The class Delaunay_triangulation_3 represents a threedimensional Delaunay triangulation.
The user is advised to use the class Triangulation_hierarchy_3 rather than this basic Delaunay triangulation class: it offers the same functionalities but is much more efficient for large data sets.
#include <CGAL/Delaunay_triangulation_3.h>
The first template argument must be a model of the DelaunayTriangulationTraits_3 concept.
The second template argument must be a model of the TriangulationDataStructure_3 concept. It has the default value Triangulation_data_structure_3<Triangulation_vertex_base_3<DelaunayTriangulationTraits_3>, Triangulation_cell_base_3<DelaunayTriangulationTraits_3> >.
In addition to those inherited, the following types are defined, for use by the construction of the Voronoi diagram:
 


 


 


 

 
Creates an empty Delaunay triangulation, possibly specifying a traits class
traits.
 
 
Copy constructor.
 
 
 
Creates a Delaunay triangulation of the points specified by the iterator range
[first,last) of value type Point, possibly specifying a traits
class traits.

The following methods overload the corresponding methods of triangulations to ensure the empty sphere property of Delaunay triangulations.
In the degenerate case when there are cospherical points, the Delaunay triangulation is known not to be uniquely defined. In this case, CGAL chooses a particular Delaunay triangulation using a symbolic perturbation scheme [DT03].

 
Inserts point p in the triangulation and returns the corresponding vertex. Similar to the insertion in a triangulation, but ensures in addition the empty sphere property of all the created faces. The optional argument start is used as a starting place for the search.  

 
Inserts point p in the triangulation and returns the corresponding vertex. Similar to the above insert() function, but takes as additional parameter the return values of a previous location query. See description of Triangulation_3::locate(). 
The following method allows one to insert several points. It returns the number of inserted points.
 

 
Inserts the points in the iterator range $$[.first, last$$.), of value type Point. 
When a vertex v is removed from a triangulation, all the cells incident to v must be removed, and the polyhedral region consisting of all the tetrahedra that are incident to v must be retriangulated. So, the problem reduces to triangulating a polyhedral region, while preserving its boundary, or to compute a constrained triangulation. This is known to be sometimes impossible: the Schönhardt polyhedron cannot be triangulated [She98].
However, when dealing with Delaunay triangulations, the case of such polyhedra that cannot be retriangulated cannot happen, so CGAL proposes a vertex removal.

 
Returns a value indicating on which side of the circumscribed sphere
of c the point p lies. More precisely, it returns:  ON_BOUNDED_SIDE if p is inside the sphere. For an infinite cell this means that p lies strictly either in the half space limited by its finite facet and not containing any other point of the triangulation, or in the interior of the disk circumscribing the finite facet.  ON_BOUNDARY if p on the boundary of the sphere. For an infinite cell this means that p lies on the circle circumscribing the finite facet.  ON_UNBOUNDED_SIDE if p lies outside the sphere. For an infinite cell this means that p does not satisfy either of the two previous conditions. Precondition: dt.dimension() $$=3.  

 
Returns a value indicating on which side of the circumscribed circle
of f the point p lies. More precisely, it returns:  in dimension 3:  For a finite facet, ON_BOUNDARY if p lies on the circle, ON_UNBOUNDED_SIDE when it lies in the exterior of the disk, ON_BOUNDED_SIDE when it lies in its interior.  For an infinite facet, it considers the plane defined by the finite facet of the same cell, and does the same as in dimension 2 in this plane.  in dimension 2:  For a finite facet, ON_BOUNDARY if p lies on the circle, ON_UNBOUNDED_SIDE when it lies in the exterior of the disk, ON_BOUNDED_SIDE when it lies in its interior.  For an infinite facet, ON_BOUNDARY if the point lies on the finite edge of f (endpoints included), ON_BOUNDED_SIDE for a point in the open half plane defined by f and not containing any other point of the triangulation, ON_UNBOUNDED_SIDE elsewhere. Precondition: dt.dimension() $$ 2 and in dimension 3, p is coplanar with f.  

 
Same as the previous method for facet i of cell c.  

 
Returns any nearest vertex to the point p, or the default constructed
handle if the triangulation is empty. The optional argument c is a hint
specifying where to start the search. Precondition: c is a cell of dt.  

 
Returns the vertex of the cell c that is nearest to $$p. 
A point p is said to be in conflict with a cell c in dimension 3 (resp. a facet f in dimension 2) iff dt.side_of_sphere(c, p) (resp. dt.side_of_circle(f, p)) returns ON_BOUNDED_SIDE. The set of cells (resp. facets in dimension 2) which are in conflict with p is connected, and it forms a hole.
 
 
 
Computes the conflict hole induced by p. The starting cell
(resp. facet) c must be in conflict. Then this function returns
respectively in the output iterators:  cit: the cells (resp. facets) in conflict.  bfit: the facets (resp. edges) on the boundary, that is, the facets (resp. edges) (t, i) where the cell (resp. facet) t is in conflict, but t>neighbor(i) is not. This function can be used in conjunction with insert_in_hole() in order to decide the insertion of a point after seeing which elements of the triangulation are affected. Returns the pair composed of the resulting output iterators. Precondition: dt.dimension() $$ 2, and c is in conflict with p.  
 
 
 
Same as the other find_conflicts() function, except that it also
computes the internal facets, i.e. the facets common to two cells which
are in conflict with p.
Then this function returns respectively in the output iterators:  cit: the cells (resp. facets) in conflict.  bfit: the facets (resp. edges) on the boundary, that is, the facets (resp. edges) (t, i) where the cell (resp. facet) t is in conflict, but t>neighbor(i) is not.  ifit: the facets (resp. edges) inside the hole, that is, delimiting two cells (resp facets) in conflict. Returns the Triple composed of the resulting output iterators. Precondition: dt.dimension() $$ 2, and c is in conflict with p.  
 

 
Similar to find_conflicts(), but reports the vertices which are on the
boundary of the conflict hole of p, in the output iterator res.
Returns the resulting output iterator. Precondition: dt.dimension() $$ 2, and c is in conflict with p. 
A face (cell, facet or edge) is said to be a Gabriel face iff its smallest circumscribing sphere do not enclose any vertex of the triangulation. Any Gabriel face belongs to the Delaunay triangulation, but the reciprocal is not true. The following member functions test the Gabriel property of Delaunay faces.

 

 

 


Note that the user should use a kernel with exact constructions in order to guarantee the computation of the Voronoi diagram (as opposed to computing the triangulation only, which requires only exact predicates).

 
Returns the circumcenter of the four vertices of c. Precondition: dt.dimension()$$=3 and c is not infinite.  


Returns the dual of facet f, which is in dimension 3: either a segment, if the two cells incident to f are finite, or a ray, if one of them is infinite; in dimension 2: a point. Precondition: dt.dimension() $$ 2 and f is not infinite. 

 
same as the previous method for facet (c,i).  
 

 
Sends the set of duals to all the facets of dt into os. 
advanced 

 
Checks the combinatorial validity of the triangulation and the
validity of its geometric embedding (see
Section 22.1). Also checks that all the
circumscribing spheres (resp. circles in dimension 2) of cells
(resp. facets in dimension 2) are empty. When verbose is set to true, messages describing the first invalidity encountered are printed.  

 
Checks the combinatorial and geometric validity of the cell (see
Section 22.1). Also checks that the
circumscribing sphere (resp. circle in dimension 2) of cells
(resp. facet in dimension 2) is empty. When verbose is set to true, messages are printed to give a precise indication of the kind of invalidity encountered. 
These methods are mainly a debugging help for the users of advanced features.
advanced 
CGAL::Triangulation_hierarchy_3.