#include <CGAL/Periodic_3_Delaunay_triangulation_3.h>

Inherits from

CGAL::Periodic_3_triangulation_3< PT, TDS >.

Definition

template<typename PT, typename TDS>
class CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS >

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

Template Parameters

PT must be a model of the concept Periodic_3DelaunayTriangulationTraits_3.

TDS

must be a model of the concept TriangulationDataStructure_3 whose vertex and cell are models of Periodic_3TriangulationDSVertexBase_3 and Periodic_3TriangulationDSCellBase_3, respectively. It defaults to:

CGAL::Triangulation_data_structure_3<
  CGAL::Triangulation_vertex_base_3<PT, Periodic_3_triangulation_ds_vertex_base_3<> >,
  CGAL::Triangulation_cell_base_3<PT,Periodic_3_triangulation_ds_cell_base_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`.

	Periodic_3_Delaunay_triangulation_3 (const Periodic_3_Delaunay_triangulation_3 &dt1)
	Copy constructor.

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)`.

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.

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.

The following method allows the insertion of several points and 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` \( \left.\right)\).

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.

Removal
The following methods remove points in the triangulation. 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. The problem thus 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 [5]. 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.

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`.

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.

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.

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`).

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`.

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`.

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	canonical_dual (Cell_handle c) const
	Returns the representative of the circumcenter of the four vertices of c that lies in the original domain.

Point	dual (Cell_handle c) const
	Returns the circumcenter of the four vertices of c.

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

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

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`.

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

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

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

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

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

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).

bool	is_valid (Cell_handle c, bool verbose=false) const
	Checks the combinatorial and geometric validity of the cell (see Section Representation).

Additional Inherited Members
Public Types inherited from CGAL::Periodic_3_triangulation_3< PT, TDS >
enum	Locate_type
	The enum `Locate_type` is defined by `Periodic_3_triangulation_3` to specify which case occurs when locating a point in the triangulation. More...

enum	Iterator_type
	The enum `Iterator_type` is defined by `Periodic_3_triangulation_3` to specify the behavior of geometric iterators. More...

typedef PT	Geom_traits

typedef TDS	Triangulation_data_structure

typedef Geom_traits::Periodic_3_offset_3	Offset

typedef Geom_traits::Iso_cuboid_3	Iso_cuboid
	A type representing an axis-aligned cuboid.

typedef array< int, 3 >	Covering_sheets
	Integer triple to store the number of sheets in each direction of space.

typedef TDS::Vertex::Point	Point
	The point type of the triangulation.

typedef Geom_traits::Point_3	Point_3
	The geometric basic 3D point type.

typedef Geom_traits::Segment_3	Segment

typedef Geom_traits::Triangle_3	Triangle

typedef Geom_traits::Tetrahedron_3	Tetrahedron

typedef std::pair< Point, Offset >	Periodic_point
	Represents a point-offset pair.

typedef std::pair< Point_3, Offset >	Periodic_point_3
	Represents a point-offset pair.

typedef array< Periodic_point, 2 >	Periodic_segment
	A periodic segment.

typedef array< Periodic_point_3, 2 >	Periodic_segment_3
	A periodic segment.

typedef array< Periodic_point, 3 >	Periodic_triangle
	A periodic triangle.

typedef array< Periodic_point_3, 3 >	Periodic_triangle_3
	A periodic triangle.

typedef array< Periodic_point, 4 >	Periodic_tetrahedron
	A periodic tetrahedron.

typedef array< Periodic_point_3, 4 >	Periodic_tetrahedron_3
	A periodic tetrahedron.

typedef Triangulation_data_structure::Vertex	Vertex

typedef Triangulation_data_structure::Cell	Cell

typedef Triangulation_data_structure::Edge	Edge

typedef Triangulation_data_structure::Facet	Facet

typedef Triangulation_data_structure::Vertex_handle	Vertex_handle
	handle to a vertex

typedef Triangulation_data_structure::Cell_handle	Cell_handle
	handle to a cell

typedef Triangulation_data_structure::size_type	size_type
	Size type (an unsigned integral type)

typedef Triangulation_data_structure::difference_type	difference_type
	Difference type (a signed integral type)

typedef Triangulation_data_structure::Cell_iterator	Cell_iterator
	iterator over cells

typedef Triangulation_data_structure::Facet_iterator	Facet_iterator
	iterator over facets

typedef Triangulation_data_structure::Edge_iterator	Edge_iterator
	iterator over edges

typedef Triangulation_data_structure::Vertex_iterator	Vertex_iterator
	iterator over vertices

typedef unspecified_type	Unique_vertex_iterator
	iterator over the vertices whose corresponding points lie in the original domain, i.e. for each set of periodic copies the `Unique_vertex_iterator` iterates over exactly one representative.

typedef Triangulation_data_structure::Cell_circulator	Cell_circulator
	circulator over all cells incident to a given edge

typedef Triangulation_data_structure::Facet_circulator	Facet_circulator
	circulator over all facets incident to a given edge

typedef unspecified_type	Periodic_tetrahedron_iterator
	iterator over the tetrahedra corresponding to cells of the triangulation.

typedef unspecified_type	Periodic_triangle_iterator
	iterator over the triangles corresponding to facets of the triangulation.

typedef unspecified_type	Periodic_segment_iterator
	iterator over the segments corresponding to edges of the triangulation.

typedef unspecified_type	Periodic_point_iterator
	iterator over the points corresponding to vertices of the triangulation.

Public Member Functions inherited from CGAL::Periodic_3_triangulation_3< PT, TDS >
	Periodic_3_triangulation_3 (const Iso_cuboid &domain=Iso_cuboid(0, 0, 0, 1, 1, 1), const Geom_traits &traits=Geom_traits())
	Introduces an empty triangulation `t` with `domain` as original domain.

	Periodic_3_triangulation_3 (const Periodic_3_triangulation_3 &tr)
	Copy constructor.

Periodic_3_triangulation_3 &	operator= (const Periodic_3_triangulation_3 &tr)
	The triangulation `tr` is duplicated, and modifying the copy after the duplication does not modify the original.

void	swap (Periodic_3_triangulation_3 &tr)
	The triangulations `tr` and `t` are swapped.

void	clear ()
	Deletes all vertices and all cells of `t`.

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

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

Iso_cuboid	domain () const
	Returns the original domain.

Covering_sheets	number_of_sheets () const
	This is an advanced function.

Offset	neighbor_offset (Cell_handle ch, int i) const
	Get the offset between the origins of the internal offset coordinate systems of two neighboring cells with respect from ch to its i-th neighbor.

Triangulation_data_structure &	tds ()
	This is an advanced function.

void	set_domain (const Iso_cuboid dom)
	This is an advanced function.

bool	is_extensible_triangulation_in_1_sheet_h1 () const
	The current triangulation remains a triangulation in the 1-sheeted covering space even after adding points if this method returns `true`.

bool	is_extensible_triangulation_in_1_sheet_h2 () const
	The same as `is_extensible_triangulation_in_1_sheet_h1()` but with a more precise heuristic, i.e. it might answer `true` in cases in which `is_extensible_triangulation_in_1_sheet_h1()` would not.

bool	is_triangulation_in_1_sheet () const
	Returns `true` if the current triangulation would still be a triangulation in the 1-sheeted covering space, returns `false` otherwise.

void	convert_to_1_sheeted_covering () const
	Converts the current triangulation into the same periodic triangulation in the 1-sheeted covering space.

void	convert_to_27_sheeted_covering () const
	Converts the current triangulation into the same periodic triangulation in the 27-sheeted covering space.

size_type	number_of_vertices () const
	Returns the number of vertices.

size_type	number_of_cells () const
	Returns the number of cells.

size_type	number_of_stored_vertices () const
	Returns the number of vertices in the data structure.

size_type	number_of_stored_cells () const
	Returns the number of cells in the data structure.

size_type	number_of_edges () const
	Returns the number of edges.

size_type	number_of_facets () const
	Returns the number of facets.

size_type	number_of_stored_edges () const
	Returns the number of edges in the data structure.

size_type	number_of_stored_facets () const
	Returns the number of facets in the data structure.

Periodic_point	periodic_point (const Vertex_handle v) const
	Returns the periodic point given by vertex `v`.

Periodic_point	periodic_point (const Cell_handle c, int i) const
	If `t` is represented in the 1-sheeted covering space, this function returns the periodic point given by the \( i\)-th vertex of cell `c`, that is the point in the original domain and the offset of the vertex in `c`.

Periodic_segment	periodic_segment (const Cell_handle c, int i, int j) const
	Returns the periodic segment formed by the two point-offset pairs corresponding to the two vertices of edge `(c,i,j)`.

Periodic_segment	periodic_segment (const Cell_handle c, Offset offset, int i, int j) const
	Returns the periodic segment formed by the two point-offset pairs corresponding to the two vertices of edge `(c,i,j)`.

Periodic_segment	periodic_segment (const Edge &e) const
	Same as the previous method for edge `e`.

Periodic_triangle	periodic_triangle (const Cell_handle c, int i) const
	Returns the periodic triangle formed by the three point-offset pairs corresponding to the three vertices of facet `(c,i)`.

Periodic_triangle	periodic_triangle (const Cell_handle c, Offset offset, int i) const
	Returns the periodic triangle formed by the three point-offset pairs corresponding to the three vertices of facet `(c,i)`.

Periodic_triangle	periodic_triangle (const Facet &f) const
	Same as the previous method for facet `f`.

Periodic_tetrahedron	periodic_tetrahedron (const Cell_handle c) const
	Returns the periodic tetrahedron formed by the four point-offset pairs corresponding to the four vertices of `c`.

Periodic_tetrahedron	periodic_tetrahedron (const Cell_handle c, Offset offset) const
	Returns the periodic tetrahedron formed by the four point-offset pairs corresponding to the four vertices of `c`.

Point_3	construct_point (const PP &pp) const
	Converts the periodic point of type `PP` to a `Point_3`.

Point_3	construct_point (const Point &p) const
	Converts the `Point` `p` to a `Point_3`.

Point_3	construct_point (const P &p1, const Offset &o1) const
	Same as above, with offsets.

Segment	construct_segment (const PS &s) const
	Converts the periodic segment of type `PS` to a `Segment`.

Segment	construct_segment (const P &p1, const P &p2) const
	Creates a segment from two points.

Segment	construct_segment (const P &p1, const P &p2, const Offset &o1, const Offset &o2) const
	Same as above, with offsets.

Triangle	construct_triangle (const PT &t) const
	Converts the periodic triangle of type `PT` to a `Triangle`.

Triangle	construct_triangle (const P &p1, const P &p2, const P &p3) const
	Creates a triangle from three points.

Triangle	construct_triangle (const P &p1, const P &p2, const P &p3, const Offset &o1, const Offset &o2, const Offset &o3) const
	Same as above, with offsets.

Tetrahedron	construct_tetrahedron (const PT &t) const
	Converts the periodic tetrahedron of type `PT` to a `Tetrahedron`.

Tetrahedron	construct_tetrahedron (const P &p1, const P &p2, const P &p3, const P &p4) const
	Creates a tetrahedron from four points.

Tetrahedron	construct_tetrahedron (const P &p1, const P &p2, const P &p3, const P &p4, const Offset &o1, const Offset &o2, const Offset &o3, const Offset &o4) const
	Same as above, with offsets.

bool	is_vertex (const Point &p, Vertex_handle &v) const
	Tests whether `p` is a vertex of `t` by locating `p` in the triangulation.

bool	is_vertex (Vertex_handle v) const
	Tests whether `v` is a vertex of `t`.

bool	is_edge (Vertex_handle u, Vertex_handle v, Cell_handle &c, int &i, int &j) const
	Tests whether `(u,v)` is an edge of `t`.

bool	is_edge (Vertex_handle u, const Offset &offu, Vertex_handle v, const Offset &offv, Cell_handle &c, int &i, int &j) const
	Tests whether `((u,offu),(v,offu))` is an edge of `t`.

bool	is_facet (Vertex_handle u, Vertex_handle v, Vertex_handle w, Cell_handle &c, int &i, int &j, int &k) const
	Tests whether `(u,v,w)` is a facet of `t`.

bool	is_facet (Vertex_handle u, const Offset &offu, Vertex_handle v, const Offset &offv, Vertex_handle w, const Offset &offw, Cell_handle &c, int &i, int &j, int &k) const
	Tests whether `((u,offu),(v,offv),(w,offw))` is a facet of `t`.

bool	is_cell (Cell_handle c) const
	Tests whether `c` is a cell of `t`.

bool	is_cell (Vertex_handle u, Vertex_handle v, Vertex_handle w, Vertex_handle x, Cell_handle &c, int &i, int &j, int &k, int &l) const
	Tests whether `(u,v,w,x)` is a cell of `t`.

bool	is_cell (Vertex_handle u, Vertex_handle v, Vertex_handle w, Vertex_handle x, Cell_handle &c) const
	Tests whether `(u,v,w,x)` is a cell of `t` and computes this cell `c`.

bool	is_cell (Vertex_handle u, const Offset &offu, Vertex_handle v, const Offset &offv, Vertex_handle w, const Offset &offw, Vertex_handle x, const Offset &offx, Cell_handle &c, int &i, int &j, int &k, int &l) const
	Tests whether `((u,offu),(v,offv),(w,offv),(x,offx))` is a cell of `t`.

bool	is_cell (Vertex_handle u, const Offset &offu, Vertex_handle v, const Offset &offv, Vertex_handle w, const Offset &offw, Vertex_handle x, const Offset &offx, Cell_handle &c) const
	Tests whether `((u,offu),(v,offv),(w,offv),(x,offx))` is a cell of `t` and computes this cell `c`.

bool	has_vertex (const Facet &f, Vertex_handle v, int &j) const
	If `v` is a vertex of `f`, then `j` is the index of `v` in the cell `f.first`, and the method returns `true`.

bool	has_vertex (Cell_handle c, int i, Vertex_handle v, int &j) const
	Same for facet `(c,i)`.

bool	has_vertex (const Facet &f, Vertex_handle v) const

bool	has_vertex (Cell_handle c, int i, Vertex_handle v) const
	Same as the first two methods, but these two methods do not return the index of the vertex.

bool	are_equal (Cell_handle c, int i, Cell_handle n, int j) const

bool	are_equal (const Facet &f, const Facet &g) const

bool	are_equal (const Facet &f, Cell_handle n, int j) const

Cell_handle	locate (const Point &query, Cell_handle start=Cell_handle()) const
	Returns the cell that contains the query in its interior.

Cell_handle	locate (const Point &query, Offset &locate_offset, Cell_handle start=Cell_handle()) const
	Returns the cell that contains the query in its interior.

Cell_handle	locate (const Point &query, Locate_type &lt, int &li, int &lj, Cell_handle start=Cell_handle()) const
	The \( k\)-face that contains `query` in its interior is returned, by means of the cell returned together with `lt`, which is set to the locate type of the query (`VERTEX, EDGE, FACET, CELL`) and two indices `li` and `lj` that specify the \( k\)-face of the cell containing `query`.

Cell_handle	locate (const Point &query, Offset &locate_offset, Locate_type &lt, int &li, int &lj, Cell_handle start=Cell_handle()) const
	The \( k\)-face that contains `query` in its interior is returned, by means of the cell returned together with `lt`, which is set to the locate type of the query (`VERTEX, EDGE, FACET, CELL`) and two indices `li` and `lj` that specify the \( k\)-face of the cell containing `query`.

Cell_handle	inexact_locate (const Point &query, Cell_handle start=Cell_handle()) const
	Same as `locate()` but uses inexact predicates.

Bounded_side	side_of_cell (const Point &p, Cell_handle c, Locate_type &lt, int &li, int &lj) const
	Returns a value indicating on which side of the oriented boundary of `c` the point `p` lies.

Vertex_iterator	vertices_begin () const
	Starts at an arbitrary vertex.

Vertex_iterator	vertices_end () const
	Past-the-end iterator.

Edge_iterator	edges_begin () const
	Starts at an arbitrary edge.

Edge_iterator	edges_end () const
	Past-the-end iterator.

Facet_iterator	facets_begin () const
	Starts at an arbitrary facet.

Facet_iterator	facets_end () const
	Past-the-end iterator.

Cell_iterator	cells_begin () const
	Starts at an arbitrary cell.

Cell_iterator	cells_end () const
	Past-the-end iterator.

Unique_vertex_iterator	unique_vertices_begin () const
	Starts at an arbitrary vertex.

Unique_vertex_iterator	unique_vertices_end () const
	Past-the-end iterator.

Periodic_point_iterator	periodic_points_begin (Iterator_type it=STORED) const
	Iterates over the points of the triangulation.

Periodic_point_iterator	periodic_points_end (Iterator_type it=STORED) const
	Past-the-end iterator.

Periodic_segment_iterator	periodic_segments_begin (Iterator_type it=STORED) const
	Iterates over the segments of the triangulation.

Periodic_segment_iterator	periodic_segments_end (Iterator_type it=STORED) const
	Past-the-end iterator.

Periodic_triangle_iterator	periodic_triangles_begin (Iterator_type it=STORED) const
	Iterates over the triangles of the triangulation.

Periodic_triangle_iterator	periodic_triangles_end (Iterator_type it=STORED) const
	Past-the-end iterator.

Periodic_tetrahedron_iterator	periodic_tetrahedra_begin (Iterator_type it=STORED) const
	Iterates over the tetrahedra of the triangulation.

Periodic_tetrahedron_iterator	periodic_tetrahedra_end (Iterator_type it=STORED) const
	Past-the-end iterator.

Cell_circulator	incident_cells (Edge e) const
	Starts at an arbitrary cell incident to `e`.

Cell_circulator	incident_cells (Cell_handle c, int i, int j) const
	As above for edge `(i,j)` of `c`.

Cell_circulator	incident_cells (Edge e, Cell_handle start) const
	Starts at cell `start`.

Cell_circulator	incident_cells (Cell_handle c, int i, int j, Cell_handle start) const
	As above for edge `(i,j)` of `c`.

Facet_circulator	incident_facets (Edge e) const
	Starts at an arbitrary facet incident to `e`.

Facet_circulator	incident_facets (Cell_handle c, int i, int j) const
	As above for edge `(i,j)` of `c`.

Facet_circulator	incident_facets (Edge e, Facet start) const
	Starts at facet `start`.

Facet_circulator	incident_facets (Edge e, Cell_handle start, int f) const
	Starts at facet of index `f` in `start`.

Facet_circulator	incident_facets (Cell_handle c, int i, int j, Facet start) const
	As above for edge `(i,j)` of `c`.

Facet_circulator	incident_facets (Cell_handle c, int i, int j, Cell_handle start, int f) const
	As above for edge `(i,j)` of `c` and facet `(start,f)`.

OutputIterator	incident_cells (Vertex_handle v, OutputIterator cells) const
	Copies the `Cell_handle`s of all cells incident to `v` to the output iterator `cells`.

OutputIterator	incident_facets (Vertex_handle v, OutputIterator facets) const
	Copies the `Facet`s incident to `v` to the output iterator `facets`.

OutputIterator	incident_edges (Vertex_handle v, OutputIterator edges) const
	Copies the `Edge`s incident to `v` to the output iterator `edges`.

OutputIterator	adjacent_vertices (Vertex_handle v, OutputIterator vertices) const
	Copies the `Vertex_handle`s of all vertices adjacent to `v` to the output iterator `vertices`.

size_type	degree (Vertex_handle v) const
	Returns the degree of a vertex, that is, the number of adjacent vertices.

int	mirror_index (Cell_handle c, int i) const
	Returns the index of `c` in its \( i^{th}\) neighbor.

Vertex_handle	mirror_vertex (Cell_handle c, int i) const
	Returns the vertex of the \( i^{th}\) neighbor of `c` that is opposite to `c`.

Facet	mirror_facet (Facet f) const
	Returns the same facet viewed from the other adjacent cell.

bool	is_valid (bool verbose=false) const
	Checks the combinatorial validity of the triangulation.

bool	is_valid (Cell_handle c, bool verbose=false) const
	Checks the combinatorial validity of the cell by calling the `is_valid` method of the `Triangulation_data_structure` cell class.

Related Functions inherited from CGAL::Periodic_3_triangulation_3< PT, TDS >
istream &	operator>> (istream &is, Periodic_3_triangulation_3 &t)
	Reads a triangulation from `is` and stores it in `t`.

ostream &	operator<< (ostream &os, const Periodic_3_triangulation_3 &t)
	Writes the triangulation `t` into `os`.

bool	operator== (const Periodic_3_triangulation_3< PT, TDS1 > &t1, const Periodic_3_triangulation_3< PT, TDS2 > &t2)
	Equality operator.

bool	operator!= (const Periodic_3_triangulation_3< PT, TDS1 > &t1, const Periodic_3_triangulation_3< PT, TDS2 > &t2)
	The opposite of `operator==`.

Constructor & Destructor Documentation

◆ Periodic_3_Delaunay_triangulation_3() [1/2]

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.

◆ Periodic_3_Delaunay_triangulation_3() [2/2]

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

◆ dual() [1/4]

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 the facet (c,i).

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

◆ dual() [2/4]

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 the edge (c,i,j).

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

◆ dual() [3/4]

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.

◆ dual() [4/4]

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 the 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.

◆ find_conflicts()

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.

◆ insert() [1/3]

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.

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.

◆ insert() [2/3]

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.

◆ insert() [3/3]

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 the original domain.

◆ is_valid() [1/2]

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 (if any) are printed.

◆ is_valid() [2/2]

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 (if any).

◆ move_point()

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.

◆ nearest_vertex()

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 the original domain, even if computing in a multiply sheeted covering space.

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

◆ nearest_vertex_in_cell()

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.

◆ remove() [1/2]

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.

◆ remove() [2/2]

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.

◆ side_of_sphere()

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

◆ vertices_in_conflict()

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.

Inherits from

Definition

Creation

Insertion

Point moving

Removal

Queries

Voronoi diagram

Checking

Additional Inherited Members

Constructor & Destructor Documentation

◆ Periodic_3_Delaunay_triangulation_3() [1/2]

◆ Periodic_3_Delaunay_triangulation_3() [2/2]

Member Function Documentation

◆ dual() [1/4]

◆ dual() [2/4]

◆ dual() [3/4]

◆ dual() [4/4]

◆ find_conflicts()

◆ insert() [1/3]

◆ insert() [2/3]

◆ insert() [3/3]

◆ is_valid() [1/2]

◆ is_valid() [2/2]

◆ move_point()

◆ nearest_vertex()

◆ nearest_vertex_in_cell()

◆ remove() [1/2]

◆ remove() [2/2]

◆ side_of_sphere()

◆ vertices_in_conflict()