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changes after Andreas' comments

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Monique Teillaud 2018-11-22 11:34:04 +01:00
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Hyperbolic_triangulation_2/doc/Hyperbolic_triangulation_2/Hyperbolic_triangulation_2.txt
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@ -9,7 +9,7 @@ namespace CGAL {
\anchor chapterHTriangulation2 \anchor chapterHTriangulation2
\cgalAutoToc \cgalAutoToc
\author Mikhail Bogdanov, and Iordan Iordanov, and Monique Teillaud \author Mikhail Bogdanov, Iordan Iordanov, and Monique Teillaud
<center> <center>
<img src="header.png" style="max-width:60%; width=60%;"/> <img src="header.png" style="max-width:60%; width=60%;"/>
@ -46,22 +46,8 @@ As hyperbolic circles coincide with Euclidean circles contained in the
unit disk, the combinatorial structure of the hyperbolic Delaunay unit disk, the combinatorial structure of the hyperbolic Delaunay
triangulation of a set \f$\mathcal P\f$ of points in \f$\mathbb H^2\f$ triangulation of a set \f$\mathcal P\f$ of points in \f$\mathbb H^2\f$
is a subset of the Euclidean Delaunay triangulation of \f$\mathcal is a subset of the Euclidean Delaunay triangulation of \f$\mathcal
P\f$. Of course, the hyperbolic and Euclidean geometric embeddings of a P\f$. More precisely, the hyperbolic Delaunay triangulation of \f$\mathcal
given Delaunay face are different. See P\f$ is a connected simplicial complex that only
\cgalFigureRef{Hyperbolic_triangulation_2Euclidean_vs_hyperbolic}.
\cgalFigureAnchor{Hyperbolic_triangulation_2Euclidean_vs_hyperbolic}
<center>
<img src="hyperbolic-vs-euclidean.png" style="max-width:35%; width=35%;"/>
</center>
\cgalFigureCaptionBegin{Hyperbolic_triangulation_2Euclidean_vs_hyperbolic}
Comparison of the Euclidean (green) and hyperbolic (black) Delaunay triangulations
of a given set of points in the unit disk. Note that only the colored faces
are faces of the hyperbolic Delaunay triangulation.
\cgalFigureCaptionEnd
More precisely, the hyperbolic Delaunay triangulation of \f$\mathcal
P\f$ is a connected simplicial complex. It only
contains the simplices of the Euclidean Delaunay triangulation that contains the simplices of the Euclidean Delaunay triangulation that
are <i>hyperbolic</i>: are <i>hyperbolic</i>:
<ul> <ul>
@ -72,6 +58,20 @@ are <i>hyperbolic</i>:
P\f$) passing through its endpoints is contained in \f$\mathbb P\f$) passing through its endpoints is contained in \f$\mathbb
H^2\f$. H^2\f$.
</ul> </ul>
See \cgalFigureRef{Hyperbolic_triangulation_2Euclidean_vs_hyperbolic}.
\cgalFigureAnchor{Hyperbolic_triangulation_2Euclidean_vs_hyperbolic}
<center>
<img src="hyperbolic-vs-euclidean.png" style="max-width:35%; width=35%;"/>
</center>
\cgalFigureCaptionBegin{Hyperbolic_triangulation_2Euclidean_vs_hyperbolic}
Comparison of the Euclidean (green) and hyperbolic (black) Delaunay triangulations
of a given set of points in the unit disk. Only the colored faces
are faces of the hyperbolic Delaunay triangulation. The hyperbolic and Euclidean geometric embeddings of a
Delaunay face that exists in both triangulations are different.
\cgalFigureCaptionEnd
In the Euclidean Delaunay triangulation, there is a bijection between In the Euclidean Delaunay triangulation, there is a bijection between
non-hyperbolic faces and non-hyperbolic edges non-hyperbolic faces and non-hyperbolic edges
\cgalCite{cgal:bdt-hdcvd-14}. See \cgalCite{cgal:bdt-hdcvd-14}. See
@ -91,7 +91,7 @@ as no empty circle through its endpoints is contained in
\section HT2_Software_design Software Design \section HT2_Software_design Software Design
From what was said above, it is natural that the class From what was said above, it is natural that the class
`Hyperbolic_Delaunay_triangulation_2` privately inherits from the class `Hyperbolic_Delaunay_triangulation_2` privately inherits from the class
`Delaunay_triangulation_2`. So, users are encouraged to look at the chapter `Delaunay_triangulation_2`. So, users are encouraged to look at Chapter
\ref Chapter_2D_Triangulations "2D Triangulation" of the CGAL manual to \ref Chapter_2D_Triangulations "2D Triangulation" of the CGAL manual to
know more in particular about the representation of triangulations in know more in particular about the representation of triangulations in
CGAL and the flexibility of the design. CGAL and the flexibility of the design.