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Andreas Fabri
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@ -4,34 +4,36 @@ namespace CGAL {
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/*!
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\ingroup PkgTriangulation2TriangulationClasses
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A constrained Delaunay triangulation is a triangulation with
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constrained edges which tries to be as much Delaunay as possible.
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Constrained edges are not necessarily Delaunay edges,
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therefore a constrained Delaunay triangulation is not a Delaunay
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triangulation. A constrained Delaunay is a triangulation
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whose faces do not
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necessarily fulfill the empty circle property
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but fulfill a weaker property called the
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<I>constrained empty circle</I>.
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To state this property,
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it is convenient to think of constrained
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edges as blocking the view. Then, a triangulation is
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constrained Delaunay if
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the circumscribing circle
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of any of its triangular faces includes in its interior
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no vertex that is visible
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from the interior of the triangle.
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The class `Constrained_Delaunay_triangulation_2` is designed to represent
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constrained Delaunay triangulations.
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A constrained Delaunay triangulation is a triangulation with
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constrained edges which tries to be as much Delaunay as possible.
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Constrained edges are not necessarily Delaunay edges, therefore a
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constrained Delaunay triangulation is not a Delaunay triangulation. A
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constrained Delaunay is a triangulation whose faces do not necessarily
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fulfill the empty circle property but fulfill a weaker property called
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the <I>constrained empty circle</I>. To state this property, it is
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convenient to think of constrained edges as blocking the view. Then, a
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triangulation is constrained Delaunay if the circumscribing circle of
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any of its triangular faces includes in its interior no vertex that is
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visible from the interior of the triangle.
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As in the case of constrained triangulations, three different versions
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of Delaunay constrained triangulations are offered
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depending on whether the user wishes to handle
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intersecting input constraints or not.
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The desired version can be selected through the instantiation of the
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third template parameter `Itag` which can be one of the
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following :
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\tparam Traits is the geometric traits of a constrained Delaunay triangulation.
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It must be a model of `DelaunayTriangulationTraits_2`, providing the `side_of_oriented_circle` test
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of a Delaunay triangulation.
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When intersection of input constraints are supported,
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the geometric traits class
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is required to provide additional function object types
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to compute the intersection of two segments.
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and has then to be also a model of the concept
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`ConstrainedTriangulationTraits_2`.
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\tparam Tds must be a model of `TriangulationDataStructure_2`.
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\tparam Itag allows to select if intersecting constraints are supported and how they are handled.
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- `No_intersection_tag` if intersections of
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input constraints are disallowed,
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- `Exact_predicates_tag` allows intersections between input
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@ -41,21 +43,6 @@ predicates but approximate constructions of the intersection points,
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constraints and is to be used in conjunction with an exact arithmetic
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type.
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The template parameters `Tds`
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has to be instantiate with a model of `TriangulationDataStructure_2`.
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The geometric traits
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of a constrained Delaunay triangulation is required
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to provide the `side_of_oriented_circle` test as the geometric traits
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of a Delaunay triangulation and the `Traits`
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parameter has
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to be instantiated with a model
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`DelaunayTriangulationTraits_2`.
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When intersection of input constraints are supported,
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the geometric traits class
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is required to provide additional function object types
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to compute the intersection of two segments.
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and has then to be also a model of the concept
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`ConstrainedTriangulationTraits_2`.
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A constrained Delaunay triangulation is not a Delaunay
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triangulation but it is a constrained triangulation.
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@ -77,10 +77,8 @@ Section \ref Section_2D_Triangulations_Constrained_Plus.
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\image html constraints.gif
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The class `Constrained_triangulation_2` of the \cgal library
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implements constrained triangulations.
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The template parameter `Traits`
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stands for a geometric traits class. It has to be a model
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\tparam Traits is a geometric traits class and must be a model
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of the concept `TriangulationTraits_2`.
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When intersection of input constraints are supported,
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the geometric traits class
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@ -88,15 +86,15 @@ is required to provide additional function object types
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to compute the intersection of two segments.
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It has then to be a model of the concept
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`ConstrainedTriangulationTraits_2`.
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The template parameter `Tds`
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stands for
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a triangulation data structure class that has to be a model
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\tparam Tds must be a model
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of the concept `TriangulationDataStructure_2`.
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The third parameter `Itag` is the intersection tag
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\tparam Itag is the intersection tag
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which serves to choose between the different
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strategies to deal with constraints intersections.
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\cgal provides three valid types for this parameter:
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- `No_intersection_tag` disallows intersections of
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input constraints,
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- `Exact_predicates_tag` is to be used when the traits
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@ -17,22 +17,15 @@ of more than three points,
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the Delaunay triangulation is unique, it is the dual
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of the Voronoi diagram of the points.
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### Parameters ###
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The template parameter `Tds`
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is to be instantiated with a model of
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`TriangulationDataStructure_2`.
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\tparam Tds must be a model of `TriangulationDataStructure_2`.
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\cgal provides a default instantiation for this parameter,
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which is the class
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`CGAL::Triangulation_data_structure_2 < CGAL::Triangulation_vertex_base_2<Traits>, CGAL::Triangulation_face_base_2<Traits> >`.
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The geometric traits `Traits`
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is to be instantiated with a model of
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`DelaunayTriangulationTraits_2`.
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\tparam Traits must be a model of `DelaunayTriangulationTraits_2`.
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The concept `DelaunayTriangulationTraits_2` refines the
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concept `TriangulationTraits_2`, providing
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a predicate type
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to check the empty circle property.
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a predicate type to check the empty circle property.
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Changing this predicate type
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allows to build Delaunay triangulations for different metrics
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@ -107,10 +107,7 @@ the list of hidden vertices whose points are located in the facet.
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When a facet is removed,
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points hidden by this facet are reinserted in the triangulation.
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### Parameters ###
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The geometric traits parameter `Traits` has
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to be instantiated with a model of the concept
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\tparam Traits is the geometric traits parameter and must be a model of the concept
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`RegularTriangulationTraits_2`.
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The concept `RegularTriangulationTraits_2` refines the
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concept `TriangulationTraits_2` by adding the type
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@ -118,15 +115,14 @@ concept `TriangulationTraits_2` by adding the type
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and the type `Power_test_2` to perform
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power tests on weighted points.
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The `Tds` parameter has to be instantiated by a model of
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`TriangulationDataStructure_2`. The face base of a regular
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triangulation has to be a model of the concept
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\tparam Tds must be a model of `TriangulationDataStructure_2`.
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The face base of a regular triangulation has to be a model of the concept
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`RegularTriangulationFaceBase_2`. while
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the vertex base class has to be a model
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of `RegularTriangulationVertexBase_2`.
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\cgal provides a default
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instantiation for the `Tds` parameter by the class
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`CGAL::Triangulation_data_structure_2 < CGAL::Regular_triangulation_vertex_base_2<Traits>, CGAL::Regular_triangulation_face_base_2<Traits> >`.
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`Triangulation_data_structure_2 < Regular_triangulation_vertex_base_2<Traits>, Regular_triangulation_face_base_2<Traits> >`.
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\sa `CGAL::Triangulation_2<Traits,Tds>`
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\sa `TriangulationDataStructure_2`
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@ -193,7 +189,7 @@ typedef Hidden_type Hidden_vertices_iterator;
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/// @{
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/*!
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Introduces an empty regular triangulation `rt`.
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Introduces an empty regular triangulation.
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*/
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Regular_triangulation_2(const Traits& gt = Traits());
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@ -70,30 +70,26 @@ of `f` (see Figure \ref Triangulation_ref_Fig_neighbors).
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\anchor Triangulation_ref_Fig_neighbors
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\image html neighbors.gif "Vertices and neighbors."
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### Parameters ###
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The class `Triangulation_2` has two template parameters. The first one
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`Traits` is the geometric traits, it is to be instantiated by
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\tparam Traits is the geometric traits which must be
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a model of the concept `TriangulationTraits_2`.
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The second parameter is the triangulation data structure,
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it has to be instantiated by a model of the concept
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`TriangulationDataStructure_2`.
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\tparam Tds is the triangulation data structure which
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must be a model of the concept `TriangulationDataStructure_2`.
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By default, the triangulation data structure is instantiated by
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`CGAL::Triangulation_data_structure_2 < CGAL::Triangulation_vertex_base_2<Gt>, CGAL::Triangulation_face_base_2<Gt> >`.
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`Triangulation_data_structure_2 < Triangulation_vertex_base_2<Gt>, Triangulation_face_base_2<Gt> >`.
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### Traversal of the Triangulation ###
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A triangulation can be seen as a container of faces and vertices.
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Therefore the triangulation provides several iterators and circulators
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that allow to traverse it (completely or partially).
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that allow to traverse it completely or partially.
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### Traversal of the Convex Hull ###
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Applied on the `infinite_vertex` the above functions allow to visit
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Applied on the infinite vertex the above functions allow to visit
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the vertices on the convex hull and the infinite edges and faces. Note
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that a counterclockwise traversal of the vertices adjacent to the
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`infinite_vertex` is a clockwise traversal of the convex hull.
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infinite vertex is a clockwise traversal of the convex hull.
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\code
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typedef CGAL::Triangulation_2<Traits, Tds> Triangulation_2;
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@ -30,9 +30,9 @@ queries on real
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data. As proved in \cite d-iirdt-98, this structure has an optimal behavior
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when it is built for Delaunay triangulations.
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However it can be used as well for other triangulations.
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The class `Triangulation_hierarchy_2` is templated by a parameter
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which can be instantiated by any of the \cgal triangulation
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classes.
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\tparam Tr may be any of the \cgal triangulation classes.
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### Types ###
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@ -161,10 +161,6 @@ Construct_circumcenter_2 construct_circumcenter_2_object();
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*/
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Construct_bisector_2 construct_bisector_2_object();
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/*!
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*/
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Construct_direction_2 construct_direction_2_object();
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/*!
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