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

CGAL 4.8 - 2D Triangulation

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CGAL::Constrained_triangulation_2< Traits, Tds, Itag > Class Template Reference
Triangulation Classes

`#include <CGAL/Constrained_triangulation_2.h>`

## Inherits from

CGAL::Triangulation_2< Traits, Tds >.

Inherited by CGAL::Constrained_Delaunay_triangulation_2< Traits, Tds, Itag >.

## 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. 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 Constaints 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`.
  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_intersection_tag` disallows intersections of input 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.

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`.

CGAL provides default instantiations for the template parameters `Tds` and `Itag`, and for the `ConstrainedTriangulationFaceBase_2`. If `Gt` is the geometric traits class parameter, the default for `ConstrainedTriangulationFaceBase_2` is the class `Constrained_triangulation_face_base_2<Gt>` and the default for the triangulation data structure parameter is the class `Triangulation_data_structure_2 < Triangulation_vertex_base_2<Gt>, Constrained_triangulation_face_base_2<Gt> >`. The default intersection tag is `No_intersection_tag`.

See Also
  `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.)
ostream & | operator<< (ostream &os, const Constrained_triangulation_2< Traits, Tds > &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.
istream & | operator>> (istream &is, Constrained_triangulation_2< Traits, Tds > Ct &t)
| Reads a triangulation from stream `is` and assigns it to `t`. More...
Related Functions inherited from CGAL::Triangulation_2< Traits, Tds >
ostream & | operator<< (ostream &os, const Triangulation_2< Traits, Tds > &T)
| Inserts the triangulation into the stream `os`. More...
istream & | operator>> (istream &is, const Triangulation_2< Traits, Tds > &T)
| Reads a triangulation from stream `is` and assigns it to the triangulation. More...

## Types
typedef std::pair< Point, Point > | Constraint
typedef unspecified_type | Constrained_edges_iterator
| A bidirectional iterator to visit all the edges `e` of the triangulation which are constrained. More...
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.

## 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)
| Make the edges incident to vertex `v` unconstrained edges.
void | remove_constrained_edge (Face_handle f, int i)
| Make 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.

## Additional Inherited Members
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 the finite vertices of the triangulation.
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...
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_traits & | geom_traits () const
| Returns a const reference to the triangulation traits object.
const TriangulationDataStructure_2 & | tds () const
| Returns a const reference to the triangulation data structure.
TriangulationDataStructure_2 & | tds ()
| 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 InputIterator >
std::ptrdiff_t | insert (InputIterator first, InputIterator last)
| Inserts the points in the range `[first,last)`. 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)
| same as above if there is 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.
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.
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.

## Member Typedef Documentation

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`.

template<typename Traits , typename Tds , typename Itag >

typedef std::pair<Point,Point> CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::Constraint

Deprecated:
  The type of constraints.

## Member Function Documentation

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.

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`.

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)`.

See Also
  `Triangulation_2::locate()`

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.

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.

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`.

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.

## Friends And Related Function Documentation

template<typename Traits , typename Tds , typename Itag >

istream & operator>> | ( | istream & | is,
|  | Constrained_triangulation_2< Traits, Tds > Ct & | t
| ) |  |
| related

Reads a triangulation from stream `is` and assigns it to `t`.

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