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CGAL 6.1 - 2D Boolean Operations on Nef Polygons
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CGAL 6.1 - 2D Boolean Operations on Nef Polygons
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#include <CGAL/Nef_polyhedron_2.h>
Inherited by CGAL::Nef_polyhedron_2< T >::Explorer.
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: 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. | |
| typedef unspecified_type | Isolated_vertex_const_iterator |
iterating all isolated vertices of a face f. | |
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. | |
| 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. | |
| 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. | |
| 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. | |
| 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 | |
| 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. | |
| 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.
| 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.
| Face_const_handle CGAL::Nef_polyhedron_2< T >::Topological_explorer::face | ( | Vertex_const_handle | v | ) |
returns the face incident to v.
is_isolated(v). | 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.
!is_isolated(v). | 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.
| 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.
!is_isolated(v). 
1.9.6