#include <CGAL/Regular_triangulation.h>

Inherits from

CGAL::Triangulation< Regular_triangulation_traits_adapter< RegularTriangulationTraits_ >, TriangulationDataStructure_ >.

Definition

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

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.

Warning: The removal of points is not supported yet.

Template Parameters

RegularTriangulationTraits_ is the geometric traits class that provides the geometric types and predicates needed by regular triangulations. RegularTriangulationTraits_ must be a model of the concept RegularTriangulationTraits.

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:

RegularTriangulationTraits_::Dimension
Triangulation_vertex<CGAL::Regular_triangulation_traits_adapter<RegularTriangulationTraits_> >
Triangulation_full_cell<CGAL::Regular_triangulation_traits_adapter<RegularTriangulationTraits_> >.

Regular_triangulation can be defined by specifying only the first parameter, or by using the tag CGAL::Default as the second parameter.

See also: Delaunay_triangulation; Triangulation_data_structure; Regular_triangulation_traits_adapter

Examples:: regular_triangulation.cpp.

Types
typedef RegularTriangulationTraits_::Weighted_point_d	Weighted_point
	A point in Euclidean space with an associated weight.

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

Point Insertion
Vertex_handle	insert (const Weighted_point &p, Full_cell_handle hint=Full_cell_handle())
	Inserts weighted point `p` in the triangulation and returns the corresponding vertex. More...

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

Vertex_handle	insert (const Weighted_point &p, Locate_type lt, const Face &f, const Facet &ft, Full_cell_handle c)
	Inserts weighted point `p` in the triangulation. More...

template<typename ForwardIterator >
std::ptrdiff_t	insert (ForwardIterator s, ForwardIterator e)
	Inserts the weighted points found in range `[s,e)` in the regular triangulation. More...

Vertex_handle	insert_if_in_star (const Weighted_point &p, Vertex_handle star_center, Full_cell_handle start=Full_cell_handle())
	inserts the weighted point `p` in the triangulation on the condition that the vertex `star_center` appears in the conflict zone of `p` (that is, `p` would appear in the star of `star_center` after the insertion). More...

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

Vertex_handle	insert_if_in_star (const Weighted_point &p, Vertex_handle star_center, Locate_type lt, const Face &f, const Facet &ft, Full_cell_handle s)
	inserts the weighted point `p` in the triangulation. More...

Queries
bool	is_in_conflict (const Weighted_point &p, Full_cell_const_handle c) const
	Returns `true` if and only if the point `p` is in conflict with full cell `c` (A weighted point `p` is said to be in conflict with a cell `c` if it has a negative power distance to the power sphere of `c`.)

template<typename OutputIterator >
Facet	compute_conflict_zone (const Weighted_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...

Access Functions
size_type	number_of_vertices () const
	Returns the number of finite vertices that are not hidden.

size_type	number_of_hidden_vertices () const
	Returns the number of hidden vertices.

Additional Inherited Members
Public Types inherited from CGAL::Triangulation< Regular_triangulation_traits_adapter< RegularTriangulationTraits_ >, 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 Regular_triangulation_traits_adapter< RegularTriangulationTraits_ >	Geom_traits
	Type for the model of the `TriangulationTraits_` concept.

typedef Regular_triangulation_traits_adapter< RegularTriangulationTraits_ > ::Point_d	Point
	A point in Euclidean space. More...

typedef Regular_triangulation_traits_adapter< RegularTriangulationTraits_ > ::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< Regular_triangulation_traits_adapter< RegularTriangulationTraits_ >, 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

◆ Regular_triangulation()

template<typename RegularTriangulationTraits_ , typename TriangulationDataStructure_ >

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

Instantiates a regular 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 triangulation stores a copy of the geometric traits gt.

Member Function Documentation

◆ compute_conflict_zone()

template<typename RegularTriangulationTraits_ , typename TriangulationDataStructure_ >

template<typename OutputIterator >

Facet CGAL::Regular_triangulation< RegularTriangulationTraits_, TriangulationDataStructure_ >::compute_conflict_zone	(	const Weighted_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 rt.current_dimension() \( \geq 1\).

◆ insert() [1/3]

template<typename RegularTriangulationTraits_ , typename TriangulationDataStructure_ >

Vertex_handle CGAL::Regular_triangulation< RegularTriangulationTraits_, TriangulationDataStructure_ >::insert	(	const Weighted_point &	p,
		Full_cell_handle	hint = `Full_cell_handle()`
	)

Inserts weighted point p in the triangulation and returns the corresponding vertex.

If this insertion creates a vertex, this vertex is returned.

If p coincides with an existing vertex and has a greater weight, then the existing weighted point becomes hidden and p replaces it as vertex of the triangulation.

If p coincides with an already existing vertex (both point and weights being equal), then this vertex is returned and the triangulation remains unchanged.

Otherwise if p does not appear as a vertex of the triangulation, then it is stored as a hidden point and this method returns the default constructed handle.

Prior to the actual insertion, p is located in the triangulation; hint is used as a starting place for locating p.

◆ insert() [2/3]

template<typename RegularTriangulationTraits_ , typename TriangulationDataStructure_ >

Vertex_handle CGAL::Regular_triangulation< RegularTriangulationTraits_, TriangulationDataStructure_ >::insert	(	const Weighted_point &	p,
		Locate_type	lt,
		const Face &	f,
		const Facet &	ft,
		Full_cell_handle	c
	)

Inserts weighted point p in the triangulation.

Similar to the above insert() function, but takes as additional parameter the return values of a previous location query. See description of Triangulation::locate().

◆ insert() [3/3]

template<typename RegularTriangulationTraits_ , typename TriangulationDataStructure_ >

template<typename ForwardIterator >

std::ptrdiff_t CGAL::Regular_triangulation< RegularTriangulationTraits_, TriangulationDataStructure_ >::insert	(	ForwardIterator	s,
		ForwardIterator	e
	)

Inserts the weighted points found in range [s,e) in the regular triangulation.

Returns the difference between the number of vertices after and before the insertions (it may be negative due to hidden points). Note that this function is not guaranteed to insert the points following the order of ForwardIterator because spatial_sort() is used to improve efficiency.

Template Parameters

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

◆ insert_if_in_star() [1/2]

template<typename RegularTriangulationTraits_ , typename TriangulationDataStructure_ >

Vertex_handle CGAL::Regular_triangulation< RegularTriangulationTraits_, TriangulationDataStructure_ >::insert_if_in_star	(	const Weighted_point &	p,
		Vertex_handle	star_center,
		Full_cell_handle	start = `Full_cell_handle()`
	)

inserts the weighted point p in the triangulation on the condition that the vertex star_center appears in the conflict zone of p (that is, p would appear in the star of star_center after the insertion).

If the insertion of p creates a new vertex, this vertex is returned. Otherwise, a default constructed handle is returned.

If the dimension of the triangulation is 0 and if p coincides with an existing vertex and has a greater weight, then p replaces it as vertex of the triangulation and this vertex is returned.

If p coincides with an already existing vertex, with both point and weights being equal, then this vertex is returned and the triangulation remains unchanged.

By convention, if the insertion of p would increase the dimension of the triangulation, it is inserted.

Prior to the actual insertion, p is located in the triangulation; start is used as a starting place for locating p.

See also: Regular_triangulation::compute_conflict_zone()

◆ insert_if_in_star() [2/2]

template<typename RegularTriangulationTraits_ , typename TriangulationDataStructure_ >

Vertex_handle CGAL::Regular_triangulation< RegularTriangulationTraits_, TriangulationDataStructure_ >::insert_if_in_star	(	const Weighted_point &	p,
		Vertex_handle	star_center,
		Locate_type	lt,
		const Face &	f,
		const Facet &	ft,
		Full_cell_handle	s
	)

inserts the weighted point p in the triangulation.

This function is similar to the above insert_if_in_star() function, but takes as additional parameters the return values of the location query of p.

See also: Triangulation::locate().

Inherits from

Definition

Types

Creation

Point Insertion

Queries

Access Functions

Additional Inherited Members

Constructor & Destructor Documentation

◆ Regular_triangulation()

Member Function Documentation

◆ compute_conflict_zone()

◆ insert() [1/3]

◆ insert() [2/3]

◆ insert() [3/3]

◆ insert_if_in_star() [1/2]

◆ insert_if_in_star() [2/2]