mirror of https://github.com/CGAL/cgal
small fixes, new comments and todos
This commit is contained in:
Monique Teillaud
parent
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e3bc22b104
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@ -14,7 +14,7 @@ namespace CGAL {
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\image html new-triangulation-350px.png
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This package allows to compute Delaunay triangulations of the Bolza surface, which is the
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most symmetric surface of genus 2. The Bolza surface is a hyperbolic closed compact orientable surface of genus 2.
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most symmetric surface of genus 2. The Bolza surface is a hyperbolic closed compact orientable surface.
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A triangulation is defined as a <i>simplicial complex</i>, i.e.:
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<ul>
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@ -31,8 +31,13 @@ A triangulations of the Bolza surface can be seen as a periodic triangulation of
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Let \f$\mathbb{H}^2\f$ denote the hyperbolic plane, represented in the Poincaré disk model.
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Consider the regular hyperbolic octagon \f$ \mathcal D_O \f$ centered at the origin, with
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all angles equal to \f$ \pi/4\f$. Note that \f$\mathcal D_O\f$ is unique up to rotation, and cannot
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be scaled, since this operation would change its angle sum.
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all angles equal to \f$ \pi/4\f$ that is shown in
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\cgalFigureRef{P4HTriangulationOctagonId} - Left.
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Note that \f$\mathcal D_O\f$ is unique up to rotation, and cannot
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be scaled, since this operation would change its angle sum.
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\cgalModifBegin
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shouldn't it be more simply: its angles ?
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\cgalModifEnd
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Now, consider the
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four hyperbolic translations \f$ a,b,c,d\f$ with their respective inverses \f$\overline{a},
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\overline{b}, \overline{c}, \overline{d}\f$ that identify the opposite sides of
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@ -40,7 +45,7 @@ four hyperbolic translations \f$ a,b,c,d\f$ with their respective inverses \f$\o
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See \cgalFigureRef{P4HTriangulationOctagonId} - Left.
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The four translations
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\f$a, b, c, d\f$ generate a discrete group of orientation-preserving isometries, with
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\f$a, b, c, d\f$ generate a (non-commutative) discrete group of orientation-preserving isometries, with
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finite presentation
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\f[ \mathcal{G} = \left< a,b,c,d \; \bigg| \;
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abcd\overline{a}\overline{b}\overline{c}\overline{d} \right>. \f]
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@ -104,17 +109,11 @@ D\f$ under the action of \f$\mathcal G\f$.
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\section P4HT2_representation Representation
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\cgalModifBegin
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begin: ?
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\cgalModifEnd
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When there is no risk
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of confusion, the same notation will be used for a point on the
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surface \f$\mathcal M\f$ and its representative in \f$\mathcal
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D\f$. Similarly, \f$\mathcal{P}\f$ denotes both a set of points on the
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surface and the set of their representatives in \f$\mathcal D\f$.
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\cgalModifBegin
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end: ?
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\cgalModifEnd
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The Delaunay triangulation of \f$\mathcal{M}\f$ defined by a point set
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\f$\mathcal{P}\f$ is defined as the projection by \f$\pi\f$ of the
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@ -128,7 +127,7 @@ orbits of points, all faces of the Delaunay triangulation of
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\f$\mathcal{G}\f$ project by \f$\pi\f$ onto the same face on the
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surface \f$\mathcal{M}\f$; the data structure stores a unique
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<i>canonical</i> representative of each orbit, which has at least one
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vertex in \f$\mathcal D\f$. Some faces have vertices both inside and
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vertex in \f$\mathcal D\f$. Some canonical faces have vertices both inside and
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outside \f$\mathcal D\f$. Such faces can be uniquely specified by
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three pairs of points in \f$\mathcal D\f$ and (reduced) translations
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of \f$\mathcal{G}\f$; points in the original domain are paired with
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@ -150,10 +149,10 @@ three translations that are paired with the point in each vertex. See
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More precisely, the translations are elements of the subset
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\f$\mathcal N\f$ of \f$\mathcal G\f$ for which the image of
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\f$\mathcal D_O\f$ has at least one vertex in common with
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\f$\mathcal D_O\f$.
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See \cgalFigureRef{P4HTriangulationCanonicalRepExample}.
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We consider the images of \f$\mathcal D_O\f$ under translations of
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\f$\mathcal N\f$ to be ordered counterclockwise around
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\f$\mathcal D_O\f$. These images of \f$\mathcal D_O\f$ by
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translations in \f$\mathcal N\f$ are shaded in
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\cgalFigureRef{P4HTriangulationCanonicalRepExample}; we consider them
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to be ordered counterclockwise around
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\f$\mathcal D_O\f$, arbitrarily starting with the one corresponding to
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translation \f$abcd\f$. The canonical representative in
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\f$\mathbb H^2\f$ of a face on \f$\mathcal M\f$ is such that
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@ -193,7 +192,7 @@ A periodic hyperbolic triangulation is said to be `valid` if and only if
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complex. </li>
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</ol>
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Some point sets do not admit a triangulation of \f$\mathcal M\f$. For
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Some point sets do not define a valid triangulation of \f$\mathcal M\f$. For
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instance, a single point does not define a valid triangulation of
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\f$\mathcal M\f$, as the induced subdivision would not be a simplicial
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complex. Another example is shown in
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@ -218,7 +217,7 @@ projection by \f$\pi\f$ of the Delaunay triangulation of \f$\mathcal{G}
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\f$\mathcal M\f$ is a simplicial
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complex for any set of input points \f$\mathcal{P}\f$
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\cgalCite{cgal:btv-dtosl-16}.
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See \cgalFigureRef{P4HTriangulationDummyPoints}
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See \cgalFigureRef{P4HTriangulationDummyPoints}.
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\cgalFigureAnchor{P4HTriangulationDummyPoints}
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<center>
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@ -239,14 +238,6 @@ If the set of input points is not sufficiently large, or if the points
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are not well-distributed on the surface, then the final triangulation will contain the
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dummy points necessary to guarantee that the triangulation is a simplicial complex.
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\cgalModifBegin
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tout ce qui concerne les experiences pourrait eventuellement
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descendre dans la sec Performance [modified, texte proposé ci-dessous]
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See \ref P4HT2_Performance for experiments showing the average number of random points
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needed in order to remove all dummy points from the triangulation.
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\cgalModifEnd
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\section P4HT2_design Software Design
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@ -254,8 +245,8 @@ needed in order to remove all dummy points from the triangulation.
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TODO enlever les redondances dans cette section et améliorer la structure [modified]
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\cgalModifEnd
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The main class of this package is `Periodic_4_hyperbolic_Delaunay_triangulation_2`, and
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it implements Delaunay triangulations of the Bolza surface \f$\mathcal M\f$. The prefix
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The main class of this package is `Periodic_4_hyperbolic_Delaunay_triangulation_2`, which
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implements Delaunay triangulations of the Bolza surface \f$\mathcal M\f$. The prefix
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"Periodic_4" emphasizes that the triangulation in the universal covering \f$\mathbb H^2\f$
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is periodic in the four directions defined by the hyperbolic translations \f$ a,b,c\f$,
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and \f$d\f$.
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@ -270,7 +261,7 @@ point sets (i.e., more than three co-circular points) are handled with symbolic
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The implementation is fully dynamic, supporting both point insertion and vertex removal.
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However, a vertex can be removed only if the triangulation remains valid (as defined above).
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The main class `Periodic_4_hyperbolic_Delaunay_triangulation_2` provides high-level geometric
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The class `Periodic_4_hyperbolic_Delaunay_triangulation_2` provides high-level geometric
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functionality specific to Delaunay triangulations, such as point insertion and vertex removal,
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the side-of-circle test, finding the conflicting region of a given point, and dual functions.
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It inherits from a base class called `Periodic_4_hyperbolic_triangulation_2` that provides
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@ -279,36 +270,35 @@ and access functions, but supports neither point insertion nor vertex removal.
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Both classes `Periodic_4_hyperbolic_triangulation_2` and `Periodic_4_hyperbolic_Delaunay_triangulation_2`
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require two template parameters that reflect the separation between the geometric and combinatorial
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structure of the triangulation:
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structures of the triangulation:
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- a <B>geometric traits</B> class, which provides all objects, predicates, and constructions
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necessary for the computation of Delaunay triangulations of the Bolza surface. It also provides
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the hyperbolic translations generating the group \f$\mathcal G\f$ and the translations in the
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set \f$\mathcal N\f$. This parameter must meet the requirements described in the concept
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`Periodic_4HyperbolicDelaunayTriangulationTraits_2`. For more details, see
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necessary for the computation of Delaunay triangulations of the Bolza surface.For more details, see
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Section \ref P4HT2_traits.
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- a <B>triangulation data structure</B> class, which encodes the combinatorial structure of
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the triangulation. This parameter must meet the requirements described in the concept
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`TriangulationDataStructure_2`, and for the purposes of this package is described in
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Section \ref P4HT2_datastructure.
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the triangulation. See Section \ref P4HT2_datastructure.
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\subsection P4HT2_traits The Geometric Traits Parameter
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The geometric traits class must fulfill the requirements described in the concept
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`Periodic_4HyperbolicDelaunayTriangulationTraits_2`. It must provide all necessary objects,
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predicates, and constructions, as well as the hyperbolic translations of the group
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\f$\mathcal G\f$.
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predicates, and constructions,
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\cgalModifBegin
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TODO corriger la fin de la phrase suite aux discussions
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"as well as the hyperbolic translations of the group
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\f$\mathcal G\f$."
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\cgalModifEnd
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A model for the concept `Periodic_4HyperbolicDelaunayTriangulationTraits_2` offered by this
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package is the class `Periodic_4_hyperbolic_Delaunay_triangulation_traits_2`. The class requires
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two template parameters:
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\cgalModifBegin
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1- BUG dans le ref manual : les paramètres templates ne sont pas
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documentes. TODO fix [modified]
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TODO un seul template est documente - enlever le concept
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translations
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\cgalModifEnd
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2- il y a une raison pour ne pas mettre Core-expr comme default ? [non; c'est ajouté]
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\cgalModifEnd
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<ul>
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<li> A `Kernel` with a `number type` that guarantees exact computations on algebraic numbers
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with nested square roots. The default value for this parameter is `CGAL::Cartesian<CORE::Expr>`.</li>
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@ -316,45 +306,16 @@ two template parameters:
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be the same as the `number type` of the `Kernel`. This template parameter defaults to the class
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`Hyperbolic_octagon_translation<Kernel::NT>`. </li>
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</ul>
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\cgalModifBegin
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D'habitude on n'impose pas un parametre template dans un
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concept. Dans le concept on dit seulement que le NT doit être fourni
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comme nested type. En pratique dans le modele il y a un template
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parameter. Dans ton ref manual c'est en fait ce que tu fais, tu ne
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demandes pas le template. TOD rendre tout ca coherent.
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[modified]
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\cgalModifEnd
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\cgalModifBegin
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If the field type used in `Kernel` provides exact operations with algebraic numbers involving
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nested square roots, then the traits class provides exact predicates
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and exact constructions.
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sinon, tout le code plante, c'est-ce pas ? ce n'est pas vraiment ce qu'on veut... de
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plus c'est contradictoire avec ce qui est écrit plus haut : A
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`Kernel` with a `number type` that guarantees
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exact computations on algebraic numbers with nested square
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roots. TODO rendre le texte coherent.
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[d'après discussion, à enlever]
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\cgalModifEnd
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\cgalModifBegin
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In addition, inexact constructions using `double` number type are also provided.
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pour quoi faire ?
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[En fait, ce n'est pas documenté. Je propose d'enlever cette phrase.]
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\cgalModifEnd
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TODO reflechir NT vs FT et assurer la coherence
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\cgalModifEnd
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\subsection P4HT2_datastructure The Triangulation Data Structure Parameter
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The triangulation data structure class is the second template parameter of the periodic hyperbolic
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triangulation classes. It is a container for the faces and vertices and maintains
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incidence and adjacency relations. For details, see Chapter
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\ref Chapter_2D_Triangulation_Data_Structure.
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incidence and adjacency relations.
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This parameter must meet the requirements described in the concept `TriangulationDataStructure_2`,
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for which \cgal offers the model `Triangulation_data_structure_2`.
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This model is itself parameterized by a vertex
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@ -368,44 +329,52 @@ The default value for the triangulation data structure parameter in the hyperbol
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triangulation classes is
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`Triangulation_data_structure_2< Periodic_4_hyperbolic_triangulation_face_base_2<Traits>, Periodic_4_hyperbolic_triangulation_vertex_base_2<Traits> >`,
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where `Traits` is a model of `Periodic_4HyperbolicDelaunayTriangulationTraits_2`.
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\cgalModifBegin
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est-ce que c'est vraiment le meme concept de traits pour les deux hyperbolic periodic
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triangulation classes ? a priori non... ne parler que de la classe
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Delaunay corrigerait le probleme.
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\cgalModifEnd
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\section P4HT2_examples Example
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This example shows the initialization of a periodic hyperbolic 2D Delaunay triangulation, the
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insertion of random points uniformly distributed in the unit disk for the Euclidean metric, and the
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properties of the triangulation after the insertion. It uses the default parameter of the
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`Periodic_4_hyperbolic_Delaunay_triangulation_2` class for the triangulation data structure.
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Most functionalities of periodic hyperbolic triangulations are similar
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to those of Euclidean 2D triangulations.
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\cgalModifBegin
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We refer the reader to selected examples of the \ref Chapter_2D_Triangulations "2D Triangulation" package:
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<ul>
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<li> \ref Subsection_2D_Triangulations_Voronoi
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<li> \ref Triangulation_2AddingColors
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</ul>
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\cgalModifBegin
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TODO corriger le premier ref pour enlever le mot Example: pour assurer
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la coherence avec le deuxieme ref
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\cgalModifEnd
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\cgalModifBegin
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TODO mettre le lien précis, envoyer directement sur la sec examples de T2
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[modified: added two links; implement here instead?]
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\cgalModifEnd
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The example below shows the initialization of a periodic hyperbolic 2D Delaunay triangulation, the
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insertion of random points uniformly distributed in the unit disk for the Euclidean metric, and the
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properties of the triangulation after the insertion. It uses the default parameter of the
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`Periodic_4_hyperbolic_Delaunay_triangulation_2` class for the triangulation data structure.
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\cgalExample{Periodic_4_hyperbolic_triangulation_2/p4ht2_example_insertion.cpp}
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\cgalModifBegin
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TODO ajouter dans le code 1 ou 2 commentaires bien places qui
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rappellent que les dummy points sont enleves automatiquement (si
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possible) quand on insere un "range" de points mais qu'ils ne le sont
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pas par defaut quand n insere les points un par un.
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Illustrer aussi le booleen qui permet de choisir de les enlever ?
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\cgalModifEnd
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\section P4HT2_Performance Performance
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We perform tests to measure the insertion execution time of our implementation against existing
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\cgal code. We generate random points, uniformly distributed in the unit disk with respect to
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the Euclidean metric. From the generated points, we select 1 million points that lie in the original
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We have measured the insertion execution time of our implementation
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against other \cgal triangulations. Random points, uniformly distributed in the unit disk with respect to
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the Euclidean metric, are generated. From these generated points, we select 1 million points that lie in the original
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octagon \f$\mathcal D_O\f$. We insert the same set of points in three triangulations:
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<ul>
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<li> a periodic hyperbolic Delaunay triangulation with `CORE::Expr` as `number type`;
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<li> a non-periodic Euclidean Delaunay triangulation with `CORE::Expr` as `number type`;
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<li> a non-periodic Euclidean Delaunay triangulation with `double` as `number_type`.
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<li> a (non-periodic) Euclidean Delaunay triangulation with `CORE::Expr` as `number type`;
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<li> a (non-periodic) Euclidean Delaunay triangulation with `double` as `number_type`.
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</ul>
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We report results averaged over 10 executions of this experiment in \ref Table_1 "Table 1" below. The experiments
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have been executed on two machines:
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@ -425,13 +394,10 @@ have been executed on two machines:
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</table>
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</center>
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\cgalModifBegin
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Another experiment we have conducted shows that, on average, all dummy points are removed
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Another experiment shows that, on average, all dummy points are removed
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from the triangulation with the insertion of less than 200 random points uniformly distributed
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in the unit disk with respect to the Euclidean metric \cgalCite{cgal:it-idtbs-17}.
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The experiment we conduct is the following: We start with an empty triangulation of the Bolza
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We start with an empty triangulation of the Bolza
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surface (i.e., initialized with only the dummy points), and we start inserting random points
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one by one. After each insertion, the unnecessary dummy points (if any) are removed from the
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triangulation. As soon as the last dummy point has been removed, we stop the process and record
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@ -448,7 +414,6 @@ the number of random points inserted. Results are shown in \cgalFigureRef{P4HDum
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points needed to remove all dummy points was 23, the maximum 188, and on average 64 random
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points were needed.
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\cgalFigureCaptionEnd
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\cgalModifEnd
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\section P4HT2_Design Design and Implementation History
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