iterator_range< Finite_edges_iteratorFinite_edges
range type for iterating over finite edges.

typedef Iterator_range< unspecified_typeFinite_vertex_handles
range type for iterating over finite vertices, with a nested type iterator that has as value type Vertex_handle.

typedef Iterator_range< Point_iteratorPoints
range type for iterating over the points of the finite vertices.

typedef unspecified_type Line_face_circulator
circulator over all faces intersected by a line.

typedef unspecified_type Face_circulator
circulator over all faces incident to a given vertex.

typedef unspecified_type Edge_circulator
circulator over all edges incident to a given vertex.

typedef unspecified_type Vertex_circulator
circulator over all vertices incident to a given vertex.

typedef Traits Geom_traits
the traits class.

typedef Tds Triangulation_data_structure
the triangulation data structure type.

typedef Traits::Point_2 Point
the point type.

typedef Traits::Segment_2 Segment
the segment type.

typedef Traits::Triangle_2 Triangle
the triangle type.

typedef Tds::Vertex Vertex
the vertex type.

typedef Tds::Face Face
the face type.

typedef Tds::Edge Edge
the edge type.

typedef Tds::size_type size_type
Size type (an unsigned integral type).

typedef Tds::difference_type difference_type
Difference type (a signed integral type). Public Member Functions inherited from CGAL::Triangulation_2< Traits, Tds >
Triangulation_2 (const Traits &gt=Traits())
Introduces an empty triangulation.

Triangulation_2 (const Triangulation_2 &tr)
Copy constructor. More...

template<class InputIterator >
Triangulation_2 (InputIterator first, InputIterator last, const Traits &gt=Traits())
Equivalent to constructing an empty triangulation with the optional traits class argument and calling insert(first,last).

Triangulation_2 operator= (const Triangulation_2< Traits, Tds > &tr)
Assignment. More...

void swap (Triangulation_2 &tr)
The triangulations tr and *this are swapped. More...

void clear ()
Deletes all faces and finite vertices resulting in an empty triangulation.

int dimension () const
Returns the dimension of the convex hull.

size_type number_of_vertices () const
Returns the number of finite vertices.

size_type number_of_faces () const
Returns the number of finite faces.

Face_handle infinite_face () const
a face incident to the infinite vertex.

Vertex_handle infinite_vertex () const
the infinite vertex.

Vertex_handle finite_vertex () const
a vertex distinct from the infinite vertex.

const Geom_traitsgeom_traits () const
Returns a const reference to the triangulation traits object.

const TriangulationDataStructure_2tds () const
Returns a const reference to the triangulation data structure.

TriangulationDataStructure_2tds ()
Returns a reference to the triangulation data structure.

bool is_infinite (Vertex_handle v) const
true iff v is the infinite vertex.

bool is_infinite (Face_handle f) const
true iff face f is infinite.

bool is_infinite (Face_handle f, int i) const
true iff edge (f,i) is infinite.

bool is_infinite (Edge e) const
true iff edge e is infinite.

bool is_infinite (Edge_circulator ec) const
true iff edge *ec is infinite.

bool is_infinite (All_edges_iterator ei) const
true iff edge *ei is infinite.

bool is_edge (Vertex_handle va, Vertex_handle vb)
true if there is an edge having va and vb as vertices.

bool is_edge (Vertex_handle va, Vertex_handle vb, Face_handle &fr, int &i)
as above. More...

bool includes_edge (Vertex_handle va, Vertex_handle vb, Vertex_handle &vbr, Face_handle &fr, int &i)
true if the line segment from va to vb includes an edge e incident to va. More...

bool is_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3)
true if there is a face having v1, v2 and v3 as vertices.

bool is_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3, Face_handle &fr)
as above. More...

Face_handle locate (const Point &query, Face_handle f=Face_handle()) const
If the point query lies inside the convex hull of the points, a face that contains the query in its interior or on its boundary is returned. More...

Face_handle inexact_locate (const Point &query, Face_handle start=Face_handle()) const
Same as locate() but uses inexact predicates. More...

Face_handle locate (const Point &query, Locate_type &lt, int &li, Face_handle h=Face_handle()) const
Same as above. More...

Oriented_side oriented_side (Face_handle f, const Point &p) const
Returns on which side of the oriented boundary of f lies the point p. More...

Oriented_side side_of_oriented_circle (Face_handle f, const Point &p)
Returns on which side of the circumcircle of face f lies the point p. More...

void flip (Face_handle f, int i)
Exchanges the edge incident to f and f->neighbor(i) with the other diagonal of the quadrilateral formed by f and f->neighbor(i). More...

Vertex_handle insert (const Point &p, Face_handle f=Face_handle())
Inserts point p in the triangulation and returns the corresponding vertex. More...

Vertex_handle insert (const Point &p, Locate_type lt, Face_handle loc, int li)
Same as above except that the location of the point p to be inserted is assumed to be given by (lt,loc,i) (see the description of the locate method above.)

Vertex_handle push_back (const Point &p)
Equivalent to insert(p).

template<class PointInputIterator >
std::ptrdiff_t insert (PointInputIterator first, PointInputIterator last)
Inserts the points in the range [first,last) in the given order, and returns the number of inserted points. More...

template<class PointWithInfoInputIterator >
std::ptrdiff_t insert (PointWithInfoInputIterator first, PointWithInfoInputIterator last)
inserts the points in the iterator range [first,last) in the given order, and returns the number of inserted points. More...

void remove (Vertex_handle v)
Removes the vertex from the triangulation. More...

Vertex_handle move_if_no_collision (Vertex_handle v, const Point &p)
If there is not already another vertex placed on p, the triangulation is modified such that the new position of vertex v is p, and v is returned. More...

Vertex_handle move (Vertex_handle v, const Point &p)
If there is no collision during the move, this function is the same as move_if_no_collision . More...

Vertex_handle insert_first (const Point &p)
Inserts the first finite vertex .

Vertex_handle insert_second (const Point &p)
Inserts the second finite vertex .

Vertex_handle insert_in_face (const Point &p, Face_handle f)
Inserts vertex v in face f. More...

Vertex_handle insert_in_edge (const Point &p, Face_handle f, int i)
Inserts vertex v in edge i of f. More...

Vertex_handle insert_outside_convex_hull (const Point &p, Face_handle f)
Inserts a point which is outside the convex hull but in the affine hull. More...

Vertex_handle insert_outside_affine_hull (const Point &p)
Inserts a point which is outside the affine hull.

void remove_degree_3 (Vertex_handle v)
Removes a vertex of degree three. More...

void remove_second (Vertex_handle v)
Removes the before last finite vertex.

void remove_first (Vertex_handle v)
Removes the last finite vertex.

template<class EdgeIt >
Vertex_handle star_hole (Point p, EdgeIt edge_begin, EdgeIt edge_end)
creates a new vertex v and use it to star the hole whose boundary is described by the sequence of edges [edge_begin, edge_end). More...

template<class EdgeIt , class FaceIt >
Vertex_handle star_hole (Point p, EdgeIt edge_begin, EdgeIt edge_end, FaceIt face_begin, FaceIt face_end)
same as above, except that the algorithm first recycles faces in the sequence [face_begin, face_end) and create new ones only when the sequence is exhausted. More...

Finite_vertices_iterator finite_vertices_begin () const
Starts at an arbitrary finite vertex.

Finite_vertices_iterator finite_vertices_end () const
Past-the-end iterator.

Finite_edges_iterator finite_edges_begin () const
Starts at an arbitrary finite edge.

Finite_edges_iterator finite_edges_end () const
Past-the-end iterator.

Finite_faces_iterator finite_faces_begin () const
Starts at an arbitrary finite face.

Finite_faces_iterator finite_faces_end () const
Past-the-end iterator.

Point_iterator points_begin () const

Point_iterator points_end () const
Past-the-end iterator.

Finite_vertex_handles finite_vertex_handles () const
returns a range of iterators over finite vertices. More...

Finite_edges finite_edges () const
returns a range of iterators over finite edges.

Finite_face_handles finite_face_handles () const
returns a range of iterators over finite faces. More...

Points points () const
returns a range of iterators over the points of finite vertices.

All_vertices_iterator all_vertices_begin () const
Starts at an arbitrary vertex.

All_vertices_iterator all_vertices_end () const
Past-the-end iterator.

All_edges_iterator all_edges_begin () const
Starts at an arbitrary edge.

All_edges_iterator all_edges_end () const
Past-the-end iterator.

All_faces_iterator all_faces_begin () const
Starts at an arbitrary face.

All_faces_iterator all_faces_end () const
Past-the-end iterator.

All_vertex_handles all_vertex_handles () const
returns a range of iterators over all vertices. More...

All_edges all_edges () const
returns a range of iterators over all edges.

All_face_handles all_face_handles () const
returns a range of iterators over all faces. More...

Line_face_circulator line_walk (const Point &p, const Point &q, Face_handle f=Face_handle()) const
This function returns a circulator that allows to visit the faces intersected by the line pq. More...

Face_circulator incident_faces (Vertex_handle v) const
Starts at an arbitrary face incident to v.

Face_circulator incident_faces (Vertex_handle v, Face_handle f) const
Starts at face f. More...

Edge_circulator incident_edges (Vertex_handle v) const
Starts at an arbitrary edge incident to v.

Edge_circulator incident_edges (Vertex_handle v, Face_handle f) const
Starts at the first edge of f incident to v, in counterclockwise order around v. More...

Vertex_circulator incident_vertices (Vertex_handle v) const
Starts at an arbitrary vertex incident to v.

Vertex_circulator incident_vertices (Vertex_handle v, Face_handle f)
Starts at the first vertex of f adjacent to v in counterclockwise order around v. More...

Vertex_handle mirror_vertex (Face_handle f, int i) const
returns the vertex of the $$i^{th}$$ neighbor of f that is opposite to f. More...

int mirror_index (Face_handle f, int i) const
returns the index of f in its $$i^{th}$$ neighbor. More...

Edge mirror_edge (Edge e) const
returns the same edge seen from the other adjacent face. More...

int ccw (int i) const
Returns $$i+1$$ modulo 3. More...

int cw (int i) const
Returns $$i+2$$ modulo 3. More...

Triangle triangle (Face_handle f) const
Returns the triangle formed by the three vertices of f. More...

Segment segment (Face_handle f, int i) const
Returns the line segment formed by the vertices ccw(i) and cw(i) of face f. More...

Segment segment (const Edge &e) const
Returns the line segment corresponding to edge e. More...

Segment segment (const Edge_circulator &ec) const
Returns the line segment corresponding to edge *ec. More...

Segment segment (const Edge_iterator &ei) const
Returns the line segment corresponding to edge *ei. More...

Point circumcenter (Face_handle f) const
Compute the circumcenter of the face pointed to by f. More...

void set_infinite_vertex (const Vertex_handle &v)
This is an advanced function. More...

bool is_valid (bool verbose=false, int level=0) const
Checks the combinatorial validity of the triangulation and also the validity of its geometric embedding. More... Public Member Functions inherited from CGAL::Triangulation_cw_ccw_2
Triangulation_cw_ccw_2 ()
default constructor.

int ccw (const int i) const
returns the index of the neighbor or vertex that is next to the neighbor or vertex with index i in counterclockwise order around a face.

int cw (const int i) const
returns the index of the neighbor or vertex that is next to the neighbor or vertex with index i in counterclockwise order around a face.

## ◆ Constrained_edges_iterator

template<typename Traits , typename Tds , typename Itag >
 typedef unspecified_type CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::Constrained_edges_iterator

A bidirectional iterator to visit all the edges e of the triangulation which are constrained.

The order of visit is undefined. The value type of this iterator is Edge.

## ◆ Constraint

template<typename Traits , typename Tds , typename Itag >
 typedef std::pair CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::Constraint
Deprecated:
The type of constraints.

## ◆ incident_constraints()

template<typename Traits , typename Tds , typename Itag >
template<class OutputItEdges >
 OutputItEdges CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::incident_constraints ( Vertex_handle v, OutputItEdges out ) const

outputs the constrained edges incident to v into the output iterator out and returns the resulting output iterator.

Template Parameters
 OutputItEdges is an OutputIterator with Edge as value type.

## ◆ insert() [1/2]

template<typename Traits , typename Tds , typename Itag >
 Vertex_handle CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::insert ( Point p, Face_handle f = Face_handle() )

inserts point p and restores the status (constrained or not) of all the touched edges.

If present, f is used as an hint for the location of p.

## ◆ insert() [2/2]

template<typename Traits , typename Tds , typename Itag >
 Vertex_handle CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::insert ( const Point & p, Locate_type & lt, Face_handle loc, int li )

inserts point p in the triangulation at the location given by (lt,loc,i).

Triangulation_2::locate()

## ◆ insert_constraint() [1/3]

template<typename Traits , typename Tds , typename Itag >
 void CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::insert_constraint ( Point a, Point b )

inserts points a and b in this order, and inserts the line segment ab as a constraint.

Removes the faces crossed by segment ab and creates new faces instead. If a vertex c lies on segment ab, constraint ab is replaced by the two constraints ac and cb. Apart from the insertion of a and b, the algorithm runs in time proportional to the number of removed triangles.

## ◆ insert_constraint() [2/3]

template<typename Traits , typename Tds , typename Itag >
 void CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::insert_constraint ( const Vertex_handle & va, const Vertex_handle & vb )

inserts the line segment s whose endpoints are the vertices va and vb as a constraint.

The triangles intersected by s are removed and new ones are created.

## ◆ insert_constraint() [3/3]

template<typename Traits , typename Tds , typename Itag >
template<class PointIterator >
 void CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::insert_constraint ( PointIterator first, PointIterator last, bool close = false )

inserts a polyline defined by the points in the range [first,last).

The polyline is considered as a polygon if the first and last point are equal or if close = true. This enables for example passing the vertex range of a Polygon_2.

Template Parameters
 PointIterator must be an InputIterator with the value type Point.

## ◆ remove()

template<typename Traits , typename Tds , typename Itag >
 void CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::remove ( Vertex_handle v )

removes a vertex v.

Precondition
Vertex v is not incident to a constrained edge.

## ◆ operator>>()

template<typename Traits , typename Tds , typename Itag >
 std::istream & operator>> ( std::istream & is, Constrained_triangulation_2< Traits, Tds, Itag > Ct & ct )
related

reads a triangulation from stream is and assigns it to ct.

Data in the stream must have the same format operator<< uses. Note that ct is first cleared.