CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes > Class Template Reference

#include <CGAL/Triangulation_on_hyperbolic_surface_2.h>

Definition

template<class Traits, class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>
class CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >

represents a triangulation of a closed orientable hyperbolic surface.

The class provides functions such as the generation of the triangulation from a convex fundamental domain, the Delaunay flip algorithm, and the construction of a portion of the lift of the triangulation in the hyperbolic plane.

Template Parameters

Traits	must be a model of HyperbolicSurfaceTraits_2.
Attributes	must be a model of GenericMapItems whose edges are decorated with complex numbers to represent cross ratios.

Examples

Triangulation_on_hyperbolic_surface_2/Triangulation_on_hyperbolic_surface_2.cpp.

Classes
struct	Anchor
	stores a dart \( d \) of the combinatorial map, belonging to a triangle \( t \), and stores the three vertices of a lift of \( t \) in the hyperbolic plane. More...

Types
typedef Combinatorial_map< 2, Attributes >	Combinatorial_map_with_cross_ratios
	Type of combinatorial map whose edges are decorated with complex numbers.
typedef Combinatorial_map_with_cross_ratios::Dart_descriptor	Dart_descriptor
	Combinatorial map dart descriptor type.
typedef Combinatorial_map_with_cross_ratios::Dart_const_descriptor	Dart_const_descriptor
	Combinatorial map dart const descriptor type.
typedef Combinatorial_map_with_cross_ratios::template One_dart_per_cell_range< 0 >	Vertex_range
	Range of one dart for each vertex (that is 0-cell) of the combinatorial map.
typedef Combinatorial_map_with_cross_ratios::template One_dart_per_cell_range< 1 >	Edge_range
	Range of one dart for each edge (that is 1-cell) of the combinatorial map.
typedef Combinatorial_map_with_cross_ratios::template One_dart_per_cell_range< 2 >	Face_range
	Range of one dart for each face (that is 2-cell) of the combinatorial map.
typedef Combinatorial_map_with_cross_ratios::template One_dart_per_cell_const_range< 0 >	Vertex_const_range
	Range of one dart for each vertex (that is 0-cell) of the combinatorial map.
typedef Combinatorial_map_with_cross_ratios::template One_dart_per_cell_const_range< 1 >	Edge_const_range
	Range of one dart for each edge (that is 1-cell) of the combinatorial map.
typedef Combinatorial_map_with_cross_ratios::template One_dart_per_cell_const_range< 2 >	Face_const_range
	Range of one dart for each face (that is 2-cell) of the combinatorial map.
typedef Traits::Hyperbolic_point_2	Point
	Point type.

Creation
	Triangulation_on_hyperbolic_surface_2 ()
	Default constructor.
	Triangulation_on_hyperbolic_surface_2 (const Hyperbolic_fundamental_domain_2< Traits > &domain)
	Constructor from a convex fundamental domain: triangulates the polygon of the domain.
	Triangulation_on_hyperbolic_surface_2 (Combinatorial_map_with_cross_ratios &cmap, Anchor &an_anchor)
	Constructor from a decorated combinatorial map and an anchor.

Assignment
Triangulation_on_hyperbolic_surface_2 &	operator= (Triangulation_on_hyperbolic_surface_2 other)

Access Functions
Combinatorial_map_with_cross_ratios &	combinatorial_map ()
	returns the decorated combinatorial map.
bool	has_anchor () const
	returns whether the triangulation has an anchor or not.
Anchor &	anchor ()
	returns the anchor.
const Anchor &	anchor () const
	returns the anchor.
Vertex_range	vertices_range ()
	returns the range of vertices.
Edge_range	edges_range ()
	returns the range of edges.
Face_range	faces_range ()
	returns the range of faces.
Vertex_const_range	vertices_const_range () const
	returns the range of vertices.
Edge_const_range	edges_const_range () const
	returns the range of edges.
Face_const_range	faces_const_range () const
	returns the range of faces.

Delaunay Flip Algorithm
bool	is_Delaunay_flippable (Dart_descriptor dart) const
	returns whether the edge supported by the dart is Delaunay flippable or not.
void	flip (Dart_descriptor dart)
	flips the edge supported by the dart.
bool	is_Delaunay () const
	determines if the triangulation is a valid Delaunay triangulation.
int	make_Delaunay ()
	applies the Delaunay flip algorithm: flips Delaunay-flippable edges until there is no such edge anymore.

Lifting
std::vector< std::tuple< Dart_const_descriptor, Point, Point, Point > >	lift (bool center=true) const
	lifts the triangulation in the hyperbolic plane.

Validity
bool	is_valid () const
	checks that the underlying combinatorial map \( M \) has no boundary and calls the is_valid method of \( M \).

Constructor & Destructor Documentation

Triangulation_on_hyperbolic_surface_2()

template<class Traits , class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>

CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >::Triangulation_on_hyperbolic_surface_2

(

const Hyperbolic_fundamental_domain_2< Traits > &

domain

)

Constructor from a convex fundamental domain: triangulates the polygon of the domain.

(The triangulation is defined by adding an internal edge between domain.vertex(size-1) and the other vertices. The anchor has the three vertices of indices size-1, 0 and 1 and the dart is the one between the vertices of indices size-1 and 0.)

Member Function Documentation

anchor() [1/2]

template<class Traits , class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>

Anchor & CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >::anchor

(

)

returns the anchor.

Precondition

is_valid() && has_anchor()

anchor() [2/2]

template<class Traits , class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>

const Anchor & CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >::anchor

(

)

const

returns the anchor.

Precondition

is_valid() && has_anchor()

flip()

template<class Traits , class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>

void CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >::flip

(

Dart_descriptor

dart

)

flips the edge supported by the dart.

Precondition

is_valid()

has_anchor()

template<class Traits , class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>

bool CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >::has_anchor

(

)

const

returns whether the triangulation has an anchor or not.

Precondition

is_valid()

is_Delaunay_flippable()

template<class Traits , class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>

bool CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >::is_Delaunay_flippable

(

Dart_descriptor

dart

)

const

returns whether the edge supported by the dart is Delaunay flippable or not.

An edge \( e \) is Delaunay flippable if the imaginary part of its cross ratio is positive.

Precondition

is_valid()

is_valid()

template<class Traits , class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>

bool CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >::is_valid

(

)

const

checks that the underlying combinatorial map \( M \) has no boundary and calls the is_valid method of \( M \).

If there is an anchor, then checks that the dart descriptor of the anchor does indeed point to a dart of \( M \), and checks that the three vertices of the anchor lie within the open unit disk.

lift()

template<class Traits , class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>

std::vector< std::tuple< Dart_const_descriptor, Point, Point, Point > > CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >::lift

(

bool

center = true

)

const

lifts the triangulation in the hyperbolic plane.

Returns, for every triangle \( t \) of the triangulation, one of the darts of \( t \) in the combinatorial map of the triangulation, together with a triple \( p,q,r \) of points in the hyperbolic plane. The points \( p,q,r \) are the vertices of a lift of \( t \) in the hyperbolic plane. If the center parameter is set to true, then one of the vertices of the anchor is translated to the origin \( 0 \).

Precondition

is_valid() && has_anchor()

make_Delaunay()

template<class Traits , class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>

int CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >::make_Delaunay

(

)

applies the Delaunay flip algorithm: flips Delaunay-flippable edges until there is no such edge anymore.

Precondition

is_valid()

operator=()

template<class Traits , class Attributes = Combinatorial_map_with_cross_ratios_item<Traits>>

Triangulation_on_hyperbolic_surface_2 & CGAL::Triangulation_on_hyperbolic_surface_2< Traits, Attributes >::operator=

(

Triangulation_on_hyperbolic_surface_2< Traits, Attributes >

other

)

Precondition

other.is_valid()