#include <CGAL/Periodic_3_Delaunay_triangulation_3.h>

Inherits from

CGAL::Periodic_3_triangulation_3< Periodic_3DelaunayTriangulationTraits_3, TriangulationDataStructure_3 >.

Definition

The class Periodic_3_Delaunay_triangulation_3 represents a Delaunay triangulation in three-dimensional periodic space.

Template Parameters

The first template argument PT must be a model of the Periodic_3DelaunayTriangulationTraits_3 concept.

The second template argument TDS must be a model of the TriangulationDataStructure_3 concept with some additional functionality in cells and vertices. Its default value is Triangulation_data_structure_3<Triangulation_vertex_base_3<PT,Periodic_3_triangulation_ds_vertex_base_3<>>,Triangulation_cell_base_3<PT,Periodic_3_triangulation_ds_cell_base_3<>>>.

See Also: Periodic_3_triangulation_3

Examples:: Periodic_3_triangulation_3/colored_vertices.cpp, Periodic_3_triangulation_3/covering.cpp, Periodic_3_triangulation_3/geometric_access.cpp, Periodic_3_triangulation_3/large_point_set.cpp, Periodic_3_triangulation_3/periodic_adding_handles.cpp, and Periodic_3_triangulation_3/simple_example.cpp.

Creation
	Periodic_3_Delaunay_triangulation_3 (const Iso_cuboid &domain=Iso_cuboid(0, 0, 0, 1, 1, 1), const Geom_traits &traits=Geom_traits())
	Creates an empty periodic Delaunay triangulation `dt`, with `domain` as original domain and possibly specifying a traits class `traits`. More...

	Periodic_3_Delaunay_triangulation_3 (const Periodic_3_Delaunay_triangulation_3 &dt1)
	Copy constructor. More...

template<class InputIterator >
	Periodic_3_Delaunay_triangulation_3 (InputIterator first, InputIterator last, const Iso_cuboid &domain=Iso_cuboid(0, 0, 0, 1, 1, 1), const Geom_traits &traits=Geom_traits())
	Equivalent to constructing an empty triangulation with the optional domain and traits class arguments and calling `insert(first,last)`. More...

Insertion
The following methods insert points in the triangulation ensuring the empty sphere property of Delaunay triangulations. The inserted points need to lie in the original domain (see Section The Flat Torus of the user manual). In the degenerate case when there are co-spherical 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 [2]. Note that insertion of a new point can cause a switch from computing in the 27-sheeted covering space to computing in the 1-sheeted covering space, which invalidates some `Vertex_handle`s and `Cell_handle`s.
Vertex_handle	insert (const Point &p, Cell_handle start=Cell_handle())
	Inserts point `p` in the triangulation and returns the corresponding vertex. More...

Vertex_handle	insert (const Point &p, Locate_type lt, Cell_handle loc, int li, int lj)
	Inserts point `p` in the triangulation and returns the corresponding vertex. More...

The following method allows one to insert several points. It returns the number of inserted points.
template<class InputIterator >
std::ptrdiff_t	insert (InputIterator first, InputIterator last, bool is_large_point_set=false)
	Inserts the points in the iterator range \( \left[\right.\)`first`, `last` \( \left.\right)\). More...

Point moving
Vertex_handle	move_point (Vertex_handle v, const Point &p)
	Moves the point stored in `v` to `p`, while preserving the Delaunay property. More...

Removal
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 re-triangulated. 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 [4]. However, when dealing with Delaunay triangulations, the case of such polyhedra that cannot be re-triangulated cannot happen, so CGAL proposes a vertex removal.
void	remove (Vertex_handle v)
	Removes the vertex `v` from the triangulation. More...

template<class InputIterator >
std::ptrdiff_t	remove (InputIterator first, InputIterator beyond)
	Removes the vertices specified by the iterator range (`first, beyond`) of value type `Vertex_handle`. More...

Queries
Bounded_side	side_of_sphere (Cell_handle c, const Point &p, const Offset &off=Offset(0, 0, 0)) const
	Returns a value indicating on which side of the circumscribed sphere of `c` the point-offset pair (`p`,`off`) lies. More...

Vertex_handle	nearest_vertex (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 (Cell_handle c, Point p, Offset off=Offset(0, 0, 0)) const
	Returns the vertex of the cell `c` that is nearest to the point-offset pair (`p`,`off`). More...

A point-offset pair (`p`,`off`) is said to be in conflict with a cell `c` iff `dt.side_of_sphere(c, p, off)` returns `ON_BOUNDED_SIDE`. The set of cells that are in conflict with (`p`,`off`) is star-shaped.
template<class OutputIteratorBoundaryFacets , class OutputIteratorCells , class OutputIteratorInternalFacets >
Triple < OutputIteratorBoundaryFacets, OutputIteratorCells, OutputIteratorInternalFacets >	find_conflicts (Point p, Cell_handle c, OutputIteratorBoundaryFacets bfit, OutputIteratorCells cit, OutputIteratorInternalFacets ifit)
	Computes the conflict hole induced by `p`. More...

template<class OutputIterator >
OutputIterator	vertices_in_conflict (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 functions to display the Voronoi diagram of a set of points in 3D. Note that a traits class providing exact constructions should be used 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 representative of the circumcenter of the four vertices of c that lies in the original domain `domain`. More...

Periodic_segment	dual (Facet f) const
	Returns the dual of facet `f`, which is a periodic segment. More...

Periodic_segment	dual (Cell_handle c, int i) const
	same as the previous method for facet `(c,i)`. More...

template<class OutputIterator >
OutputIterator	dual (Edge e, OutputIterator pts) const
	Returns in the output iterator the points of the dual polygon of edge `e` in the same order as the `Facet_circulator` returns facets incident to the edge `e`. More...

template<class OutputIterator >
OutputIterator	dual (Cell_handle c, int i, int j, OutputIterator pts) const
	same as the previous method for edge `(c,i,j)`. More...

template<class OutputIterator >
OutputIterator	dual (Vertex_handle v, OutputIterator pts) const
	Returns in the output iterator the points of the dual polyhedron of vertex `v` in no particular order. More...

template<class Stream >
Stream &	draw_dual (Stream &os)
	Sends the set of duals to all the facets of `dt` into `os`. More...

Geom_traits::FT	dual_volume (Vertex_handle v) const
	Returns the volume of the Voronoi cell dual to `v`. More...

Point	dual_centroid (Vertex_handle v) const
	Returns the centroid of the Voronoi cell dual to `v`. More...

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
	Checks the combinatorial and geometric validity of the cell (see Section Representation). More...

Constructor & Destructor Documentation

template<typename PT , typename TDS >

CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::Periodic_3_Delaunay_triangulation_3	(	const Iso_cuboid &	domain = `Iso_cuboid(0, 0, 0, 1, 1, 1)`,
		const Geom_traits &	traits = `Geom_traits()`
	)

Creates an empty periodic Delaunay triangulation dt, with domain as original domain and possibly specifying a traits class traits.

Precondition: domain is a cube.

template<typename PT , typename TDS >

CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::Periodic_3_Delaunay_triangulation_3 ( const Periodic_3_Delaunay_triangulation_3< PT, TDS > & dt1 )

Copy constructor.

template<typename PT , typename TDS >

template<class InputIterator >

CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::Periodic_3_Delaunay_triangulation_3	(	InputIterator	first,
		InputIterator	last,
		const Iso_cuboid &	domain = `Iso_cuboid(0, 0, 0, 1, 1, 1)`,
		const Geom_traits &	traits = `Geom_traits()`
	)

Equivalent to constructing an empty triangulation with the optional domain and traits class arguments and calling insert(first,last).

Precondition: The value_type of first and last are Points lying inside domain and domain is a cube.

Member Function Documentation

template<typename PT , typename TDS >

template<class Stream >

Stream& CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::draw_dual ( Stream & os )

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

template<typename PT , typename TDS >

Point CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::dual ( Cell_handle c ) const

Returns the representative of the circumcenter of the four vertices of c that lies in the original domain domain.

template<typename PT , typename TDS >

Periodic_segment CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::dual ( Facet f ) const

Returns the dual of facet f, which is a periodic segment.

template<typename PT , typename TDS >

Periodic_segment CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::dual	(	Cell_handle	c,
		int	i
	)		const

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

Precondition: \( i\in\{0,1,2,3\}\)

template<typename PT , typename TDS >

template<class OutputIterator >

OutputIterator CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::dual	(	Edge	e,
		OutputIterator	pts
	)		const

Returns in the output iterator the points of the dual polygon of edge e in the same order as the Facet_circulator returns facets incident to the edge e.

The points form the dual polygon in \( \mathbb R^3\), so they do not necessarily all lie inside the original domain.

template<typename PT , typename TDS >

template<class OutputIterator >

OutputIterator CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::dual	(	Cell_handle	c,
		int	i,
		int	j,
		OutputIterator	pts
	)		const

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

Precondition: \( i,j\in\{0,1,2,3\}, i\neq j\)

template<typename PT , typename TDS >

template<class OutputIterator >

OutputIterator CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::dual	(	Vertex_handle	v,
		OutputIterator	pts
	)		const

Returns in the output iterator the points of the dual polyhedron of vertex v in no particular order.

The points form the dual polyhedron in \( \mathbb R^3\), so they do not necessarily lie all inside the original domain.

template<typename PT , typename TDS >

Point CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::dual_centroid ( Vertex_handle v ) const

Returns the centroid of the Voronoi cell dual to v.

template<typename PT , typename TDS >

Geom_traits::FT CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::dual_volume ( Vertex_handle v ) const

Returns the volume of the Voronoi cell dual to v.

template<typename PT , typename TDS >

template<class OutputIteratorBoundaryFacets , class OutputIteratorCells , class OutputIteratorInternalFacets >

Triple<OutputIteratorBoundaryFacets, OutputIteratorCells, OutputIteratorInternalFacets> CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::find_conflicts	(	Point	p,
		Cell_handle	c,
		OutputIteratorBoundaryFacets	bfit,
		OutputIteratorCells	cit,
		OutputIteratorInternalFacets	ifit
	)

Computes the conflict hole induced by p.

The starting cell c must be in conflict. Then this function returns respectively in the output iterators:

cit: the cells in conflict.
bfit: the facets on the boundary, that is, the facets (t, i) where the cell t is in conflict, but t->neighbor(i) is not.
ifit: the facets inside the hole, that is, delimiting two cells in conflict.

Returns the pair composed of the resulting output iterators.

Precondition: c is in conflict with p and p lies in the original domain domain.

template<typename PT , typename TDS >

Vertex_handle CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::insert	(	const Point &	p,
		Cell_handle	start = `Cell_handle()`
	)

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

The optional argument start is used as a starting place for the point location.

Precondition: p lies in the original domain domain.

Examples:: Periodic_3_triangulation_3/colored_vertices.cpp, Periodic_3_triangulation_3/covering.cpp, Periodic_3_triangulation_3/geometric_access.cpp, and Periodic_3_triangulation_3/periodic_adding_handles.cpp.

template<typename PT , typename TDS >

Vertex_handle CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::insert	(	const Point &	p,
		Locate_type	lt,
		Cell_handle	loc,
		int	li,
		int	lj
	)

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 Periodic_3_triangulation_3::locate().

Precondition: p lies in the original domain domain.

template<typename PT , typename TDS >

template<class InputIterator >

std::ptrdiff_t CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::insert	(	InputIterator	first,
		InputIterator	last,
		bool	is_large_point_set = `false`
	)

Inserts the points in the iterator range \( \left[\right.\)first, last \( \left.\right)\).

Returns the number of inserted points. This function uses spatial sorting (cf. chapter Spatial Sorting) and therefore is not guaranteed to insert the points following the order of InputIterator. If the third argument is_large_point_set is set to true a heuristic for optimizing the insertion of large point sets is applied.

Precondition: The value_type of first and last are Points lying inside domain.

template<typename PT , typename TDS >

bool CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::is_Gabriel	(	Cell_handle	c,
		int	i
	)

template<typename PT , typename TDS >

bool CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::is_Gabriel	(	Cell_handle	c,
		int	i,
		int	j
	)

template<typename PT , typename TDS >

bool CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::is_Gabriel ( const Facet & f )

template<typename PT , typename TDS >

bool CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::is_Gabriel ( const Edge & e )

template<typename PT , typename TDS >

bool CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::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 of cells are empty.

When verbose is set to true, messages describing the first invalidity encountered are printed.

template<typename PT , typename TDS >

bool CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::is_valid	(	Cell_handle	c,
		bool	verbose = `false`
	)		const

Checks the combinatorial and geometric validity of the cell (see Section Representation).

Also checks that the circumscribing sphere of cells is empty.

When verbose is set to true, messages are printed to give a precise indication of the kind of invalidity encountered.

template<typename PT , typename TDS >

Vertex_handle CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::move_point	(	Vertex_handle	v,
		const Point &	p
	)

Moves the point stored in v to p, while preserving the Delaunay property.

This performs an action semantically equivalent to remove(v) followed by insert(p), but is supposedly faster when the point has not moved much. Returns the handle to the new vertex.

Precondition: p lies in the original domain domain.

template<typename PT , typename TDS >

Vertex_handle CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::nearest_vertex	(	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. It always returns a vertex corresponding to a point inside domain even if computing in a multiply sheeted covering space.

Precondition: c is a cell of dt and p lies in the original domain domain.

template<typename PT , typename TDS >

Vertex_handle CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::nearest_vertex_in_cell	(	Cell_handle	c,
		Point	p,
		Offset	off = `Offset(0, 0, 0)`
	)		const

Returns the vertex of the cell c that is nearest to the point-offset pair (p,off).

Precondition: p lies in the original domain domain.

template<typename PT , typename TDS >

void CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::remove ( Vertex_handle v )

Removes the vertex v from the triangulation.

When computing in the 27-sheeted covering space it removes all 27 copies of v.

template<typename PT , typename TDS >

template<class InputIterator >

std::ptrdiff_t CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::remove	(	InputIterator	first,
		InputIterator	beyond
	)

Removes the vertices specified by the iterator range (first, beyond) of value type Vertex_handle.

remove() is called for each element of the range. The number of vertices removed is returned; this number does not account for periodic copies of removed vertices.

Precondition: The iterator must not iterate over several periodic copies of the same vertex, use e.g. the Unique_vertex_iterator.

template<typename PT , typename TDS >

Bounded_side CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::side_of_sphere	(	Cell_handle	c,
		const Point &	p,
		const Offset &	off = `Offset(0, 0, 0)`
	)		const

Returns a value indicating on which side of the circumscribed sphere of c the point-offset pair (p,off) lies.

More precisely, it returns:

ON_BOUNDED_SIDE if (p,off) is inside the sphere.
ON_BOUNDARY if (p,off) on the boundary of the sphere.
ON_UNBOUNDED_SIDE if (p,off) lies outside the sphere.
Precondition
p lies in the original domain domain.

template<typename PT , typename TDS >

template<class OutputIterator >

OutputIterator CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >::vertices_in_conflict	(	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.

Precondition: c is in conflict with p and p lies in the original domain domain.

Inherits from

Definition

Creation

Insertion

Point moving

Removal

Queries

Voronoi diagram

Checking

Constructor & Destructor Documentation

Member Function Documentation