CGAL 4.12  2D Triangulation

#include <CGAL/Triangulation_2.h>
Inherited by CGAL::Constrained_triangulation_2< Traits, Tds, Itag >, CGAL::Delaunay_triangulation_2< Traits, Tds >, and CGAL::Regular_triangulation_2< Traits, Tds >.
The class Triangulation_2
is the basic class designed to handle triangulations of set of points \( { A}\) in the plane.
Such a triangulation has vertices at the points of \( { A}\) and its domain covers the convex hull of \( { A}\). It can be viewed as a planar partition of the plane whose bounded faces are triangular and cover the convex hull of \( { A}\). The single unbounded face of this partition is the complementary of the convex hull of \( { A}\).
In many applications, it is convenient to deal only with triangular faces. Therefore, we add to the triangulation a fictitious vertex, called the infinite vertex
and we make each convex hull edge incident to an infinite
face having as third vertex the infinite vertex
. In that way, each edge is incident to exactly two faces and special cases at the boundary of the convex hull are simpler to deal with.
Triangulation_2
implements this point of view and therefore considers the triangulation of the set of points as a set of triangular, finite and infinite faces. Although it is convenient to draw a triangulation as in figure Triangulation_ref_Fig_infinite_vertex, note that the infinite vertex
has no significant coordinates and that no geometric predicate can be applied on it or on an infinite face.
A triangulation is a collection of vertices and faces that are linked together through incidence and adjacency relations. Each face give access to its three incident vertices and to its three adjacent faces. Each vertex give access to one 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.
The triangulation class offers two functions int cw(int i)
and int ccw(int i)
which, given the index of a vertex in a face, compute the index of the next vertex of the same face in clockwise or counterclockwise order. Thus, for example the neighbor neighbor(cw(i))
is the neighbor of f
which is next to neighbor(i)
turning clockwise around f
. The face neighbor(cw(i))
is also the first face encountered after f
when turning clockwise around vertex i
of f
(see Figure Triangulation_ref_Fig_neighbors).
Traits  is the geometric traits which must be a model of the concept TriangulationTraits_2 . 
Tds  is the triangulation data structure which must be a model of the concept TriangulationDataStructure_2 . By default, the triangulation data structure is instantiated by Triangulation_data_structure_2 < Triangulation_vertex_base_2<Gt>, Triangulation_face_base_2<Gt> > . 
Traversal of the Triangulation
A triangulation can be seen as a container of faces and vertices. Therefore the triangulation provides several iterators and circulators that allow to traverse it completely or partially.
Traversal of the Convex Hull
Applied on the infinite vertex the above functions allow to visit the vertices on the convex hull and the infinite edges and faces. Note that a counterclockwise traversal of the vertices adjacent to the infinite vertex is a clockwise traversal of the convex hull.
I/O
The I/O operators are defined for iostream
. The format for the iostream is an internal format.
The information output in the iostream
is:
The index of an item (vertex of face) is the rank of this item in the output order. When dimension \( <\) 2, the same information is output for faces of maximal dimension instead of faces.
Implementation
Locate is implemented by a line walk from a vertex of the face given as optional parameter (or from a finite vertex of infinite_face()
if no optional parameter 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.
Insertion of a point is done by locating a face that contains the point, and then splitting this face. If the point falls outside the convex hull, the triangulation is restored by flips. Apart from the location, insertion takes a time 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 time \( O(d^2)\) in the worst case, if \( d\) 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.
Related Functions  
(Note that these are not member functions.)  
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 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).  
Handles, Iterators, and Circulators  
The vertices and faces of the triangulations are accessed through handles, iterators and circulators. The handles are models of the concept The edges of the triangulation can also be visited through iterators and circulators, the edge circulators and iterators are also bidirectional and non mutable. In the following, we called infinite any face or edge incident to the infinite vertex and the infinite vertex itself. Any other feature (face, edge or vertex) of the triangulation is said to be finite. Some iterators (the  
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.  
Creation  
Triangulation_2 (const Traits >=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.  
Access Functions  
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.  
Non const access  
 
TriangulationDataStructure_2 &  tds () 
Returns a reference to the triangulation data structure.  
Predicates  
The class  
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...  
Queries  
The class It also provides methods to locate a point with respect to a given finite face of the triangulation.  
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...  
Modifiers  
The following operations are guaranteed to lead to a valid triangulation when they are applied on a valid triangulation.  
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...  
Specialized Modifiers  
The following member functions offer more specialized versions of the insertion or removal operations to be used when one knows to be in the corresponding case.  
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 Face, Edge and Vertex Iterators  
The following iterators allow respectively to visit finite faces, finite edges and finite vertices of the triangulation. These iterators are non mutable, bidirectional and their value types are respectively  
Finite_vertices_iterator  finite_vertices_begin () const 
Starts at an arbitrary finite vertex.  
Finite_vertices_iterator  finite_vertices_end () const 
Pasttheend iterator.  
Finite_edges_iterator  finite_edges_begin () const 
Starts at an arbitrary finite edge.  
Finite_edges_iterator  finite_edges_end () const 
Pasttheend iterator.  
Finite_faces_iterator  finite_faces_begin () const 
Starts at an arbitrary finite face.  
Finite_faces_iterator  finite_faces_end () const 
Pasttheend iterator.  
Point_iterator  points_begin () const 
Point_iterator  points_end () const 
Pasttheend iterator.  
All Face, Edge and Vertex Iterators  
The following iterators allow respectively to visit all (finite or infinite) faces, edges and vertices of the triangulation. These iterators are non mutable, bidirectional and their value types are respectively  
All_vertices_iterator  all_vertices_begin () const 
Starts at an arbitrary vertex.  
All_vertices_iterator  all_vertices_end () const 
Pasttheend iterator.  
All_edges_iterator  all_edges_begin () const 
Starts at an arbitrary edge.  
All_edges_iterator  all_edges_end () const 
Pasttheend iterator.  
All_faces_iterator  all_faces_begin () const 
Starts at an arbitrary face.  
All_faces_iterator  all_faces_end () const 
Pasttheend iterator.  
Line Face Circulator  
The triangulation defines a circulator that allows to visit all faces that are intersected by a line. A face
The circulator has a singular value if the line  
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, Edge and Vertex Circulators  
The triangulation also provides circulators that allows to visit respectively all faces or edges incident to a given vertex or all vertices adjacent to a given vertex. These circulators are nonmutable and bidirectional. The  
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...  
Traversal Between Adjacent Faces  
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...  
Miscellaneous  
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...  
Setting  
void  set_infinite_vertex (const Vertex_handle &v) 
This is an advanced function. More...  
Checking  
Advanced
The responsibility of keeping a valid triangulation belongs to the users if advanced operations are used. Obviously the advanced user, who implements higher levels operations may have to make a triangulation invalid at some times. The following method is provided to help the debugging.  
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...  
Additional Inherited Members  
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.  
enum CGAL::Triangulation_2::Locate_type 
specifies which case occurs when locating a point in the triangulation.
CGAL::Triangulation_2<Traits,Tds>
CGAL::Triangulation_2< Traits, Tds >::Triangulation_2  (  const Triangulation_2< Traits, Tds > &  tr  ) 
Copy constructor.
All the vertices and faces are duplicated. After the copy, *this
and tr
refer to different triangulations: if tr
is modified, *this
is not.
int CGAL::Triangulation_2< Traits, Tds >::ccw  (  int  i  )  const 
Returns \( i+1\) modulo 3.
Point CGAL::Triangulation_2< Traits, Tds >::circumcenter  (  Face_handle  f  )  const 
Compute the circumcenter of the face pointed to by f.
This function is available only if the corresponding function is provided in the geometric traits.
int CGAL::Triangulation_2< Traits, Tds >::cw  (  int  i  )  const 
Returns \( i+2\) modulo 3.
void CGAL::Triangulation_2< Traits, Tds >::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)
.
f
and f>neighbor(i)
are finite faces and their union form a convex quadrilateral. Edge_circulator CGAL::Triangulation_2< Traits, Tds >::incident_edges  (  Vertex_handle  v, 
Face_handle  f  
)  const 
Starts at the first edge of f
incident to v
, in counterclockwise order around v
.
f
is incident to vertex v
. Face_circulator CGAL::Triangulation_2< Traits, Tds >::incident_faces  (  Vertex_handle  v, 
Face_handle  f  
)  const 
Starts at face f
.
f
is incident to vertex v
. Vertex_circulator CGAL::Triangulation_2< Traits, Tds >::incident_vertices  (  Vertex_handle  v, 
Face_handle  f  
) 
Starts at the first vertex of f
adjacent to v
in counterclockwise order around v
.
f
is incident to vertex v
. bool CGAL::Triangulation_2< Traits, Tds >::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
.
If true
, vbr
becomes the other vertex of e
, e
is the edge (fr,i)
where fr
is a handle to the face incident to e
and on the right side e
oriented from va
to vb
.
Face_handle CGAL::Triangulation_2< Traits, Tds >::inexact_locate  (  const Point &  query, 
Face_handle  start = Face_handle() 

)  const 
Same as locate()
but uses inexact predicates.
This function returns a handle on a face that is a good approximation of the exact location of query
, while being faster. Note that it may return a handle on a face whose interior does not contain query
. When the triangulation has dimension smaller than 2, start
is returned.
Vertex_handle CGAL::Triangulation_2< Traits, Tds >::insert  (  const Point &  p, 
Face_handle  f = Face_handle() 

) 
Inserts point p
in the triangulation and returns the corresponding vertex.
If point p
coincides with an already existing vertex, this vertex is returned and the triangulation remains unchanged.
If point p
is on an edge, the two incident faces are split in two.
If point p
is strictly inside a face of the triangulation, the face is split in three.
If point p
is strictly outside the convex hull, p
is linked to all visible points on the convex hull to form the new triangulation.
At last, if p
is outside the affine hull (in case of degenerate 1dimensional or 0dimensional triangulations), p
is linked all the other vertices to form a triangulation whose dimension is increased by one. The last argument f
is an indication to the underlying locate algorithm of where to start.
std::ptrdiff_t CGAL::Triangulation_2< Traits, Tds >::insert  (  InputIterator  first, 
InputIterator  last  
) 
Inserts the points in the range [first,last)
.
Returns the number of inserted points.
value_type
of InputIterator
is Point
. Vertex_handle CGAL::Triangulation_2< Traits, Tds >::insert_in_edge  (  const Point &  p, 
Face_handle  f,  
int  i  
) 
Inserts vertex v in edge i
of f
.
v
lies on the edge opposite to the vertex i
of face f
. Vertex_handle CGAL::Triangulation_2< Traits, Tds >::insert_in_face  (  const Point &  p, 
Face_handle  f  
) 
Inserts vertex v
in face f
.
Face f
is modified, two new faces are created.
v
lies inside face f
. Vertex_handle CGAL::Triangulation_2< Traits, Tds >::insert_outside_convex_hull  (  const Point &  p, 
Face_handle  f  
) 
Inserts a point which is outside the convex hull but in the affine hull.
f
points to a face which is a proof of the location ofp
, see the description of the locate
method above. bool CGAL::Triangulation_2< Traits, Tds >::is_edge  (  Vertex_handle  va, 
Vertex_handle  vb,  
Face_handle &  fr,  
int &  i  
) 
as above.
In addition, if true
is returned, the edge with vertices va
and vb
is the edge e=(fr,i)
where fr
is a handle to the face incident to e
and on the right side of e
oriented from va
to vb
.
bool CGAL::Triangulation_2< Traits, Tds >::is_face  (  Vertex_handle  v1, 
Vertex_handle  v2,  
Vertex_handle  v3,  
Face_handle &  fr  
) 
as above.
In addition, if true
is returned, fr is a handle to the face with v1
, v2
and v3
as vertices.
bool CGAL::Triangulation_2< Traits, Tds >::is_valid  (  bool  verbose = false , 
int  level = 0 

)  const 
Checks the combinatorial validity of the triangulation and also the validity of its geometric embedding.
This method is mainly a debugging help for the users of advanced features.
Line_face_circulator CGAL::Triangulation_2< Traits, Tds >::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
.
If there is no such face the circulator has a singular value.
The starting point of the circulator is the face f
, or the first finite face traversed by l
, if f
is omitted.
The circulator wraps around the infinite vertex: after the last traversed finite face, it steps through the infinite face adjacent to this face then through the infinite face adjacent to the first traversed finite face then through the first finite traversed face again.
p
and q
must be different points. f != NULL
, it must point to a finite face and the point p
must be inside or on the boundary of f
. Face_handle CGAL::Triangulation_2< Traits, Tds >::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.
If the point query
lies outside the convex hull of the triangulation but in the affine hull, the returned face is an infinite face which is a proof of the point's location:
query
lies to the left of the oriented line \( pq\) (the rest of the triangulation lying to the right of this line).query
and the triangulation lie on either side of p
.If the point query
lies outside the affine hull, the returned Face_handle
is NULL
.
The optional Face_handle
argument, if provided, is used as a hint of where the locate process has to start its search.
Face_handle CGAL::Triangulation_2< Traits, Tds >::locate  (  const Point &  query, 
Locate_type &  lt,  
int &  li,  
Face_handle  h = Face_handle() 

)  const 
Same as above.
Additionally, the parameters lt
and li
describe where the query point is located. The variable lt
is set to the locate type of the query. If lt==VERTEX
the variable li
is set to the index of the vertex, and if lt==EDGE
li
is set to the index of the vertex opposite to the edge. Be careful that li
has no meaning when the query type is FACE
, OUTSIDE_CONVEX_HULL
, or OUTSIDE_AFFINE_HULL
or when the triangulation is \( 0\)dimensional.
Edge CGAL::Triangulation_2< Traits, Tds >::mirror_edge  (  Edge  e  )  const 
returns the same edge seen from the other adjacent face.
int CGAL::Triangulation_2< Traits, Tds >::mirror_index  (  Face_handle  f, 
int  i  
)  const 
returns the index of f
in its \( i^{th}\) neighbor.
Vertex_handle CGAL::Triangulation_2< Traits, Tds >::mirror_vertex  (  Face_handle  f, 
int  i  
)  const 
returns the vertex of the \( i^{th}\) neighbor of f
that is opposite to f
.
Vertex_handle CGAL::Triangulation_2< Traits, Tds >::move  (  Vertex_handle  v, 
const Point &  p  
) 
same as above if there is no collision.
Otherwise, v
is deleted and the vertex placed on p
is returned.
v
must be finite. Vertex_handle CGAL::Triangulation_2< Traits, Tds >::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.
Otherwise, the triangulation is not modified and the vertex at point p
is returned.
v
must be finite. Triangulation_2 CGAL::Triangulation_2< Traits, Tds >::operator=  (  const Triangulation_2< Traits, Tds > &  tr  ) 
Assignment.
All the vertices and faces are duplicated. After the assignment, *this
and tr
refer to different triangulations: if tr
is modified, *this
is not.
Oriented_side CGAL::Triangulation_2< Traits, Tds >::oriented_side  (  Face_handle  f, 
const Point &  p  
)  const 
Returns on which side of the oriented boundary of f
lies the point p
.
f
is finite. void CGAL::Triangulation_2< Traits, Tds >::remove  (  Vertex_handle  v  ) 
Removes the vertex from the triangulation.
The created hole is retriangulated.
v
must be finite. void CGAL::Triangulation_2< Traits, Tds >::remove_degree_3  (  Vertex_handle  v  ) 
Removes a vertex of degree three.
Two of the incident faces are destroyed, the third one is modified.
v
is a finite vertex with degree three. Segment CGAL::Triangulation_2< Traits, Tds >::segment  (  Face_handle  f, 
int  i  
)  const 
Returns the line segment formed by the vertices ccw(i)
and cw(i)
of face f
.
ccw(i)
and cw(i)
of f
are finite. Segment CGAL::Triangulation_2< Traits, Tds >::segment  (  const Edge &  e  )  const 
Returns the line segment corresponding to edge e
.
e
is a finite edge. Segment CGAL::Triangulation_2< Traits, Tds >::segment  (  const Edge_circulator &  ec  )  const 
Returns the line segment corresponding to edge *ec
.
*ec
is a finite edge. Segment CGAL::Triangulation_2< Traits, Tds >::segment  (  const Edge_iterator &  ei  )  const 
Returns the line segment corresponding to edge *ei
.
*ei
is a finite edge. void CGAL::Triangulation_2< Traits, Tds >::set_infinite_vertex  (  const Vertex_handle &  v  ) 
This is an advanced function.
This method is meant to be used only if you have done a lowlevel operation on the underlying tds that invalidated the infinite vertex. Sets the infinite vertex.
Oriented_side CGAL::Triangulation_2< Traits, Tds >::side_of_oriented_circle  (  Face_handle  f, 
const Point &  p  
) 
Returns on which side of the circumcircle of face f
lies the point p
.
The circle is assumed to be counterclockwise oriented, so its positive side correspond to its bounded side. This predicate is available only if the corresponding predicates on points is provided in the geometric traits class.
Vertex_handle CGAL::Triangulation_2< Traits, Tds >::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)
.
Returns a handle to the new vertex.
This function is intended to be used in conjunction with the find_conflicts()
member functions of Delaunay and constrained Delaunay triangulations to perform insertions.
Vertex_handle CGAL::Triangulation_2< Traits, Tds >::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.
This function is intended to be used in conjunction with the find_conflicts()
member functions of Delaunay and constrained Delaunay triangulations to perform insertions.
void CGAL::Triangulation_2< Traits, Tds >::swap  (  Triangulation_2< Traits, Tds > &  tr  ) 
The triangulations tr
and *this
are swapped.
This method should be used instead of assignment of copy construtor. if tr
is deleted after that.
Triangle CGAL::Triangulation_2< Traits, Tds >::triangle  (  Face_handle  f  )  const 
Returns the triangle formed by the three vertices of f
.

related 
Inserts the triangulation into the stream os
.
Point
.

related 
Reads a triangulation from stream is
and assigns it to the triangulation.
Point
.