#include <CGAL/Regular_triangulation_3.h>

Inherits from

CGAL::Triangulation_3< Traits, TDS, SLDS >.

Definition

Let \( {S}^{(w)}\) be a set of weighted points in \( \mathbb{R}^3\).

Let \( {p}^{(w)}=(p,w_p), p\in\mathbb{R}^3, w_p\in\mathbb{R}\) and \( {z}^{(w)}=(z,w_z), z\in\mathbb{R}^3, w_z\in\mathbb{R}\) be two weighted points. A weighted point \( {p}^{(w)}=(p,w_p)\) can also be seen as a sphere of center \( p\) and radius \( \sqrt{w_p}\). The power product (or power distance ) between \( {p}^{(w)}\) and \( {z}^{(w)}\) is defined as

\[ \Pi({p}^{(w)},{z}^{(w)}) = {\|{p-z}\|^2-w_p-w_z} \]

where \( \|{p-z}\|\) is the Euclidean distance between \( p\) and \( z\). \( {p}^{(w)}\) and \( {z}^{(w)}\) are said to be orthogonal if \( \Pi{({p}^{(w)}-{z}^{(w)})} = 0\) (see Figure 42.2).

Four weighted points have a unique common orthogonal weighted point called the power sphere. A sphere \( {z}^{(w)}\) is said to be regular if \( \forall {p}^{(w)}\in{S}^{(w)}, \Pi{({p}^{(w)}-{z}^{(w)})}\geq 0\).

A triangulation of \( {S}^{(w)}\) is regular if the power spheres of all simplices are regular.

Template Parameters

Traits	is the geometric traits class, and must be a model of `RegularTriangulationTraits_3`
TDS	is the triangulation data structure and must be a model of `TriangulationDataStructure_3`. `Default` may be used with default value `Triangulation_data_structure_3<Regular_triangulation_vertex_base_3<Traits>, Regular_triangulation_cell_base_3<Traits> >`. Any custom type can be used instead of `Regular_triangulation_vertex_base_3` and `Regular_triangulation_cell_base_3`, provided that they are models of the concepts `RegularTriangulationVertexBase_3` and `RegularTriangulationCellBase_3`, respectively.
SLDS	is an optional parameter to specify the type of the spatial lock data structure. It must be a model of the `SurjectiveLockDataStructure` concept, with `Object` being a `Point`. It is only used if the triangulation data structure used is concurrency-safe (i.e. when `TDS::Concurrency_tag` is `Parallel_tag`). The default value is `Spatial_lock_grid_3<Tag_priority_blocking>` if the triangulation data structure is concurrency-safe, and `void` otherwise. In order to use concurrent operations, the user must provide a reference to a `SLDS` instance via the constructor or `Triangulation_3::set_lock_data_structure`.

If TDS::Concurrency_tag is Parallel_tag, some operations, such as insertion/removal of a range of points, are performed in parallel. See the documentation of the operations for more details.

See also: CGAL::Triangulation_3; CGAL::Delaunay_triangulation_3

Examples:: Triangulation_3/parallel_insertion_and_removal_in_regular_3.cpp, Triangulation_3/regular_3.cpp, and Triangulation_3/regular_with_info_3.cpp.

Types
typedef Traits::Point_3	Bare_point
	The type for points `p` of weighted points \( {p}^{(w)}=(p,w_p)\).

typedef Traits::Weighted_point_3	Weighted_point
	The type for weighted points.

typedef SLDS	Lock_data_structure

Creation
	Regular_triangulation_3 (const Traits &traits=Traits(), Lock_data_structure *lock_ds=NULL)
	Creates an empty regular triangulation, possibly specifying a traits class `traits`. More...

	Regular_triangulation_3 (const Regular_triangulation_3 &rt1)
	Copy constructor. More...

template<class InputIterator >
	Regular_triangulation_3 (InputIterator first, InputIterator last, const Traits &traits=Traits(), Lock_data_structure *lock_ds=NULL)
	Equivalent to constructing an empty triangulation with the optional traits class argument and calling `insert(first,last)`. More...

template<class InputIterator >
	Regular_triangulation_3 (InputIterator first, InputIterator last, Lock_data_structure *lock_ds, const Traits &traits=Traits())
	Same as before, with last two parameters in reverse order.

Insertion
The following methods, which already exist in `Triangulation_3`, are overloaded to ensure the property that all power spheres are regular. The following method allows one to insert several points.
Vertex_handle	insert (const Weighted_point &p, Cell_handle start=Cell_handle(), bool *could_lock_zone=NULL)
	Inserts the weighted point `p` in the triangulation. More...

Vertex_handle	insert (const Weighted_point &p, Vertex_handle hint, bool *could_lock_zone=NULL)
	Same as above but uses `hint` as a starting place for the search.

Vertex_handle	insert (const Weighted_point &p, Locate_type lt, Cell_handle loc, int li, int lj, bool *could_lock_zone=NULL)
	Inserts the weighted point `p` in the triangulation and returns the corresponding vertex. More...

template<class InputIterator >
std::ptrdiff_t	insert (InputIterator first, InputIterator last)
	Inserts the weighted points in the range `[first,last)`. More...

template<class WeightedPointWithInfoInputIterator >
std::ptrdiff_t	insert (WeightedPointWithInfoInputIterator first, WeightedPointWithInfoInputIterator last)
	Inserts the weighted points in the iterator range `[first,last)`. More...

The following methods, which already exist in `Triangulation_3`, are overloaded to ensure that hidden points are well created and maintained.
template<class CellIt >
Vertex_handle	insert_in_hole (const Weighted_point &p, CellIt cell_begin, CellIt cell_end, Cell_handle begin, int i)
	Creates a new vertex by starring a hole. More...

template<class CellIt >
Vertex_handle	insert_in_hole (const Weighted_point &p, CellIt cell_begin, CellIt cell_end, Cell_handle begin, int i, Vertex_handle newv)
	Same as above, except that `newv` will be used as the new vertex, which must have been allocated previously with, e.g. `create_vertex`.

Removal
void	remove (Vertex_handle v)
	Removes the vertex `v` from the triangulation.

bool	remove (Vertex_handle v, bool *could_lock_zone)
	Removes the vertex `v` from the triangulation. More...

template<typename InputIterator >
int	remove (InputIterator first, InputIterator beyond)
	Removes the vertices specified by the iterator range `[first, beyond)`. More...

Queries
Let us remark that \( \Pi({p}^{(w)}-{z}^{(w)}) > 0 \) is equivalent to `p` lies outside the sphere with center `z` and radius \( \sqrt{w_p^2+w_z^2}\). This remark helps provide an intuition about the following predicates. side_of_power_circle
Bounded_side	side_of_power_sphere (Cell_handle c, const Weighted_point &p) const
	Returns the position of the weighted point \( p\) with respect to the power sphere of `c`. More...

Bounded_side	side_of_power_circle (const Facet &f, const Weighted_point &p) const
	Returns the position of the point `p` with respect to the power circle of `f`. More...

Bounded_side	side_of_power_circle (Cell_handle c, int i, const Weighted_point &p) const
	Same as the previous method for facet `i` of cell `c`.

Bounded_side	side_of_power_segment (Cell_handle c, const Weighted_point &p) const
	In dimension 1, returns. More...

Vertex_handle	nearest_power_vertex (const Bare_point &p, Cell_handle c=Cell_handle())
	Returns the vertex of the triangulation which is nearest to \( p\) with respect to the power distance. More...

Vertex_handle	nearest_power_vertex_in_cell (const Bare_point &p, Cell_handle c)
	Returns the vertex of the cell `c` that is nearest to \( p\) with respect to the power distance.

A weighted point `p` is said to be in conflict with a cell `c` in dimension 3 (resp. with a facet `f` in dimension 2) if it has a negative power distance to the power sphere of `c` (resp. to the power circle of `f`). The set of cells (resp. facets in dimension 2) which are in conflict with `p` is connected.
template<class OutputIteratorBoundaryFacets , class OutputIteratorCells , class OutputIteratorInternalFacets >
Triple< OutputIteratorBoundaryFacets, OutputIteratorCells, OutputIteratorInternalFacets >	find_conflicts (const Weighted_point p, Cell_handle c, OutputIteratorBoundaryFacets bfit, OutputIteratorCells cit, OutputIteratorInternalFacets ifit, bool could_lock_zone=NULL, const Facet this_facet_must_be_in_the_cz=NULL, bool *the_facet_is_in_its_cz=NULL)
	Compute the conflicts with `p`. More...

template<class OutputIterator >
OutputIterator	vertices_in_conflict (const Weighted_point &p, Cell_handle c, OutputIterator res)

template<class OutputIterator >
OutputIterator	vertices_on_conflict_zone_boundary (const Weighted_point &p, Cell_handle c, OutputIterator res)
	Similar to `find_conflicts()`, but reports the vertices which are on the boundary of the conflict zone of `p`, in the output iterator `res`. More...

template<class OutputIterator >
OutputIterator	vertices_inside_conflict_zone (const Weighted_point &p, Cell_handle c, OutputIterator res)
	Similar to `find_conflicts()`, but reports the vertices which are in the interior of the conflict zone of `p`, in the output iterator `res`. More...

In the weighted setting, a face (cell, facet, edge or vertex) is said to be a Gabriel face iff the smallest sphere orthogonal to the weighted points associated to its vertices, has a positive power product with the weighted point of any other vertex of the triangulation. Any weighted Gabriel face belongs to the regular triangulation, but the reciprocal is not true. The following member functions test the Gabriel property of the faces of the regular triangulation.
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)

bool	is_Gabriel (Vertex_handle v)

Power Diagram
CGAL offers several functionalities to display the power diagram of a set of points in 3D. Note that the user should use a kernel with exact constructions in order to guarantee the computation of the Voronoi diagram (as opposed to computing the triangulation only, which requires only exact predicates).
Bare_point	dual (Cell_handle c) const
	Returns the weighted circumcenter of the four vertices of c. More...

Object	dual (Facet f) const
	Returns the dual of facet `f`, which is. More...

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

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

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

Additional Inherited Members
Public Types inherited from CGAL::Triangulation_3< Traits, TDS, SLDS >
enum	Locate_type { VERTEX =0, EDGE, FACET, CELL, OUTSIDE_CONVEX_HULL, OUTSIDE_AFFINE_HULL }
	The enum `Locate_type` is defined by `Triangulation_3` to specify which case occurs when locating a point in the triangulation. More...

typedef Traits	Geom_traits

typedef TDS	Triangulation_data_structure

typedef SLDS	Lock_data_structure

typedef Triangulation_data_structure::Vertex::Point	Point

typedef Geom_traits::Segment_3	Segment

typedef Geom_traits::Triangle_3	Triangle

typedef Geom_traits::Tetrahedron_3	Tetrahedron

typedef Triangulation_data_structure::Vertex	Vertex

typedef Triangulation_data_structure::Cell	Cell

typedef Triangulation_data_structure::Facet	Facet

typedef Triangulation_data_structure::Edge	Edge

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_simplex_3< Self >	Simplex
	Reference to a simplex (vertex, edge, facet or cell) of the triangulation.

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	All_cells_iterator
	iterator over cells

typedef Triangulation_data_structure::Facet_iterator	All_facets_iterator
	iterator over facets

typedef Triangulation_data_structure::Edge_iterator	All_edges_iterator
	iterator over edges

typedef Triangulation_data_structure::Vertex_iterator	All_vertices_iterator
	iterator over vertices

typedef unspecified_type	Finite_cells_iterator
	iterator over finite cells

typedef unspecified_type	Finite_facets_iterator
	iterator over finite facets

typedef unspecified_type	Finite_edges_iterator
	iterator over finite edges

typedef unspecified_type	Finite_vertices_iterator
	iterator over finite vertices

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

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 Triangulation_data_structure::Concurrency_tag	Concurrency_tag
	Concurrency tag (from the TDS).

Public Member Functions inherited from CGAL::Triangulation_3< Traits, TDS, SLDS >
	Triangulation_3 (const Geom_traits &traits=Geom_traits(), Lock_data_structure *lock_ds=NULL)
	Introduces a triangulation `t` having only one vertex which is the infinite vertex. More...

	Triangulation_3 (Lock_data_structure *lock_ds=NULL, const Geom_traits &traits=Geom_traits())
	Same as the previous one, but with parameters in reverse order.

	Triangulation_3 (const Triangulation_3 &tr)
	Copy constructor. More...

template<class InputIterator >
	Triangulation_3 (InputIterator first, InputIterator last, const Geom_traits &traits=Geom_traits(), Lock_data_structure *lock_ds=NULL)
	Equivalent to constructing an empty triangulation with the optional traits class argument and calling `insert(first,last)`.

Triangulation_3 &	operator= (const Triangulation_3 &tr)
	The triangulation `tr` is duplicated, and modifying the copy after the duplication does not modify the original. More...

void	swap (Triangulation_3 &tr)
	The triangulations `tr` and `t` are swapped. More...

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

template<class GT , class Tds1 , class Tds2 >
bool	operator== (const Triangulation_3< GT, Tds1 > &t1, const Triangulation_3< GT, Tds2 > &t2)
	Equality operator. More...

template<class GT , class Tds1 , class Tds2 >
bool	operator!= (const Triangulation_3< GT, Tds1 > &t1, const Triangulation_3< GT, Tds2 > &t2)
	The opposite of `operator==`.

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.

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

int	dimension () const
	Returns the dimension of the affine hull.

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

size_type	number_of_cells () const
	Returns the number of cells or 0 if `t.dimension() < 3`.

Vertex_handle	infinite_vertex ()
	Returns the infinite vertex.

void	set_infinite_vertex (Vertex_handle v)
	This is an advanced function. More...

Cell_handle	infinite_cell () const
	Returns a cell incident to the infinite vertex.

size_type	number_of_facets () const
	The number of facets. More...

size_type	number_of_edges () const
	The number of edges. More...

size_type	number_of_finite_cells () const
	The number of finite cells. More...

size_type	number_of_finite_facets () const
	The number of finite facets. More...

size_type	number_of_finite_edges () const
	The number of finite edges. More...

Tetrahedron	tetrahedron (Cell_handle c) const
	Returns the tetrahedron formed by the four vertices of `c`. More...

Triangle	triangle (Cell_handle c, int i) const
	Returns the triangle formed by the three vertices of facet `(c,i)`. More...

Triangle	triangle (const Facet &f) const
	Same as the previous method for facet `f`. More...

Segment	segment (const Edge &e) const
	Returns the line segment formed by the vertices of `e`. More...

Segment	segment (Cell_handle c, int i, int j) const
	Same as the previous method for edge `(c,i,j)`. More...

const Point &	point (Cell_handle c, int i) const
	Returns the point given by vertex `i` of cell `c`. More...

const Point &	point (Vertex_handle v) const
	Same as the previous method for vertex `v`. More...

bool	is_infinite (Vertex_handle v) const
	`true`, iff vertex `v` is the infinite vertex.

bool	is_infinite (Cell_handle c) const
	`true`, iff `c` is incident to the infinite vertex. More...

bool	is_infinite (Cell_handle c, int i) const
	`true`, iff the facet `i` of cell `c` is incident to the infinite vertex. More...

bool	is_infinite (const Facet &f) const
	`true` iff facet `f` is incident to the infinite vertex. More...

bool	is_infinite (Cell_handle c, int i, int j) const
	`true`, iff the edge `(i,j)` of cell `c` is incident to the infinite vertex. More...

bool	is_infinite (const Edge &e) const
	`true` iff edge `e` is incident to the infinite vertex. More...

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_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_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	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`. More...

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
	For these three methods: More...

Cell_handle	locate (const Point &query, Cell_handle start=Cell_handle(), bool *could_lock_zone=NULL) const
	If the point `query` lies inside the convex hull of the points, the cell that contains the query in its interior is returned. More...

Cell_handle	locate (const Point &query, Vertex_handle hint, bool *could_lock_zone=NULL) const
	Same as above but uses `hint` as the starting place for the search.

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

Cell_handle	locate (const Point &query, Locate_type &lt, int &li, int &lj, Cell_handle start=Cell_handle(), bool *could_lock_zone=NULL) const
	If `query` lies inside the affine hull of the points, the \( k\)-face (finite or infinite) 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`, or `OUTSIDE_CONVEX_HULL` if the cell is infinite and `query` lies strictly in it) and two indices `li` and `lj` that specify the \( k\)-face of the cell containing `query`. More...

Cell_handle	locate (const Point &query, Locate_type &lt, int &li, int &lj, Vertex_handle hint, bool *could_lock_zone=NULL) const
	Same as above but uses `hint` as the starting place for the search.

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...

Bounded_side	side_of_facet (const Point &p, const Facet &f, Locate_type &lt, int &li, int &lj) const
	Returns a value indicating on which side of the oriented boundary of `f` the point `p` lies: More...

Bounded_side	side_of_facet (const Point &p, Cell_handle c, Locate_type &lt, int &li, int &lj) const
	Same as the previous method for the facet `(c,3)`.

Bounded_side	side_of_edge (const Point &p, const Edge &e, Locate_type &lt, int &li) const
	Returns a value indicating on which side of the oriented boundary of `e` the point `p` lies: More...

Bounded_side	side_of_edge (const Point &p, Cell_handle c, Locate_type &lt, int &li) const
	Same as the previous method for edge \( (c,0,1)\).

bool	flip (Edge e)

bool	flip (Cell_handle c, int i, int j)
	Before flipping, these methods check that edge `e=(c,i,j)` is flippable (which is quite expensive). More...

void	flip_flippable (Edge e)

void	flip_flippable (Cell_handle c, int i, int j)
	Should be preferred to the previous methods when the edge is known to be flippable. More...

bool	flip (Facet f)

bool	flip (Cell_handle c, int i)
	Before flipping, these methods check that facet `f=(c,i)` is flippable (which is quite expensive). More...

void	flip_flippable (Facet f)

void	flip_flippable (Cell_handle c, int i)
	Should be preferred to the previous methods when the facet is known to be flippable. More...

Vertex_handle	insert (const Point &p, Cell_handle start=Cell_handle())
	Inserts the point `p` in the triangulation and returns the corresponding vertex. More...

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

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

template<class InputIterator >
std::ptrdiff_t	insert (InputIterator first, InputIterator last)
	Inserts the points in the range `[first,last)`. More...

Vertex_handle	insert_in_cell (const Point &p, Cell_handle c)
	Inserts the point `p` in the cell `c`. More...

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

Vertex_handle	insert_in_facet (const Point &p, Cell_handle c, int i)
	As above, insertion in the facet `(c,i)`. More...

Vertex_handle	insert_in_edge (const Point &p, const Edge &e)
	Inserts `p` in the edge `e`. More...

Vertex_handle	insert_in_edge (const Point &p, Cell_handle c, int i, int j)
	As above, inserts `p` in the edge \( (i, j)\) of `c`. More...

Vertex_handle	insert_outside_convex_hull (const Point &p, Cell_handle c)
	The cell `c` must be an infinite cell containing `p`. More...

Vertex_handle	insert_outside_affine_hull (const Point &p)
	`p` is linked to all the points, and the infinite vertex is linked to all the points (including `p`) to triangulate the new infinite face, so that all the points now belong to the boundary of the convex hull. More...

template<class CellIt >
Vertex_handle	insert_in_hole (const Point &p, CellIt cell_begin, CellIt cell_end, Cell_handle begin, int i)
	Creates a new vertex by starring a hole. More...

template<class CellIt >
Vertex_handle	insert_in_hole (const Point &p, CellIt cell_begin, CellIt cell_end, Cell_handle begin, int i, Vertex_handle newv)
	Same as above, except that `newv` will be used as the new vertex, which must have been allocated previously with e.g. `create_vertex`.

Finite_vertices_iterator	finite_vertices_begin () const
	Starts at an arbitrary finite vertex. More...

Finite_vertices_iterator	finite_vertices_end () const
	Past-the-end iterator.

Finite_edges_iterator	finite_edges_begin () const
	Starts at an arbitrary finite edge. More...

Finite_edges_iterator	finite_edges_end () const
	Past-the-end iterator.

Finite_facets_iterator	finite_facets_begin () const
	Starts at an arbitrary finite facet. More...

Finite_facets_iterator	finite_facets_end () const
	Past-the-end iterator.

Finite_cells_iterator	finite_cells_begin () const
	Starts at an arbitrary finite cell. More...

Finite_cells_iterator	finite_cells_end () const
	Past-the-end iterator.

All_vertices_iterator	all_vertices_begin () const
	Starts at an arbitrary vertex. More...

All_vertices_iterator	all_vertices_end () const
	Past-the-end iterator.

All_edges_iterator	all_edges_begin () const
	Starts at an arbitrary edge. More...

All_edges_iterator	all_edges_end () const
	Past-the-end iterator.

All_facets_iterator	all_facets_begin () const
	Starts at an arbitrary facet. More...

All_facets_iterator	all_facets_end () const
	Past-the-end iterator.

All_cells_iterator	all_cells_begin () const
	Starts at an arbitrary cell. More...

All_cells_iterator	all_cells_end () const
	Past-the-end iterator.

Point_iterator	points_begin () const
	Iterates over the points of the triangulation.

Point_iterator	points_end () const
	Past-the-end iterator.

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

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`. More...

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)`.

template<class OutputIterator >
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`. More...

bool	try_lock_and_get_incident_cells (Vertex_handle v, std::vector< Cell_handle > &cells) const
	Try to lock and copy the `Cell_handle`s of all cells incident to `v` into `cells`. More...

template<class OutputIterator >
OutputIterator	finite_incident_cells (Vertex_handle v, OutputIterator cells) const
	Copies the `Cell_handle`s of all finite cells incident to `v` to the output iterator `cells`. More...

template<class OutputIterator >
OutputIterator	incident_facets (Vertex_handle v, OutputIterator facets) const
	Copies all `Facet`s incident to `v` to the output iterator `facets`. More...

template<class OutputIterator >
OutputIterator	finite_incident_facets (Vertex_handle v, OutputIterator facets) const
	Copies all finite `Facet`s incident to `v` to the output iterator `facets`. More...

template<class OutputIterator >
OutputIterator	incident_edges (Vertex_handle v, OutputIterator edges) const
	Copies all `Edge`s incident to `v` to the output iterator `edges`. More...

template<class OutputIterator >
OutputIterator	finite_incident_edges (Vertex_handle v, OutputIterator edges) const
	Copies all finite `Edge`s incident to `v` to the output iterator `edges`. More...

template<class OutputIterator >
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`. More...

template<class OutputIterator >
OutputIterator	finite_adjacent_vertices (Vertex_handle v, OutputIterator vertices) const
	Copies the `Vertex_handle`s of all finite 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 incident 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 seen from the other adjacent cell.

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

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

istream &	operator>> (istream &is, Triangulation_3 &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 cell classes (such as point coordinates), which are provided by overloading the stream operators of the vertex and cell types. More...

ostream &	operator<< (ostream &os, const Triangulation_3 &t)
	Writes the triangulation `t` into `os`.

void	set_lock_data_structure (Lock_data_structure *lock_ds) const
	Set the pointer to the lock data structure.

Static Public Member Functions inherited from CGAL::Triangulation_utils_3
static unsigned int	next_around_edge (unsigned int i, unsigned int j)

static int	vertex_triple_index (const int i, const int j)

static unsigned int	ccw (unsigned int i)

static unsigned int	cw (unsigned int i)

Constructor & Destructor Documentation

◆ Regular_triangulation_3() [1/3]

template<typename Traits , typename TDS , typename SLDS >

CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::Regular_triangulation_3	(	const Traits &	traits = `Traits()`,
		Lock_data_structure *	lock_ds = `NULL`
	)

Creates an empty regular triangulation, possibly specifying a traits class traits.

lock_ds is an optional pointer to the lock data structure for parallel operations. It must be provided if concurrency is enabled.

◆ Regular_triangulation_3() [2/3]

template<typename Traits , typename TDS , typename SLDS >

CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::Regular_triangulation_3 ( const Regular_triangulation_3< Traits, TDS, SLDS > & rt1 )

Copy constructor.

The pointer to the lock data structure is not copied. Thus, the copy won't be concurrency-safe as long as the user has not called Triangulation_3::set_lock_data_structure.

◆ Regular_triangulation_3() [3/3]

template<typename Traits , typename TDS , typename SLDS >

template<class InputIterator >

CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::Regular_triangulation_3	(	InputIterator	first,
		InputIterator	last,
		const Traits &	traits = `Traits()`,
		Lock_data_structure *	lock_ds = `NULL`
	)

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

If parallelism is enabled, the points will be inserted in parallel.

Template Parameters

InputIterator must be an input iterator with value type Weighted_point.

Member Function Documentation

◆ dual() [1/2]

template<typename Traits , typename TDS , typename SLDS >

Bare_point CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::dual ( Cell_handle c ) const

Returns the weighted circumcenter of the four vertices of c.

Precondition: rt.dimension() \( =3\) and c is not infinite.

◆ dual() [2/2]

template<typename Traits , typename TDS , typename SLDS >

Object CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::dual ( Facet f ) const

Returns the dual of facet f, which is.

in dimension 3: either a segment, if the two cells incident to f are finite, or a ray, if one of them is infinite;

in dimension 2: a point.

Precondition: rt.dimension() \( \geq2\) and f is not infinite.

◆ find_conflicts()

template<typename Traits , typename TDS , typename SLDS >

template<class OutputIteratorBoundaryFacets , class OutputIteratorCells , class OutputIteratorInternalFacets >

Triple<OutputIteratorBoundaryFacets, OutputIteratorCells, OutputIteratorInternalFacets> CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::find_conflicts	(	const Weighted_point	p,
		Cell_handle	c,
		OutputIteratorBoundaryFacets	bfit,
		OutputIteratorCells	cit,
		OutputIteratorInternalFacets	ifit,
		bool *	could_lock_zone = `NULL`,
		const Facet *	this_facet_must_be_in_the_cz = `NULL`,
		bool *	the_facet_is_in_its_cz = `NULL`
	)

Compute the conflicts with p.

Parameters

p	The query point.
c	The starting cell.
cit	The cells (resp. facets) in conflict with `p`.
bfit	The facets (resp. edges) on the boundary of the conflict zone, that is, the facets (resp. edges) `(t, i)` where the cell (resp.. facet) `t` is in conflict, but `t->neighbor(i)` is not.
ifit	The facets (resp. edges) inside the conflict zone, that facets incident to two cells (resp. facets) in conflict.
could_lock_zone	The optional argument `could_lock_zone` is used by the concurrency-safe version of the triangulation. If the pointer is not null, the algorithm will try to lock all the cells of the conflict zone, i.e. all the vertices that are inside or on the boundary of the conflict zone (as a result, the boundary cells become partially locked). If it succeeds, `*could_lock_zone` is true, otherwise it is false (and the returned conflict zone is only partial). In any case, the locked cells are not unlocked by the function, leaving this choice to the user.
this_facet_must_be_in_the_cz	If the optional argument `this_facet_must_be_in_the_cz` is not null, the algorithm will check if this facet is in the conflict zone (it may be internal as well as boundary).
the_facet_is_in_its_cz	This argument must be not null if the previous `this_facet_must_be_in_the_cz` argument is not null. The boolean value pointed by this pointer is set to true if *`this_facet_must_be_in_the_cz` is among the internal or boundary facets of the conflict zone, and false otherwise.

Precondition: The starting cell (resp. facet) c must be in conflict with p.; rt.dimension() \( \geq2\), and c is in conflict with p.

Returns: the Triple composed of the resulting output iterators.

◆ insert() [1/4]

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::insert	(	const Weighted_point &	p,
		Cell_handle	start = `Cell_handle()`,
		bool *	could_lock_zone = `NULL`
	)

Inserts the weighted point p in the triangulation.

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

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 (see RegularTriangulationCellBase_3) 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.

The optional argument could_lock_zone is used by the concurrency-safe version of the triangulation. If the pointer is not null, the insertion will try to lock all the cells of the conflict zone, i.e. all the vertices that are inside or on the boundary of the conflict zone. If it succeeds, *could_lock_zone is true, otherwise it is false (and the point is not inserted). In any case, the locked cells are not unlocked by the function, leaving this choice to the user.

◆ insert() [2/4]

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::insert	(	const Weighted_point &	p,
		Locate_type	lt,
		Cell_handle	loc,
		int	li,
		int	lj,
		bool *	could_lock_zone = `NULL`
	)

Inserts the weighted 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 Triangulation_3::locate().

◆ insert() [3/4]

template<typename Traits , typename TDS , typename SLDS >

template<class InputIterator >

std::ptrdiff_t CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::insert	(	InputIterator	first,
		InputIterator	last
	)

Inserts the weighted points in the range [first,last).

It returns the difference of the number of vertices between 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 InputIterator, as spatial_sort() is used to improve efficiency. If parallelism is enabled, the points will be inserted in parallel.

Template Parameters

InputIterator must be an input iterator with value type Weighted_point.

◆ insert() [4/4]

template<typename Traits , typename TDS , typename SLDS >

template<class WeightedPointWithInfoInputIterator >

std::ptrdiff_t CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::insert	(	WeightedPointWithInfoInputIterator	first,
		WeightedPointWithInfoInputIterator	last
	)

Inserts the weighted points in the iterator range [first,last).

It returns the difference of the number of vertices between after and before the insertions (it may be negative due to hidden points). Note that this function is not guaranteed to insert the weighted points following the order of WeightedPointWithInfoInputIterator, as spatial_sort() is used to improve efficiency. If parallelism is enabled, the points will be inserted in parallel. Given a pair (p,i), the vertex v storing p also stores i, that is v.point() == p and v.info() == i. If several pairs have the same point, only one vertex is created, one of the objects of type Vertex::Info will be stored in the vertex.

Precondition: Vertex must be model of the concept TriangulationVertexBaseWithInfo_3.

Template Parameters

(WeightedPointWithInfoInputIterator must be an input iterator with value type std::pair<Weighted_point,Vertex::Info>.

◆ insert_in_hole()

template<typename Traits , typename TDS , typename SLDS >

template<class CellIt >

Vertex_handle CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::insert_in_hole	(	const Weighted_point &	p,
		CellIt	cell_begin,
		CellIt	cell_end,
		Cell_handle	begin,
		int	i
	)

Creates a new vertex by starring a hole.

It takes an iterator range [cell_begin,cell_end) of Cell_handles which specifies a hole: a set of connected cells (resp. facets in dimension 2) which is star-shaped wrt p. (begin, i) is a facet (resp. an edge) on the boundary of the hole, that is, begin belongs to the set of cells (resp. facets) previously described, and begin->neighbor(i) does not. Then this function deletes all the cells (resp. facets) describing the hole, creates a new vertex v, and for each facet (resp. edge) on the boundary of the hole, creates a new cell (resp. facet) with v as vertex. Then v->set_point(p) is called and v is returned.

If the hole contains interior vertices, each of them is hidden by the insertion of p and is stored in the new cell which contains it.

Precondition: rt.dimension() \( \geq2\), the set of cells (resp. facets in dimension 2) is connected, not empty, its boundary is connected, and p lies inside the hole, which is star-shaped wrt p.

◆ is_valid()

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::is_valid ( bool verbose = false ) const

This is a function for debugging purpose.

Debugging Support

Checks the combinatorial validity of the triangulation and the validity of its geometric embedding (see Section Representation). Also checks that all the power spheres (resp. power circles in dimension 2, power segments in dimension 1) of cells (resp. facets in dimension 2, edges in dimension 1) are regular. When verbose is set to true, messages describing the first invalidity encountered are printed. This method is mainly a debugging help for the users of advanced features.

◆ nearest_power_vertex()

template<typename Traits , typename TDS , typename SLDS >

Vertex_handle CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::nearest_power_vertex	(	const Bare_point &	p,
		Cell_handle	c = `Cell_handle()`
	)

Returns the vertex of the triangulation which is nearest to \( p\) with respect to the power distance.

This means that the power of the query point p with respect to the weighted point in the returned vertex is smaller than the power of p with respect to the weighted point for any other vertex. Ties are broken arbitrarily. The default constructed handle is returned if the triangulation is empty. The optional argument c is a hint specifying where to start the search.

Precondition: c is a cell of rt.

◆ remove() [1/2]

template<typename Traits , typename TDS , typename SLDS >

bool CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::remove	(	Vertex_handle	v,
		bool *	could_lock_zone
	)

Removes the vertex v from the triangulation.

This function is concurrency-safe if the triangulation is concurrency-safe. It will first try to lock all the cells adjacent to v. If it succeeds, *could_lock_zone is true, otherwise it is false (and the point is not removed). In any case, the locked cells are not unlocked by the function, leaving this choice to the user.

This function will try to remove v only if the removal does not decrease the dimension. The return value is only meaningful if *could_lock_zone is true:

returns true if the vertex was removed
returns false if the vertex wasn't removed since it would decrease the dimension.

Precondition: v is a finite vertex of the triangulation.; dt.dimension() \( =3\).

◆ remove() [2/2]

template<typename Traits , typename TDS , typename SLDS >

template<typename InputIterator >

int CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::remove	(	InputIterator	first,
		InputIterator	beyond
	)

Removes the vertices specified by the iterator range [first, beyond).

The number of vertices removed is returned. If parallelism is enabled, the points will be removed in parallel. Note that if at some step, the triangulation dimension becomes lower than 3, the removal of the remaining points will go on sequentially.

Precondition: (i) all vertices of the range are finite vertices of the triangulation; and (ii) no vertices are repeated in the range.

Template Parameters

InputIterator must be an input iterator with value type Vertex_handle.

◆ side_of_power_circle()

template<typename Traits , typename TDS , typename SLDS >

Bounded_side CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::side_of_power_circle	(	const Facet &	f,
		const Weighted_point &	p
	)		const

Returns the position of the point p with respect to the power circle of f.

More precisely, it returns:

in dimension 3:
For a finite facet,

ON_BOUNDARY if p is orthogonal to the power circle in the plane of the facet,

ON_UNBOUNDED_SIDE when their angle is less than \( \pi/2\),

ON_BOUNDED_SIDE when it is greater than \( \pi/2\) (see Figure Triangulation3figsidedim2).

For an infinite facet, it considers the plane defined by the finite facet of the cell f.first, and does the same as in dimension 2 in this plane.
in dimension 2:
For a finite facet,

ON_BOUNDARY if p is orthogonal to the circle,

ON_UNBOUNDED_SIDE when the angle between p and the power circle of f is less than \( \pi/2\), ON_BOUNDED_SIDE when it is greater than \( \pi/2\).

For an infinite facet,

ON_BOUNDED_SIDE for a point in the open half plane defined by f and not containing any other point of the triangulation,

ON_UNBOUNDED_SIDE in the other open half plane.

If the point p is collinear with the finite edge e of f, it returns:

ON_BOUNDED_SIDE if \( \Pi({p}^{(w)}-{z(e)}^{(w)})<0\), where \( {z(e)}^{(w)}\) is the power segment of e in the line supporting e,

ON_BOUNDARY if \( \Pi({p}^{(w)}-{z(e)}^{(w)})=0\),

ON_UNBOUNDED_SIDE if \( \Pi({p}^{(w)}-{z(e)}^{(w)})>0\) .

Precondition: rt.dimension() \( \geq2\).

◆ side_of_power_segment()

template<typename Traits , typename TDS , typename SLDS >

Bounded_side CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::side_of_power_segment	(	Cell_handle	c,
		const Weighted_point &	p
	)		const

In dimension 1, returns.

ON_BOUNDED_SIDE if \( \Pi({p}^{(w)}-{z(c)}^{(w)})<0\), where \( {z(c)}^{(w)}\) is the power segment of the edge represented by c,

ON_BOUNDARY if \( \Pi({p}^{(w)}-{z(c)}^{(w)})=0\),

ON_UNBOUNDED_SIDE if \( \Pi({p}^{(w)}-{z(c)}^{(w)})>0\) .

Precondition: rt.dimension() \( = 1\).

◆ side_of_power_sphere()

template<typename Traits , typename TDS , typename SLDS >

Bounded_side CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::side_of_power_sphere	(	Cell_handle	c,
		const Weighted_point &	p
	)		const

Returns the position of the weighted point \( p\) with respect to the power sphere of c.

More precisely, it returns:

ON_BOUNDED_SIDE if \( \Pi({p}^{(w)}-{z(c)}^{(w)})<0\) where \( {z(c)}^{(w)}\) is the power sphere of c. For an infinite cell this means either that p lies strictly in the half space limited by its finite facet and not containing any other point of the triangulation, or that the angle between p and the power circle of the finite facet of c is greater than \( \pi/2\).
ON_BOUNDARY if p is orthogonal to the power sphere of c i.e. \( \Pi({p}^{(w)}-{z(c)}^{(w)})=0\). For an infinite cell this means that p is orthogonal to the power circle of its finite facet.
ON_UNBOUNDED_SIDE if \( \Pi({p}^{(w)}-{z(c)}^{(w)})>0\) i.e. the angle between the weighted point p and the power sphere of c is less than \( \pi/2\) or if these two spheres do not intersect. For an infinite cell this means that p does not satisfy either of the two previous conditions.
Precondition
rt.dimension() \( =3\).

◆ vertices_in_conflict()

template<typename Traits , typename TDS , typename SLDS >

template<class OutputIterator >

OutputIterator CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::vertices_in_conflict	(	const Weighted_point &	p,
		Cell_handle	c,
		OutputIterator	res
	)

Deprecated:: This function is renamed vertices_on_conflict_zone_boundary since CGAL-3.8.

◆ vertices_inside_conflict_zone()

template<typename Traits , typename TDS , typename SLDS >

template<class OutputIterator >

OutputIterator CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::vertices_inside_conflict_zone	(	const Weighted_point &	p,
		Cell_handle	c,
		OutputIterator	res
	)

Similar to find_conflicts(), but reports the vertices which are in the interior of the conflict zone of p, in the output iterator res.

The vertices that are on the boundary of the conflict zone are not reported. Returns the resulting output iterator.

Precondition: rt.dimension() \( \geq2\), and c is a cell containing p.

◆ vertices_on_conflict_zone_boundary()

template<typename Traits , typename TDS , typename SLDS >

template<class OutputIterator >

OutputIterator CGAL::Regular_triangulation_3< Traits, TDS, SLDS >::vertices_on_conflict_zone_boundary	(	const Weighted_point &	p,
		Cell_handle	c,
		OutputIterator	res
	)

Similar to find_conflicts(), but reports the vertices which are on the boundary of the conflict zone of p, in the output iterator res.

Returns the resulting output iterator.

Precondition: rt.dimension() \( \geq2\), and c is a cell containing p.

Inherits from

Definition

Types

Creation

Insertion

Removal

Queries

Power Diagram

Checking

Additional Inherited Members

Constructor & Destructor Documentation

◆ Regular_triangulation_3() [1/3]

◆ Regular_triangulation_3() [2/3]

◆ Regular_triangulation_3() [3/3]

Member Function Documentation

◆ dual() [1/2]

◆ dual() [2/2]

◆ find_conflicts()

◆ insert() [1/4]

◆ insert() [2/4]

◆ insert() [3/4]

◆ insert() [4/4]

◆ insert_in_hole()

◆ is_valid()

◆ nearest_power_vertex()

◆ remove() [1/2]

◆ remove() [2/2]

◆ side_of_power_circle()

◆ side_of_power_segment()

◆ side_of_power_sphere()

◆ vertices_in_conflict()

◆ vertices_inside_conflict_zone()

◆ vertices_on_conflict_zone_boundary()