CGAL 4.7 - 3D Periodic Triangulations
CGAL::Periodic_3_Delaunay_triangulation_3< PT, TDS > Class Template Reference

#include <CGAL/Periodic_3_Delaunay_triangulation_3.h>

## 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<>>>.

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.

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_handles and Cell_handles.

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.

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

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.

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). More...

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

Public Types inherited from CGAL::Periodic_3_triangulation_3< Periodic_3DelaunayTriangulationTraits_3, TriangulationDataStructure_3 >
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
Periodic_3DelaunayTriangulationTraits_3
Geometric_traits

typedef
TriangulationDataStructure_3
Triangulation_data_structure

typedef
Geometric_traits::Periodic_3_offset_3
Offset

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

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

typedef Geometric_traits::Point_3 Point

typedef Geometric_traits::Segment_3 Segment

typedef
Geometric_traits::Triangle_3
Triangle

typedef
Geometric_traits::Tetrahedron_3
Tetrahedron

typedef std::pair< Point, OffsetPeriodic_point
Represents a point-offset pair. More...

typedef array< Periodic_point, 2 > Periodic_segment

typedef array< Periodic_point, 3 > Periodic_triangle

typedef array< Periodic_point, 4 > 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< Periodic_3DelaunayTriangulationTraits_3, TriangulationDataStructure_3 >
Periodic_3_triangulation_3 (const Iso_cuboid &domain=Iso_cuboid(0, 0, 0, 1, 1, 1), const Geometric_traits &traits=Geometric_traits())
Introduces an empty triangulation t with domain as original domain. More...

Periodic_3_triangulation_3 (const Periodic_3_triangulation_3 &tr)
Copy constructor. More...

Periodic_3_triangulation_3operator= (const Periodic_3_triangulation_3 &tr)
The triangulation tr is duplicated, and modifying the copy after the duplication does not modify the original. More...

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

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

const Geometric_traitsgeom_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. More...

Triangulation_data_structuretds ()
This is an advanced function. More...

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

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. More...

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. More...

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. More...

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

size_type number_of_vertices () const
Returns the number of vertices. More...

size_type number_of_cells () const
Returns the number of cells. More...

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

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

size_type number_of_edges () const
Returns the number of edges. More...

size_type number_of_facets () const
Returns the number of facets. More...

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

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

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

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. More...

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). More...

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). More...

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.

Point point (const Periodic_point &p) const
Converts the Periodic_point s (point-offset pair) to the corresponding Point in $$\mathbb R^3$$.

Segment segment (const Periodic_segment &s) const
Converts the Periodic_segment s to a Segment.

Triangle triangle (const Periodic_triangle &t) const
Converts the Periodic_triangle t to a Triangle.

Tetrahedron tetrahedron (const Periodic_tetrahedron &t) const
Converts the Periodic_tetrahedron t to a Tetrahedron.

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

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. More...

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. More...

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. More...

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. More...

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. More...

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. More...

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. More...

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. More...

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). More...

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. More...

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. More...

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

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. More...

Vertex_iterator vertices_begin () const
Starts at an arbitrary vertex. More...

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

Edge_iterator edges_begin () const
Starts at an arbitrary edge. More...

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

Facet_iterator facets_begin () const
Starts at an arbitrary facet. More...

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

Cell_iterator cells_begin () const
Starts at an arbitrary cell. More...

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

Unique_vertex_iterator unique_vertices_begin () const
Starts at an arbitrary vertex. More...

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. More...

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

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

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

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

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

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

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

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. More...

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. More...

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_handles of all cells incident to v to the output iterator cells. More...

OutputIterator incident_facets (Vertex_handle v, OutputIterator facets) const
Copies the Facets incident to v to the output iterator facets. More...

OutputIterator incident_edges (Vertex_handle v, OutputIterator edges) const
Copies the Edges incident to v to the output iterator edges. More...

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

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

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

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. More...

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. More...

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. 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 >
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 >
 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 >
template<class OutputIteratorBoundaryFacets , class OutputIteratorCells , class OutputIteratorInternalFacets >
 Triple 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:
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_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.