CGAL 5.1.3  3D Triangulations

#include <CGAL/Delaunay_triangulation_3.h>
The class Delaunay_triangulation_3
represents a threedimensional Delaunay triangulation.
Traits  is the geometric traits class and must be a model of DelaunayTriangulationTraits_3 . 
TDS  is the triangulation data structure and must be a model of TriangulationDataStructure_3 . Default may be used, with default type Triangulation_data_structure_3<Triangulation_vertex_base_3<Traits>, Delaunay_triangulation_cell_base_3<Traits> > . Any custom type can be used instead of Triangulation_vertex_base_3 and Delaunay_triangulation_cell_base_3 , provided that they are models of the concepts TriangulationVertexBase_3 and DelaunayTriangulationCellBase_3 , respectively. 
LP  is a tag which must be a Location_policy<Tag> : either CGAL::Fast_location or CGAL::Compact_location . CGAL::Fast_location offers faster ( \( O(\log n)\) time) point location, which can be beneficial when performing point locations or random point insertions (with no good location hint) in large data sets. It is currently implemented using an additional triangulation hierarchy data structure [12]. The default is CGAL::Compact_location , which saves memory (35%) by avoiding the need for this separate data structure, and point location is then performed roughly in \( O(n^{1/3})\) time. If the triangulation is parallel (see user manual), the default compact location policy must be used. Note that this argument can also come in second position, which can be useful when the default value for the TDS parameter is satisfactory (this is achieved using socalled deduced parameters). Note that this argument replaces the functionality provided before CGAL 3.6 by Triangulation_hierarchy_3 . An example of use can be found in the user manual Fast Point Location for Delaunay Triangulations. 
SLDS  is an optional parameter to specify the type of the spatial lock data structure. It must be a model of the SurjectiveLockDataStructure concept, with Object being a Point (as defined below). It is only used if the triangulation data structure used is concurrencysafe (i.e. when TDS::Concurrency_tag is CGAL::Parallel_tag ). The default value is Spatial_lock_grid_3<Tag_priority_blocking> if the triangulation data structure is concurrencysafe, and void otherwise. In order to use concurrent operations, the user must provide a reference to a SurjectiveLockDataStructure instance via the constructor or Triangulation_3::set_lock_data_structure() . 
If TDS::Concurrency_tag
is CGAL::Parallel_tag
, some operations, such as insertion/removal of a range of points, are performed in parallel. See the documentation of the operations for more details.
Types  
typedef Traits  Geom_traits 
typedef TDS  Triangulation_data_structure 
typedef LP  Location_policy 
typedef SLDS  Lock_data_structure 
In addition to those inherited, the following types are defined, for use by the construction of the Voronoi diagram:  
typedef Geom_traits::Line_3  Line 
typedef Geom_traits::Ray_3  Ray 
typedef Geom_traits::Plane_3  Plane 
typedef Geom_traits::Object_3  Object 
Creation  
Delaunay_triangulation_3 (const Geom_traits &traits=Geom_traits(), Lock_data_structure *lock_ds=nullptr)  
Creates an empty Delaunay triangulation, possibly specifying a traits class traits . More...  
Delaunay_triangulation_3 (const Delaunay_triangulation_3 &dt1)  
Copy constructor. More...  
template<class InputIterator >  
Delaunay_triangulation_3 (InputIterator first, InputIterator last, const Geom_traits &traits=Geom_traits(), Lock_data_structure *lock_ds=nullptr)  
Equivalent to constructing an empty triangulation with the optional traits class argument and calling insert(first,last) . More...  
template<class InputIterator >  
Delaunay_triangulation_3 (InputIterator first, InputIterator last, Lock_data_structure *lock_ds, const Geom_traits &traits=Geom_traits())  
Same as before, with last two parameters in reverse order.  
Insertion  
Vertex_handle  insert (const Point &p, Cell_handle start=Cell_handle(), bool *could_lock_zone=nullptr) 
Inserts the point p in the triangulation and returns the corresponding vertex. More...  
Vertex_handle  insert (const Point &p, Vertex_handle hint, bool *could_lock_zone=nullptr) 
Same as above but uses hint as a starting place for the search.  
Vertex_handle  insert (const Point &p, Locate_type lt, Cell_handle loc, int li, int lj, bool *could_lock_zone=nullptr) 
Inserts the point p in the triangulation and returns the corresponding vertex. More...  
template<class PointInputIterator >  
std::ptrdiff_t  insert (PointInputIterator first, PointInputIterator last) 
Inserts the points in the iterator range [first,last) . More...  
template<class PointWithInfoInputIterator >  
std::ptrdiff_t  insert (PointWithInfoInputIterator first, PointWithInfoInputIterator last) 
Inserts the points in the iterator range [first,last) . More...  
Displacement  
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) 
If there is no collision during the move, this function is the same as move_if_no_collision . More...  
Removal  
When a vertex 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 [19]. However, when dealing with Delaunay triangulations, the case of such polyhedra that cannot be retriangulated cannot happen, so CGAL proposes a vertex removal. If,due to some point removals, the size of the Delaunay triangulation decreases drastically, it might be interesting to defragment the  
void  remove (Vertex_handle v) 
Removes the vertex v from the triangulation. More...  
bool  remove (Vertex_handle v, bool *could_lock_zone) 
Removes the vertex v from the triangulation. More...  
template<typename InputIterator >  
int  remove (InputIterator first, InputIterator beyond) 
Removes the vertices specified by the iterator range [first, beyond) . More...  
template<typename InputIterator >  
int  remove_cluster (InputIterator first, InputIterator beyond) 
This function has exactly the same result and the same preconditions as remove(first, beyond) . More...  
Queries  
Bounded_side  side_of_sphere (Cell_handle c, const Point &p) const 
Returns a value indicating on which side of the circumscribed sphere of c the point p lies. More...  
Bounded_side  side_of_circle (const Facet &f, const Point &p) const 
Returns a value indicating on which side of the circumscribed circle of f the point p lies. More...  
Bounded_side  side_of_circle (Cell_handle c, int i, const Point &p) 
Same as the previous method for facet i of cell c .  
Vertex_handle  nearest_vertex (const Point &p, Cell_handle c=Cell_handle()) 
Returns any nearest vertex to the point p , or the default constructed handle if the triangulation is empty. More...  
Vertex_handle  nearest_vertex_in_cell (const Point &p, Cell_handle c) 
Returns the vertex of the cell c that is nearest to \( p\).  
A point The set of cells (resp. facets in dimension 2) which are in conflict with  
template<class OutputIteratorBoundaryFacets , class OutputIteratorCells >  
std::pair< OutputIteratorBoundaryFacets, OutputIteratorCells >  find_conflicts (const Point &p, Cell_handle c, OutputIteratorBoundaryFacets bfit, OutputIteratorCells cit, bool *could_lock_zone=nullptr) 
Computes the conflict hole induced by p . More...  
template<class OutputIteratorBoundaryFacets , class OutputIteratorCells , class OutputIteratorInternalFacets >  
Triple< OutputIteratorBoundaryFacets, OutputIteratorCells, OutputIteratorInternalFacets >  find_conflicts (const Point &p, Cell_handle c, OutputIteratorBoundaryFacets bfit, OutputIteratorCells cit, OutputIteratorInternalFacets ifit, bool *could_lock_zone=nullptr) 
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 . More...  
template<class OutputIterator >  
OutputIterator  vertices_in_conflict (const Point &p, Cell_handle c, OutputIterator res) 
template<class OutputIterator >  
OutputIterator  vertices_on_conflict_zone_boundary (const Point &p, Cell_handle c, OutputIterator res) 
Similar to find_conflicts() , but reports the vertices which are on the boundary of the conflict hole of p , in the output iterator res . More...  
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.  
bool  is_Gabriel (Cell_handle c, int i) 
bool  is_Gabriel (Cell_handle c, int i, int j) 
bool  is_Gabriel (const Facet &f) 
bool  is_Gabriel (const Edge &e) 
Voronoi Diagram  
CGAL offers several functionalities to display the Voronoi diagram of a set of points in 3D. 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).  
Point  dual (Cell_handle c) const 
Returns the circumcenter of the four vertices of c. More...  
Object  dual (Facet f) const 
Returns the dual of facet f , which is. More...  
Object  dual (Cell_handle c, int i) const 
same as the previous method for facet (c,i) .  
Line  dual_support (Cell_handle c, int i) const 
returns the line supporting the dual of the facet. More...  
template<class Stream >  
Stream &  draw_dual (Stream &os) 
Sends the set of duals to all the facets of dt into os .  
Checking  
These methods are mainly a debugging help for the users of advanced features.  
bool  is_valid (bool verbose=false) const 
Checks the combinatorial validity of the triangulation and the validity of its geometric embedding (see Section Representation). More...  
bool  is_valid (Cell_handle c, bool verbose=false) const 
This is a function for debugging purpose. More...  
Additional Inherited Members  
Static Public Member Functions inherited from CGAL::Triangulation_utils_3  
static unsigned int  next_around_edge (unsigned int i, unsigned int j) 
static int  vertex_triple_index (const int i, const int j) 
static unsigned int  ccw (unsigned int i) 
static unsigned int  cw (unsigned int i) 
CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::Delaunay_triangulation_3  (  const Geom_traits &  traits = Geom_traits() , 
Lock_data_structure *  lock_ds = nullptr 

) 
Creates an empty Delaunay triangulation, possibly specifying a traits class traits
.
lock_ds
is an optional pointer to the lock data structure for parallel operations. It must be provided if concurrency is enabled.
CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::Delaunay_triangulation_3  (  const Delaunay_triangulation_3< Traits, TDS, LP, SLDS > &  dt1  ) 
Copy constructor.
The pointer to the lock data structure is not copied. Thus, the copy won't be concurrencysafe as long as the user has not called Triangulation_3::set_lock_data_structure()
.
CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::Delaunay_triangulation_3  (  InputIterator  first, 
InputIterator  last,  
const Geom_traits &  traits = Geom_traits() , 

Lock_data_structure *  lock_ds = nullptr 

) 
Equivalent to constructing an empty triangulation with the optional traits class argument and calling insert(first,last)
.
If parallelism is enabled, the points will be inserted in parallel.
Point CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::dual  (  Cell_handle  c  )  const 
Returns the circumcenter of the four vertices of c.
dt
.dimension()
\( =3\) and c
is not infinite. Object CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::dual  (  Facet  f  )  const 
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.
dt
.dimension()
\( \geq2\) and f
is not infinite. Line CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::dual_support  (  Cell_handle  c, 
int  i  
)  const 
returns the line supporting the dual of the facet.
dt
.dimension()
\( \geq2\) and f
is not infinite. std::pair<OutputIteratorBoundaryFacets, OutputIteratorCells> CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::find_conflicts  (  const Point &  p, 
Cell_handle  c,  
OutputIteratorBoundaryFacets  bfit,  
OutputIteratorCells  cit,  
bool *  could_lock_zone = nullptr 

) 
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.could_lock_zone
: The optional argument could_lock_zone
is used by the concurrencysafe version of the triangulation. If the pointer is not null, the algorithm will try to lock all the cells of the conflict zone, i.e. all the vertices that are inside or on the boundary of the conflict zone (as a result, the boundary cells become partially locked). If it succeeds, *could_lock_zone
is true, otherwise it is false (and the returned conflict zone is only partial). In any case, the locked cells are not unlocked by the function, leaving this choice to the user.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.
dt
.dimension()
\( \geq2\), and c
is in conflict with p
. Triple<OutputIteratorBoundaryFacets, OutputIteratorCells, OutputIteratorInternalFacets> CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::find_conflicts  (  const Point &  p, 
Cell_handle  c,  
OutputIteratorBoundaryFacets  bfit,  
OutputIteratorCells  cit,  
OutputIteratorInternalFacets  ifit,  
bool *  could_lock_zone = nullptr 

) 
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.could_lock_zone
: The optional argument could_lock_zone
is used by the concurrencysafe version of the triangulation. If the pointer is not null, the algorithm will try to lock all the cells of the conflict zone, i.e. all the vertices that are inside or on the boundary of the conflict zone (as a result, the boundary cells become partially locked). If it succeeds, *could_lock_zone
is true, otherwise it is false (and the returned conflict zone is only partial). In any case, the locked cells are not unlocked by the function, leaving this choice to the user.Returns the Triple
composed of the resulting output iterators.
dt
.dimension()
\( \geq2\), and c
is in conflict with p
. Vertex_handle CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::insert  (  const Point &  p, 
Cell_handle  start = Cell_handle() , 

bool *  could_lock_zone = nullptr 

) 
Inserts the 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.
The optional argument could_lock_zone
is used by the concurrencysafe version of the triangulation. If the pointer is not null, the insertion will try to lock all the cells of the conflict zone, i.e. all the vertices that are inside or on the boundary of the conflict zone. If it succeeds, *could_lock_zone
is true, otherwise it is false and the return value is Vertex_handle() (the point is not inserted). In any case, the locked cells are not unlocked by the function, leaving this choice to the user.
Vertex_handle CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::insert  (  const Point &  p, 
Locate_type  lt,  
Cell_handle  loc,  
int  li,  
int  lj,  
bool *  could_lock_zone = nullptr 

) 
Inserts the 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()
.
std::ptrdiff_t CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::insert  (  PointInputIterator  first, 
PointInputIterator  last  
) 
Inserts the points in the iterator range [first,last)
.
Returns the number of inserted points. Note that this function is not guaranteed to insert the points following the order of PointInputIterator
, as spatial_sort()
is used to improve efficiency. If parallelism is enabled, the points will be inserted in parallel.
PointInputIterator  must be an input iterator with the value type Point . 
std::ptrdiff_t CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::insert  (  PointWithInfoInputIterator  first, 
PointWithInfoInputIterator  last  
) 
Inserts the points in the iterator range [first,last)
.
Returns the number of inserted points. Note that this function is not guaranteed to insert the points following the order of PointWithInfoInputIterator
, as spatial_sort()
is used to improve efficiency. If parallelism is enabled, the points will be inserted in parallel. Given a pair (p,i)
, the vertex v
storing p
also stores i
, that is v.point() == p
and v.info() == i
. If several pairs have the same point, only one vertex is created, and one of the objects of type Vertex::Info
will be stored in the vertex.
Vertex
must be model of the concept TriangulationVertexBaseWithInfo_3
.PointWithInfoInputIterator  must be an input iterator with the value type std::pair<Point,Vertex::Info> . 
bool CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::is_valid  (  bool  verbose = false  )  const 
Checks the combinatorial validity of the triangulation and the validity of its geometric embedding (see Section Representation).
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.
bool CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::is_valid  (  Cell_handle  c, 
bool  verbose = false 

)  const 
This is a function for debugging purpose.
Checks the combinatorial and geometric validity of the cell (see Section Representation). 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.
Vertex_handle CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::move  (  Vertex_handle  v, 
const Point &  p  
) 
If there is no collision during the move, this function is the same as move_if_no_collision
.
Otherwise, v
is removed and the vertex at point p
is returned.
v
must be finite. Vertex_handle CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::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. Vertex_handle CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::nearest_vertex  (  const Point &  p, 
Cell_handle  c = Cell_handle() 

) 
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.
c
is a cell of dt
. void CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::remove  (  Vertex_handle  v  ) 
Removes the vertex v
from the triangulation.
v
is a finite vertex of the triangulation. bool CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::remove  (  Vertex_handle  v, 
bool *  could_lock_zone  
) 
Removes the vertex v
from the triangulation.
This function is concurrencysafe if the triangulation is concurrencysafe. It will first try to lock all the cells adjacent to v
. If it succeeds, *could_lock_zone
is true, otherwise it is false (and the point is not removed). In any case, the locked cells are not unlocked by the function, leaving this choice to the user.
This function will try to remove v
only if the removal does not decrease the dimension.
The return value is only meaningful if *could_lock_zone
is true
:
v
is a finite vertex of the triangulation. dt
.dimension()
\( =3\). int CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::remove  (  InputIterator  first, 
InputIterator  beyond  
) 
Removes the vertices specified by the iterator range [first, beyond)
.
The number of vertices removed is returned. If parallelism is enabled, the points will be removed in parallel. Note that if at some step, the triangulation dimension becomes lower than 3, the removal of the remaining points will go on sequentially.
InputIterator  must be an input iterator with value type Vertex_handle . 
int CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::remove_cluster  (  InputIterator  first, 
InputIterator  beyond  
) 
This function has exactly the same result and the same preconditions as remove(first, beyond)
.
The difference is in the implementation and efficiency. This version does not retriangulate the hole after each point removal but only after removing all vertices. This is more efficient if (and only if) the removed points are organized in a small number of connected components of the Delaunay triangulation. Another difference is that there is no parallel version of this function.
InputIterator  must be an input iterator with value type Vertex_handle . 
Bounded_side CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::side_of_circle  (  const Facet &  f, 
const Point &  p  
)  const 
Returns a value indicating on which side of the circumscribed circle of f
the point p
lies.
More precisely, it returns:
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.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.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. dt
.dimension()
\( \geq2\) and in dimension 3, p
is coplanar with f
. Bounded_side CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::side_of_sphere  (  Cell_handle  c, 
const Point &  p  
)  const 
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. dt
.dimension()
\( =3\). OutputIterator CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::vertices_in_conflict  (  const Point &  p, 
Cell_handle  c,  
OutputIterator  res  
) 
vertices_on_conflict_zone_boundary
since CGAL3.8.OutputIterator CGAL::Delaunay_triangulation_3< Traits, TDS, LP, SLDS >::vertices_on_conflict_zone_boundary  (  const Point &  p, 
Cell_handle  c,  
OutputIterator  res  
) 
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.
dt
.dimension()
\( \geq2\), and c
is in conflict with p
.