%% Copyright (c) 2004 SciSoft. All rights reserved. %% %% This file is part of CGAL (www.cgal.org); you may redistribute it under %% the terms of the Q Public License version 1.0. %% See the file LICENSE.QPL distributed with CGAL. %% %% Licensees holding a valid commercial license may use this file in %% accordance with the commercial license agreement provided with the software. %% %% This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE %% WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. %% %% %% %% Author(s) : Fernando Cacciola \begin{ccRefConcept}{HalfedgeGraph} %% \ccHtmlCrossLink{} %% add further rules for cross referencing links %% \ccHtmlIndexC[concept]{} %% add further index entries \ccDefinition The concept \ccRefName\ 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 {\em left} of the halfedge. \ccHeading{Requirements} For each {\em directed edge} $e=(v,w)$ its opposite edge $e'=(w,v)$ must be part of the graph. The incoming edges of a vertex $v$ have a fixed order, that is all calls of \ccc{in_edges(v,g)} must return the same iterator range, modulo a cyclic permutation. The order must be {\em clockwise}. As the \ccRefName\ is equivalent to a polyhedral surface there must exist an embedding for the vertices and edges such that the ordered edges do not intersect. \ccRefines \ccAnchor{http://www.boost.org/libs/graph/doc/IncidenceGraph.html}{IncidenceGraph}\\ \ccAnchor{http://www.boost.org/libs/graph/doc/PropertyGraph.html}{PropertyGraph} A model of \ccRefName\ must have the {\em interior properties} \ccc{edge_is_border} attached to its edges, and it must have \ccc{vertex_is_border} and \ccc{vertex_point} attached to its vertices. \ccHeading{Associated Types} \ccNestedType{halfedge_graph_traits::Point} {The type of the geometric location of a vertex.} Because (directed) edges must come in pairs, there is the additional notion of an {\em undirected edge}\footnote{ The directed edges are not called \ccc{halfedges} (as in a \ccc{HalfedgeDS}) because from the point of view of this graph, being a refinement of a {\sc 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 \ccc{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. \ccNestedType{halfedge_graph_traits::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 \ccc{boost::graph_traits::edge_descriptor}.}{} \ccHeading{Valid Expressions} Following the {\sc Bgl} design, the following graph operations are defined as free rather than member functions. \ccFunction {template std::pair::undirected_edge_iterator, typename halfedge_graph_traits::undirected_edge_iterator> undirected_edges(Graph const& g ); } {Returns the undirected edges of \ccc{g}.} An edge $e=(v,w)$ is said to be the {\em opposite edge} of edge $e'=(w,v)$. \ccFunction {template typename boost::graph_traits::edge_descriptor opposite_edge(typename boost::graph_traits::edge_descriptor e, Graph const& g ); } {Returns the opposite edge of \ccc{e}.} An edge $e'=(v,w)$ is called the {\em clockwise neighbor} of edge $e=(u,w)$, and $e$ the {\em counterclockwise neighbor} of $e'$, iff there exist two iterators $it$ and $it'$ in the iterator range \ccc{in_edges(w,g)} such that \ccc{**it == e} and \ccc{**it' == e'}, and \ccc{it' == it++} or \ccc{it} is the last and \ccc{it'} the first iterator of the iterator range. \ccFunction {template typename boost::graph_traits::edge_descriptor next_edge_cw(typename boost::graph_traits::edge_descriptor e, Graph const& g ); } {Returns the clockwise neighbor of \ccc{e}.} \ccFunction {template typename boost::graph_traits::edge_descriptor next_edge_ccw(typename boost::graph_traits::edge_descriptor e, Graph const& g ); } {Returns the counterclockwise neighbor of \ccc{e}.} A composition of these access functions yields an access function for the edge cycle adjacent to the same face. An edge $e'=(v,w)$ is called the {\em successor} of edge $e=(u,v)$, and $e$ the {\em predecessor} of $e'$, iff $e'$ is the clockwise neighbor of the opposite edge of $e$. \ccFunction {template typename boost::graph_traits::edge_descriptor next_edge(typename boost::graph_traits::edge_descriptor e, Graph const& g ); } {Returns the successor of \ccc{e}.} \ccFunction {template typename boost::graph_traits::edge_descriptor prev_edge(typename boost::graph_traits::edge_descriptor e, Graph const& g ); } {Returns the predecessor of \ccc{e}.} \ccHasModels \ccRefIdfierPage{CGAL::Polyhedron_3} \end{ccRefConcept} % +------------------------------------------------------------------------+ %%RefPage: end of main body, begin of footer % EOF % +------------------------------------------------------------------------+