From dfad7cc34f93bdba547e5454f2aaf6d7caab8200 Mon Sep 17 00:00:00 2001 From: Iordan Iordanov Date: Fri, 16 Nov 2018 23:28:08 +0100 Subject: [PATCH] Edits to account for Mael's comment from last revision --- .../Periodic_4_hyperbolic_triangulation_2.txt | 29 +++++++------------ 1 file changed, 10 insertions(+), 19 deletions(-) diff --git a/Periodic_4_hyperbolic_triangulation_2/doc/Periodic_4_hyperbolic_triangulation_2/Periodic_4_hyperbolic_triangulation_2.txt b/Periodic_4_hyperbolic_triangulation_2/doc/Periodic_4_hyperbolic_triangulation_2/Periodic_4_hyperbolic_triangulation_2.txt index 4705cf3a508..12ed8bc9493 100644 --- a/Periodic_4_hyperbolic_triangulation_2/doc/Periodic_4_hyperbolic_triangulation_2/Periodic_4_hyperbolic_triangulation_2.txt +++ b/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 +
+ +
+ 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 representative 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: 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