CGAL 6.1.1 - dD Triangulations

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CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ > Class Template Reference
Reference Manual » Triangulation Classes

`#include <CGAL/Delaunay_triangulation.h>`

## Inherits from

CGAL::Triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >.

## Definition

template<typename DelaunayTriangulationTraits_, typename TriangulationDataStructure_>
class CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >

This class is used to maintain the Delaunay triangulation of a set of points in \( \mathbb{R}^D \).

It permits point insertion and removal. The maximal dimension \( D\) can be specified at compile-time or run-time. It should be kept reasonably small, see the performance section in the user manual for what reasonable means.

In a Delaunay triangulation, each face has the so-called Delaunay or empty-ball property: there exists a circumscribing ball whose interior does not contain any vertex of the triangulation. A circumscribing ball of a simplex is a ball having all vertices of the simplex on its boundary.

Template Parameters

  DelaunayTriangulationTraits_ | is the geometric traits class that provides the geometric types and predicates needed by Delaunay triangulations. `DelaunayTriangulationTraits_` must be a model of the concept `DelaunayTriangulationTraits`.
  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: `DelaunayTriangulationTraits_::Dimension` `Triangulation_vertex<DelaunayTriangulationTraits_>` `Triangulation_full_cell<DelaunayTriangulationTraits_>`.
  Delaunay_triangulation | can be defined by specifying only the first parameter, or by using the tag `CGAL::Default` as the second parameter.

See also
  `Regular_triangulation`
  `Triangulation_data_structure`

Examples
  delaunay_triangulation.cpp.

## Creation
| Delaunay_triangulation (int dim, const Geom_traits &gt=Geom_traits())
| Instantiates a Delaunay triangulation with one vertex (the vertex at infinity).

## Point Removal
Full_cell_handle | remove (Vertex_handle v)
| Remove the vertex `v` from the Delaunay triangulation.
template<typename ForwardIterator >
void | remove (ForwardIterator start, ForwardIterator end)
| Remove the vertices pointed by the vertex handles in the range `[start, end)`.

## Point Insertion
template<typename ForwardIterator >
size_type | insert (ForwardIterator s, ForwardIterator e)
| Inserts the points found in range `[s,e)` in the Delaunay triangulation and ensures that the empty-ball property is preserved.
Vertex_handle | insert (const Point &p, Full_cell_handle hint=Full_cell_handle())
| Inserts point `p` in the Delaunay triangulation and ensures that the empty-ball property is preserved.
Vertex_handle | insert (const Point &p, Vertex_handle hint)
| Same as above but uses a vertex as starting place for the search.
Vertex_handle | insert (const Point &p, Locate_type lt, const Face &f, const Facet &ft, Full_cell_handle c)
| Inserts the point `p` in the Delaunay triangulation and ensures that the empty-ball property is preserved.

## Queries
bool | is_in_conflict (const Point &p, Full_cell_const_handle c) const
| Returns `true` if and only if the point `p` is in (Delaunay) conflict with full cell `c` (i.e., the circumscribing ball of \( c\) contains \( p\) in its interior).
template<typename OutputIterator >
Facet | compute_conflict_zone (const Point &p, Full_cell_handle c, OutputIterator out) const
| Outputs handles to the full cells in conflict with point `p` into the `OutputIterator out`.

## Additional Inherited Members
Public Types inherited from CGAL::Triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >
enum | Locate_type
| Used by `Triangulation` to specify which case occurs when locating a point in the triangulation.
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)
typedef DelaunayTriangulationTraits_ | Geom_traits
| Type for the model of the `TriangulationTraits_` concept.
typedef TriangulationTraits_::Point_d | Point
| A point in Euclidean space.
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`).
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`.
Public Member Functions inherited from CGAL::Triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >
| Triangulation (int dim, const Geom_traits &gt=Geom_traits())
| Instantiates a triangulation with one vertex (the vertex at infinity).
| Triangulation (const Triangulation &t2)
| The copy constructor.
const Triangulation_ds & | tds () const
| Returns a const reference to the underlying triangulation data structure.
Triangulation_ds & | tds ()
| This is an advanced function.
const Geom_traits & | geom_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.
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.
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.
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.
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`.
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.
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.
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.
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.
Vertex_handle | collapse_face (const Point &p, const Face &f)
| Contracts the `Face f` to a single vertex at position `p`.
size_type | insert (ForwardIterator s, ForwardIterator e)
| Inserts the points found in range `[s,e)` in the triangulation.
Vertex_handle | insert (const Point &p, Full_cell_handle hint=Full_cell_handle())
| Inserts point `p` in the triangulation.
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.
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`.
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.
Vertex_handle | insert_in_facet (const Point &p, const Facet &ft)
| Inserts point `p` in the triangulation.
Vertex_handle | insert_in_full_cell (const Point &p, Full_cell_handle c)
| Inserts point `p` in the triangulation.
Vertex_handle | insert_outside_convex_hull (const Point &, Full_cell_handle c)
| Inserts point `p` in the triangulation.
Vertex_handle | insert_outside_affine_hull (const Point &)
| Inserts point `p` in the triangulation.
bool | is_valid (bool verbose=false) const
| This is a function for debugging purpose.
bool | are_incident_full_cells_valid (Vertex_const_handle v, bool verbose=false) const
| This is a function for debugging purpose.
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.
std::ostream & | operator<< (std::ostream &os, const Triangulation &t)
| Writes the triangulation `t` into `os`.

## Constructor & Destructor Documentation

## ◆ Delaunay_triangulation()

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::Delaunay_triangulation | ( | int | dim,
|  | const Geom_traits & | gt = `Geom_traits()`
| ) |  |

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

See the description of the inherited nested type `Triangulation::Maximal_dimension` for an explanation of the use of the parameter `dim`. The complex stores a copy of the geometric traits `gt`.

## Member Function Documentation

## ◆ compute_conflict_zone()

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >
template<typename OutputIterator >

Facet CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::compute_conflict_zone | ( | const Point & | p,
|  | Full_cell_handle | c,
|  | OutputIterator | out
| ) |  | const

Outputs handles to the full cells in conflict with point `p` into the `OutputIterator out`.

The full cell `c` is used as a starting point for gathering the full cells in conflict with `p`. A facet `(cc,i)` on the boundary of the conflict zone with `cc` in conflict is returned.

Precondition
  `c` is in conflict with `p` and `dt`.`current_dimension()` \( \geq2\).

## ◆ insert() [1/3]

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

Vertex_handle CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::insert | ( | const Point & | p,
|  | Full_cell_handle | hint = `Full_cell_handle()`
| ) |  |

Inserts point `p` in the Delaunay triangulation and ensures that the empty-ball property is preserved.

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() [2/3]

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

Vertex_handle CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::insert | ( | const Point & | p,
|  | Locate_type | lt,
|  | const Face & | f,
|  | const Facet & | ft,
|  | Full_cell_handle | c
| ) |  |

Inserts the point `p` in the Delaunay triangulation and ensures that the empty-ball property is preserved.

Returns a handle to the (possibly newly created) vertex at that position. The behavior depends on the value of `lt`:

`OUTSIDE_AFFINE_HULL`
  Point `p` is inserted so as to increase the current dimension of the Delaunay triangulation.
`ON_VERTEX`
  The triangulation remains unchanged.
Anything else
  The point `p` is inserted. the full cell `c` is assumed to be in conflict with `p`.

The parameters `lt`, `f`, `ft` and `c` must be consistent with the localization of point `p` in the Delaunay triangulation e.g. by a call to `Triangulation::locate(const Point &, Locate_type &, Face &, Vertex_handle) const`.

## ◆ insert() [3/3]

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >
template<typename ForwardIterator >

size_type CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::insert | ( | ForwardIterator | s,
|  | ForwardIterator | e
| ) |  |

Inserts the points found in range `[s,e)` in the Delaunay triangulation and ensures that the empty-ball property is preserved.

Returns the number of vertices actually inserted. (If more than one vertex share the same position in space, only one insertion is counted.)

Template Parameters

  ForwardIterator | must be an input iterator with the value type `Point`.

## ◆ remove() [1/2]

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >
template<typename ForwardIterator >

void CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::remove | ( | ForwardIterator | start,
|  | ForwardIterator | end
| ) |  |

Remove the vertices pointed by the vertex handles in the range `[start, end)`.

Template Parameters

  ForwardIterator | must be an input iterator with the value type `Vertex_handle`.

## ◆ remove() [2/2]

template<typename DelaunayTriangulationTraits_ , typename TriangulationDataStructure_ >

Full_cell_handle CGAL::Delaunay_triangulation< DelaunayTriangulationTraits_, TriangulationDataStructure_ >::remove | ( | Vertex_handle | v | ) |

Remove the vertex `v` from the Delaunay triangulation.

If the current dimension of the triangulation has not changed after the removal, then the returned full cell `c` geometrically contains the removed vertex `v` (`c` can be finite or infinite). Otherwise, the default-constructed `Full_cell_handle` is returned.

Precondition
  `v` is a vertex of the triangulation, different from the `infinite_vertex()`.