CGAL 5.5.2 - dD Triangulations
CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ > Class Template Reference

#include <CGAL/Triangulation.h>

Definition

This class implements triangulations of point sets in dimension $$d$$.

The triangulation covers the convex hull of the input points (the embedded vertices of the triangulation).

To store this triangulation in a triangulation data structure, we turn the set of its faces into a topological sphere by adding a fictitious vertex, called the infinite vertex, as well as infinite simplices incident to boundary faces of the convex hull. Each infinite $$i$$-simplex is incident to the infinite vertex and to an $$(i-1)$$-simplex of the convex hull boundary.

Template Parameters
 TriangulationTraits_ is the geometric traits class that provides the geometric types and predicates needed by triangulations. TriangulationTraits_ must be a model of the concept TriangulationTraits. TriangulationDataStructure_ must be a model of the concept TriangulationDataStructure. This model is used to store the faces of the triangulation. The parameter TriangulationDataStructure_ defaults to Triangulation_data_structure whose template parameters are instantiated as follows: TriangulationTraits_::Dimension Triangulation_vertex Triangulation_full_cell.

The triangulation deduces its maximal dimension from the type TriangulationTraits_::Dimension. This dimension has to match the dimension returned by TriangulationDataStructure_::maximal_dimension().

Input/Output

The information in the iostream is: the current dimension, the number of finite vertices, the non-combinatorial information about vertices (point, etc.), the number of full cells, the indices of the vertices of each full cell, plus the non-combinatorial information about each full cell, then the indices of the neighbors of each full cell, where the index corresponds to the preceding list of full cells.

Triangulation_data_structure<Dimensionality, TriangulationDSVertex_, TriangulationDSFullCell_>
Delaunay_triangulation<DelaunayTriangulationTraits_, TriangulationDataStructure_>
Examples:
triangulation.cpp.

Types

typedef TriangulationTraits_ Geom_traits
Type for the model of the TriangulationTraits_ concept.

typedef TriangulationTraits_::Point_d Point
A point in Euclidean space. More...

typedef TriangulationTraits_::Dimension Maximal_dimension
This indicates whether the maximal dimension is static (i.e. if the type of Maximal_dimension is CGAL::Dimension_tag<int dim>) or dynamic (i.e. if the type of Maximal_dimension is CGAL::Dynamic_dimension_tag). More...

typedef TriangulationDataStructure_ Triangulation_ds
The second template parameter: the triangulation data structure.

typedef TriangulationDataStructure_::Vertex Vertex
A model of the concept TriangulationVertex.

typedef TriangulationDataStructure_::Full_cell Full_cell
A model of the concept TriangulationFullCell.

typedef TriangulationDataStructure_::Facet Facet
The facet class.

typedef TriangulationDataStructure_::Face Face
A model of the concept TriangulationDSFace.

Handles and Iterators

The vertices and full cells of triangulations are accessed through handles and iterators.

A handle is a model of the Handle concept, and supports the two dereference operators: operator* and operator->. Iterators are bidirectional and non-mutable. They are convertible to the corresponding handles, thus the user can pass them directly as arguments to the functions. All handles are model of LessThanComparable and Hashable, that is they can be used as keys in containers such as std::map and boost::unordered_map.

enum  Locate_type {
ON_VERTEX =0, IN_FACE, IN_FACET, IN_FULL_CELL,
OUTSIDE_CONVEX_HULL, OUTSIDE_AFFINE_HULL
}
Used by Triangulation to specify which case occurs when locating a point in the triangulation. More...

typedef TriangulationDataStructure_::Vertex_handle Vertex_handle
handle to a a vertex

typedef TriangulationDataStructure_::Vertex_const_handle Vertex_const_handle
const handle to a a vertex

typedef TriangulationDataStructure_::Vertex_iterator Vertex_iterator
iterator over all vertices (including the infinite one)

typedef TriangulationDataStructure_::Vertex_const_iterator Vertex_const_iterator
const iterator over all vertices (including the infinite one)

typedef unspecified_type Finite_vertex_iterator
iterator over finite vertices

typedef unspecified_type Finite_vertex_const_iterator
const iterator over finite vertices

typedef TriangulationDataStructure_::Full_cell_handle Full_cell_handle
handle to a full cell

typedef TriangulationDataStructure_::Full_cell_const_handle Full_cell_const_handle
const handle to a full cell

typedef TriangulationDataStructure_::Full_cell_iterator Full_cell_iterator
iterator over all full cells (including the infinite ones)

typedef TriangulationDataStructure_::Full_cell_const_iterator Full_cell_const_iterator
const iterator over all full cells (including the infinite ones)

typedef unspecified_type Finite_full_cell_iterator
iterator over finite full cells

typedef unspecified_type Finite_full_cell_const_iterator
const iterator over finite full cells

typedef TriangulationDataStructure_::Facet_iterator Facet_iterator
iterator over all facets (including the infinite ones)

typedef unspecified_type Finite_facet_iterator
iterator over finite facets

typedef TriangulationDataStructure_::size_type size_type
size type (an unsigned integral type)

typedef TriangulationDataStructure_::difference_type difference_type
difference type (a signed integral type)

Creation

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

Triangulation (const Triangulation &t2)
The copy constructor.

Access Functions

const Triangulation_dstds () const
Returns a const reference to the underlying triangulation data structure.

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

const Geom_traitsgeom_traits () const
Returns a const reference to the geometric traits instance.

int maximal_dimension () const
Returns the maximal dimension of the full dimensional cells that can be stored in the triangulation.

int current_dimension () const
Returns the dimension of the triangulation (as an embedded manifold).

bool empty () const
Returns true if the triangulation has no finite vertex. More...

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

size_type number_of_full_cells () const
Returns the number of full cells of maximal dimension in the triangulation (full cells incident to the vertex at infinity are counted).

Vertex_handle infinite_vertex () const
Returns a handle to the vertex at infinity.

Full_cell_handle infinite_full_cell () const
Returns a handle to some full cell incident to the vertex at infinity.

Non-Constant-Time Access Functions

size_type number_of_finite_full_cells () const
Returns the number of full cells of maximal dimension that are not incident to the vertex at infinity.

Tests for Finite and Infinite Elements

bool is_infinite (Vertex_handle v) const
Returns true if and only if the vertex v is the infinite vertex.

bool is_infinite (Full_cell_handle c) const
Returns true if and only if c is incident to the infinite vertex.

bool is_infinite (const Facet &ft) const
Returns true if and only if facet ft is incident to the infinite vertex.

bool is_infinite (const Face &f) const
Returns true if and only if the face f is incident to the infinite vertex.

Faces and Facets

Full_cell_handle full_cell (const Facet &f) const
Returns a full cell containing the facet f

int index_of_covertex (const Facet &f) const
Returns the index of the vertex of the full cell c=tr.full_cell(f) which does not belong to c.

Triangulation Traversal

Vertex_iterator vertices_begin ()
The first vertex of tr.

Vertex_iterator vertices_end ()
The beyond vertex of tr.

Finite_vertex_iterator finite_vertices_begin ()
The first finite vertex of tr.

Finite_vertex_iterator finite_vertices_end ()
The beyond finite vertex of tr.

Full_cell_iterator full_cells_begin ()
The first full cell of tr.

Full_cell_iterator full_cells_end ()
The beyond full cell of tr.

Finite_full_cell_iterator finite_full_cells_begin ()
The first finite full cell of tr.

Finite_full_cell_iterator finite_full_cells_end ()
The beyond finite full cell of tr.

Facet_iterator facets_begin ()
Iterator to the first facet of the triangulation.

Facet_iterator facets_end ()
Iterator to the beyond facet of the triangulation.

Finite_facet_iterator finite_facets_begin ()
Iterator to the first finite facet of the triangulation.

Finite_facet_iterator finite_facets_end ()
Iterator to the beyond finite facet of the triangulation.

Point Location

The class Triangulation provides methods to locate a query point with respect to the triangulation:

Full_cell_handle locate (const Point &query, Full_cell_const_handle hint=Full_cell_handle()) const
The optional argument hint is used as a starting place for the search. More...

Full_cell_handle locate (const Point &query, Vertex_handle hint) const
Same as above but hint is a vertex and not a full cell.

Full_cell_handle locate (const Point &query, Locate_type &loc_type, Face &f, Facet &ft, Full_cell_handle hint=Full_cell_handle()) const
The optional argument hint is used as a starting place for the search. More...

Full_cell_handle locate (const Point &query, Locate_type &loc_type, Face &f, Vertex_handle hint) const
Same as above but hint, the starting place for the search, is a vertex. More...

Removal

Vertex_handle collapse_face (const Point &p, const Face &f)
Contracts the Face f to a single vertex at position p. More...

Point Insertion

The class Triangulation provides functions to insert a given point in the triangulation:

template<typename ForwardIterator >
size_type insert (ForwardIterator s, ForwardIterator e)
Inserts the points found in range [s,e) in the triangulation. More...

Vertex_handle insert (const Point &p, Full_cell_handle hint=Full_cell_handle())
Inserts point p in the triangulation. More...

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

Vertex_handle insert (const Point &p, Locate_type loc_type, Face &f, Facet &ft, Full_cell_handle c)
Inserts point p into the triangulation and returns a handle to the Vertex at that position. More...

template<typename ForwardIterator , typename OutputIterator >
Vertex_handle insert_in_hole (const Point &p, ForwardIterator s, ForwardIterator e, const Facet &ft, OutputIterator out)
Removes the full cells in the range $$C=$$[s, e), inserts a vertex at position p and fills the hole by connecting each face of the boundary to p. More...

template<typename ForwardIterator >
Vertex_handle insert_in_hole (const Point &p, ForwardIterator s, ForwardIterator e, const Facet &ft)
Same as above, but the newly created full cells are not retrieved.

Vertex_handle insert_in_face (const Point &p, const Face &f)
Inserts point p in the triangulation. More...

Vertex_handle insert_in_facet (const Point &p, const Facet &ft)
Inserts point p in the triangulation. More...

Vertex_handle insert_in_full_cell (const Point &p, Full_cell_handle c)
Inserts point p in the triangulation. More...

Vertex_handle insert_outside_convex_hull (const Point &, Full_cell_handle c)
Inserts point p in the triangulation. More...

Vertex_handle insert_outside_affine_hull (const Point &)
Inserts point p in the triangulation. More...

Validity Check

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

bool are_incident_full_cells_valid (Vertex_const_handle v, bool verbose=false) const
This is a function for debugging purpose. More...

Input/Output

std::istream & operator>> (std::istream &is, Triangulation &t)
Reads the underlying combinatorial triangulation from is by calling the corresponding input operator of the triangulation data structure class (note that the infinite vertex is numbered 0), and the non-combinatorial information by calling the corresponding input operators of the vertex and the full cell classes (such as point coordinates), which are provided by overloading the stream operators of the vertex and full cell types. More...

std::ostream & operator<< (std::ostream &os, const Triangulation &t)
Writes the triangulation t into os.

◆ Maximal_dimension

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 typedef TriangulationTraits_::Dimension CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::Maximal_dimension

This indicates whether the maximal dimension is static (i.e. if the type of Maximal_dimension is CGAL::Dimension_tag<int dim>) or dynamic (i.e. if the type of Maximal_dimension is CGAL::Dynamic_dimension_tag).

In the latter case, the dim parameter passed to the constructor of the class is used.

◆ Point

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 typedef TriangulationTraits_::Point_d CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::Point

A point in Euclidean space.

Note that in the context of a Regular_triangulation class (which derives from this class), TriangulationTraits_::Point_d is a weighted point.

◆ Locate_type

template<typename TriangulationTraits_, typename TriangulationDataStructure_>

Used by Triangulation to specify which case occurs when locating a point in the triangulation.

Enumerator
ON_VERTEX

when the located point coincides with a vertex of the triangulation

IN_FACE

when the point is in the interior of a face of dimension equal or less than maximal_dimension() - 2

IN_FACET

when the point is in the interior of a facet

IN_FULL_CELL

when the point is in the interior of a full cell

OUTSIDE_CONVEX_HULL

when the point is outside the convex hull but in the affine hull of the current triangulation

OUTSIDE_AFFINE_HULL

when the point is outside the affine hull of the current triangulation.

◆ Triangulation()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::Triangulation ( int dim, const Geom_traits & gt = Geom_traits() )

Instantiates a triangulation with one vertex (the vertex at infinity).

See the description of the nested type Maximal_dimension above for an explanation of the use of the parameter dim. The triangulation stores a copy of the geometric traits gt.

◆ are_incident_full_cells_valid()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 bool CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::are_incident_full_cells_valid ( Vertex_const_handle v, bool verbose = false ) const

This is a function for debugging purpose.

Debugging Support

Returns true if and only if all finite full cells incident to v have positive orientation. The verbose parameter is not used.

◆ collapse_face()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Vertex_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::collapse_face ( const Point & p, const Face & f )

Contracts the Face f to a single vertex at position p.

Returns a handle to that vertex.

Precondition
The boundary of the union of full cells incident to f must be a triangulation of a sphere of dimension tr.current_dimension()).

◆ empty()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 bool CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::empty ( ) const

Returns true if the triangulation has no finite vertex.

Returns false otherwise.

◆ insert() [1/3]

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
template<typename ForwardIterator >
 size_type CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::insert ( ForwardIterator s, ForwardIterator e )

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

Returns the number of vertices actually inserted. (If several vertices share the same position in space, only the vertex that was actually inserted is counted.)

Template Parameters
 ForwardIterator must be an input iterator with the value type Point.

◆ insert() [2/3]

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Vertex_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::insert ( const Point & p, Full_cell_handle hint = Full_cell_handle() )

Inserts point p in the triangulation.

Returns a Vertex_handle to the vertex of the triangulation with position p. Prior to the actual insertion, p is located in the triangulation; hint is used as a starting place for locating p.

◆ insert() [3/3]

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Vertex_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::insert ( const Point & p, Locate_type loc_type, Face & f, Facet & ft, Full_cell_handle c )

Inserts point p into the triangulation and returns a handle to the Vertex at that position.

The action taken depends on the value of loc_type:

ON_VERTEX
The point of the pVertex described by f is set to p.
IN_FACE
The point p is inserted in the Face f.
IN_FACET
The point p is inserted in the Facet ft.
Anything else
The point p is inserted in the triangulation according to the value of loc_type, using the full cell c.

This method is used internally by the other insert() methods.

◆ insert_in_face()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Vertex_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::insert_in_face ( const Point & p, const Face & f )

Inserts point p in the triangulation.

Precondition
p must lie in the relative interior of f.

◆ insert_in_facet()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Vertex_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::insert_in_facet ( const Point & p, const Facet & ft )

Inserts point p in the triangulation.

Precondition
p must lie in the relative interior of ft.

◆ insert_in_full_cell()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Vertex_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::insert_in_full_cell ( const Point & p, Full_cell_handle c )

Inserts point p in the triangulation.

Precondition
p must lie in the interior of c.

◆ insert_in_hole()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
template<typename ForwardIterator , typename OutputIterator >
 Vertex_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::insert_in_hole ( const Point & p, ForwardIterator s, ForwardIterator e, const Facet & ft, OutputIterator out )

Removes the full cells in the range $$C=$$[s, e), inserts a vertex at position p and fills the hole by connecting each face of the boundary to p.

A Vertex_handle to the new Vertex is returned. The facet ft must lie on the boundary of $$C$$ and its defining full cell, tr.full_cell(ft) must lie inside $$C$$. Handles to the newly created full cells are output in the out output iterator.

Precondition
$$C$$ must be a topological ball, must contain p in its interior and must not contain any vertex of the triangulation.

◆ insert_outside_affine_hull()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Vertex_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::insert_outside_affine_hull ( const Point & )

Inserts point p in the triangulation.

Precondition
p must lie outside the affine hull of tr.

◆ insert_outside_convex_hull()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Vertex_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::insert_outside_convex_hull ( const Point & , Full_cell_handle c )

Inserts point p in the triangulation.

Precondition
p must lie outside the convex hull of tr. The half-space defined by the infinite full cell c must contain p.

◆ is_valid()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 bool CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::is_valid ( bool verbose = false ) const

This is a function for debugging purpose.

Debugging Support

Partially checks whether tr is a triangulation. This function returns true if the combinatorial triangulation data structure's is_valid() test returns true and if some geometric tests are passed with success. It is checked that the orientation of each finite full cell is positive and that the orientation of each infinite full cell is consistent with their finite adjacent full cells. The verbose parameter is not used.

◆ locate() [1/3]

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Full_cell_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::locate ( const Point & query, Full_cell_const_handle hint = Full_cell_handle() ) const

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

If the query point lies outside the affine hull of the points (which can happen when tr.current_dimension() < tr.maximal_dimension()) or if there is no finite vertex yet in the triangulation, then locate returns a default constructed Full_cell_handle().

If the point query lies in the interior of a bounded (finite) full cell of tr, the latter full cell is returned.

If query lies on the boundary of some finite full cells, one of the cells is returned.

Let $$d=$$tr.current_dimension(). If the point query lies outside the convex hull of the points, an infinite full cell with vertices $$\{ p_1, p_2, \ldots, p_d, \infty\}$$ is returned such that the full cell $$(p_1, p_2, \ldots, p_d, query)$$ is positively oriented (the rest of the triangulation lies on the other side of facet $$(p_1, p_2, \ldots, p_d)$$).

◆ locate() [2/3]

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Full_cell_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::locate ( const Point & query, Locate_type & loc_type, Face & f, Facet & ft, Full_cell_handle hint = Full_cell_handle() ) const

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

If the query point lies outside the affine hull of the points (which can happen when tr.current_dimension() < tr.maximal_dimension()) or if there is no finite vertex yet in the triangulation, then loc_type is set to OUTSIDE_AFFINE_HULL, and locate returns Full_cell_handle(). If the query point lies inside the affine hull of the points, the function finds the $$k$$-face that contains query in its relative interior (if the $$k$$-face is finite, it is unique) and the result is returned as follows:

$$k=0$$
loc_type is set to ON_VERTEX, f is set to the vertex v the query lies on and a full cell having v as a vertex is returned.
$$0<k<$$c.current_dimension()-1
loc_type is set to IN_FACE, f is set to the unique finite face containing the query point. A full cell having f on its boundary is returned.
$$k=$$c.current_dimension()-1
loc_type is set to IN_FACET, ft is set to one of the two representation of the finite facet containing the query point. The full cell of ft is returned.
$$k=$$c.current_dimension()
If the query point lies outside the convex hull of the points in the triangulation, then loc_type is set to OUTSIDE_CONVEX_HULL and a full cell is returned as in the locate method above. If the query point lies inside the convex hull of the points in the triangulation, then loc_type is set to IN_FULL_CELL and the unique full cell containing the query point is returned.

◆ locate() [3/3]

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Full_cell_handle CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::locate ( const Point & query, Locate_type & loc_type, Face & f, Vertex_handle hint ) const

Same as above but hint, the starting place for the search, is a vertex.

The parameter hint is ignored if it is a default constructed Vertex_handle().

◆ operator>>()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 std::istream& CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::operator>> ( std::istream & is, Triangulation< TriangulationTraits_, TriangulationDataStructure_ > & t )

Reads the underlying combinatorial triangulation from is by calling the corresponding input operator of the triangulation data structure class (note that the infinite vertex is numbered 0), and the non-combinatorial information by calling the corresponding input operators of the vertex and the full cell classes (such as point coordinates), which are provided by overloading the stream operators of the vertex and full cell types.

Assigns the resulting triangulation to t.

◆ tds()

template<typename TriangulationTraits_, typename TriangulationDataStructure_>
 Triangulation_ds& CGAL::Triangulation< TriangulationTraits_, TriangulationDataStructure_ >::tds ( )