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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
\cgalAutoToc
\author Mikhail Bogdanov, and Iordan Iordanov, and Monique Teillaud
\author Mikhail Bogdanov, Iordan Iordanov, and Monique Teillaud
<center>
<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
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$\mathcal
P\f$. Of course, the hyperbolic and Euclidean geometric embeddings of a
given Delaunay face are different. 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. 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
P\f$. More precisely, the hyperbolic Delaunay triangulation of \f$\mathcal
P\f$ is a connected simplicial complex that only
contains the simplices of the Euclidean Delaunay triangulation that
are <i>hyperbolic</i>:
<ul>
@ -72,6 +58,20 @@ are <i>hyperbolic</i>:
P\f$) passing through its endpoints is contained in \f$\mathbb
H^2\f$.
</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
non-hyperbolic faces and non-hyperbolic edges
\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
From what was said above, it is natural that 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
know more in particular about the representation of triangulations in
CGAL and the flexibility of the design.