Definition

template<class GT, class TDS>
class CGAL::Periodic_4_hyperbolic_triangulation_2< GT, TDS >

The class Periodic_4_hyperbolic_triangulation_2 offers base functionalities needed by Periodic_4_hyperbolic_Delaunay_triangulation_2.

Note that this class does not support modification of the triangulation (insertion or removal).

The class expects two template parameters.

Template Parameters

GT Geometric traits parameter. Must be a model of the concept Periodic_4HyperbolicTriangulationTraits_2. This parameter has no default value.

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> >

Enums
The following enumeration type indicates where a point is located in the triangulation.
enum	Locate_type { VERTEX = 0 , EDGE , FACE }

Types
typedef GT	Geometric_traits

typedef TDS	Triangulation_data_structure

The following types represent geometric objects.
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.

The following type represents hyperbolic translations specific for the Bolza surface.
typedef Geometric_traits::Hyperbolic_translation	Hyperbolic_translation

The following types represent images of points, segments, and triangles under the action of hyperbolic translations.
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.

The following types give access to the elements of the triangulation.
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

The following iterator and circulator types are defined to give access over the vertices, edges, and faces of the triangulation.
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

Creation
	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.

Assignment
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.

Access functions
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.

Geometric access functions
The functions below return periodic points, segments, and triangles.
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`.

The functions below return non-periodic points, segments and triangles.
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`.

Point location
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.

Predicate functions
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)`.

Queries
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, Edge, and Face iterators
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, Edge and Face circulators
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`.

Traversal of the Vertices, Edges and Faces incident to a vertex
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`.

Traversal between adjacent faces
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.

Validity checking
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`.