CGAL 6.2 - 2D Hyperbolic Delaunay Triangulations

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CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds > Class Template Reference
Reference Manual » Main Classes

`#include <CGAL/Hyperbolic_Delaunay_triangulation_2.h>`

## Inherits from

CGAL::Delaunay_triangulation_2< Gt, Tds >.

## Definition

template<class Gt, class Tds>
class CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >

The class `Hyperbolic_Delaunay_triangulation_2` is the main class of the 2D Hyperbolic Delaunay Triangulations package.

It is designed to represent Delaunay triangulations of sets of points in the hyperbolic plane. The hyperbolic plane is represented in the Poincaré disk model.

Template Parameters

  Gt | is the geometric traits class and must be a model of `HyperbolicDelaunayTriangulationTraits_2`.
  Tds | is the triangulation graph data structure and must be a model of `TriangulationDataStructure_2` whose vertex and face are models of `TriangulationVertexBase_2` and `HyperbolicTriangulationFaceBase_2`, respectively. It defaults to: CGAL::Triangulation_data_structure_2< CGAL::Triangulation_vertex_base_2<Gt>, CGAL::Hyperbolic_triangulation_face_base_2<Gt> > CGAL::Hyperbolic_triangulation_face_base_2 The class Hyperbolic_triangulation_face_base_2 is designed as the default model for the face concept ... Definition: Hyperbolic_triangulation_face_base_2.h:22 CGAL::Triangulation_data_structure_2 CGAL::Triangulation_vertex_base_2

See also
  `Delaunay_triangulation_2`

Examples
  Hyperbolic_triangulation_2/ht2_example.cpp, and Hyperbolic_triangulation_2/ht2_example_color.cpp.

## Enums
The enumeration `Locate_type` is defined to specify which case occurs when locating a point in the triangulation.
enum | Locate_type { VERTEX = 0 , EDGE , FACE , OUTSIDE_CONVEX_HULL , OUTSIDE_AFFINE_HULL }

## Types
typedef Gt | Geom_traits
typedef Tds | Triangulation_data_structure
typedef Triangulation_data_structure::size_type | size_type
| Size type (integral unsigned).
typedef Geom_traits::Hyperbolic_point_2 | Point
typedef Geom_traits::Hyperbolic_triangle_2 | Hyperbolic_triangle
The following types are defined to give access to the elements of the triangulation:
typedef Triangulation_data_structure::Vertex_handle | Vertex_handle
typedef Triangulation_data_structure::Face_handle | Face_handle
typedef Triangulation_data_structure::Edge | Edge
The following types are defined for use in the construction of the Voronoi diagram:
typedef Geom_traits::Hyperbolic_Voronoi_point_2 | Hyperbolic_Voronoi_point
typedef Geom_traits::Hyperbolic_segment_2 | Hyperbolic_segment
The following iterator and circulator types are defined to give access over hyperbolic faces and edges:
typedef Triangulation_data_structure::Face_iterator | All_faces_iterator
typedef Triangulation_data_structure::Edge_iterator | All_edges_iterator
typedef Triangulation_data_structure::Vertex_iterator | All_vertices_iterator
typedef Triangulation_data_structure::Vertex_circulator | Vertex_circulator

## Creation
| Hyperbolic_Delaunay_triangulation_2 (const Geom_traits &gt=Geom_traits())
| Default constructor
| Hyperbolic_Delaunay_triangulation_2 (const Hyperbolic_Delaunay_triangulation_2< Gt, Tds > &tr)
| Copy constructor.
template<class InputIterator >
| Hyperbolic_Delaunay_triangulation_2 (InputIterator first, InputIterator last, const Geom_traits &gt=Geom_traits())
| Equivalent to creating an empty triangulation and calling `insert(first, last)`.

## Assignment
Hyperbolic_Delaunay_triangulation_2 & | operator= (Hyperbolic_Delaunay_triangulation_2 tr)
| The triangulation `tr` is duplicated, and modifying the copy after the duplication does not modify the original.
void | swap (Hyperbolic_Delaunay_triangulation_2 &tr)
| The triangulation is swapped with `tr`.
void | clear ()
| Deletes all vertices and faces of the triangulation.
bool | operator== (const Hyperbolic_Delaunay_triangulation_2< Gt, Tds > &t1, const Hyperbolic_Delaunay_triangulation_2< Gt, Tds > &t2)
| Equality operator.
bool | operator!= (const Hyperbolic_Delaunay_triangulation_2< Gt, Tds > &t1, const Hyperbolic_Delaunay_triangulation_2< Gt, Tds > &t2)
| Inequality operator.

## Access functions
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.
bool | is_valid ()
| Checks the combinatorial validity of the triangulation, the validity of its geometric embedding, and also that all edges and faces are Delaunay hyperbolic.
int | dimension () const
| Returns the dimension of the affine hull.
size_type | number_of_vertices () const
| Returns the number of vertices.
size_type | number_of_hyperbolic_edges () const
| Returns the number of hyperbolic edges.
size_type | number_of_hyperbolic_faces () const
| Returns the number of hyperbolic faces.

## Geometric access functions
Hyperbolic_triangle | hyperbolic_triangle (const Face_handle f) const
Hyperbolic_segment | hyperbolic_segment (const Face_handle f, const int i) const
| Returns the hyperbolic segment formed by the vertices of the edge `(f, i)`.
Hyperbolic_segment | hyperbolic_segment (const Edge &e) const
| Returns the hyperbolic segment formed by the vertices of edge `e`.
Point | point (const Vertex_handle vh) const
| Returns the hyperbolic point given by the finite vertex `vh`.
Point | point (const Face_handle fh, const int i) const
| Returns the point given by vertex `i` of face `fh`.

## Insertion
Vertex_handle | insert (const Point &p, Face_handle start=Face_handle())
| Inserts the point `p` in the triangulation.
template<class InputIterator >
std::ptrdiff_t | insert (InputIterator first, InputIterator last)
| Inserts the points in the range [first,last) into the triangulation.

## Removal
void | remove (Vertex_handle v)
| Removes the vertex `v` from the triangulation.
template<class VertexRemoveIterator >
void | remove (VertexRemoveIterator first, VertexRemoveIterator last)
| Removes the vertices in the iterator range `[first, last)` from the triangulation.

## Point Location
Face_handle | locate (const Point &query, const Face_handle hint=Face_handle()) const
| Locates the point `query` in the triangulation.
Face_handle | locate (const Point &query, Locate_type &lt, int &li, Face_handle hint=Face_handle()) const
| Same as above.

## Queries
template<class OutputItFaces >
OutputItFaces | find_conflicts (const Point &p, OutputItFaces fit, Face_handle start=Face_handle()) const
| Computes the conflict zone induced by `p`.

## Vertex, Face and Edge iterators and circulators
All_vertices_iterator | all_vertices_begin () const
All_vertices_iterator | all_vertices_end () const
All_edges_iterator | all_edges_begin () const
All_edges_iterator | all_edges_end () const
All_faces_iterator | all_faces_begin () const
All_faces_iterator | all_faces_end () const
Vertex_circulator | adjacent_vertices (Vertex_handle v) const

## Voronoi Diagram
Users 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).
Hyperbolic_Voronoi_point | dual (Face_handle f) const
| Returns the hyperbolic center of the circumdisk of `f`.
Hyperbolic_segment | dual (const Edge &e) const
| Returns the hyperbolic segment that is dual to `e`.
Hyperbolic_segment | dual (Face_handle f, int i) const
| Returns the hyperbolic segment that is dual to the edge `(f,i)`.

## Member Enumeration Documentation

## ◆ Locate_type

template<class Gt , class Tds >

enum CGAL::Hyperbolic_Delaunay_triangulation_2::Locate_type

Enumerator
VERTEX |
EDGE |
FACE |
OUTSIDE_CONVEX_HULL |
OUTSIDE_AFFINE_HULL |

## Member Function Documentation

## ◆ dual() [1/2]

template<class Gt , class Tds >

Hyperbolic_Voronoi_point CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::dual | ( | Face_handle | f | ) | const

Returns the hyperbolic center of the circumdisk of `f`.

Precondition
  `f` is hyperbolic

## ◆ dual() [2/2]

template<class Gt , class Tds >

Hyperbolic_segment CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::dual | ( | Face_handle | f,
|  | int | i
| ) |  | const

Returns the hyperbolic segment that is dual to the edge `(f,i)`.

Precondition
  `f` is hyperbolic

## ◆ find_conflicts()

template<class Gt , class Tds >
template<class OutputItFaces >

OutputItFaces CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::find_conflicts | ( | const Point & | p,
|  | OutputItFaces | fit,
|  | Face_handle | start = `Face_handle()`
| ) |  | const

Computes the conflict zone induced by `p`.

If the optional parameter `start` is given, then it must be a face in conflict with `p`. Returns an iterator on the faces of the triangulation in conflict with `p`.

## ◆ insert() [1/2]

template<class Gt , class Tds >

Vertex_handle CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::insert | ( | const Point & | p,
|  | Face_handle | start = `Face_handle()`
| ) |  |

Inserts the point `p` in the triangulation.

If the point `p` coincides with an existing vertex, then the vertex is returned and the triangulation is not modified. The optional parameter `start` is used to initialize the location of `p`.

## ◆ insert() [2/2]

template<class Gt , class Tds >
template<class InputIterator >

std::ptrdiff_t CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::insert | ( | InputIterator | first,
|  | InputIterator | last
| ) |  |

Inserts the points in the range [first,last) into the triangulation.

Returns the number of inserted 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.

Template Parameters

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

## ◆ locate() [1/2]

template<class Gt , class Tds >

Face_handle CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::locate | ( | const Point & | query,
|  | const Face_handle | hint = `Face_handle()`
| ) |  | const

Locates the point `query` in the triangulation.

If the point `query` lies inside the hyperbolic convex hull of the points of the triangulation, then the hyperbolic face that contains the query in its interior is returned.

If `query` lies on a vertex or on an edge, then one of the faces having `query` on its boundary is returned.

If `query` lies outside of the convex hull of the points of the triangulation, then a default-constructed `Face_handle()` is returned.

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

## ◆ locate() [2/2]

template<class Gt , class Tds >

Face_handle CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::locate | ( | const Point & | query,
|  | Locate_type & | lt,
|  | int & | li,
|  | Face_handle | hint = `Face_handle()`
| ) |  | const

Same as above.

The variable `lt` contains information about the element in which `query` has been located. See the enumeration `Locate_type` for details.

If `lt` is `Locate_type::VERTEX`, then the variable `li` contains the index of the vertex in the returned face. If `lt` is `Locate_type::EDGE`, then `li` is the index of the edge in the returned face.

## ◆ operator!=()

template<class Gt , class Tds >

bool CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::operator!= | ( | const Hyperbolic_Delaunay_triangulation_2< Gt, Tds > & | t1,
|  | const Hyperbolic_Delaunay_triangulation_2< Gt, Tds > & | t2
| ) |  |

Inequality operator.

## ◆ operator==()

template<class Gt , class Tds >

bool CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::operator== | ( | const Hyperbolic_Delaunay_triangulation_2< Gt, Tds > & | t1,
|  | const Hyperbolic_Delaunay_triangulation_2< Gt, Tds > & | t2
| ) |  |

Equality operator.

## ◆ point()

template<class Gt , class Tds >

Point CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::point | ( | const Face_handle | fh,
|  | const int | i
| ) |  | const

Returns the point given by vertex `i` of face `fh`.

Precondition
  `t.dimension()` \( \geq0\) and \( i \in\{0,1,2\}\) in dimension 2, \( i \in\{0,1\}\) in dimension 1, \( i = 0\) in dimension 0, and the vertex is finite.

## ◆ remove() [1/2]

template<class Gt , class Tds >

void CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::remove | ( | Vertex_handle | v | ) |

Removes the vertex `v` from the triangulation.

Precondition
  `v` is a vertex of the triangulation.

## ◆ remove() [2/2]

template<class Gt , class Tds >
template<class VertexRemoveIterator >

void CGAL::Hyperbolic_Delaunay_triangulation_2< Gt, Tds >::remove | ( | VertexRemoveIterator | first,
|  | VertexRemoveIterator | last
| ) |  |

Removes the vertices in the iterator range `[first, last)` from the triangulation.

Precondition
  all vertices in `[first, last)` are vertices of the triangulation.