\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.12.2 - 2D Boolean Operations on Nef Polygons
List of all members
CGAL::Nef_polyhedron_2< T >::Topological_explorer Class Reference
2D Boolean Operations on Nef Polygons 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).