Definition

A Delaunay triangulation is a special triangulation of a set of points. So it is natural to derive the class CGAL_Delaunay_triangulation_2<Gt,Tds> from the basic class CGAL_Triangulation_2<Gt,Tds>. The template parameters Gt and Tds stand respectively for a model of geometric traits and for a model of triangulation data structure. The requirements for the triangulation data structure are those described previously in section  and the same models can be used to instantiate the triangulation data structure of either a CGAL_Triangulation_2<Gt,Tds> or a CGAL_Delaunay_triangulation_2<Gt,Tds>. On the contrary, because the concept of Delaunay triangulation relies on the notions of empty circles and of distance, the geometric traits is required to provide some additional predicates namely the famous in_circle test and also a Distance class. The additional requirements to be fulfilled by the geometric traits class of a CGAL_Delaunay_triangulation_2<Gt,Tds> are described in subsection . Changing the Distance class and the in_circle test allows to build Delaunay triangulation for different metrics such that $L$₁ or $L$_infinity or any metric defined by a convex object. However, the user of an exotic metric must be carefull 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 $L$_infinity) by the addition to the point set of well chosen sentinel points.

Inherits From

Types

Inherits all the types of the CGAL_Triangulation_2<Traits> . In addition, we need a type for the distance object function, and some types to represent the geometric dual features.

typedef Gt::Distance
	Distance;
typedef Gt::Line	Line;
typedef Gt::Direction
	Direction;
typedef Gt::Ray	Ray;

Creation

CGAL_Delaunay_triangulation_2<Gt,Tds> dt ( Gt gt = Gt());
	Introduces an empty Delaunay triangulation dt.
CGAL_Delaunay_triangulation_2<Gt,Tds> dt ( Vertex_handle v; Traits t = Traits());
	Introduces a Delaunay triangulation dt that is initialized with the vertices and faces that are linked to vertex v. If v has no incident face the triangulation consists only of v. Otherwise v must be the vertex at infinity.

Insertion and Removal

The following insertion and removal functions overwrite the functions inherited from the class CGAL_Triangulation_2<Gt,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 initialized the location.
Vertex_handle	dt.insert ( Point p, Locate_type& lt. Face_handle f=Face_handle())
		Same as above. Additionally, parameter lt describes where point p was located before updating the triangulation.
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 CGAL_Triangulation_2<Gt,Tds> are not overwritten. Thus a call to insert_in_face insert_in_edge, insert_outside or flip on a valid Delaunay triangulation might lead to a valid 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.

Duality

Point	dt.dual ( Face_handle f)
		Returns the center of the circle circumscribed to face f. Precondition: f is not infinite
CGAL_Object	dt.dual ( Edge e)	If both incident faces are finite, returns a segment whose endpoints are the duals of each incident face. If only one incident face is finite, returns a ray whose endpoint is the dual of the finite incident face and supported by the line which is the bisector of the edge's endpoints. If both incident faces are infinite, returns the line which is the bisector of the edge's endpoints otherwise.
CGAL_Object	dt.dual ( Edge_circulator ec)
		Idem
CGAL_Object	dt.dual ( Edge_iterator ei)
		Idem

Geometric Predicates

CGAL_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

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 CGAL_Triangulation_2 and additionally test the Delaunay property. This method is mainly useful for debugging Delaunay triangulation algorithms designed by the user.

Example

#include <CGAL/Homogeneous.h>
#include <CGAL/Triangulation_euclidean_traits_xy_3.h>
#include <CGAL/Delaunay_triangulation_2.h>
#include <CGAL/IO/Geomview_stream.h>

typedef CGAL_Homogeneous<leda_integer>  Rep;
typedef CGAL_Triangulation_euclidean_traits_xy_3<Rep>  Terrain;
typedef CGAL_Triangulation_vertex_base_2<Gt> Vb;
typedef CGAL_Triangulation_face_base_2<Gt> Fb;
typedef CGAL_Triangulation_default_data_structure_2<Gt,Vb,Fb > Tds;
typedef CGAL_Delaunay_triangulation_2<Terrain, Tds> Delaunay;

{
    Delaunay dt(Terrain());

    CGAL_Point_3<Rep> p;
    while(cin >> p){
        DT.insert(p);
    }
    CGAL_Geomview_stream G;
    G << DT;
}

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 but this time using the Delaunay criterion. Removal of a vertex of degree d takes time O(d^2), which is O(1) for a random vertex in the triangulation.

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