mirror of https://github.com/CGAL/cgal
95 lines
3.7 KiB
TeX
95 lines
3.7 KiB
TeX
%\RCSdef{\RCSTriangulationRev}{$Id$}
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%\RCSdefDate{\RCSTriangulationDate}{$Date$}
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\ccRefChapter{Triangulations\label{chap:triangulation_ref}}
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\ccChapterAuthor{Samuel Hornus \and Olivier Devillers}
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A triangulation is a pure simplicial complex, connected and without
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singularities. Its faces are such that two of them either do not
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intersect or share a common face.
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The basic triangulation class of \cgal\ is primarily designed to
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represent the triangulations of a set of points $A$ in $\R^d$.
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It can be
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viewed as a partition of the convex hull of $A$ into simplices whose
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vertices are the points of $A$. Together with the unbounded full cells having
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the convex hull boundary as its frontier, the triangulation forms a
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partition of $\R^d$.
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In order to deal only with full dimensional simplices (full cells),
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which is convenient for many
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applications, the space outside the convex hull is subdivided into
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full cells by
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considering that each convex hull facet is incident to an infinite
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full cell having as vertex an auxiliary vertex called the infinite
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vertex. In that way, each facet is incident to exactly two full cells and
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special cases at the boundary of the convex hull are simple to deal
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with.
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A triangulation is a collection of vertices and full cells that are linked
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together through incidence and adjacency relations. Each full cell gives
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access to its its incident vertices and to its its adjacent
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full cells. Each vertex gives access to one of its incident full cells.
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The vertices of a full cell are indexed in positive
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orientation, the positive orientation being defined by the orientation
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of the underlying Euclidean space $\R^d$. The neighbors of a full cell are also
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indexed in such a way that the neighbor indexed by $i$
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is opposite to the vertex with the same index.
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\section{Reference Pages Sorted by Type}
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\subsection{Concepts}
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\subsubsection*{Triangulation data structure}
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\ccRefConceptPage{TriangulationDataStructure}
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The above concept is also abbreviated as \ccc{TriangulationDataStructure}. It defines three types,
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\ccc{Vertex}, \ccc{Full_cell} and \ccc{Face}, that must respectively fulfill the
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following concepts:
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\ccRefConceptPage{TriangulationDSVertex}\\
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\ccRefConceptPage{TriangulationDSFullCell}\\
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\ccRefConceptPage{TriangulationFace}
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\subsubsection*{(Geometric) triangulations}
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\ccRefConceptPage{TriangulationTraits}\\
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\ccRefConceptPage{DelaunayTriangulationTraits}\\
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%\ccRefConceptPage{RegularTriangulationTraits}
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\ccRefConceptPage{TriangulationVertex}\\
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\ccRefConceptPage{TriangulationFullCell}
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The above concepts are also abbreviated respectively as \ccc{TriangulationTraits},
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\ccc{DelaunayTriangulationTraits},
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%\ccc{RTTraits},
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\ccc{TrVertex} and \ccc{TrFullCell}.
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\subsection{Classes}
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\subsubsection*{Triangulation data structure}
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\ccRefIdfierPage{CGAL::Triangulation_data_structure<Dimensionality, TriangulationDSVertex, TriangulationDSFullCell>}\\
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\ccRefIdfierPage{CGAL::Triangulation_ds_vertex<TriangulationDataStructure>}\\
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\ccRefIdfierPage{CGAL::Triangulation_ds_full_cell<TriangulationDataStructure, TDSFullCellStoragePolicy>}
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\ccRefIdfierPage{CGAL::Triangulation_face<TriangulationDataStructure>}
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\subsubsection*{(Geometric) triangulations}
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\ccRefIdfierPage{CGAL::Triangulation<TriangulationTraits, TriangulationDataStructure>}\\
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\ccRefIdfierPage{CGAL::Delaunay_triangulation<DelaunayTriangulationTraits, TriangulationDataStructure>}
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%\ccRefIdfierPage{CGAL::Regular_triangulation<RCTraits, TriangulationDataStructure>}
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\ccRefIdfierPage{CGAL::Triangulation_vertex<TriangulationTraits, Data, TriangulationDSVertex>}\\
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\ccRefIdfierPage{CGAL::Triangulation_full_cell<TriangulationTraits, Data, TriangulationDSFullCell>}
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\subsection{Enums}
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\ccRefIdfierPage{CGAL::Triangulation::Locate_type}
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