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.
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 $$L_{1} or $$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 $$L_{2}) and can be ensured for other metrics (like $$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>
 
default constructor.
 
 
copy constructor. All the vertices and faces are duplicated.

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

 
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.  

 
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>.  

 
equivalent to insert(p).  
 

 
inserts the points in the range
$$[.first, last$$.).
Returns the number of inserted points. Precondition: The value_type of first and last is Point.  

 
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.

 
Returns the center of the circle circumscribed to face f. Precondition: f is not infinite  

 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. 

 
Idem  

 
Idem  
 

 
output the dual voronoi diagram to stream ps. 

 
Returns the side of p with respect to the circle circumscribing the triangle associated with f 
advanced 
The checking function is_valid() is also overwritten to additionally test the empty circle property.

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