mirror of https://github.com/CGAL/cgal
removed concept for hyperbolic translaitons; imposed translation type; changed Point_2 to Hyperbolic_point_2; added missing constructor objects
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Iordan Iordanov
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// Copyright (c) 2016-2017 INRIA Nancy - Grand Est (France).
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// All rights reserved.
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/*!
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\ingroup PkgPeriodic4HyperbolicTriangulation2Concepts
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\cgalConcept
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\cgalModifBegin
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The concept `HyperbolicOctagonTranslation` describes the requirements for an object that represents
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a hyperbolic translation in the fundamental group \f$\mathcal G\f$ of the the Bolza surface.
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Generally, a hyperbolic translation \f$g\f$ is an orientation-preserving isometry acting
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on the hyperbolic plane, and is specified by two complex numbers \f$\alpha\f$ and \f$\beta\f$:
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\f[
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g(z) = \frac{\alpha \cdot z + \beta}{\overline{\beta} \cdot z + \overline{\alpha}},
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\qquad |\alpha|^2 - |\beta|^2 = 1,
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\f]
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where \f$\overline{a}\f$ and \f$\overline{b}\f$ denote the complex conjugates of \f$a\f$
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and \f$b\f$ respectively. A hyperbolic translation is also a word on the generators of the
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fundamental group of a hyperbolic surface.
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Specifically for the group \f$\mathcal G\f$, let \f$[g_0, g_1, \ldots, g_7]\f$ be the set of its
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generators, ordered counter-clockwise around the origin, with \f$g_k\f$ translating the origin
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\f$O\f$ along the real axis in the positive direction. The coefficients \f$\alpha_k\f$ and \f$\beta_k\f$
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of each generator of this ordered set are given as
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\f[
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\alpha_k = 1 + \sqrt{2}, \qquad \beta_k = \exp\left(\tfrac{ik\pi}{4}\right)\sqrt{2}\sqrt{1+\sqrt{2}}.
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\f]
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Note that with the notation given in Section \ref P4HT2_thespace of the User manual, the ordered
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set of generators can be written also as
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\f[
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[g_k, k = 0, \ldots, 7] = [ a, \overline{b}, c, \overline{d}, \overline{a}, b, \overline{c}, d ].
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\f]
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\cgalModifEnd
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\cgalHasModel CGAL::Hyperbolic_octagon_translation
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*/
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class HyperbolicOctagonTranslation {
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public:
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/// \name Types
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/// Number type of the coefficients of \f$\alpha\f$ and \f$\beta\f$.
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/// This number type must be compatible with the field type `FT` of the concept
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/// `Periodic_4HyperbolicDelaunayTriangulationTraits_2`.
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typedef unspecified_type NT;
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/// \cgalModifBegin Represents a single letter of the word corresponding to the translation. \cgalModifEnd
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typedef unsigned short int Word_letter;
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/// Word representation of the translation, a sequence of elements of type `Word_letter`.
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typedef std::vector<Word_letter> Word;
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/// \cgalModifBegin Represents a complex number with coefficients of type `NT`.\cgalModifEnd
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typedef std::pair<NT,NT> Complex;
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/// \cgalModifBegin Represents the coefficients \f$\alpha\f$ and \f$\beta\f$ of a hyperbolic translation.\cgalModifEnd
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typedef std::pair<Complex,Complex> Coefficients;
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/// \name Creation
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/// Default constructor. Creates the identity translation.
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HyperbolicOctagonTranslation();
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/// Creates the translation described by the word `w`.
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HyperbolicOctagonTranslation(Word w);
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/// Creates the translation described by the one-letter word `l`.
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HyperbolicOctagonTranslation(Word_letter l);
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/// Creates the translation described by the two-letter word `lm`.
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HyperbolicOctagonTranslation(Word_letter l, Word_letter m);
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/// Creates the translation described by the three-letter word `lmn`.
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HyperbolicOctagonTranslation(Word_letter l, Word_letter m, Word_letter n);
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/// Creates the translation described by the four-letter word `lmno`.
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HyperbolicOctagonTranslation(Word_letter l, Word_letter m, Word_letter n, Word_letter o);
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/// \name Operations
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/// Multiplication operator; composes the current translation with `other`.
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HyperbolicOctagonTranslation operator*(const HyperbolicOctagonTranslation& other) const;
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/// Difference operator; the difference of two translations \f$v\f$ and \f$w\f$ is defined as \f$v * w^{-1}\f$.
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HyperbolicOctagonTranslation operator-(const HyperbolicOctagonTranslation& other) const;
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/// Assignment operator; modifying the translation after the assignment leaves `other` unaffected.
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HyperbolicOctagonTranslation& operator=(const HyperbolicOctagonTranslation& other);
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/// Equality operator.
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bool operator==(const Hyperbolic_octagon_translation<NT>& other) const;
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/// Inequality operator.
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bool operator!=(const Hyperbolic_octagon_translation<NT>& other) const;
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/// Comparison operator.
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/// The comparison is done on the ordering of the translations around the original domain.
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bool operator<(const Hyperbolic_octagon_translation<NT>& other) const;
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/// Returns the inverse of the current translation.
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HyperbolicOctagonTranslation inverse() const;
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/// \name Access Functions
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/// Returns the coefficient \f$\alpha\f$ of the translation's matrix. The first element of the returned `pair`
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/// contains the real part of \f$\alpha\f$. The second element contains its imaginary part.
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Complex alpha() const;
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/// Returns the coefficient \f$\beta\f$ of the translation's matrix. The first element of the returned `pair`
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/// contains the real part of \f$\beta\f$. The second element contains its imaginary part.
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Complex beta() const;
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/// Returns `true` if the current translation represents the identity element of the group.
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bool is_identity();
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/// \name Static Access Functions
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/*!
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\cgalModifBegin
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Returns a vector containing the coefficients of the generators of \f$\mathcal G\f$ and of their inverses.
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The translations must be given in the order:
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\f[ [g_0, g_1, \ldots, g_7]. \f]
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\cgalModifEnd
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*/
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static void get_generator_coefficients(std::vector< Coefficients >& gens);
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/*!
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\cgalModifBegin
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Returns a vector containing the generators of \f$\mathcal G\f$ and their inverses.
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The translations must be given in the order:
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\f[ [g_0, g_1, \ldots, g_7]. \f]
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\cgalModifEnd
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*/
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static void get_generators(std::vector<HyperbolicOctagonTranslation>& gens);
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}; // end of class HyperbolicOctagonTranslation
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@ -7,23 +7,13 @@
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\cgalRefines Periodic_4HyperbolicTriangulationTraits_2
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The concept `Periodic_4HyperbolicDelaunayTriangulationTraits_2` refines the concept
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`Periodic_4HyperbolicDelaunayTriangulationTraits_2`. It adds a requirement that needs
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to be fulfilled by any class used to instantiate the first template parameter of the class
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The concept `Periodic_4HyperbolicDelaunayTriangulationTraits_2` adds a requirement
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to `Periodic_4HyperbolicDelaunayTriangulationTraits_2` that needs to be fulfilled
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by any class used to instantiate the first template parameter of the class
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`CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2`. The added requirement is
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essential for the removal of existing vertices in a Delaunay triangulation of the
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Bolza surface.
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\cgalModifBegin
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The concept requires that two nested types are provided:
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<ul>
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<li> A field number type `FT` that must support exact computations with algebraic
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numbers, in particular with nested square roots;
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<li> A `Hyperbolic_translation` type, which satisfies the requirements described
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in the concept `HyperbolicOctagonTranslation`.
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</ul>
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\cgalModifEnd
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\cgalHasModel CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_traits_2
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*/
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@ -36,13 +26,15 @@ public:
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/// \name Computation Types
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/// @{
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/*!
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\cgalModifBegin
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Type that permits to compute the hyperbolic diameter of a circle.
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Must provide the function operator
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`double operator()(Circle_2 c),`
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`double operator()(Hyperbolic_point_2 p1, Hyperbolic_point_2 p2, Hyperbolic_point_2 p3),`
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which returns a floating-point approximation of the hyperbolic diameter
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of the circle `c`.
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of the circle defined by the points `p1, p2,` and `p3`.
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\cgalModifEnd
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*/
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typedef unspecified_type Compute_approximate_hyperbolic_diameter;
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/// @}
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@ -16,9 +16,11 @@ Compare with Section \ref Section_2D_Triangulations_Software_Design of the 2D Tr
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package. The vertices and neighbors are indexed counter-clockwise 0, 1, and 2. Neighbor `i` lies
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opposite to vertex `i`.
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For periodic hyperbolic triangulations, the face base class needs to store three hyperbolic translations,
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one for each vertex. Applying each translation to the point stored in the corresponding vertex produces
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the canonical representative of the face in the hyperbolic plane.
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For periodic hyperbolic triangulations, the face base class needs to store three hyperbolic
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translations, one for each vertex. Applying each translation to the point stored in the
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corresponding vertex produces the canonical representative of the face in the hyperbolic plane.
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Hyperbolic translations are represented by a nested type which is provided by the concept
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`Periodic_4HyperbolicDelaunayTriangulationTraits_2`.
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\cgalHasModel CGAL::Periodic_4_hyperbolic_triangulation_face_base_2
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@ -33,11 +35,8 @@ public:
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/// \name Types
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/// @{
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/*!
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This must be a model of the concept HyperbolicOctagonTranslation.
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*/
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typedef undefined_type Hyperbolic_translation;
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typedef Periodic_4HyperbolicDelaunayTriangulationTraits_2 Geometric_traits;
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typedef Geometric_traits::Hyperbolic_translation Hyperbolic_translation;
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///@}
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@ -7,22 +7,16 @@
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\cgalRefines HyperbolicDelaunayTriangulationTraits_2
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The concept `Periodic_4HyperbolicTriangulationTraits_2` refines the concept
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`HyperbolicDelaunayTriangulationTraits_2`. It describes the set of requirements to be
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fulfilled by any class used to instantiate the first template parameter of the class
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`CGAL::Periodic_4_hyperbolic_triangulation_2`. In addition to the geometric
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types and the operations defined on them in `HyperbolicDelaunayTriangulationTraits_2`,
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it defines the hyperbolic translations that allow to encode the periodicity of the
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triangulation.
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The concept `Periodic_4HyperbolicTriangulationTraits_2` describes the set of requirements
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to be fulfilled by any class used to instantiate the first template parameter of the class
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`CGAL::Periodic_4_hyperbolic_triangulation_2`. In addition to the geometric types and the
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operations defined on them in `HyperbolicDelaunayTriangulationTraits_2`, it defines the
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hyperbolic translations that allow to encode the periodicity of the triangulation.
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\cgalModifBegin
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The concept requires that two nested types are provided:
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<ul>
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<li> A field number type `FT` that must support exact computations with algebraic
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numbers, in particular with nested square roots;
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<li> A `Hyperbolic_translation` type, which satisfies the requirements described
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in the concept `HyperbolicOctagonTranslation`.
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</ul>
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The concept requires that a nested field number type `FT` is provided. The type `FT`
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must support exact computations with algebraic numbers, in particular with nested
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square roots.
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\cgalModifEnd
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*/
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@ -37,18 +31,10 @@ public:
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/// \name Types
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/// @{
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/*!
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\cgalModifBegin
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Represents a hyperbolic circle, i.e., a circle contained in the unit disk.
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\cgalModifEnd
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*/
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typedef unspecified_type Circle_2;
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/*!
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Represents a hyperbolic translation.
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Must be a model of the concept `HyperbolicOctagonTranslation`.
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*/
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typedef unspecified_type Hyperbolic_translation;
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typedef CGAL::Hyperbolic_octagon_translation<FT> Hyperbolic_translation;
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/// @}
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@ -57,7 +43,7 @@ public:
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/*!
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Predicate type. Must provide the function operator
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`Bounded_side operator()(Point_2 p),`
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`Bounded_side operator()(Hyperbolic_point_2 p),`
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which returns the position of point `p` relative to the half-open
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regular hyperbolic octagon.
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@ -71,12 +57,12 @@ public:
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/*!
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Construction type. Must provide the function operator
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`Point_2 operator() ( const Point_2& pt, const HyperbolicOctagonTranslation& tr ) const,`
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`Hyperbolic_point_2 operator() ( const Hyperbolic_point_2& pt, const Hyperbolic_translation& tr ) const,`
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which returns the image of the point `pt` under the action of the
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hyperbolic translation `tr`.
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*/
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typedef unspecified_type Construct_point_2;
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typedef unspecified_type Construct_hyperbolic_point_2;
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/// @}
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@ -92,8 +78,8 @@ public:
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/// \name
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/// The following functions give access to the construction objects:
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/// @{
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Construct_point_2
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construct_point_2_object() const;
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Construct_hyperbolic_point_2
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construct_hyperbolic_point_2_object() const;
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/// @}
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};
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