CGAL 6.1.1 - 2D Periodic Hyperbolic Triangulations

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CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2< GT, TDS > Class Template Reference
Reference Manual » Main Classes

`#include <CGAL/Periodic_4_hyperbolic_Delaunay_triangulation_2.h>`

## Inherits from

CGAL::Periodic_4_hyperbolic_triangulation_2< GT, TDS >.

## Definition

template<class GT, class TDS>
class CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2< GT, TDS >

The class `Periodic_4_hyperbolic_Delaunay_triangulation_2` enables the construction and handling of Delaunay triangulations of the Bolza surface.

The class expects two template parameters.

Template Parameters

  GT | Geometric Traits parameter. Must be a model of the concept `Periodic_4HyperbolicDelaunayTriangulationTraits_2`. The default value for this parameter is `Periodic_4_hyperbolic_Delaunay_triangulation_traits_2<>`.
  TDS | Triangulation Data Structure parameter. Must be a model of the concept `TriangulationDataStructure_2` with some additional functionality in the vertices and faces, following the concepts `Periodic_4HyperbolicTriangulationVertexBase_2` and `Periodic_4HyperbolicTriangulationFaceBase_2`, respectively. The default value for this parameter is CGAL::Triangulation_data_structure_2< CGAL::Periodic_4_hyperbolic_triangulation_vertex_base_2<GT>, CGAL::Periodic_4_hyperbolic_triangulation_face_base_2<GT> > CGAL::Periodic_4_hyperbolic_triangulation_face_base_2 The class Periodic_4_hyperbolic_triangulation_face_base_2 is the default model for the concept Period... Definition: Periodic_4_hyperbolic_triangulation_face_base_2.h:32 CGAL::Periodic_4_hyperbolic_triangulation_vertex_base_2 The class Periodic_4_hyperbolic_triangulation_vertex_base_2 is the default model for the concept Peri... Definition: Periodic_4_hyperbolic_triangulation_vertex_base_2.h:31 CGAL::Triangulation_data_structure_2

Examples
  Periodic_4_hyperbolic_triangulation_2/p4ht2_example_insertion.cpp.

## Creation
| Periodic_4_hyperbolic_Delaunay_triangulation_2 (const Geom_traits &gt=Geom_traits())
| Default constructor with an optional parameter for the geometric traits object.
| Periodic_4_hyperbolic_Delaunay_triangulation_2 (const Periodic_4_hyperbolic_Delaunay_triangulation_2 &tr)
| Copy constructor.
template<class InputIterator >
| Periodic_4_hyperbolic_Delaunay_triangulation_2 (InputIterator first, InputIterator last, const Geom_traits &gt=Geom_traits())
| Point range constructor.

## Clearing the triangulation
void | clear ()
| Deletes all faces and vertices of the triangulation, and re-initializes the triangulation data structure with the set of dummy points.

## Queries
template<class OutputFaceIterator >
void | find_conflicts (const Point &p, OutputFaceIterator it, Face_handle start=Face_handle(), Hyperbolic_translation ltr=Hyperbolic_translation()) const
| Computes the conflict zone induced by `p`.
bool | is_dummy_vertex (Vertex_handle vh) const
| Checks if the vertex `vh` is part of the set of vertices storing dummy points.
int | number_of_dummy_points () const
| Returns the number of dummy points currently existing in the triangulation.
bool | is_valid (bool verbose=false) const
| Checks the combinatorial validity of the triangulation, and also the validity of its geometric embedding.

## Point 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, const bool flag_try_to_remove_dummy_vertices=true)
| Inserts all points in the input iterator into the triangulation.

## Vertex removal
bool | 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.

## Removal of dummy points
int | try_to_remove_dummy_vertices ()
| Tries to remove the dummy points from the triangulation one by one.

## Voronoi diagram
Voronoi_point | dual (Face_handle f, Hyperbolic_translation nbtr=Hyperbolic_translation()) const
| Returns the dual of face `f`, i.e., the hyperbolic center of the disk defined by the vertices of `f`.
Segment | dual (const Edge &e) const
| Returns the hyperbolic segment that is dual to the edge `e`.

## Additional Inherited Members
Public Types inherited from CGAL::Periodic_4_hyperbolic_triangulation_2< GT, TDS >
enum | Locate_type { VERTEX = 0 , EDGE , FACE }
typedef GT | Geometric_traits
typedef TDS | Triangulation_data_structure
typedef Triangulation_data_structure::Vertex::Point | Point
| Represents a point in the hyperbolic plane.
typedef Geometric_traits::Hyperbolic_segment_2 | Hyperbolic_segment
| Represents a hyperbolic segment in the hyperbolic plane.
typedef Geometric_traits::Hyperbolic_triangle_2 | Hyperbolic_triangle
| Represents a triangle contained in the Poincaré disk.
typedef Geometric_traits::Hyperbolic_translation | Hyperbolic_translation
typedef std::pair< Point, Hyperbolic_translation > | Periodic_point
| Represents a periodic point, i.e., a pair of a point in the original octagon and a hyperbolic translation.
typedef std::array< std::pair< Point, Hyperbolic_translation >, 2 > | Periodic_segment
| Represents a periodic segment, defined by two periodic points.
typedef std::array< std::pair< Point, Hyperbolic_translation >, 3 > | Periodic_triangle
| Represents a periodic triangle, defined by three periodic points.
typedef Triangulation_data_structure::Edge | Edge
typedef Triangulation_data_structure::Face | Face
typedef Triangulation_data_structure::Vertex_handle | Vertex_handle
typedef Triangulation_data_structure::Face_handle | Face_handle
typedef Triangulation_data_structure::Face_iterator | Face_iterator
typedef Triangulation_data_structure::Edge_iterator | Edge_iterator
typedef Triangulation_data_structure::Vertex_iterator | Vertex_iterator
typedef Triangulation_data_structure::Face_circulator | Face_circulator
typedef Triangulation_data_structure::Edge_circulator | Edge_circulator
typedef Triangulation_data_structure::Vertex_circulator | Vertex_circulator
Public Member Functions inherited from CGAL::Periodic_4_hyperbolic_triangulation_2< GT, TDS >
| Periodic_4_hyperbolic_triangulation_2 (const Geometric_traits &gt=Geometric_traits())
| Default constructor, with an optional parameter for the geometric traits object.
| Periodic_4_hyperbolic_triangulation_2 (const Periodic_4_hyperbolic_triangulation_2 &tr)
| Copy constructor.
Periodic_4_hyperbolic_triangulation_2 & | operator= (Periodic_4_hyperbolic_triangulation_2 tr)
| The triangulation `tr` is duplicated, and modifying the copy after the duplication does not modify the original.
void | swap (Periodic_4_hyperbolic_triangulation_2 &tr)
| The triangulation is swapped with `tr`.
void | clear ()
| Deletes all faces and vertices of the triangulation.
const Geometric_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.
size_type | number_of_vertices () const
| Returns the number of vertices in the triangulation.
size_type | number_of_edges () const
| Returns the number of edges in the triangulation.
size_type | number_of_faces () const
| Returns the number of faces in the triangulation.
Periodic_point | periodic_point (const Face_handle f, int i) const
| Returns the periodic point given by the `i`-th vertex of face `f`, that is the point in the original domain and the translation of the vertex in `f`.
Periodic_segment | periodic_segment (const Point &p1, const Point &p2, const Hyperbolic_translation &tr1, const Hyperbolic_translation &tr2) const
| Returns the periodic segment formed by the two point-translation pairs `(p1, tr1)` and `(p2, tr2)`.
Periodic_segment | periodic_segment (const Point &p1, const Point &p2) const
| Returns the periodic segment formed by the two point-translation pairs `(p1, Id)` and `(p2, Id)`, where `Id` is the identity translation.
Periodic_segment | periodic_segment (const Face_handle c, int i, int j) const
| Returns the periodic segment formed by the endpoints of edge `(f,i,j)`.
Periodic_segment | periodic_segment (const Edge &e) const
| Returns the periodic segment formed by the endpoints of edge `e`.
Periodic_segment | periodic_segment (const Edge &e, const Hyperbolic_translation &tr) const
| Returns the periodic segment formed by the endpoints of edge `e` translated by `tr`.
Periodic_triangle | periodic_triangle (const Face &f) const
| Returns the periodic triangle formed by the pairs of points and translations corresponding to each vertex of `f`.
Periodic_triangle | periodic_triangle (const Face &f, const Hyperbolic_translation &tr) const
| Returns the periodic triangle formed by the pairs of points and translations corresponding to each vertex of `f`, translated by `tr`.
Point | construct_point (const Periodic_point &pp) const
| Converts the periodic point `pp` into a point in the hyperbolic plane by applying `pp.second` to `pp.first`.
Hyperbolic_segment | construct_hyperbolic_segment (const Face_handle &fh, int idx) const
| Constructs the hyperbolic segment formed by the endpoints of edge `(fh, idx)`.
Hyperbolic_segment | construct_hyperbolic_segment (const Point &p1, const Point &p2) const
| Returns the hyperbolic segment with endpoints `p1` and `p2`.
Hyperbolic_segment | construct_hyperbolic_segment (const pair< Face_handle, int > &e) const
| Returns the hyperbolic segment formed by the endpoints of `e`.
Hyperbolic_segment | construct_hyperbolic_segment (const Periodic_segment &ps) const
| Returns the hyperbolic segment formed by the endpoints of `ps`.
Triangle | construct_triangle (const Face_handle &fh) const
| Returns the triangle formed by the vertices of `fh`.
Triangle | construct_triangle (const Periodic_triangle &pt) const
| Returns the triangle formed by the vertices of `pt`.
Face_handle | hyperbolic_periodic_locate (const Point &p, Hyperbolic_translation &lo, const Face_handle start=Face_handle()) const
| Returns the face `rf` for which the periodic triangle `(rf, lo)` contains the query point `p`.
Face_handle | hyperbolic_periodic_locate (const Point &p, Locate_type &lt, int &li, Hyperbolic_translation &lo, const Face_handle start=Face_handle()) const
| Same as above.
Face_handle | hyperbolic_locate (const Point &p, Face_handle start=Face_handle()) const
| Returns the canonical representative of the face that contains the query point `p`.
Face_handle | hyperbolic_locate (const Point &p, Locate_type &lt, int &li, Face_handle start=Face_handle()) const
| Same as above.
Orientation | orientation (const Point &p1, const Point &p2, const Point &p3) const
| Returns the `Orientation` of the points `p1, p2, p3`.
Orientation | orientation (const Point &p1, const Point &p2, const Point &p3, const Hyperbolic_translation &tr1, const Hyperbolic_translation &tr2, const Hyperbolic_translation &tr3) const
| Returns the `Orientation` of the three periodic points `(p1, tr1), (p2, tr2)` and `(p3, tr3)`.
Oriented_side | side_of_oriented_circle (const Point &p1, const Point &p2, const Point &p3, const Point &q) const
| Returns the `Oriented_side` on which `q` is located with respect to the circle defined by the points `p1,p2,p3`.
Oriented_side | side_of_oriented_circle (const Point &p1, const Point &p2, const Point &p3, const Point &q, const Hyperbolic_translation &tr1, const Hyperbolic_translation &tr2, const Hyperbolic_translation &tr3, const Hyperbolic_translation &trq) const
| Returns the `Oriented_side` on which the periodic point `(q, trq)` is located with respect to the circle defined by the periodic points `(p1, tr1), (p2, tr2)` and `(p3, tr3)`.
bool | is_vertex (Vertex_handle v) const
| Tests whether `v` is a vertex of the triangulation.
bool | is_edge (Vertex_handle u, Vertex_handle v, Face_handle &fh, int &i) const
| Tests whether `(u, v)` is an edge of the triangulation.
bool | is_face (Vertex_handle u, Vertex_handle v, Vertex_handle w, Face_handle &fh) const
| Tests whether there exists a face in the triangulation with vertices `u, v` and `w`.
bool | has_vertex (const Face_handle f, const Vertex_handle v, const int i) const
| Tests whether face `f` has `v` as a vertex.
Vertex_iterator | vertices_begin () const
| Starts at an arbitrary vertex.
Vertex_iterator | vertices_end () const
| Past-the-end iterator.
Edge_iterator | edges_begin () const
| Starts at an arbitrary edge.
Edge_iterator | edges_end () const
| Past-the-end iterator.
Face_iterator | faces_begin () const
| Starts at an arbitrary face.
Face_iterator | faces_end () const
| Past-the-end iterator.
Vertex_circulator | adjacent_vertices (Vertex_handle v) const
| Starts at an arbitrary vertex incident to `v`.
Vertex_circulator | adjacent_vertices (Vertex_handle v, Face_handle f) const
| Starts at the first vertex of `f` adjacent to `v` in counterclockwise order around `v`.
Edge_circulator | incident_edges (Vertex_handle v) const
| Starts at an arbitrary edge incident to `v`.
Edge_circulator | incident_edges (Vertex_handle v, Face_handle f) const
| Starts at the first edge of `f` incident to `v`, in counterclockwise order around `v`.
Face_circulator | incident_faces (Vertex_handle v) const
| Starts at an arbitrary face incident to `v`.
Face_circulator | incident_faces (Vertex_handle v, Face_handle f) const
| Starts at face `f`.
template<class OutputIterator >
OutputIterator | incident_faces (Vertex_handle v, OutputIterator faces) const
| Copies the faces incident to `v` into the output iterator `faces`.
template<class OutputIterator >
OutputIterator | incident_edges (Vertex_handle v, OutputIterator edges) const
| Copies the edges incident to `v` into the output iterator `edges`.
template<class OutputIterator >
OutputIterator | adjacent_vertices (Vertex_handle v, OutputIterator vertices) const
| Copies the vertices incident to `v` into the output iterator `vertices`.
size_type | degree (Vertex_handle v) const
| Returns the degree of `v`, i.e., the number of edges incident to `v`.
int | mirror_index (Face_handle f, int i) const
| Returns the index of `f` in its `i`-th neighbor.
Vertex_handle | mirror_vertex (Face_handle f, int i) const
| Returns the vertex of the `i`-th neighbor of `f` that is opposite to `f`.
Edge | mirror_edge (Edge e) const
| Returns the same edge seen from the other adjacent face.
Hyperbolic_translation | neighbor_translation (const Face_handle fh, int i) const
| Returns the hyperbolic translation for which the `i`-th neighbor of `fh` is adjacent to (the canonical representative of) `fh` in the hyperbolic plane.
bool | is_valid (bool verbose=false) const
| Checks the combinatorial validity of the triangulation, and also the validity of its geometric embedding.
bool | is_valid (Face_handle f, bool verbose=false) const
| Checks the combinatorial validity of face `f`.

## Constructor & Destructor Documentation

## ◆ Periodic_4_hyperbolic_Delaunay_triangulation_2()

template<class GT , class TDS >
template<class InputIterator >

CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2< GT, TDS >::Periodic_4_hyperbolic_Delaunay_triangulation_2 | ( | InputIterator | first,
|  | InputIterator | last,
|  | const Geom_traits & | gt = `Geom_traits()`
| ) |  |

Point range constructor.

Initializes a triangulation and then inserts the points in the iterator range `[first, last)`.

## Member Function Documentation

## ◆ dual()

template<class GT , class TDS >

Voronoi_point CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2< GT, TDS >::dual | ( | Face_handle | f,
|  | Hyperbolic_translation | nbtr = `Hyperbolic_translation()`
| ) |  | const

Returns the dual of face `f`, i.e., the hyperbolic center of the disk defined by the vertices of `f`.

If the optional parameter `nbtr` is given, then this function returns the dual of the periodic triangle `(f, nbtr)`.

## ◆ find_conflicts()

template<class GT , class TDS >
template<class OutputFaceIterator >

void CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2< GT, TDS >::find_conflicts | ( | const Point & | p,
|  | OutputFaceIterator | it,
|  | Face_handle | start = `Face_handle()`,
|  | Hyperbolic_translation | ltr = `Hyperbolic_translation()`
| ) |  | const

Computes the conflict zone induced by `p`.

The output iterator `it` contains all faces in conflict with `p`. The optional parameters `start` and `ltr`, if given, must be such that `start` translated by `ltr` is in conflict with `p`.

## ◆ insert() [1/2]

template<class GT , class TDS >

Vertex_handle CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2< GT, TDS >::insert | ( | const Point & | p,
|  | Face_handle | start = `Face_handle()`
| ) |  |

Inserts the point `p` in the triangulation.

The face `start`, if given, is used as a starting place for the location of the point. Note that this function does not remove unnecessary dummy points. The removal, if desired, should be done by manually calling the function `try_to_remove_dummy_vertices()`.

## ◆ insert() [2/2]

template<class GT , class TDS >
template<class InputIterator >

std::ptrdiff_t CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2< GT, TDS >::insert | ( | InputIterator | first,
|  | InputIterator | last,
|  | const bool | flag_try_to_remove_dummy_vertices = `true`
| ) |  |

Inserts all points in the input iterator into the triangulation.

Note that this function by default tries to remove unnecessary dummy points at the end of the insertion process. This behavior is controlled by the optional boolean parameter `flag_try_to_remove_dummy_vertices`; if automatic removal of the dummy points is not desired, set the flag to `false`.

## ◆ is_valid()

template<class GT , class TDS >

bool CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2< GT, TDS >::is_valid | ( | bool | verbose = `false` | ) | const

Checks the combinatorial validity of the triangulation, and also the validity of its geometric embedding.

In more detail, this function verifies that:

- The underlying triangulation data structure, is `valid` (see the function is_valid())
- Each face of the triangulation is positively oriented and has the Delaunay property.
- The Euler relation for genus-2 surfaces is satisfied, i.e., \(V-E+F = -2\), where \(V, E, F\) are the number of vertices, edges and faces of the triangulation, respectively.

## ◆ remove() [1/2]

template<class GT , class TDS >

bool CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2< GT, TDS >::remove | ( | Vertex_handle | v | ) |

Removes the vertex `v` from the triangulation.

Note that `v` is removed from the triangulation only if the resulting triangulation remains a simplicial complex. The function returns `true` if the vertex `v` is removed from the triangulation, otherwise it returns `false`.

## ◆ remove() [2/2]

template<class GT , class TDS >
template<class VertexRemoveIterator >

void CGAL::Periodic_4_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.

## ◆ try_to_remove_dummy_vertices()

template<class GT , class TDS >

int CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2< GT, TDS >::try_to_remove_dummy_vertices | ( |  | ) |

Tries to remove the dummy points from the triangulation one by one.

Returns the number of dummy points that could not be removed and are still present in the triangulation.