CGAL 5.1.3 - 2D Apollonius Graphs (Delaunay Graphs of Disks)
CGAL::Apollonius_graph_2< Gt, Agds > Class Template Reference

#include <CGAL/Apollonius_graph_2.h>

Inherited by CGAL::Apollonius_graph_hierarchy_2< Gt, Agds >.

## Definition

The class Apollonius_graph_2 represents the Apollonius graph.

It supports insertions and deletions of sites. It is templated by two template arguments Gt, which must be a model of ApolloniusGraphTraits_2, and Agds, which must be a model of ApolloniusGraphDataStructure_2. The second template argument defaults to CGAL::Triangulation_data_structure_2< CGAL::Apollonius_graph_vertex_base_2<Gt,true>, CGAL::Triangulation_face_base_2<Gt> >.

Is Model Of:
DelaunayGraph_2

Traversal of the Apollonius Graph

An Apollonius graph can be seen as a container of faces and vertices. Therefore the Apollonius graph provides several iterators and circulators that allow to traverse it (completely or partially).

Traversal of the Convex Hull

Applied on the infinite_vertex the incident_* functions allow to visit the vertices on the convex hull and the infinite edges and faces. Note that a counterclockwise traversal of the vertices adjacent to the infinite_vertex is a clockwise traversal of the convex hull.

DelaunayGraph_2
ApolloniusGraphTraits_2
ApolloniusGraphDataStructure_2
ApolloniusGraphVertexBase_2
TriangulationFaceBase_2
CGAL::Apollonius_graph_traits_2<K,Method_tag>
CGAL::Apollonius_graph_filtered_traits_2<CK,CM,EK,EM,FK,FM>
CGAL::Triangulation_data_structure_2<Vb,Fb>
CGAL::Apollonius_graph_vertex_base_2<Gt,StoreHidden>
CGAL::Triangulation_face_base_2<Gt>
Examples:
Apollonius_graph_2/ag2_exact_traits.cpp, Apollonius_graph_2/ag2_exact_traits_sqrt.cpp, and Apollonius_graph_2/ag2_filtered_traits_no_hidden.cpp.

## Types

typedef Agds Data_structure
A type for the underlying data structure.

typedef Data_structure Triangulation_data_structure
Same as the Data_structure type. More...

typedef Gt Geom_traits
A type for the geometric traits.

typedef Gt::Point_2 Point_2
A type for the point defined in the geometric traits.

typedef Gt::Site_2 Site_2
A type for the Apollonius site, defined in the geometric traits.

## Handles And Iterators

The vertices and faces of the Apollonius graph are accessed through handles, iterators and circulators.

The iterators and circulators are all bidirectional and non-mutable. The circulators and iterators are assignable to the corresponding handle types, and they are also convertible to the corresponding handles. The edges of the Apollonius graph can also be visited through iterators and circulators, the edge circulators and iterators are also bidirectional and non-mutable. In the following, we call infinite any face or edge incident to the infinite vertex and the infinite vertex itself. Any other feature (face, edge or vertex) of the Apollonius graph is said to be finite. Some iterators (the All iterators ) allow to visit finite or infinite features while the others (the Finite iterators) visit only finite features. Circulators visit both infinite and finite features.

typedef Data_structure::Edge Edge
the edge type. More...

typedef Data_structure::Vertex Vertex
A type for a vertex.

typedef Data_structure::Face Face
A type for a face.

typedef Data_structure::Vertex_handle Vertex_handle
A type for a handle to a vertex.

typedef Data_structure::Face_handle Face_handle
A type for a handle to a face.

typedef Data_structure::Vertex_circulator Vertex_circulator
A type for a circulator over vertices incident to a given vertex.

typedef Data_structure::Face_circulator Face_circulator
A type for a circulator over faces incident to a given vertex.

typedef Data_structure::Edge_circulator Edge_circulator
A type for a circulator over edges incident to a given vertex.

typedef Data_structure::Vertex_iterator All_vertices_iterator
A type for an iterator over all vertices.

typedef Data_structure::Face_iterator All_faces_iterator
A type for an iterator over all faces.

typedef Data_structure::Edge_iterator All_edges_iterator
A type for an iterator over all edges.

typedef Data_structure::size_type size_type
An unsigned integral type.

typedef unspecified_type Finite_vertices_iterator
A type for an iterator over finite vertices.

typedef unspecified_type Finite_faces_iterator
A type for an iterator over finite faces.

typedef unspecified_type Finite_edges_iterator
A type for an iterator over finite edges.

## Site Iterators

The following iterators allow respectively to visit all sites, the visible sites and the hidden sites.

These iterators are non-mutable, bidirectional and their value type is Site_2. They are all invalidated by any change in the Apollonius graph.

typedef unspecified_type Sites_iterator
A type for an iterator over all sites.

typedef unspecified_type Visible_sites_iterator
A type for an iterator over all visible sites.

typedef unspecified_type Hidden_sites_iterator
A type for an iterator over all hidden sites.

Sites_iterator sites_begin ()
Starts at an arbitrary site.

Sites_iterator sites_end ()
Past-the-end iterator.

Visible_sites_iterator visible_sites_begin ()
Starts at an arbitrary visible site.

Visible_sites_iterator visible_sites_end ()
Past-the-end iterator.

Hidden_sites_iterator hidden_sites_begin ()
Starts at an arbitrary hidden site.

Hidden_sites_iterator hidden_sites_end ()
Past-the-end iterator.

## Creation

Apollonius_graph_2 (Gt gt=Gt())
Creates an Apollonius graph ag using gt as geometric traits.

template<class Input_iterator >
Apollonius_graph_2 (Input_iterator first, Input_iterator beyond, Gt gt=Gt())
Creates an Apollonius graph ag using gt as geometric traits and inserts all sites in the range [first, beyond). More...

Apollonius_graph_2 (const Apollonius_graph_2< Gt, Agds > &other)
Copy constructor. More...

Apollonius_graph_2< Gt, Agds > operator= (const Apollonius_graph_2< Gt, Agds > &other)
Assignment. More...

## Access Functions

Geom_traits geom_traits ()
Returns a reference to the Apollonius graph traits object.

Data_structure data_structure ()
Returns a reference to the underlying data structure.

Data_structure tds ()
Same as data_structure(). More...

int dimension ()
Returns the dimension of the Apollonius graph.

size_type number_of_vertices ()
Returns the number of finite vertices.

size_type number_of_visible_sites ()
Returns the number of visible sites.

size_type number_of_hidden_sites ()
Returns the number of hidden sites.

size_type number_of_faces ()
Returns the number of faces (both finite and infinite) of the Apollonius graph.

Face_handle infinite_face ()
Returns a face incident to the infinite_vertex.

Vertex_handle infinite_vertex ()
Returns the infinite_vertex.

Vertex_handle finite_vertex ()
Returns a vertex distinct from the infinite_vertex. More...

## Face, Edge and Vertex Iterators

The following iterators allow respectively to visit finite faces, finite edges and finite vertices of the Apollonius graph.

These iterators are non-mutable, bidirectional and their value types are respectively Face, Edge and Vertex. They are all invalidated by any change in the Apollonius graph. The following iterators allow respectively to visit all (both finite and infinite) faces, edges and vertices of the Apollonius graph. These iterators are non-mutable, bidirectional and their value types are respectively Face, Edge and Vertex. They are all invalidated by any change in the Apollonius graph.

Finite_vertices_iterator finite_vertices_begin ()
Starts at an arbitrary finite vertex.

Finite_vertices_iterator finite_vertices_end ()
Past-the-end iterator.

Finite_edges_iterator finite_edges_begin ()
Starts at an arbitrary finite edge.

Finite_edges_iterator finite_edges_end ()
Past-the-end iterator.

Finite_faces_iterator finite_faces_begin ()
Starts at an arbitrary finite face.

Finite_faces_iterator finite_faces_end () const
Past-the-end iterator.

All_vertices_iterator all_vertices_begin ()
Starts at an arbitrary vertex.

All_vertices_iterator all_vertices_end ()
Past-the-end iterator.

All_edges_iterator all_edges_begin ()
Starts at an arbitrary edge.

All_edges_iterator all_edges_end ()
Past-the-end iterator.

All_faces_iterator all_faces_begin ()
Starts at an arbitrary face.

All_faces_iterator all_faces_end ()
Past-the-end iterator.

## Face, Edge and Vertex Circulators

The Apollonius graph also provides circulators that allow to visit respectively all faces or edges incident to a given vertex or all vertices adjacent to a given vertex.

These circulators are non-mutable and bidirectional. The operator operator++ moves the circulator counterclockwise around the vertex while the operator-- moves clockwise. A face circulator is invalidated by any modification of the face pointed to. An edge circulator is invalidated by any modification in one of the two faces incident to the edge pointed to. A vertex circulator is invalidated by any modification in any of the faces adjacent to the vertex pointed to.

Face_circulator incident_faces (Vertex_handle v)
Starts at an arbitrary face incident to v.

Face_circulator incident_faces (Vertex_handle v, Face_handle f)
Starts at face f. More...

Edge_circulator incident_edges (Vertex_handle v)
Starts at an arbitrary edge incident to v.

Edge_circulator incident_edges (Vertex_handle v, Face_handle f)
Starts at the first edge of f incident to v, in counterclockwise order around v. More...

Vertex_circulator incident_vertices (Vertex_handle v)
Starts at an arbitrary vertex incident to v.

Vertex_circulator incident_vertices (Vertex_handle v, Face_handle f)
Starts at the first vertex of f adjacent to v in counterclockwise order around v. More...

## Predicates

The class Apollonius_graph_2 provides methods to test the finite or infinite character of any feature.

bool is_infinite (Vertex_handle v) const
true, iff v is the infinite_vertex.

bool is_infinite (Face_handle f) const
true, iff face f is infinite.

bool is_infinite (Face_handle f, int i) const
true, iff edge (f,i) is infinite.

bool is_infinite (Edge e) const
true, iff edge e is infinite.

bool is_infinite (Edge_circulator ec) const
true, iff edge *ec is infinite.

## Insertion

template<class Input_iterator >
unsigned int insert (Input_iterator first, Input_iterator beyond)
Inserts the sites in the range [first,beyond). More...

Vertex_handle insert (Site_2 s)
Inserts the site s in the Apollonius graph. More...

Vertex_handle insert (Site_2 s, Vertex_handle vnear)
Inserts s in the Apollonius graph using the site associated with vnear as an estimate for the nearest neighbor of the center of s. More...

## Removal

void remove (Vertex_handle v)
Removes the site associated to the vertex handle v from the Apollonius graph. More...

## Nearest Neighbor Location

Vertex_handle nearest_neighbor (Point_2 p)
Finds the nearest neighbor of the point p. More...

Vertex_handle nearest_neighbor (Point_2 p, Vertex_handle vnear)
Finds the nearest neighbor of the point p using the site associated with vnear as an estimate for the nearest neighbor of p. More...

The Apollonius_graph_2 class provides access to the duals of the faces of the graph.

The dual of a face of the Apollonius graph is a site. If the originating face is infinite, its dual is a site with center at infinity (or equivalently with infinite weight), which means that it can be represented geometrically as a line. If the originating face is finite, its dual is a site with finite center and weight. In the following three methods the returned object is assignable to either Site_2 or Gt::Line_2, depending on whether the corresponding face of the Apollonius graph is finite or infinite, respectively.

Gt::Object_2 dual (Face_handle f) const
Returns the dual corresponding to the face handle f. More...

Gt::Object_2 dual (All_faces_iterator it) const
Returns the dual of the face to which it points to. More...

Gt::Object_2 dual (Finite_faces_iterator it) const
Returns the dual of the face to which it points to. More...

## I/O

template<class Stream >
Stream & draw_primal (Stream &str)
Draws the Apollonius graph to the stream str. More...

template<class Stream >
Stream & draw_dual (Stream &str)
Draws the dual of the Apollonius graph, i.e., the Apollonius diagram, to the stream str. More...

template<class Stream >
Stream & draw_primal_edge (Edge e, Stream &str)
Draws the edge e of the Apollonius graph to the stream str. More...

template<class Stream >
Stream & draw_dual_edge (Edge e, Stream &str)
Draws the dual of the edge e to the stream str. More...

void file_output (std::ostream &os)
Writes the current state of the Apollonius graph to an output stream. More...

void file_input (std::istream &is)
Reads the state of the Apollonius graph from an input stream.

std::ostream & operator<< (std::ostream &os, Apollonius_graph_2< Gt, Agds > ag)
Writes the current state of the Apollonius graph to an output stream.

std::istream & operator>> (std::istream &is, Apollonius_graph_2< Gt, Agds > ag)
Reads the state of the Apollonius graph from an input stream.

## Validity Check

bool is_valid (bool verbose=false, int level=1)
Checks the validity of the Apollonius graph. More...

## Miscellaneous

void clear ()
Clears all contents of the Apollonius graph.

void swap (Apollonius_graph_2< Gt, Agds > other)
The Apollonius graphs other and ag are swapped. More...

## ◆ Edge

template<typename Gt, typename Agds>
 typedef Data_structure::Edge CGAL::Apollonius_graph_2< Gt, Agds >::Edge

the edge type.

The Edge(f,i) is the edge common to faces f and f.neighbor(i). It is also the edge joining the vertices vertex(cw(i)) and vertex(ccw(i)) of f.

Precondition
i must be 0, 1 or 2.

## ◆ Triangulation_data_structure

template<typename Gt, typename Agds>
 typedef Data_structure CGAL::Apollonius_graph_2< Gt, Agds >::Triangulation_data_structure

Same as the Data_structure type.

This type has been introduced in order for the Apollonius_graph_2 class to be a model of the DelaunayGraph_2 concept.

## ◆ Apollonius_graph_2() [1/2]

template<typename Gt, typename Agds>
template<class Input_iterator >
 CGAL::Apollonius_graph_2< Gt, Agds >::Apollonius_graph_2 ( Input_iterator first, Input_iterator beyond, Gt gt = Gt() )

Creates an Apollonius graph ag using gt as geometric traits and inserts all sites in the range [first, beyond).

Precondition
Input_iterator must be a model of InputIterator. The value type of Input_iterator must be Site_2.

## ◆ Apollonius_graph_2() [2/2]

template<typename Gt, typename Agds>
 CGAL::Apollonius_graph_2< Gt, Agds >::Apollonius_graph_2 ( const Apollonius_graph_2< Gt, Agds > & other )

Copy constructor.

All faces and vertices are duplicated. After the construction, ag and other refer to two different Apollonius graphs : if other is modified, ag is not.

## ◆ draw_dual()

template<typename Gt, typename Agds>
template<class Stream >
 Stream& CGAL::Apollonius_graph_2< Gt, Agds >::draw_dual ( Stream & str )

Draws the dual of the Apollonius graph, i.e., the Apollonius diagram, to the stream str.

Precondition
The following operators must be defined: Stream& operator<<(Stream&, Gt::Segment_2), Stream& operator<<(Stream&, Gt::Ray_2), Stream& operator<<(Stream&, Gt::Line_2).

## ◆ draw_dual_edge()

template<typename Gt, typename Agds>
template<class Stream >
 Stream& CGAL::Apollonius_graph_2< Gt, Agds >::draw_dual_edge ( Edge e, Stream & str )

Draws the dual of the edge e to the stream str.

The dual of e is an edge of the Apollonius diagram.

Precondition
The following operators must be defined: Stream& operator<<(Stream&, Gt::Segment_2), Stream& operator<<(Stream&, Gt::Ray_2), Stream& operator<<(Stream&, Gt::Line_2).

## ◆ draw_primal()

template<typename Gt, typename Agds>
template<class Stream >
 Stream& CGAL::Apollonius_graph_2< Gt, Agds >::draw_primal ( Stream & str )

Draws the Apollonius graph to the stream str.

Precondition
The following operators must be defined: Stream& operator<<(Stream&, Gt::Segment_2), Stream& operator<<(Stream&, Gt::Ray_2).

## ◆ draw_primal_edge()

template<typename Gt, typename Agds>
template<class Stream >
 Stream& CGAL::Apollonius_graph_2< Gt, Agds >::draw_primal_edge ( Edge e, Stream & str )

Draws the edge e of the Apollonius graph to the stream str.

Precondition
The following operators must be defined: Stream& operator<<(Stream&, Gt::Segment_2), Stream& operator<<(Stream&, Gt::Ray_2).

## ◆ dual() [1/3]

template<typename Gt, typename Agds>
 Gt::Object_2 CGAL::Apollonius_graph_2< Gt, Agds >::dual ( Face_handle f ) const

Returns the dual corresponding to the face handle f.

The returned object can be assignable to one of the following: Site_2, Gt::Line_2.

## ◆ dual() [2/3]

template<typename Gt, typename Agds>
 Gt::Object_2 CGAL::Apollonius_graph_2< Gt, Agds >::dual ( All_faces_iterator it ) const

Returns the dual of the face to which it points to.

The returned object can be assignable to one of the following: Site_2, Gt::Line_2.

## ◆ dual() [3/3]

template<typename Gt, typename Agds>
 Gt::Object_2 CGAL::Apollonius_graph_2< Gt, Agds >::dual ( Finite_faces_iterator it ) const

Returns the dual of the face to which it points to.

The returned object can be assignable to one of the following: Site_2, Gt::Line_2.

## ◆ file_output()

template<typename Gt, typename Agds>
 void CGAL::Apollonius_graph_2< Gt, Agds >::file_output ( std::ostream & os )

Writes the current state of the Apollonius graph to an output stream.

In particular, all visible and hidden sites are written as well as the underlying combinatorial data structure.

## ◆ finite_vertex()

template<typename Gt, typename Agds>
 Vertex_handle CGAL::Apollonius_graph_2< Gt, Agds >::finite_vertex ( )

Returns a vertex distinct from the infinite_vertex.

Precondition
The number of (visible) vertices in the Apollonius graph must be at least one.

## ◆ incident_edges()

template<typename Gt, typename Agds>
 Edge_circulator CGAL::Apollonius_graph_2< Gt, Agds >::incident_edges ( Vertex_handle v, Face_handle f )

Starts at the first edge of f incident to v, in counterclockwise order around v.

Precondition
Face f is incident to vertex v.

## ◆ incident_faces()

template<typename Gt, typename Agds>
 Face_circulator CGAL::Apollonius_graph_2< Gt, Agds >::incident_faces ( Vertex_handle v, Face_handle f )

Starts at face f.

Precondition
Face f is incident to vertex v.

## ◆ incident_vertices()

template<typename Gt, typename Agds>
 Vertex_circulator CGAL::Apollonius_graph_2< Gt, Agds >::incident_vertices ( Vertex_handle v, Face_handle f )

Starts at the first vertex of f adjacent to v in counterclockwise order around v.

Precondition
Face f is incident to vertex v.

## ◆ insert() [1/3]

template<typename Gt, typename Agds>
template<class Input_iterator >
 unsigned int CGAL::Apollonius_graph_2< Gt, Agds >::insert ( Input_iterator first, Input_iterator beyond )

Inserts the sites in the range [first,beyond).

The number of sites in the range [first, beyond) is returned.

Precondition
Input_iterator must be a model of InputIterator and its value type must be Site_2.

## ◆ insert() [2/3]

template<typename Gt, typename Agds>
 Vertex_handle CGAL::Apollonius_graph_2< Gt, Agds >::insert ( Site_2 s )

Inserts the site s in the Apollonius graph.

If s is visible then the vertex handle of s is returned, otherwise Vertex_handle(nullptr) is returned.

## ◆ insert() [3/3]

template<typename Gt, typename Agds>
 Vertex_handle CGAL::Apollonius_graph_2< Gt, Agds >::insert ( Site_2 s, Vertex_handle vnear )

Inserts s in the Apollonius graph using the site associated with vnear as an estimate for the nearest neighbor of the center of s.

If s is visible then the vertex handle of s is returned, otherwise Vertex_handle(nullptr) is returned.

## ◆ is_valid()

template<typename Gt, typename Agds>
 bool CGAL::Apollonius_graph_2< Gt, Agds >::is_valid ( bool verbose = false, int level = 1 )

Checks the validity of the Apollonius graph.

If verbose is true a short message is sent to std::cerr. If level is 0, only the data structure is validated. If level is 1, then both the data structure and the Apollonius graph are validated. Negative values of level always return true, and values greater then 1 are equivalent to level being 1.

## ◆ nearest_neighbor() [1/2]

template<typename Gt, typename Agds>
 Vertex_handle CGAL::Apollonius_graph_2< Gt, Agds >::nearest_neighbor ( Point_2 p )

Finds the nearest neighbor of the point p.

In other words it finds the site whose Apollonius cell contains p. Ties are broken arbitrarily and one of the nearest neighbors of p is returned. If there are no visible sites in the Apollonius diagram Vertex_handle(nullptr) is returned.

## ◆ nearest_neighbor() [2/2]

template<typename Gt, typename Agds>
 Vertex_handle CGAL::Apollonius_graph_2< Gt, Agds >::nearest_neighbor ( Point_2 p, Vertex_handle vnear )

Finds the nearest neighbor of the point p using the site associated with vnear as an estimate for the nearest neighbor of p.

Ties are broken arbitrarily and one of the nearest neighbors of p is returned. If there are no visible sites in the Apollonius diagram Vertex_handle(nullptr) is returned.

## ◆ operator=()

template<typename Gt, typename Agds>
 Apollonius_graph_2 CGAL::Apollonius_graph_2< Gt, Agds >::operator= ( const Apollonius_graph_2< Gt, Agds > & other )

Assignment.

If ag and other are the same object nothing is done. Otherwise, all the vertices and faces are duplicated. After the assignment, ag and other refer to different Apollonius graphs : if other is modified, ag is not.

## ◆ remove()

template<typename Gt, typename Agds>
 void CGAL::Apollonius_graph_2< Gt, Agds >::remove ( Vertex_handle v )

Removes the site associated to the vertex handle v from the Apollonius graph.

Precondition
v must correspond to a valid finite vertex of the Apollonius graph.

## ◆ swap()

template<typename Gt, typename Agds>
 void CGAL::Apollonius_graph_2< Gt, Agds >::swap ( Apollonius_graph_2< Gt, Agds > other )

The Apollonius graphs other and ag are swapped.

ag.swap(other) should be preferred to ag= other or to ag(other) if other is deleted afterwards.

## ◆ tds()

template<typename Gt, typename Agds>
 Data_structure CGAL::Apollonius_graph_2< Gt, Agds >::tds ( )

Same as data_structure().

This method has been added in compliance with the DelaunayGraph_2 concept.