CGAL 5.2.1 - 2D Triangulation
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#include <CGAL/Constrained_triangulation_2.h>
CGAL::Triangulation_2< Traits, Tds >.
Inherited by CGAL::Constrained_Delaunay_triangulation_2< Traits, Tds, Itag >.
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
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. 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.
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:
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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... | |
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 Iterator_range< Constrained_edges_iterator > | Constrained_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 <, 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. | |
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 Iterator_range< unspecified_type > | All_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_iterator > | All_edges |
range type for iterating over all edges (including infinite ones). | |
typedef Iterator_range< unspecified_type > | All_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_type > | Finite_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_iterator > | Finite_edges |
range type for iterating over finite edges. | |
typedef Iterator_range< unspecified_type > | Finite_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_iterator > | Points |
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 >=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 >=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_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 <, 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. | |
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
.
typedef std::pair<Point,Point> CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::Constraint |
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.
OutputItEdges | is an OutputIterator with Edge as value type. |
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
.
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()
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.
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.
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
.
PointIterator | must be an InputIterator with the value type Point . |
void CGAL::Constrained_triangulation_2< Traits, Tds, Itag >::remove | ( | Vertex_handle | v | ) |
removes a vertex v
.
v
is not incident to a constrained edge.
|
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.