#include <CGAL/Delaunay_triangulation.h>

Inherits from

CGAL::Triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >.

Definition

This class is used to maintain the Delaunay triangulation of a set of points in \( \mathbb{R}^D \).

It permits point insertion and removal. The maximal dimension \( D\) can be specified at compile-time or run-time. It should be kept reasonably small, see the performance section in the user manual for what reasonable means.

In a Delaunay triangulation, each face has the so-called Delaunay or empty-ball property: there exists a circumscribing ball whose interior does not contain any vertex of the triangulation. A circumscribing ball of a simplex is a ball having all vertices of the simplex on its boundary.

Template Parameters

DelaunayTriangulationTraits_	is the geometric traits class that provides the geometric types and predicates needed by Delaunay triangulations. `DelaunayTriangulationTraits_` must be a model of the concept `DelaunayTriangulationTraits`.
TriangulationDataStructure_	must be a model of the concept `TriangulationDataStructure`. This model is used to store the faces of the triangulation. The parameter `TriangulationDataStructure_` defaults to `Triangulation_data_structure` whose template parameters are instantiated as follows: `DelaunayTriangulationTraits_::Dimension` `Triangulation_vertex<DelaunayTriangulationTraits_>` `Triangulation_full_cell<DelaunayTriangulationTraits_>`.
Delaunay_triangulation	can be defined by specifying only the first parameter, or by using the tag `CGAL::Default` as the second parameter.

See also: Regular_triangulation; Triangulation_data_structure

Examples:: delaunay_triangulation.cpp.

Creation
	Delaunay_triangulation (int dim, const Geom_traits &gt=Geom_traits())
	Instantiates a Delaunay triangulation with one vertex (the vertex at infinity). More...

Point Removal
Full_cell_handle	remove (Vertex_handle v)
	Remove the vertex `v` from the Delaunay triangulation. More...

template<typename ForwardIterator >
void	remove (ForwardIterator start, ForwardIterator end)
	Remove the vertices pointed by the vertex handles in the range `[start, end)`. More...

Point Insertion
template<typename ForwardIterator >
size_type	insert (ForwardIterator s, ForwardIterator e)
	Inserts the points found in range `[s,e)` in the Delaunay triangulation and ensures that the empty-ball property is preserved. More...

Vertex_handle	insert (const Point &p, Full_cell_handle hint=Full_cell_handle())
	Inserts point `p` in the Delaunay triangulation and ensures that the empty-ball property is preserved. More...

Vertex_handle	insert (const Point &p, Vertex_handle hint)
	Same as above but uses a vertex as starting place for the search.

Vertex_handle	insert (const Point &p, Locate_type lt, const Face &f, const Facet &ft, Full_cell_handle c)
	Inserts the point `p` in the Delaunay triangulation and ensures that the empty-ball property is preserved. More...

Queries
bool	is_in_conflict (const Point &p, Full_cell_const_handle c) const
	Returns `true` if and only if the point `p` is in (Delaunay) conflict with full cell `c` (i.e., the circumscribing ball of \( c\) contains \( p\) in its interior).

template<typename OutputIterator >
Facet	compute_conflict_zone (const Point &p, Full_cell_handle c, OutputIterator out) const
	Outputs handles to the full cells in conflict with point `p` into the `OutputIterator out`. More...

Additional Inherited Members
Public Types inherited from CGAL::Triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >
enum	Locate_type
	Used by `Triangulation` to specify which case occurs when locating a point in the triangulation.

typedef TriangulationDataStructure_::Vertex_handle	Vertex_handle
	handle to a a vertex

typedef TriangulationDataStructure_::Vertex_const_handle	Vertex_const_handle
	const handle to a a vertex

typedef TriangulationDataStructure_::Vertex_iterator	Vertex_iterator
	iterator over all vertices (including the infinite one)

typedef TriangulationDataStructure_::Vertex_const_iterator	Vertex_const_iterator
	const iterator over all vertices (including the infinite one)

typedef unspecified_type	Finite_vertex_iterator
	iterator over finite vertices

typedef unspecified_type	Finite_vertex_const_iterator
	const iterator over finite vertices

typedef TriangulationDataStructure_::Full_cell_handle	Full_cell_handle
	handle to a full cell

typedef TriangulationDataStructure_::Full_cell_const_handle	Full_cell_const_handle
	const handle to a full cell

typedef TriangulationDataStructure_::Full_cell_iterator	Full_cell_iterator
	iterator over all full cells (including the infinite ones)

typedef TriangulationDataStructure_::Full_cell_const_iterator	Full_cell_const_iterator
	const iterator over all full cells (including the infinite ones)

typedef unspecified_type	Finite_full_cell_iterator
	iterator over finite full cells

typedef unspecified_type	Finite_full_cell_const_iterator
	const iterator over finite full cells

typedef TriangulationDataStructure_::Facet_iterator	Facet_iterator
	iterator over all facets (including the infinite ones)

typedef unspecified_type	Finite_facet_iterator
	iterator over finite facets

typedef TriangulationDataStructure_::size_type	size_type
	size type (an unsigned integral type)

typedef TriangulationDataStructure_::difference_type	difference_type
	difference type (a signed integral type)

typedef DelaunayTriangulationTraits_	Geom_traits
	Type for the model of the `TriangulationTraits_` concept.

typedef DelaunayTriangulationTraits_ ::Point_d	Point
	A point in Euclidean space. More...

typedef DelaunayTriangulationTraits_ ::Dimension	Maximal_dimension
	This indicates whether the maximal dimension is static (i.e. if the type of `Maximal_dimension` is `CGAL::Dimension_tag<int dim>`) or dynamic (i.e. if the type of `Maximal_dimension` is `CGAL::Dynamic_dimension_tag`). More...

typedef TriangulationDataStructure_	Triangulation_ds
	The second template parameter: the triangulation data structure.

typedef TriangulationDataStructure_::Vertex	Vertex
	A model of the concept `TriangulationVertex`.

typedef TriangulationDataStructure_::Full_cell	Full_cell
	A model of the concept `TriangulationFullCell`.

typedef TriangulationDataStructure_::Facet	Facet
	The facet class.

typedef TriangulationDataStructure_::Face	Face
	A model of the concept `TriangulationDSFace`.

Public Member Functions inherited from CGAL::Triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >
	Triangulation (int dim, const Geom_traits &gt=Geom_traits())
	Instantiates a triangulation with one vertex (the vertex at infinity). More...

	Triangulation (const Triangulation &t2)
	The copy constructor.

const Triangulation_ds &	tds () const
	Returns a const reference to the underlying triangulation data structure.

Triangulation_ds &	tds ()
	This is an advanced function. More...

const Geom_traits &	geom_traits () const
	Returns a const reference to the geometric traits instance.

int	maximal_dimension () const
	Returns the maximal dimension of the full dimensional cells that can be stored in the triangulation.

int	current_dimension () const
	Returns the dimension of the triangulation (as an embedded manifold).

bool	empty () const
	Returns `true` if the triangulation has no finite vertex. More...

size_type	number_of_vertices () const
	Returns the number of finite vertices in the triangulation.

size_type	number_of_full_cells () const
	Returns the number of full cells of maximal dimension in the triangulation (full cells incident to the vertex at infinity are counted).

Vertex_handle	infinite_vertex () const
	Returns a handle to the vertex at infinity.

Full_cell_handle	infinite_full_cell () const
	Returns a handle to some full cell incident to the vertex at infinity.

size_type	number_of_finite_full_cells () const
	Returns the number of full cells of maximal dimension that are not incident to the vertex at infinity.

bool	is_infinite (Vertex_handle v) const
	Returns `true` if and only if the vertex `v` is the infinite vertex.

bool	is_infinite (Full_cell_handle c) const
	Returns `true` if and only if `c` is incident to the infinite vertex.

bool	is_infinite (const Facet &ft) const
	Returns `true` if and only if facet `ft` is incident to the infinite vertex.

bool	is_infinite (const Face &f) const
	Returns `true` if and only if the face `f` is incident to the infinite vertex.

Full_cell_handle	full_cell (const Facet &f) const
	Returns a full cell containing the facet `f`

int	index_of_covertex (const Facet &f) const
	Returns the index of the vertex of the full cell `c=tr.full_cell(f)` which does not belong to `c`.

Vertex_iterator	vertices_begin ()
	The first vertex of `tr`.

Vertex_iterator	vertices_end ()
	The beyond vertex of `tr`.

Finite_vertex_iterator	finite_vertices_begin ()
	The first finite vertex of `tr`.

Finite_vertex_iterator	finite_vertices_end ()
	The beyond finite vertex of `tr`.

Full_cell_iterator	full_cells_begin ()
	The first full cell of `tr`.

Full_cell_iterator	full_cells_end ()
	The beyond full cell of `tr`.

Finite_full_cell_iterator	finite_full_cells_begin ()
	The first finite full cell of `tr`.

Finite_full_cell_iterator	finite_full_cells_end ()
	The beyond finite full cell of `tr`.

Facet_iterator	facets_begin ()
	Iterator to the first facet of the triangulation.

Facet_iterator	facets_end ()
	Iterator to the beyond facet of the triangulation.

Finite_facet_iterator	finite_facets_begin ()
	Iterator to the first finite facet of the triangulation.

Finite_facet_iterator	finite_facets_end ()
	Iterator to the beyond finite facet of the triangulation.

Full_cell_handle	locate (const Point &query, Full_cell_const_handle hint=Full_cell_handle()) const
	The optional argument `hint` is used as a starting place for the search. More...

Full_cell_handle	locate (const Point &query, Vertex_handle hint) const
	Same as above but `hint` is a vertex and not a full cell.

Full_cell_handle	locate (const Point &query, Locate_type &loc_type, Face &f, Facet &ft, Full_cell_handle hint=Full_cell_handle()) const
	The optional argument `hint` is used as a starting place for the search. More...

Full_cell_handle	locate (const Point &query, Locate_type &loc_type, Face &f, Vertex_handle hint) const
	Same as above but `hint`, the starting place for the search, is a vertex. More...

Vertex_handle	collapse_face (const Point &p, const Face &f)
	Contracts the `Face f` to a single vertex at position `p`. More...

size_type	insert (ForwardIterator s, ForwardIterator e)
	Inserts the points found in range `[s,e)` in the triangulation. More...

Vertex_handle	insert (const Point &p, Full_cell_handle hint=Full_cell_handle())
	Inserts point `p` in the triangulation. More...

Vertex_handle	insert (const Point &p, Vertex_handle hint)
	Same as above but uses a vertex `hint` as the starting place for the search.

Vertex_handle	insert (const Point &p, Locate_type loc_type, Face &f, Facet &ft, Full_cell_handle c)
	Inserts point `p` into the triangulation and returns a handle to the `Vertex` at that position. More...

Vertex_handle	insert_in_hole (const Point &p, ForwardIterator s, ForwardIterator e, const Facet &ft, OutputIterator out)
	Removes the full cells in the range \( C=\)`[s, e)`, inserts a vertex at position `p` and fills the hole by connecting each face of the boundary to `p`. More...

Vertex_handle	insert_in_hole (const Point &p, ForwardIterator s, ForwardIterator e, const Facet &ft)
	Same as above, but the newly created full cells are not retrieved.

Vertex_handle	insert_in_face (const Point &p, const Face &f)
	Inserts point `p` in the triangulation. More...

Vertex_handle	insert_in_facet (const Point &p, const Facet &ft)
	Inserts point `p` in the triangulation. More...

Vertex_handle	insert_in_full_cell (const Point &p, Full_cell_handle c)
	Inserts point `p` in the triangulation. More...

Vertex_handle	insert_outside_convex_hull (const Point &, Full_cell_handle c)
	Inserts point `p` in the triangulation. More...

Vertex_handle	insert_outside_affine_hull (const Point &)
	Inserts point `p` in the triangulation. More...

bool	is_valid (bool verbose=false) const
	This is a function for debugging purpose. More...

bool	are_incident_full_cells_valid (Vertex_const_handle v, bool verbose=false) const
	This is a function for debugging purpose. More...

std::istream &	operator>> (std::istream &is, Triangulation &t)
	Reads the underlying combinatorial triangulation from `is` by calling the corresponding input operator of the triangulation data structure class (note that the infinite vertex is numbered 0), and the non-combinatorial information by calling the corresponding input operators of the vertex and the full cell classes (such as point coordinates), which are provided by overloading the stream operators of the vertex and full cell types. More...

std::ostream &	operator<< (std::ostream &os, const Triangulation &t)
	Writes the triangulation `t` into `os`.

Constructor & Destructor Documentation

◆ Delaunay_triangulation()

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::Delaunay_triangulation	(	int	dim,
		const Geom_traits &	gt = `Geom_traits()`
	)

Instantiates a Delaunay triangulation with one vertex (the vertex at infinity).

See the description of the inherited nested type Triangulation::Maximal_dimension for an explanation of the use of the parameter dim. The complex stores a copy of the geometric traits gt.

Member Function Documentation

◆ compute_conflict_zone()

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

template<typename OutputIterator >

Facet CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::compute_conflict_zone	(	const Point &	p,
		Full_cell_handle	c,
		OutputIterator	out
	)		const

Outputs handles to the full cells in conflict with point p into the OutputIterator out.

The full cell c is used as a starting point for gathering the full cells in conflict with p. A facet (cc,i) on the boundary of the conflict zone with cc in conflict is returned.

Precondition: c is in conflict with p and dt.current_dimension() \( \geq2\).

◆ insert() [1/3]

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

template<typename ForwardIterator >

size_type CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::insert	(	ForwardIterator	s,
		ForwardIterator	e
	)

Inserts the points found in range [s,e) in the Delaunay triangulation and ensures that the empty-ball property is preserved.

Returns the number of vertices actually inserted. (If more than one vertex share the same position in space, only one insertion is counted.)

Template Parameters

ForwardIterator must be an input iterator with the value type Point.

◆ insert() [2/3]

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

Vertex_handle CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::insert	(	const Point &	p,
		Full_cell_handle	hint = `Full_cell_handle()`
	)

Inserts point p in the Delaunay triangulation and ensures that the empty-ball property is preserved.

Returns a Vertex_handle to the vertex of the triangulation with position p. Prior to the actual insertion, p is located in the triangulation; hint is used as a starting place for locating p.

◆ insert() [3/3]

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

Vertex_handle CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::insert	(	const Point &	p,
		Locate_type	lt,
		const Face &	f,
		const Facet &	ft,
		Full_cell_handle	c
	)

Inserts the point p in the Delaunay triangulation and ensures that the empty-ball property is preserved.

Returns a handle to the (possibly newly created) vertex at that position. The behavior depends on the value of lt:

OUTSIDE_AFFINE_HULL: Point p is inserted so as to increase the current dimension of the Delaunay triangulation.
ON_VERTEX: The position of the vertex v described by f is set to p. v is returned.
Anything else: The point p is inserted. the full cell c is assumed to be in conflict with p.

The parameters lt, f, ft and c must be consistent with the localization of point p in the Delaunay triangulation e.g. by a call to Triangulation::locate(const Point &, Locate_type &, Face &, Vertex_handle) const.

◆ remove() [1/2]

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

Full_cell_handle CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::remove ( Vertex_handle v )

Remove the vertex v from the Delaunay triangulation.

If the current dimension of the triangulation has not changed after the removal, then the returned full cell c geometrically contains the removed vertex v (c can be finite or infinite). Otherwise, the default-constructed Full_cell_handle is returned.

Precondition: v is a vertex of the triangulation, different from the infinite_vertex().

◆ remove() [2/2]

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

template<typename ForwardIterator >

void CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::remove	(	ForwardIterator	start,
		ForwardIterator	end
	)

Remove the vertices pointed by the vertex handles in the range [start, end).

Template Parameters

ForwardIterator must be an input iterator with the value type Vertex_handle.

Inherits from

Definition

Creation

Point Removal

Point Insertion

Queries

Additional Inherited Members

Constructor & Destructor Documentation

◆ Delaunay_triangulation()

Member Function Documentation

◆ compute_conflict_zone()

◆ insert() [1/3]

◆ insert() [2/3]

◆ insert() [3/3]

◆ remove() [1/2]

◆ remove() [2/2]