mirror of https://github.com/CGAL/cgal
Doc: changes according to Mariette's live review
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Clement Jamin
parent
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@ -13,7 +13,7 @@
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\cgalPkgDescriptionBegin{Triangulations,PkgTriangulationsSummary}
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\cgalPkgPicture{detail.png}
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\cgalPkgSummaryBegin
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\cgalPkgAuthors{Samuel Hornus and Olivier Devillers}
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\cgalPkgAuthors{Samuel Hornus, Olivier Devillers and Clément Jamin}
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\cgalPkgDesc{The package `Triangulation` provides classes for manipulating triangulations (pure simplicial complexes) in Euclidean spaces whose dimension can be specified at compile-time or at run-time. Specifically, it provides a combinatorial Triangulation data structure, which is extended into geometric triangulation and Delaunay triangulation classes. Point location and point insertion are supported. The Delaunay triangulation also supports point removal.}
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\cgalPkgManuals{Chapter_Triangulations,PkgTriangulations}
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\cgalPkgSummaryEnd
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@ -6,15 +6,15 @@ namespace CGAL {
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\anchor Chapter_Triangulations
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\cgalAutoToc
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\authors Samuel Hornus and Olivier Devillers
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\authors Samuel Hornus, Olivier Devillers and Clément Jamin.
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This package proposes data structure and algorithms to compute
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This package proposes data structures and algorithms to compute
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triangulations of points in any dimensions.
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The `Triangulation_data_structure` allows to store and manipulate the
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combinatorial part of a triangulation while the geometric classes
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The `Triangulation_data_structure` handles the
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combinatorial aspect of triangulations while the geometric classes
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`Triangulation` and `Delaunay_triangulation` allows to
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compute a (Delaunay) triangulation of a set of points and to maintain
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it under insertions (and deletions in the Delaunay case).
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compute and maintain triangulations and Delaunay triangulations of
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sets of points.
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# Introduction #
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@ -36,12 +36,11 @@ The simplicial complex is <I>pure</I> if all the maximal simplices
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have the same dimension.
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<!--- cardinality, i.e., they have the same number of vertices.--->
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In the sequel, we will call these maximal simplices <I>full cells</I>.
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A <I>face</I> of a simplex is a subset of it.
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A <I>proper face</I> of a simplex is a strict subset of it.
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A <I>face</I> of a simplex is a subset of this simplex.
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A <I>proper face</I> of a simplex is a strict subset of this simplex.
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A complex has <i>no boundaries</i> if any proper face of a simplex is also a
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proper face of another simplex. A pure complex is <i>manifold</i> if all faces
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of dimension \f$ d-1 \f$ are proper faces of exactly two simplices.
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proper face of another simplex.
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If the vertices are embedded into Euclidean space \f$ \mathbb{R}^d\f$,
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we deal with
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@ -59,33 +58,22 @@ entry</A> for more about simplicial complexes.
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## What's in this package? ##
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This \cgal package deals with pure manifold simplicial complexes which are connected
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and have no boundaries, which
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we will simply call in the sequel <I>triangulations</I>. It provides three main classes
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This \cgal package provides three main classes
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for creating and manipulating triangulations.
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The class `CGAL::Triangulation_data_structure<Dimensionality, TriangulationDSVertex, TriangulationDSFullCell>` models an <I>abstract triangulation</I>: vertices in this
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The class `CGAL::Triangulation_data_structure<Dimensionality, TriangulationDSVertex, TriangulationDSFullCell>`
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models an <I>abstract triangulation</I>: vertices in this
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class are not embedded in Euclidean space but are only of combinatorial
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nature.
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nature. It deals with simplicial complexes
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which are pure, connected and without boundaries nor singularities.
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\cgalModifBegin
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The class `CGAL::Triangulation<TriangulationTraits, TriangulationDataStructure>`
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describes an embedded triangulation that has as vertices a given set of points
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and which covers the convex hull of these points.
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Methods are
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provided for the insertion of points in the triangulation, the
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describes an embedded triangulation that has as vertices a given set of points.
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Methods are provided for the insertion of points in the triangulation, the
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traversal of various elements of the triangulation, as well as the localization of a
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query point inside the triangulation.
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The convex hull of the points is part of the triangulation.
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The fact that there is no boundary is ensured by adding a
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fictitious vertex, called the <i>infinite vertex</i>, as well as infinite
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simplices incident to it. Each infinite \f$ i\f$-simplex is
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incident to the infinite vertex and to \f$ i\f$ vertices of the convex hull.
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\cgalModifEnd
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See Chapters \ref Chapter_2D_Triangulations "2D Triangulations" and
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\ref Chapter_3D_Triangulations "3D Triangulations" for more details
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about infinite vertices and cells.
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query point inside the triangulation.
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The triangulation covers the convex hull of the set of points.
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The class `CGAL::Delaunay_triangulation<DelaunayTriangulationTraits, TriangulationDataStructure>` adds further
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constraints to a triangulation, in that all its simplices must have the
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@ -132,27 +120,25 @@ us denote the maximal dimension with \f$ D \f$ and the current dimension with \f
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The inequalities \f$ -2 \leq d \leq D\f$ and \f$ 0 \le D\f$ always hold.
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The special meaning of negative values for \f$d\f$ is explained below.
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### The data structure triangulates \f$ \mathbb{S}^d\f$ ###
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### The Set of Faces ###
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A `TriangulationDataStructure` can be viewed as
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a triangulation of the topological sphere \f$ \mathbb{S}^d\f$,
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i.e., its faces can be embedded to form a partition of
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\f$ \mathbb{S}^d\f$ into \f$d\f$-simplices.
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The set of faces of a `TriangulationDataStructure` with
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current dimension \f$ d \f$ forms a triangulation of the
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topological sphere \f$ \mathbb{S}^d\f$.
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One nice consequence of the above important fact is that a full cell has
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always exactly \f$ d+1\f$ neighbors.
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Two full cells \f$ \sigma\f$ and \f$ \sigma'\f$ sharing a facet are called
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<I>neighbors</I>.
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<I>neighbors</I>. A full cell has always exactly \f$ d+1\f$ neighbors.
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Possible values of \f$d\f$ (the <I>current dimension</I> of the triangulation) include
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<BLOCKQUOTE>
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<DL>
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<DT><B>\f$d=-2\f$</B><DD> This corresponds to the non-existence of any object in
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the `TriangulationDataStructure`.
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<DT><B>\f$d=-1\f$</B><DD> This corresponds to a single vertex.
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<DT><B>\f$d=-2\f$</B><DD> This corresponds to an empty
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`TriangulationDataStructure`.
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<DT><B>\f$d=-1\f$</B><DD> This corresponds to an abstract simplicial
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complex reduced to a single vertex.
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<!--- and a single full cell. In a geometric triangulation, this vertex corresponds to the vertex at infinity.--->
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<DT><B>\f$d=0\f$</B><DD> This corresponds to two vertices,
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each corresponding to a full cell;
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<DT><B>\f$d=0\f$</B><DD> This corresponds to an abstract simplicial
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complex including two vertices, each corresponding to a full cell;
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the two full cells being neighbors of each other. This is the unique
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triangulation of the \f$ 0\f$-sphere.
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<!--- (geometrically, the finite vertex and the infinite vertex),--->
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@ -171,8 +157,7 @@ implementing the concept `TriangulationDataStructure`.
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A `TriangulationDataStructure` explicitly stores its vertices and full cells.
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Each vertex stores a reference (a `handle`) to one of its incident
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full cells.
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Each vertex stores a reference to one of its incident full cells.
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Each full cell stores references to its \f$ d+1\f$ vertices and
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neighbors. Its vertices and neighbors are indexed from \f$ 0\f$ to \f$ d \f$. The indices
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@ -180,13 +165,22 @@ of its neighbors have the following meaning: the \f$ i\f$-th neighbor of \f$ \si
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is the unique neighbor of \f$ \sigma\f$ that does not contain the \f$ i\f$-th vertex of
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\f$ \sigma\f$; in other words, it is the neighbor of \f$ \sigma\f$ <I>opposite</I> to
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the \f$ i\f$-th vertex of \f$ \sigma\f$ (Figure \cgalFigureRef{triangulationfigfullcell}).
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Faces of dimension between 0 and \f$ d-1 \f$ can be accessed as
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subfaces of a full cell.
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The vertices and full cells of the triangulations are accessed through
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`handles` and `iterators`. A handle is a model of the
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`Handle` concept, and supports the two dereference operators and
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`operator->`.
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\cgalFigureBegin{triangulationfigfullcell,simplex-structure.png}
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Indexing the vertices and neighbors of a full cell \f$ c\f$ in dimension \f$ d=2\f$.
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\cgalFigureEnd
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Faces of dimension between 0 and \f$ d-1 \f$ can be accessed as
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subfaces of a full cell, through the nested type `Face`. The `Face` instance
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corresponding to a face \f$ f \f$ stores a reference to a full cell `c`
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containing \f$ f \f$, and the indices of the vertices of `c` that belong
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to \f$ f \f$.
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<!---
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\cgalAdvanced The index of a full cell \f$ c\f$ in the \f$ i\f$-th
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neighbor of \f$ c\f$ is called the <I>\f$ i\f$-th mirror-index</I> of
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@ -208,7 +202,7 @@ documentation of that class template for specific details.
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###Template parameters###
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The `Triangulation_data_structure<Dimensionality, TriangulationDSVertex, TriangulationDSFullCell>`
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class template is designed in such a way that its user can choose
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class is designed in such a way that its user can choose
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<UL>
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<LI>the maximal dimension of the triangulation data structure by specifying the `Dimensionality` template parameter,
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<LI>the type used to represent vertices by specifying the `TriangulationDSVertex`
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@ -294,22 +288,27 @@ Barycentric subdivision in dimension \f$ d=2\f$.
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# Triangulations #
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The class `CGAL::Triangulation<TriangulationTraits, TriangulationDataStructure>`
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maintains a geometric
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triangulation in Euclidean space. More precisely, it
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maintains a triangulation (a partition into pairwise interior-disjoint
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full cells) of the convex hull of the points (the embedded vertices) of the
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triangulation. A special vertex at infinity is added to the convex
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hull facets to create infinite full cells and make the triangulation
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homeomorphic to a sphere of one dimension higher.
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maintains a triangulation embedded in Euclidean space. The triangulation
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covers the convex hull of the input points (the embedded vertices) of the
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triangulation.
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To store this triangulation in a triangulation data structure, we turn the set
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of its faces into a topological sphere by adding a
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fictitious vertex, called the <i>infinite vertex</i>, as well as infinite
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simplices incident to boundary faces of the convex hull.
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Each infinite \f$ i\f$-simplex is
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incident to the infinite vertex and to an \f$ (i-1)\f$-simplex of the
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convex hull boundary.
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\cgalModifEnd
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See Chapters \ref Chapter_2D_Triangulations "2D Triangulations" and
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\ref Chapter_3D_Triangulations "3D Triangulations" for more details
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about infinite vertices and cells.
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Methods are provided for the insertion of points in the triangulation, the
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contraction of faces, the traversal of various elements of the triangulation
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as well as the localization of a query point inside the triangulation.
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Infinite full cells outside the convex hull are each incident to
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a finite facet on the convex hull of the triangulation and to a unique
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<I>vertex at infinity</I>.
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The ordering of the vertices of a full cell defines an orientation of
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that full cell.
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As long as no <I>advanced</I> class method is called, it is guaranteed
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@ -335,16 +334,16 @@ In a triangulation, we can distinguish three dimensions:
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## Implementation ##
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The class `CGAL::Triangulation<TriangulationTraits, TriangulationDataStructure>` stores a model
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of the concept `TriangulationDataStructure` which is instantiated with a
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vertex type that stores a point, and a full cell type that allows the retrieval
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of the point of its vertices.
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The class `CGAL::Triangulation<TriangulationTraits, TriangulationDataStructure>`
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stores a model of the concept `TriangulationDataStructure` which is
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instantiated with a vertex type that stores a point.
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The template parameter `TriangulationTraits` must be a model of the concept
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`TriangulationTraits` which provides the geometric `Point` type as well
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`TriangulationTraits` which provides the `Point` type as well
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as various geometric predicates used by the `Triangulation` class.
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`TriangulationTraits::Dimension` must match
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The `TriangulationTraits` concept includes a nested type
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`TriangulationTraits::Dimension` which must match
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the maximal dimension of the `TriangulationDataStructure`.
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The template parameter `TriangulationDataStructure` must be a model of the concept
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@ -369,15 +368,15 @@ points. This gives us a handy way to count the convex hull vertices
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### Traversing the facets of the convex hull ###
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Remember that a triangulation triangulates the convex hull of its
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Remember that a triangulation covers the convex hull of its
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vertices.
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Each
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facet of the convex hull is incident to one finite full cell and one infinite
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Each facet of the convex hull is incident
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to one finite full cell and one infinite
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full cell. In fact there is a bijection between the infinite full cells and the
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facets of the convex hull.
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If vertices are not in general position, convex hull faces that are
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not simplices are triangulated.
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So, in order to traverse the convex hull facets,
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In order to traverse the convex hull facets,
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there are (at least) two possibilities:
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The first is to iterate over the full cells of the triangulation and check if they
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@ -400,27 +399,28 @@ visits <I>only</I> the infinite full cells but stores handles to them into the
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# Delaunay Triangulations #
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The class `CGAL::Delaunay_triangulation<DelaunayTriangulationTraits, TriangulationDataStructure>` derives from
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`CGAL::Triangulation<DelaunayTriangulationTraits, TriangulationDataStructure>` and adds further constraints to a
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triangulation, in that all its full cells must have the so-called
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<I>Delaunay</I> or <I>empty-ball</I> property: the interior of the ball
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circumscribing any full cell must be free from any vertex
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of the triangulation.
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`CGAL::Triangulation<DelaunayTriangulationTraits, TriangulationDataStructure>`
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and represent Delaunay triangulations.
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A <I>circumscribing ball</I> of a simplex is a ball
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having all vertices of the simplex on its boundary.
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In a Delaunay triangulation, each face has the so-called
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<I>Delaunay</I> or <I>empty-ball</I> property: there exists a
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circumscribing ball whose interior does not contain
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any vertex of the triangulation.
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The <I>circumscribing ball</I> of a full cell is the ball
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having all vertices of the full cell on its boundary.
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In case of degeneracies (co-spherical points) the triangulation is not
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uniquely defined;
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Note however that the \cgal implementation computes a unique
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uniquely defined. Note however that the \cgal implementation computes a unique
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triangulation even in these cases.
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When a new point `p` is inserted into a Delaunay triangulation, the
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finite full cells whose circumscribing sphere contain `p` are said to
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<I>be in conflict</I> with point `p`. The set of full cells that are in
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conflict with `p` form the <I>conflict zone</I>. That conflict zone is
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augmented with the infinite full cells whose finite facet does not lie
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anymore on the convex hull of the triangulation (with `p` added). The full cells
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full cells whose circumscribing ball contains `p` are said to
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<I>be in conflict</I> with point `p`. Note that the circumscribing ball
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of an infinite full cell is the empty half-space bounded by the affine hull
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of the finite facet of this cell. The set of full cells that are in
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conflict with `p` form the <I>conflict zone</I>. The full cells
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in the conflict zone are removed, leaving a hole that contains `p`. That
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hole is ``star shaped'' around `p` and thus is easily re-triangulated using
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hole is ``star shaped'' around `p` and thus is re-triangulated using
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`p` as a center vertex.
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Delaunay triangulations also support vertex removal.
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@ -441,7 +441,7 @@ for the computation of Delaunay triangulations.
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## Examples ##
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### Access to the conflict zone and creating full cells during point insertion ###
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### Access to the conflict zone and to the full cells created during point insertion ###
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When using a full cell type containing additional custom information, it may be
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useful to get an efficient access to the full cells that are going to be erased
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@ -472,8 +472,8 @@ preferably with the new Kernel).
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This package is heavily inspired by the works of
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Monique Teillaud and Sylvain Pion (`Triangulation_3`)
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and Mariette Yvinec (`Triangulation_2`).
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The first version was written by Samuel Hornus and then
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pursued by Samuel Hornus and Olivier Devillers.
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The first version was written by Samuel Hornus. The final version is a joint
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work by Samuel Hornus, Olivier Devillers and Clément Jamin.
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*/
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} /* namespace CGAL */
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@ -10,23 +10,16 @@
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#include <vector>
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const int D=5;
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typedef CGAL::Epick_d< CGAL::Dynamic_dimension_tag > K;
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typedef CGAL::Triangulation_ds_vertex< void > TDS_vertex;
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typedef CGAL::Triangulation_vertex< K, int, TDS_vertex > Vertex;
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typedef CGAL::Triangulation_ds_full_cell
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< void, CGAL::TDS_full_cell_default_storage_policy > TDS_cell;
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typedef CGAL::Triangulation_full_cell< K, int, TDS_cell > Cell;
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typedef CGAL::Triangulation_data_structure<
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CGAL::Dimension_tag<D>, Vertex, Cell > TDS;
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typedef CGAL::Delaunay_triangulation<K, TDS> T;
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typedef CGAL::Epick_d< CGAL::Dimension_tag<D> > K;
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typedef CGAL::Delaunay_triangulation<K> T;
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// The triangulation uses the default instanciation of the
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// TriangulationDataStructure template parameter
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int main(int argc, char **argv)
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{
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int N = 100; if( argc > 2 )N = atoi(argv[1]); // number of points
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CGAL::Timer cost; // timer
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// Instanciate a random point generator
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CGAL::Random rng(0);
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typedef CGAL::Random_points_in_cube_d<T::Point> Random_points_iterator;
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@ -69,9 +62,7 @@ int main(int argc, char **argv)
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for (Full_cells::iterator it=new_full_cells.begin();
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it!=new_full_cells.end(); ++it) (*it)->data() = zone.size();
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}
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std::cout << " done in "<<cost.time()<<" seconds." << std::endl;
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return 0;
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}
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