\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.3 - CGAL and the Boost Graph Library

All Classes Namespaces Files Functions Variables Typedefs Enumerations Enumerator Groups Pages
List of all members | Related Functions
HalfedgeGraph Concept Reference
Concepts

## Definition

The concept `HalfedgeGraph` describes the requirements for a graph that is structurally equivalent to a polyhedral surface represented by a halfedge data structure, and it provides an interface for efficient access to the opposite edge of an edge, and to the successor and predecessor of an edge in the iterator range of the incoming edges of a vertex. Each vertex has a geometric position in space. As in a halfedge data structure we define the face adjacent to a halfedge to be to the left of the halfedge.

Requirements

For each directed edge `e=(v,w)` its opposite edge `e2=(w,v)` must be part of the graph.

The incoming edges of a vertex `v` have a fixed order, that is all calls of `in_edges(v,g)` must return the same iterator range, modulo a cyclic permutation. The order must be clockwise.

As the `HalfedgeGraph` is equivalent to a polyhedral surface there must exist an embedding for the vertices and edges such that the ordered edges do not intersect.

Refines:

  BidirectionalGraph

  PropertyGraph

A model of `HalfedgeGraph` must have the interior properties `edge_is_border` attached to its edges, and it must have `vertex_is_border` and `vertex_point` attached to its vertices.

Associated Types

Because (directed) edges must come in pairs, there is the additional notion of an undirected edge The directed edges are not called `halfedges` (as in a `HalfedgeDS`) because from the point of view of this graph, being a refinement of a Bgl graph, each directed edge is an edge in itself. In other words, the unqualified term edge refers to one and only one directed edge and not to a pair. for a pair of opposite directed edges. The number of undirected edges is exactly half the number of directed edges. Note that the notion of directed and undirected edges does not imply the existence of two different types. The type `edge_descriptor` is used for both. An undirected edge must be implicitly handled, and there is no requirement on which of the directed edges of the undirected edge must be used to represent it.

Associated Type | Explanation
--- | ---
halfedge_graph_traits<HalfedgeGraph>::Point; | The type of the geometric location of a vertex.
halfedge_graph_traits<HalfedgeGraph>::undirected_edge_iterator; | An iterator that iterates over one and only one of the directed edges in each pair of opposite directed edges. The value type of the iterator is `boost::graph_traits<HalfedgeGraph>::edge_descriptor`.

Valid Expressions

Following the Bgl design, the following graph operations are defined as free rather than member functions.

Has Models:
  `CGAL::Polyhedron_3<Traits>`

## Related Functions
(Note that these are not member functions.)
template<class Graph >
std::pair< typename halfedge_graph_traits < HalfedgeGraph > ::undirected_edge_iterator, typename halfedge_graph_traits < HalfedgeGraph > ::undirected_edge_iterator > | undirected_edges (const Graph &g)
| Returns the undirected edges of `g`.
template<class Graph >
boost::graph_traits< Graph const >::edge_descriptor | opposite_edge (typename boost::graph_traits< Graph const >::edge_descriptor e, Graph const &g)
| Returns the opposite edge of `e`. More...
template<class Graph >
boost::graph_traits< Graph const >::edge_descriptor | next_edge_cw (typename boost::graph_traits< Graph const >::edge_descriptor e, Graph const &g)
| Returns the clockwise neighbor of `e`. More...
template<class Graph >
boost::graph_traits< Graph const >::edge_descriptor | next_edge_ccw (typename boost::graph_traits< Graph const >::edge_descriptor e, Graph const &g)
| Returns the counterclockwise neighbor of `e`.
template<class Graph >
boost::graph_traits< Graph const >::edge_descriptor | next_edge (typename boost::graph_traits< Graph const >::edge_descriptor e, Graph const &g)
| Returns the successor of `e`. More...
template<class Graph >
boost::graph_traits< Graph const >::edge_descriptor | prev_edge (typename boost::graph_traits< Graph const >::edge_descriptor e, Graph const &g)
| Returns the predecessor of `e`. More...

## Friends And Related Function Documentation

template<class Graph >

boost::graph_traits< Graph const >::edge_descriptor next_edge | ( | typename boost::graph_traits< Graph const >::edge_descriptor | e,
|  | Graph const & | g
| ) |  |
| related

Returns the successor of `e`.

An edge `e2=(v,w)` is called the successor of edge `e=(u,v)`, and `e` the predecessor of `e2`, iff `e2` is the clockwise neighbor of the opposite edge of `e`.

template<class Graph >

boost::graph_traits< Graph const >::edge_descriptor next_edge_cw | ( | typename boost::graph_traits< Graph const >::edge_descriptor | e,
|  | Graph const & | g
| ) |  |
| related

Returns the clockwise neighbor of `e`.

An edge `e2=(v,w)` is called the clockwise neighbor of edge `e=(u,w)`, and `e` the counterclockwise neighbor of `e2`, iff there exist two iterators `it` and `it2` in the iterator range `in_edges(w,g)` such that `**it == e` and `**it2 == e2`, and `it2 == it++` or `it` is the last and `it2` the first iterator of the iterator range.

template<class Graph >

boost::graph_traits< Graph const >::edge_descriptor opposite_edge | ( | typename boost::graph_traits< Graph const >::edge_descriptor | e,
|  | Graph const & | g
| ) |  |
| related

Returns the opposite edge of `e`.

An edge `e=(v,w)` is said to be the opposite edge of edge `e2=(w,v)`.

template<class Graph >

boost::graph_traits< Graph const >::edge_descriptor prev_edge | ( | typename boost::graph_traits< Graph const >::edge_descriptor | e,
|  | Graph const & | g
| ) |  |
| related

Returns the predecessor of `e`.

An edge `e2=(v,w)` is called the successor of edge `e=(u,v)`, and `e` the predecessor of `e2`, iff `e2` is the clockwise neighbor of the opposite edge of `e`.