modified concepts according to Monique's comments

Periodic_4_hyperbolic_triangulation_2/doc/Periodic_4_hyperbolic_triangulation_2/Concepts/HyperbolicOctagonTranslation.h

@@ -4,25 +4,33 @@
 /*!
 \ingroup PkgPeriodic4HyperbolicTriangulation2Concepts
 \cgalConcept
-
-The concept `HyperbolicOctagonTranslation` describes the requirements for the class of 
-hyperbolic translations specific to the Bolza surface. 
+\cgalModifBegin
+The concept `HyperbolicOctagonTranslation` describes the requirements for an object that represents 
+a hyperbolic translation in the fundamental group \f$\mathcal G\f$ of the the Bolza surface. 
 
 Generally, a hyperbolic translation \f$g\f$ is an orientation-preserving isometry acting 
 on the hyperbolic plane, and is specified by two complex numbers \f$\alpha\f$ and \f$\beta\f$:
 \f[
 	g(z) = \frac{\alpha \cdot z + \beta}{\overline{\beta} \cdot z + \overline{\alpha}},
-	\qquad |\alpha|^2 - |\beta|^2 = 1.
+	\qquad |\alpha|^2 - |\beta|^2 = 1,
 \f]
-A hyperbolic translation is also a word on the generators of the fundamental group 
-of a hyperbolic surface.
+where \f$\overline{a}\f$ and \f$\overline{b}\f$ denote the complex conjugates of \f$a\f$
+and \f$b\f$ respectively. A hyperbolic translation is also a word on the generators of the 
+fundamental group of a hyperbolic surface.
 
-In our particular case, we consider an application specific to the Bolza surface. For our 
-application, we only need to deal with a finite set of hyperbolic translations from the 
-fundamental group of the Bolza surface. A notable property of these translations is that
-their word length is at most four; the composition of two translation always gives a
-translation that can be simplified so that its length is at most four. For more details, 
-see \cgalCite{cgal:btv-dtosl-16} and \cgalCite{cgal:it-idtbs-17}.
+Specifically for the group \f$\mathcal G\f$, let \f$[g_0, g_1, \ldots, g_7]\f$ be the set of its 
+generators, ordered counter-clockwise around the origin, with \f$g_k\f$ translating the origin 
+\f$O\f$ along the real axis in the positive direction. The coefficients \f$\alpha_k\f$ and \f$\beta_k\f$ 
+of each generator of this ordered set are given as 
+\f[
+	\alpha_k = 1 + \sqrt{2}, \qquad \beta_k = \exp\left(\tfrac{ik\pi}{4}\right)\sqrt{2}\sqrt{1+\sqrt{2}}.
+\f]
+Note that with the notation given in Section \ref P4HT2_thespace of the User manual, the ordered 
+set of generators can be written also as
+\f[
+	[g_k, k = 0, \ldots, 7] = [ a, \overline{b}, c, \overline{d}, \overline{a}, b, \overline{c}, d ].
+\f] 
+\cgalModifEnd
 
 \cgalHasModel CGAL::Hyperbolic_octagon_translation
 */
@@ -36,14 +44,22 @@ public:
 	/// \name Types
 	
 	/// Number type of the coefficients of \f$\alpha\f$ and \f$\beta\f$.
+	/// This number type must be compatible with the field type `FT` of the concept 
+	/// `Periodic_4HyperbolicDelaunayTriangulationTraits_2`.
 	typedef unspecified_type 			NT;
 
-	/// Represents a single letter of the word corresponding to the translation.
-	typedef unspecified_type 			Word_letter;
+	/// \cgalModifBegin Represents a single letter of the word corresponding to the translation. \cgalModifEnd
+	typedef unsigned short int 			Word_letter;
 
 	/// Word representation of the translation, a sequence of elements of type `Word_letter`.
 	typedef std::vector<Word_letter> 	Word;
 
+	/// \cgalModifBegin Represents a complex number with coefficients of type `NT`.\cgalModifEnd
+	typedef std::pair<NT,NT> 			Complex;
+
+	/// \cgalModifBegin Represents the coefficients \f$\alpha\f$ and \f$\beta\f$ of a hyperbolic translation.\cgalModifEnd
+	typedef std::pair<Complex,Complex> 	Coefficients;
+
 	/// \name Creation
 
 	/// Default constructor. Creates the identity translation.
@@ -94,15 +110,36 @@ public:
 
 	/// Returns the coefficient \f$\alpha\f$ of the translation's matrix. The first element of the returned `pair` 
 	/// contains the real part of \f$\alpha\f$. The second element contains its imaginary part.
-	std::pair<NT,NT> alpha() const;
+	Complex alpha() const;
 
 	/// Returns the coefficient \f$\beta\f$ of the translation's matrix. The first element of the returned `pair` 
 	/// contains the real part of \f$\beta\f$. The second element contains its imaginary part.
-	std::pair<NT,NT> beta() const;
+	Complex beta() const;
 
 	/// Returns `true` if the current translation represents the identity element of the group.
 	bool is_identity();
 
 
+	/// \name Static Access Functions
+
+	/*!
+		\cgalModifBegin
+		Returns a vector containing the coefficients of the generators of \f$\mathcal G\f$ and of their inverses.
+		The translations must be given in the order:
+		\f[ [g_0, g_1, \ldots, g_7]. \f]
+		\cgalModifEnd
+ 	*/
+	static void get_generator_coefficients(std::vector< Coefficients >& gens);
+
+
+	/*!
+		\cgalModifBegin
+		Returns a vector containing the generators of \f$\mathcal G\f$ and their inverses.
+		The translations must be given in the order:
+		\f[ [g_0, g_1, \ldots, g_7]. \f]
+		\cgalModifEnd
+ 	*/
+	static void get_generators(std::vector<HyperbolicOctagonTranslation>& gens);
+
 }; // end of class HyperbolicOctagonTranslation
 

Periodic_4_hyperbolic_triangulation_2/doc/Periodic_4_hyperbolic_triangulation_2/Concepts/Periodic_4HyperbolicDelaunayTriangulationTraits_2.h

@@ -14,6 +14,16 @@ to be fulfilled by any class used to instantiate the first template parameter of
 essential for the removal of existing vertices in a Delaunay triangulation of the
 Bolza surface.
 
+\cgalModifBegin
+	The concept requires that two nested types are provided:
+	<ul>
+		<li> A field number type `FT` that must support exact computations with algebraic 
+			 numbers, in particular with nested square roots;
+		<li> A `Hyperbolic_translation` type, which satisfies the requirements described 
+			 in the concept `HyperbolicOctagonTranslation`.
+	</ul>
+\cgalModifEnd
+
 \cgalHasModel CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_traits_2
 */
 

Periodic_4_hyperbolic_triangulation_2/doc/Periodic_4_hyperbolic_triangulation_2/Concepts/Periodic_4HyperbolicTriangulationTraits_2.h

@@ -15,6 +15,16 @@ types and the operations defined on them in `HyperbolicDelaunayTriangulationTrai
 it defines the hyperbolic translations that allow to encode the periodicity of the 
 triangulation.
 
+\cgalModifBegin
+	The concept requires that two nested types are provided:
+	<ul>
+		<li> A field number type `FT` that must support exact computations with algebraic 
+			 numbers, in particular with nested square roots;
+		<li> A `Hyperbolic_translation` type, which satisfies the requirements described 
+			 in the concept `HyperbolicOctagonTranslation`.
+	</ul>
+\cgalModifEnd
+
 */