CGAL::Nef_polyhedron_2< T >::Topological_explorer Class Reference

#include <CGAL/Nef_polyhedron_2.h>

Inherited by CGAL::Nef_polyhedron_2< T >::Explorer.

Definition

An instance D of the data type Topological_explorer is a decorator for interfacing the topological structure of a plane map P (read-only).

A plane map P consists of a triple \( (V, E, F)\) of vertices, edges, and faces. We collectively call them objects. An edge e is a pair of vertices (v,w) with incidence operations v = source(e), w = target(e). The list of all edges with source v is called the adjacency list A(v).

Edges are paired into twins. For each edge e = (v,w) there's an edge twin(e) = (w,v) and twin(twin(e)) == eThe existence of the edge pairs makes P a bidirected graph, the twin links make P a map..

An edge e = (v,w) knows two adjacent edges en = next(e) and ep = previous(e) where source(en) = w, previous(en) = e and target(ep) = v and next(ep) = e. By this symmetric previous-next relationship all edges are partitioned into face cycles. Two edges \( e\) and \( e'\) are in the same face cycle if \( e = next^*(e')\). All edges e in the same face cycle have the same incident face \( f = face(e)\). The cyclic order on the adjacency list of a vertex v = source(e) is given by cyclic_adj_succ(e) = twin(previous(e)) and cyclic_adj_pred(e) = next(twin(e)).

A vertex v is embedded via coordinates point(v). By the embedding of its source and target an edge corresponds to a segment. P has the property that the embedding is always order-preserving. This means a ray fixed in point(v) of a vertex v and swept around counterclockwise meets the embeddings of target(e) ( \( e \in A(v)\)) in the cyclic order defined by the list order of A.

The embedded face cycles partition the plane into maximal connected subsets of points. Each such set corresponds to a face. A face is bounded by its incident face cycles. For all the edges in the non-trivial face cycles it holds that the face is left of the edges. There can also be trivial face cycles in form of isolated vertices in the interior of a face. Each such vertex v knows its surrounding face f = face(v).

Plane maps are attributed, for each object \( u \in V \cup E \cup F\) we attribute an information mark(u) of type Mark. Mark fits the concepts assignable, default-constructible, and equal-comparable.

Types
typedef unspecified_type	Plane_map
	The underlying plane map type.
typedef unspecified_type	Point
	The point type of vertices.
typedef unspecified_type	Mark
	All objects (vertices, edges, faces) are attributed by a Mark object.
typedef unspecified_type	Size_type
	The size type.

Circulators
Local types are handles, iterators and circulators of the following kind: Vertex_const_handle, Vertex_const_iterator, Halfedge_const_handle, Halfedge_const_iterator, Face_const_handle, Face_const_iterator. Additionally the following circulators are defined.
typedef unspecified_type	Halfedge_around_vertex_const_circulator
	circulating the outgoing halfedges in \( A(v)\).
typedef unspecified_type	Halfedge_around_face_const_circulator
	circulating the halfedges in the face cycle of a face f.
typedef unspecified_type	Hole_const_iterator
	iterating all holes of a face f. More...
typedef unspecified_type	Isolated_vertex_const_iterator
	iterating all isolated vertices of a face f. More...

Operations
Vertex_const_handle	source (Halfedge_const_handle e)
	returns the source of e.
Vertex_const_handle	target (Halfedge_const_handle e)
	returns the target of e.
Halfedge_const_handle	twin (Halfedge_const_handle e)
	returns the twin of e.
bool	is_isolated (Vertex_const_handle v)
	returns true iff \( A(v) = \emptyset\).
Halfedge_const_handle	first_out_edge (Vertex_const_handle v)
	returns one halfedge with source v. More...
Halfedge_const_handle	last_out_edge (Vertex_const_handle v)
	returns the halfedge with source v that is the last in the circular iteration before encountering first_out_edge(v) again. More...
Halfedge_const_handle	cyclic_adj_succ (Halfedge_const_handle e)
	returns the edge after e in the cyclic ordered adjacency list of source(e).
Halfedge_const_handle	cyclic_adj_pred (Halfedge_const_handle e)
	returns the edge before e in the cyclic ordered adjacency list of source(e).
Halfedge_const_handle	next (Halfedge_const_handle e)
	returns the next edge in the face cycle containing e.
Halfedge_const_handle	previous (Halfedge_const_handle e)
	returns the previous edge in the face cycle containing e.
Face_const_handle	face (Halfedge_const_handle e)
	returns the face incident to e.
Face_const_handle	face (Vertex_const_handle v)
	returns the face incident to v. More...
Halfedge_const_handle	halfedge (Face_const_handle f)
	returns a halfedge in the bounding face cycle of f (Halfedge_const_handle() if there is no bounding face cycle).

Iteration
Vertex_const_iterator	vertices_begin ()
	iterator over vertices of the map.
Vertex_const_iterator	vertices_end ()
	past-the-end iterator for vertices.
Halfedge_const_iterator	halfedges_begin ()
	iterator over halfedges of the map.
Halfedge_const_iterator	halfedges_end ()
	past-the-end iterator for halfedges.
Face_const_iterator	faces_begin ()
	iterator over faces of the map.
Face_const_iterator	faces_end ()
	past-the-end iterator for faces
Halfedge_around_vertex_const_circulator	out_edges (Vertex_const_handle v)
	returns a circulator for the cyclic adjacency list of v.
Halfedge_around_face_const_circulator	face_cycle (Face_const_handle f)
	returns a circulator for the outer face cycle of f.
Hole_const_iterator	holes_begin (Face_const_handle f)
	returns an iterator for all holes in the interior of f. More...
Hole_const_iterator	holes_end (Face_const_handle f)
	returns the past-the-end iterator of f.
Isolated_vertex_const_iterator	isolated_vertices_begin (Face_const_handle f)
	returns an iterator for all isolated vertices in the interior of f.
Isolated_vertex_const_iterator	isolated_vertices_end (Face_const_handle f)
	returns the past the end iterator of f.

Associated Information
The type Mark is the general attribute of an object.
const Point &	point (Vertex_const_handle v)
	returns the embedding of v.
const Mark &	mark (Vertex_const_handle v)
	returns the mark of v.
const Mark &	mark (Halfedge_const_handle e)
	returns the mark of e.
const Mark &	mark (Face_const_handle f)
	returns the mark of f.

Statistics and Integrity
Size_type	number_of_vertices ()
	returns the number of vertices.
Size_type	number_of_halfedges ()
	returns the number of halfedges.
Size_type	number_of_edges ()
	returns the number of halfedge pairs.
Size_type	number_of_faces ()
	returns the number of faces.
Size_type	number_of_face_cycles ()
	returns the number of face cycles.
Size_type	number_of_connected_components ()
	calculates the number of connected components of P.
void	print_statistics (std::ostream &os=std::cout)
	print the statistics of P: the number of vertices, edges, and faces.
void	check_integrity_and_topological_planarity (bool faces=true)
	checks the link structure and the genus of P.

Member Typedef Documentation

Hole_const_iterator

template<typename T>

typedef unspecified_type CGAL::Nef_polyhedron_2< T >::Topological_explorer::Hole_const_iterator

iterating all holes of a face f.

The type is convertible to Halfedge_const_handle.

Isolated_vertex_const_iterator

template<typename T>

typedef unspecified_type CGAL::Nef_polyhedron_2< T >::Topological_explorer::Isolated_vertex_const_iterator

iterating all isolated vertices of a face f.

The type generalizes Vertex_const_handle.

Member Function Documentation

face()

template<typename T>

Face_const_handle CGAL::Nef_polyhedron_2< T >::Topological_explorer::face

(

Vertex_const_handle

v

)

returns the face incident to v.

Precondition

is_isolated(v).

first_out_edge()

template<typename T>

Halfedge_const_handle CGAL::Nef_polyhedron_2< T >::Topological_explorer::first_out_edge

(

Vertex_const_handle

v

)

returns one halfedge with source v.

It's the starting point for the circular iteration over the halfedges with source v.

Precondition

!is_isolated(v).

holes_begin()

template<typename T>

Hole_const_iterator CGAL::Nef_polyhedron_2< T >::Topological_explorer::holes_begin

(

Face_const_handle

f

)

returns an iterator for all holes in the interior of f.

A Hole_iterator can be assigned to a Halfedge_around_face_const_circulator.

last_out_edge()

template<typename T>

Halfedge_const_handle CGAL::Nef_polyhedron_2< T >::Topological_explorer::last_out_edge

(

Vertex_const_handle

v

)

returns the halfedge with source v that is the last in the circular iteration before encountering first_out_edge(v) again.

Precondition

!is_isolated(v).