mirror of https://github.com/CGAL/cgal
98 lines
4.2 KiB
TeX
98 lines
4.2 KiB
TeX
% +------------------------------------------------------------------------+
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% | Reference manual page: Triangulation/intro.tex
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% +------------------------------------------------------------------------+
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% | 27.3.2000 Monique Teillaud
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% | Package: Triangulation3
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% |
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\RCSdef{\RCSTriangulationcellRev}{$Id$}
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\RCSdefDate{\RCSTriangulationcellDate}{$Date$}
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% |
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%%RefPage: end of header, begin of main body
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% +------------------------------------------------------------------------+
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%\clearpage
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%\section{Reference pages for 3D-Triangulations}
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\ccRefChapter{3D Triangulations}
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\ccChapterAuthor{Sylvain Pion \and Monique Teillaud}
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A three-dimensional triangulation is a three-dimensional simplicial
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complex, pure connected and without singularities \cite{by-ag-98}. Its
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cells ($3$-faces) are such that two cells either do not intersect or
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share a common facet ($2$-face), edge ($1$-face) or vertex ($0$-face).
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The basic 3D-triangulation class of \cgal\ is primarily designed to
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represent the triangulations of a set of points $A$ in $\R^3$. It can
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be viewed as a partition of the convex hull of {$A$} into tetrahedra
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whose vertices are the points of {$A$}. Together with the unbounded
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cell having the convex hull boundary as its frontier, the triangulation
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forms a partition of $\R^3$.
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In order to deal
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only with tetrahedra, which is convenient for many applications, the
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unbounded cell can be subdivided into tetrahedra by considering that
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each convex hull facet is incident to an \ccc{infinite cell} having as
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fourth vertex an auxiliary vertex called the \ccc{infinite vertex}. In
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that way, each facet is incident to exactly two cells and special cases
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at the boundary of the convex hull are simple to deal with.
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A triangulation is a collection of vertices and cells that are linked
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together through incidence and adjacency relations. Each cell gives
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access to its four incident vertices and to its four adjacent
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cells. Each vertex gives access to one of its incident cells.
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The four vertices of a cell are indexed with 0, 1, 2 and 3 in positive
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orientation, the positive orientation being defined by the orientation
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of the underlying Euclidean space $\R^3$. The neighbors of a cell are also
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indexed with 0, 1, 2, 3 in such a way that the neighbor indexed by $i$
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is opposite to the vertex with the same index. See
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Figure~\ref{Triangulation3-fig-orient}.
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\section{Classified Reference Pages}
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\subsection*{Concepts}
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\ccRefConceptPage{TriangulationTraits_3}\\
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\ccRefConceptPage{DelaunayTriangulationTraits_3}\\
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\ccRefConceptPage{RegularTriangulationTraits_3}\\
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\ccRefConceptPage{TriangulationCellBase_3}\\
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\ccRefConceptPage{TriangulationVertexBase_3}\\
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\ccRefConceptPage{TriangulationVertexBaseWithInfo_3}\\
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\ccRefConceptPage{TriangulationHierarchyVertexBase_3}\\
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\ccRefConceptPage{RegularTriangulationCellBase_3}\\
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\ccRefConceptPage{TriangulationDataStructure_3}\\
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\ccRefConceptPage{WeightedPoint}\\
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\subsection*{Classes}
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\subsubsection*{Main Classes}
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\ccRefIdfierPage{CGAL::Triangulation_3<TriangulationTraits_3,TriangulationDataStructure_3>}\\
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\ccRefIdfierPage{CGAL::Delaunay_triangulation_3<DelaunayTriangulationTraits_3,TriangulationDataStructure_3,LocationPolicy>}\\
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\ccRefIdfierPage{CGAL::Triangulation_hierarchy_3<Tr>}\\
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\ccRefIdfierPage{CGAL::Regular_triangulation_3<RegularTriangulationTraits_3,TriangulationDataStructure_3>}
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\ccRefIdfierPage{CGAL::Triangulation_cell_base_3<TriangulationTraits_3, TriangulationDSCellBase_3>}\\
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\ccRefIdfierPage{CGAL::Triangulation_cell_base_with_info_3<Info, TriangulationTraits_3, TriangulationCellBase_3>}\\
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\ccRefIdfierPage{CGAL::Triangulation_cell_base_with_circumcenter_3<DelaunayTriangulationTraits_3, TriangulationCellBase_3>}\\
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\ccRefIdfierPage{CGAL::Triangulation_vertex_base_3<TriangulationTraits_3, TriangulationDSVertexBase_3>}\\
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\ccRefIdfierPage{CGAL::Triangulation_vertex_base_with_info_3<Info, TriangulationTraits_3, TriangulationVertexBase_3>}\\
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\ccRefIdfierPage{CGAL::Triangulation_hierarchy_vertex_base_3<TriangulationVertexBase_3>}\\
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\ccRefIdfierPage{CGAL::Regular_triangulation_cell_base_3<Traits,Cb>}\\
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\ccRefIdfierPage{CGAL::Triangulation_simplex_3<Triangulation_3>}
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\subsubsection*{Traits Classes}
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\ccRefIdfierPage{CGAL::Regular_triangulation_euclidean_traits_3<K,Weight>}
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\subsection*{Enums}
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\ccRefIdfierPage{CGAL::Triangulation_3::Locate_type}
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