#include <CGAL/Periodic_4_hyperbolic_Delaunay_triangulation_2.h>

Inherits from

CGAL::Periodic_4_hyperbolic_triangulation_2< GT, TDS >.

Definition

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 Triangulation_data_structure_2< Periodic_4_hyperbolic_triangulation_vertex_base_2<GT>, Periodic_4_hyperbolic_triangulation_face_base_2<GT> >

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. More...

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`. More...

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. More...

Point insertion
Vertex_handle	insert (const Point &p, Face_handle start=Face_handle())
	Inserts the point `p` in the triangulation. More...

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. More...

Vertex removal
bool	remove (Vertex_handle v)
	Removes the vertex `v` from the triangulation. More...

template<class VertexRemoveIterator >
void	remove (VertexRemoveIterator first, VertexRemoveIterator last)
	Removes the vertices in the iterator range `[firs, last)` from the triangulation. More...

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

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`. More...

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. More...

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.

bool	operator== (const Periodic_4_hyperbolic_triangulation_2< GT, TDS > &tr1, const Periodic_4_hyperbolic_triangulation_2< GT, TDS > &tr2)
	Equality operator. More...

bool	operator!= (const Periodic_4_hyperbolic_triangulation_2< GT, TDS > &tr1, const Periodic_4_hyperbolic_triangulation_2< GT, TDS > &tr2)
	Inequality operator. More...

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`. More...

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)`. More...

Periodic_segment	periodic_segment (const Edge &e) const
	Returns the periodic segment formed by the endpoints of edge `e`. More...

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`. More...

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)`. More...

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`. More...

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. More...

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`. More...

Face_handle	hyperbolic_locate (const Point &p, Locate_type &lt, int &li, Face_handle start=Face_handle()) const
	Same as above. More...

Orientation	orientation (const Point &p1, const Point &p2, const Point &p3) const
	Returns the `Orientation` of the points `p1, p2, p3`. More...

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`. More...

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. More...

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`. More...

bool	has_vertex (const Face_handle f, const Vertex_handle v, const int i) const
	Tests whether face `f` has `v` as a vertex. More...

Vertex_iterator	vertices_begin () const
	Starts at an arbitrary vertex. More...

Vertex_iterator	vertices_end () const
	Past-the-end iterator.

Edge_iterator	edges_begin () const
	Starts at an arbitrary edge. More...

Edge_iterator	edges_end () const
	Past-the-end iterator.

Face_iterator	faces_begin () const
	Starts at an arbitrary face. More...

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 [firs, 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.

Inherits from

Definition

Creation

Clearing the triangulation

Queries

Point insertion

Vertex removal

Removal of dummy points

Voronoi diagram

Additional Inherited Members

Constructor & Destructor Documentation

◆ Periodic_4_hyperbolic_Delaunay_triangulation_2()

Member Function Documentation

◆ dual()

◆ find_conflicts()

◆ insert() [1/2]

◆ insert() [2/2]

◆ is_valid()

◆ remove() [1/2]

◆ remove() [2/2]

◆ try_to_remove_dummy_vertices()