Edits to account for Mael's comment from last revision

Periodic_4_hyperbolic_triangulation_2/doc/Periodic_4_hyperbolic_triangulation_2/Periodic_4_hyperbolic_triangulation_2.txt

@@ -11,7 +11,10 @@ namespace CGAL {
 \cgalAutoToc
 \author Iordan Iordanov & Monique Teillaud
 
-\image html new-triangulation-350px.png 
+<center>
+<img src="new-triangulation-350px.png" style="max-width:50%; width=50%;"/>
+</center>
+
 
 This package allows to compute Delaunay triangulations of the Bolza surface, which is the
 most symmetric surface of genus 2. The Bolza surface is a hyperbolic closed compact orientable surface. 
@@ -34,15 +37,11 @@ Consider the regular hyperbolic octagon \f$ \mathcal D_O \f$ centered at the ori
 all angles equal to \f$ \pi/4\f$ that is shown in
 \cgalFigureRef{P4HTriangulationOctagonId} - Left.
 Note that \f$\mathcal D_O\f$ is unique up to rotation, and cannot
-be scaled, since this operation would change its angles.
-
-Now, consider the 
+be scaled, since this operation would change its angles. Consider the 
 four hyperbolic translations \f$ a,b,c,d\f$ with their respective inverses \f$\overline{a}, 
 \overline{b}, \overline{c}, \overline{d}\f$ that identify the opposite sides of 
 \f$ \mathcal D_O \f$. The axes of these translations are diameters of the Poincaré disk. 
-See \cgalFigureRef{P4HTriangulationOctagonId} - Left. 
-
-The four translations 
+See \cgalFigureRef{P4HTriangulationOctagonId} - Left. The four translations 
 \f$a, b, c, d\f$ generate a (non-commutative) discrete group of orientation-preserving isometries, with 
 finite presentation
 \f[ \mathcal{G} = \left< a,b,c,d \; \bigg| \; 
@@ -79,7 +78,6 @@ onto \f$\mathcal M\f$.
 By definition, all points of \f$\mathbb H^2\f$ that belong to the same
 orbit under the action of \f$\mathcal G\f$ project by \f$\pi\f$ onto
 the same point of the surface \f$\mathcal M\f$. 
-
 The half-open octagon \f$\mathcal D\f$ shown in
 \cgalFigureRef{P4HTriangulationOctagonId} - Right contains exactly one
 <i>representative</i> of each point of \f$\mathcal{M}\f$;
@@ -130,7 +128,7 @@ outside \f$\mathcal D\f$. Such faces can be uniquely specified by
 three pairs of points in \f$\mathcal D\f$ and (reduced) translations
 of \f$\mathcal{G}\f$; points in the original domain are paired with
 the identity translation \f$\mathbb 1\f$. The underlying combinatorial
-triangulation is a \ref PkgTDS2Summary, enriched in each face by the
+triangulation is a \ref PkgTDS2, enriched in each face by the
 three translations that are paired with the point in each vertex. See
 \cgalFigureRef{P4HTriangulationOrientationDS}. 
 
@@ -239,10 +237,6 @@ dummy points necessary to guarantee that the triangulation is a simplicial compl
 
 \section P4HT2_design Software Design 
 
-\cgalModifBegin 
-  TODO enlever les redondances dans cette section et améliorer la structure [modified, à discuter]
-\cgalModifEnd 
-
 The main class of this package is `Periodic_4_hyperbolic_Delaunay_triangulation_2`, which
 implements Delaunay triangulations of the Bolza surface \f$\mathcal M\f$. The prefix 
 "Periodic_4" emphasizes that the triangulation in the universal covering \f$\mathbb H^2\f$ 
@@ -279,14 +273,11 @@ the triangulation. See Section \ref P4HT2_datastructure.
 
 \subsection P4HT2_traits The Geometric Traits Parameter 
 
-\cgalModifBegin
 The geometric traits class must fulfill the requirements described in the concept
 `Periodic_4HyperbolicDelaunayTriangulationTraits_2`. It must provide all necessary objects, 
 predicates, and constructions for the computation of Delaunay triangulations of the Bolza surface.
 Moreover, the traits class must represent hyperbolic translations of the group \f$\mathcal G\f$
 via the class `Hyperbolic_octagon_translation`.
-\cgalModifEnd
-
 
 A model for the concept `Periodic_4HyperbolicDelaunayTriangulationTraits_2` offered by this
 package is the class `Periodic_4_hyperbolic_Delaunay_triangulation_traits_2`. The class requires 
@@ -309,8 +300,8 @@ This model is itself parameterized by a vertex
 base class and a face base class, which gives the possibility to customize the vertices and 
 faces used by the triangulation data structure. To represent periodic hyperbolic triangulations,
 the face base and vertex base classes must be models of the concepts
-`Periodic_4HyperbolicTriangulationDSFaceBase_2` and
-`Periodic_4HyperbolicTriangulationDSVertexBase_2`, respectively.
+`Periodic_4HyperbolicTriangulationFaceBase_2` and
+`Periodic_4HyperbolicTriangulationVertexBase_2`, respectively.
 
 The default value for the triangulation data structure parameter in the class 
 `Periodic_4_hyperbolic_Delaunay_triangulation_2` is
@@ -370,7 +361,7 @@ have been executed on two machines:
 </center>
 
 Another experiment shows that, on average, all dummy points are removed 
-from the triangulation with the insertion of less than 200 random points uniformly distributed 
+from the triangulation with the insertion of fewer than 200 random points uniformly distributed 
 in the unit disk with respect to the Euclidean metric \cgalCite{cgal:it-idtbs-17}. 
 We start with an empty triangulation of the Bolza 
 surface (i.e., initialized with only the dummy points), and we start inserting random points