## CGAL::Delaunay_triangulation_2<Traits,Tds>

### Definition

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

### Inherits From

Triangulation_2<Traits,Tds>.

### Parameters

The template parameter Tds is to be instantiated with a model of TriangulationDataStructure_2. CGALprovides a default instantiation for this parameter, which is the class CGAL::Triangulation_data_structure_2 < CGAL::Triangulation_vertex_base_2<Traits>, CGAL::Triangulation_face_base_2<Traits> >.

The geometric traits Traits is to be instantiated with a model of DelaunayTriangulationTraits_2. The concept DelaunayTriangulationTraits_2 refines the concept TriangulationTraits_2, providing a predicate type to check the empty circle property.

Changing this predicate type allows to build Delaunay triangulations for different metrics such that L1 or L or any metric defined by a convex object. However, the user of an exotic metric must be careful that the constructed triangulation has to be a triangulation of the convex hull which means that convex hull edges have to be Delaunay edges. This is granted for any smooth convex metric (like L2) and can be ensured for other metrics (like L ) by the addition to the point set of well chosen sentinel points The concept of DelaunayTriangulationTraits_2 is described .

When dealing with a large triangulations, the user is advised to encapsulate the Delanay triangulation class into a triangulation hierarchy, which means to use the class Triangulation_hierarchy_2<Tr> with the template parameter instantiated with Delaunay_triangulation_2<Traits,Tds> . The triangulation hierachy will then offer the same functionalities but be much more for efficient for locations and insertions.

#include <CGAL/Delaunay_triangulation_2.h>

### Inherits From

Triangulation_2<Traits,Tds>

### Types

Inherits all the types defined in Triangulation_2<Traits,Tds>.

### Creation

Delaunay_triangulation_2<Traits,Tds> dt ( Traits gt = Traits());
| default constructor.
Delaunay_triangulation_2<Traits,Tds> dt ( tr);
| copy constructor. All the vertices and faces are duplicated.

### Insertion and Removal

The following insertion and removal functions overwrite the functions inherited from the class Triangulation_2<Traits,Tds> to maintain the Delaunay property.

Vertex_handle | dt.insert ( Point p, Face_handle f=Face_handle())
|  | inserts point p. If point p coincides with an already existing vertex, this vertex is returned and the triangulation is not updated. Optional parameter f is used to initialize the location of p.

Vertex_handle

dt.insert ( | Point p, Locate_type& lt, Face_handle loc, int li)

|  | inserts a point p, the location of which is supposed to be given by (lt,loc,li), see the description of member function locate in class Triangulation_2<Traits,Tds>.
Vertex_handle | dt.push_back ( Point p)
|  | equivalent to insert(p).
template < class InputIterator >
int | dt.insert ( InputIterator first, InputIterator last)
|  | inserts the points in the range [.first, last.). Returns the number of inserted points. Precondition: The value_type of first and last is Point.
void | dt.remove ( Vertex_handle v)
|  | removes the vertex from the triangulation.

Note that the other modifier functions of Triangulation_2<Traits,Tds> are not overwritten. Thus a call to insert_in_face insert_in_edge, insert_outside_convex_hull, insert_outside_affine_hull or flip on a valid Delaunay triangulation might lead to a triangulation which is no longer a Delaunay triangulation.

### Queries

Vertex_handle

dt.nearest_vertex ( | Point p, Face_handle f=Face_handle())

|  | returns any nearest vertex of p. The implemented function begins with a location step and f may be used to initialize the location.
template <class OutputItFaces, class OutputItBoundaryEdges>
std::pair<OutputItFaces,OutputItBoundaryEdges>

dt.get_conflicts_and_boundary ( | Point p, OutputItFaces fit, OutputItBoundaryEdges eit, Face_handle start)

OutputItFaces is an output iterator with Face_handle as value type. OutputItBoundaryEdges stands for an output iterator with Edge as value type. This members function outputs in the container pointed to by fit the faces which are in conflict with point p i. e. the faces whose circumcircle contains p. It outputs in the container pointed to by eit the the boundary of the zone in conflict with p. The boundary edges of the conflict zone are ouput in counter-clockwise order and each edge is described through its incident face which is not in conflict with p. The function returns in a std::pair the resulting output iterators.

template <class OutputItFaces>

OutputItFaces

dt.get_conflicts ( | Point p, OutputItFaces fit, Face_handle start)

|  | same as above except that only the faces in conflict with p are output. The function returns the resulting output iterator.
template <class OutputItBoundaryEdges>
OutputItBoundaryEdges

dt.get_boundary_of_conflicts ( | Point p, OutputItBoundaryEdges eit, Face_handle start)

|  | OutputItBoundaryEdges stands for an output iterator with Edge as value type. This function outputs in the container pointed to by eit, the boundary of the zone in conflict with p. The boundary edges of the conflict zone are ouput in counterclockwise order and each edge is described through the incident face which is not in conflict with p. The function returns the resulting output iterator.

### Voronoi diagram

The following member functions provide the elements of the dual Voronoi diagram.

Point | dt.dual ( Face_handle f)
|  | Returns the center of the circle circumscribed to face f. Precondition: f is not infinite

Object

dt.dual ( Edge e)

returns a segment, a ray or a line supported by the bisector of the endpoints of e. If faces incident to e are both finite, a segment whose endpoints are the duals of each incident face is returned. If only one incident face is finite, a ray whose endpoint is the dual of the finite incident face is returned. Otherwise both incident faces are infinite and the bisector line is returned.

Object | dt.dual ( Edge_circulator ec)
|  | Idem
Object | dt.dual ( Edge_iterator ei)
|  | Idem
template < class Stream>
Stream& | dt.draw_dual ( Stream & ps)
|  | output the dual voronoi diagram to stream ps.

### Predicates

Oriented_side | dt.side_of_oriented_circle ( Face_handle f, Point p)
|  | Returns the side of p with respect to the circle circumscribing the triangle associated with f

| advanced |

### Miscellaneous

The checking function is_valid() is also overwritten to additionally test the empty circle property.

bool | dt.is_valid ( bool verbose = false, int level = 0)
|  | tests the validity of the triangulation as a Triangulation_2 and additionally tests the Delaunay property. This method is mainly useful for debugging Delaunay triangulation algorithms designed by the user.

| advanced |

### See Also

CGAL::Triangulation_2<Traits,Tds>,
TriangulationDataStructure_2,
DelaunayTriangulationTraits_2,
Triangulation_hierarchy_2<Tr>.

### Implementation

Insertion is implemented by inserting in the triangulation, then performing a sequence of Delaunay flips. The number of flips is O(d) if the new vertex is of degree d in the new triangulation. For points distributed uniformly at random, insertion takes time O(1) on average.

Removal calls the removal in the triangulation and then retriangulates the hole in such a way that the Delaunay criterion is satisfied. Removal of a vertex of degree d takes time O(d^2). The degree d is O(1) for a random vertex in the triangulation.

After a point location step, the nearest neighbor is found in time O(n) in the worst case, but in time O(1) for vertices distributed uniformly at random and any query point.