mirror of https://github.com/CGAL/cgal
doc update
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Loïc Dubois
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04c481cc5f
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97df9401e9
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@ -692,24 +692,27 @@ Mourrain and Monique Teillaud"
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@inproceedings{despre2020flipping,
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title={Flipping geometric triangulations on hyperbolic surfaces},
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author={Despr{\'e}, Vincent and Schlenker, Jean-Marc and Teillaud, Monique},
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booktitle={SoCG 2020-36th International Symposium on Computational Geometry},
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year={2020}
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booktitle={36th International Symposium on Computational Geometry},
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year={2020},
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url={https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.35}
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}
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@article{despre2022experimental,
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title={Experimental analysis of Delaunay flip algorithms on genus two hyperbolic surfaces},
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title={Experimental analysis of {Delaunay} flip algorithms on genus two hyperbolic surfaces},
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author={Despr{\'e}, Vincent and Dubois, Lo{\"\i}c and Kolbe, Benedikt and Teillaud, Monique},
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year={2022}
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year={2022},
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url={https://inria.hal.science/hal-03665888v1/document}
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}
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@article{aigon2005hyperbolic,
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title={Hyperbolic octagons and Teichm{\"u}ller space in genus 2},
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title={Hyperbolic octagons and {Teichm{\"u}ller} space in genus 2},
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author={Aigon-Dupuy, Aline and Buser, Peter and Cibils, Michel and K{\"u}nzle, Alfred F and Steiner, Frank},
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journal={Journal of mathematical physics},
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volume={46},
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number={3},
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year={2005},
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publisher={AIP Publishing}
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publisher={AIP Publishing},
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url = {https://doi.org/10.1063/1.1850177}
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}
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@ -1,124 +1,17 @@
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// Copyright (c) 2024 INRIA Nancy - Grand Est (France). LIGM Marne-la-Vallée (France)
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// All rights reserved.
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namespace CGAL {
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namespace CGAL{
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/*!
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\ingroup PkgHyperbolicSurfaceTriangulation2MainClasses
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Represents a complex number over a field.
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\cgalModels{CGAL::ComplexWithoutSqrt}
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\cgalModels{ComplexWithoutSqrt}
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\tparam FT is the field type and must be a model of `FieldNumberType`.
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*/
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template <class FT>
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class Complex_without_sqrt {
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public:
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/// \name Creation
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/// @{
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/*!
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Default constructor, sets the complex number to 0 + 0 * i.
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*/
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Complex_without_sqrt();
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/*!
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Constructor from the real part, sets the complex number to 'real_part' + 0 * i.
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*/
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Complex_without_sqrt(const FT& real_part);
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/*!
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Constructor from the real and imaginary parts, sets the complex number to 'real_part' + 'imaginary_part' * i.
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*/
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Complex_without_sqrt(const FT& real_part, const FT& imaginary_part);
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/// @}
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/// \name Access functions
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/// @{
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/*!
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Sets the real part to 'real_part'
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*/
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void set_real_part(const FT& real_part);
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/*!
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Sets the imaginary part to 'imaginary_part'
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*/
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void set_imaginary_part(const FT& imaginary_part);
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/*!
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Returns the real part
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*/
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FT real_part() const;
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/*!
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Returns the imaginary part
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*/
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FT imaginary_part() const;
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/// @}
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/// \name Operations
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/// @{
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/*!
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Returns the square of the modulus
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*/
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FT squared_modulus() const;
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/*!
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Returns the conjugate
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*/
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Complex_without_sqrt<FT> conjugate() const;
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/*!
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Returns the sum of itself and other
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*/
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Complex_without_sqrt<FT> operator+(const Complex_without_sqrt<FT>& other) const;
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/*!
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Returns the difference of itself and other
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*/
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Complex_without_sqrt<FT> operator-(const Complex_without_sqrt<FT>& other) const;
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/*!
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Returns the opposite of itself
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*/
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Complex_without_sqrt<FT> operator-() const;
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/*!
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Returns the multiplication of itself and other
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*/
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Complex_without_sqrt<FT> operator*(const Complex_without_sqrt<FT>& other) const;
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/*!
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Returns the division of itself by other
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*/
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Complex_without_sqrt<FT> operator/(const Complex_without_sqrt<FT>& other) const;
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/// @}
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/// \name Equality test
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/// @{
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/*!
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Equality test operator.
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*/
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template<class FT> bool operator==(const Complex_without_sqrt<FT>& z1, const Complex_without_sqrt<FT>& z2);
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/*!
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Inequality test operator.
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*/
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template<class FT> bool operator!=(const Complex_without_sqrt<FT>& z1, const Complex_without_sqrt<FT>& z2);
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/// @}
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/// \name Input/output
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/// @{
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/*!
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Outputs the complex number in a stream.
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*/
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template<class FT>std::ostream& operator<<(std::ostream& s, const Complex_without_sqrt<FT>& z);
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/*!
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Reads the complex number from a stream.
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*/
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template<class FT>void operator>>(std::istream& s, Complex_without_sqrt<FT>& z);
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/// @}
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};
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} //namespace CGAL
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}; // namespace CGAL
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@ -1,18 +1,18 @@
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// Copyright (c) 2024 INRIA Nancy - Grand Est (France). LIGM Marne-la-Vallée (France)
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// All rights reserved.
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namespace CGAL {
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namespace CGAL{
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/*!
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\ingroup PkgHyperbolicSurfaceTriangulation2MainClasses
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Represents a fundamental domain of a closed orientable hyperbolic surface.
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The domain is given as a polygon P represented by the list of its vertices in the Poincaré disk model,
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together with a pairing of the sides of P.
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The n-th side of P is the side between the n-th and the (n+1)-th vertex, where indices are modulo the number of vertices of P.
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The sides pairing are represented by a list of integers, such that if the n-th integer of the list is m, then the n-th side is paired to the m-th side.
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The domain is given as a polygon \f$ P \f$ represented by the list of its vertices in the Poincaré disk model,
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together with a pairing of the sides of \f$ P \f$.
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The \f$ n \f$-th side of \f$ P \f$ is the side between the \f$ n \f$-th and the \f$ (n+1) \f$-th vertex, where indices are modulo the number of vertices of \f$ P \f$.
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The sides pairing are represented by a list of integers, such that if the \f$ n \f$-th integer of the list is \f$ m \f$, then the \f$ n \f$-th side is paired to the \f$ m \f$-th side.
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\tparam Traits is the traits class and must be a model of `HyperbolicSurfacesTraits`.
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\tparam Traits is the traits class and must be a model of `HyperbolicSurfacesTraits_2`.
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*/
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template<class Traits>
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class Hyperbolic_fundamental_domain_2 {
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@ -50,17 +50,17 @@ public:
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int size() const;
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/*!
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Returns the index-th vertex
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Returns the index-th vertex.
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*/
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Point vertex(int index) const;
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/*!
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Returns the index of the side paired to side A, where A is the index-th side
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Returns the index of the side paired to the index-th side.
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*/
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int paired_side(int index) const;
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/*!
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Returns the isometry that maps side A to side B, where B is the index-th side, and A is the side paired to B
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Returns the isometry that maps side \f$ A \f$ to side \f$ B \f$, where \f$ B \f$ is the index-th side, and \f$ A \f$ is the side paired to \f$ B \f$.
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*/
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Hyperbolic_isometry_2<Traits> side_pairing(int index) const;
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/// @}
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@ -79,8 +79,8 @@ public:
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The format of the output is the following.
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The first line prints the number n of vertices of the domain.
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For i=0 to n-1 the index of the side paired to side i is printed on a separate line.
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For i=0 to n-1 the i-th vertex is printed on a separate line.
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For \f$ i=0 \f$ to \f$ n-1 \f$ the index of the side paired to side \f$ i \f$ is printed on a separate line.
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For \f$ i=0 \f$ to \f$ n-1 \f$ the \f$ i \f$-th vertex is printed on a separate line.
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*/
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template<class Traits> std::ostream& operator<<(std::ostream& s, const Hyperbolic_fundamental_domain_2<Traits>& domain);
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/// @}
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@ -96,4 +96,4 @@ public:
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/// @}
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};
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} //namespace CGAL
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}; // namespace CGAL
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@ -1,14 +1,14 @@
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// Copyright (c) 2024 INRIA Nancy - Grand Est (France). LIGM Marne-la-Vallée (France)
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// All rights reserved.
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namespace CGAL {
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namespace CGAL{
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/*!
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\ingroup PkgHyperbolicSurfaceTriangulation2MainClasses
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Factory class, whose only purpose is to randomly generate some convex domains of surfaces of genus two.
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Factory class, whose only purpose is to generate some convex domains of surfaces of genus two.
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\tparam Traits is the traits class and must be a model of `HyperbolicSurfacesTraits`.
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\tparam Traits is the traits class and must be a model of `HyperbolicSurfacesTraits_2`.
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*/
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template<class Traits>
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class Hyperbolic_fundamental_domain_factory_2{
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/// \name Creation
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/// @{
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/*!
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Default constructor, the seed is used for random generation.
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Default constructor, the seed is used for generation.
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*/
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Hyperbolic_fundamental_domain_factory_2(unsigned int seed);
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/// @}
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@ -31,4 +31,4 @@ public:
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};
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} //namespace CGAL
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}; // namespace CGAL
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@ -1,18 +1,18 @@
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// Copyright (c) 2024 INRIA Nancy - Grand Est (France). LIGM Marne-la-Vallée (France)
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// All rights reserved.
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namespace CGAL {
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namespace CGAL{
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/*!
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\ingroup PkgHyperbolicSurfaceTriangulation2MainClasses
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Represents an isometry in the Poincaré disk model.
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The isometry f is represented by a list (c0, c1, c2, c3) of complex numbers,
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so that f(z) = (c0 z + c1) / (c2 z + c3) holds on every complex z in the open unit disk.
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The isometry \f$ f \f$ is represented by a list \f$ (c_0, c_1, c_2, c_3) \f$ of complex numbers,
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so that \f$ f(z) = (c_0 z + c_1) / (c_2 z + c_3) \f$ holds on every complex \f$ z \f$ in the open unit disk.
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Facilities are offered to compose isometries, and apply an isometry to a point.
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\tparam Traits is the traits class and must be a model of `HyperbolicSurfacesTraits`.
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\tparam Traits is the traits class and must be a model of `HyperbolicSurfacesTraits_2`.
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*/
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template<class Traits>
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class Hyperbolic_isometry_2{
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@ -67,7 +67,7 @@ class Hyperbolic_isometry_2{
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Point evaluate(const Point& point) const;
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/*!
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Returns the composition of itself and other.
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Returns the composition of <code> other </code> by <code> itself </code>.
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*/
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Hyperbolic_isometry_2<Traits> compose(const Hyperbolic_isometry_2<Traits>& other) const;
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/// @}
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@ -79,8 +79,6 @@ class Hyperbolic_isometry_2{
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*/
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template<class Traits> std::ostream& operator<<(std::ostream& s, const Hyperbolic_isometry_2<Traits>& isometry);
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/// @}
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};
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} //namespace CGAL
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}; // namespace CGAL
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@ -1,7 +1,7 @@
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// Copyright (c) 2024 INRIA Nancy - Grand Est (France). LIGM Marne-la-Vallée (France)
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// All rights reserved.
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namespace CGAL {
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namespace CGAL{
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/*!
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\ingroup PkgHyperbolicSurfaceTriangulation2MainClasses
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@ -11,7 +11,7 @@ Represents a triangulation of a closed orientable hyperbolic surface.
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Offers facilities such as the generation of the triangulation from a convex fundamental domain,
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the Delaunay flip algorithm, and the construction of a portion of the lift of the triangulation in the hyperbolic plane.
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\tparam Traits is the traits class and must be a model of `HyperbolicSurfacesTraits`.
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\tparam Traits is the traits class and must be a model of `HyperbolicSurfacesTraits_2`.
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*/
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template<class Traits>
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class Hyperbolic_surface_triangulation_2{
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/// \name Types
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/// @{
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/*!
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Type of combinatorial map whose edges are attributed complex numbers.
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Type of combinatorial map whose edges are decorated with complex numbers.
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*/
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typedef Combinatorial_map<2,unspecified_type> Combinatorial_map_with_cross_ratios;
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/*!
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typedef typename Traits::Hyperbolic_point_2 Point;
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/*!
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Stores a dart d of the combinatorial map, belonging to a triangle t, and stores the three vertices of a lift of t in the hyperbolic plane.
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Stores a dart \f$ d \f$ of the combinatorial map, belonging to a triangle \f$ t \f$, and stores the three vertices of a lift of \f$ t \f$ in the hyperbolic plane.
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*/
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struct Anchor{
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typename Combinatorial_map_with_cross_ratios::Dart_handle dart;
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/// \name Lifting
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/// @{
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/*!
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Lifts the triangulation in the hyperbolic plane. Returns, for every triangle T of the triangulation, one of the darts of T in the combinatorial map of the triangulation, together with a triple A,B,C of points in the hyperbolic plane.
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The points A,B,C are the vertices of a lift of T in the hyperbolic plane.
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If the center parameter is set to true, then one of the lifted triangles admits the point 0 as a vertex.
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Lifts the triangulation in the hyperbolic plane.
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Returns, for every triangle \f$ t \f$ of the triangulation, one of the darts of \f$ t \f$ in the combinatorial map of the triangulation, together with a triple \f$ p,q,r \f$ of points in the hyperbolic plane.
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The points \f$ p,q,r \f$ are the vertices of a lift of \f$ t \f$ in the hyperbolic plane.
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If the center parameter is set to true, then one of the lifted triangles admits the origin \f$ 0 \f$ as a vertex.
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This method is to be used only if the triangulation has an anchor.
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\pre This method is to be used only if the triangulation has an anchor.
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*/
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std::vector<std::tuple<Dart_const_handle, Point, Point, Point>> lift(bool center=true) const;
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/// @}
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/*!
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Validity test.
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Checks that the combinatorial map has no 1,2-boundary and calls the is_valid method of the combinatorial map.
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If there is an anchor, then checks that the dart handle of the anchor does indeed point to a dart of the combinatorial map, and checks that the three vertices of the anchor lie within the open unit disk.
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Checks that the underlying combinatorial map \f$ M \f$ has no boundary and calls the is_valid method of \f$ M \f$.
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If there is an anchor, then checks that the dart handle of the anchor does indeed point to a dart of \f$ M \f$, and checks that the three vertices of the anchor lie within the open unit disk.
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*/
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bool is_valid() const;
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/// @}
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Writes the triangulation in a stream.
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The format of the output is the following.
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Each dart of the triangulation is given an index between 0 and n-1, where n is the number of darts of the triangulation.
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The first line contains the number n of darts.
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For every triangle t, the indices of the three darts of t are printed on three distinct lines.
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For every edge e, the indices of the two darts of e are printed on two distinct lines, followed by a third line on which the cross ratio of e is printed.
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Each dart of the triangulation is given an index between \f$ 0 \f$ and \f$ n-1 \f$, where \f$ n \f$ is the number of darts of the triangulation.
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The first line contains the number \f$ n \f$ of darts.
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For every triangle \f$ t \f$, the indices of the three darts of \f$ t \f$ are printed on three distinct lines.
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For every edge \f$ e \f$, the indices of the two darts of \f$ e \f$ are printed on two distinct lines, followed by a third line on which the cross ratio of \f$ e \f$ is printed.
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The next line contains either 'yes' or 'no' and tells whether the triangulation has an anchor.
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If the triangulation has anchor A, then the two next lines print respectively the index of the dart of A, and the three complex numbers of A.
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If the triangulation has anchor, then the two next lines print respectively the index of the dart of the anchor, and the three vertices of the anchor.
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*/
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template<class Traits> std::ostream& operator<<(std::ostream& s, Hyperbolic_surface_triangulation_2<Traits>& Hyperbolic_surface_triangulation_2);
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std::ostream& operator<<(std::ostream& s, Hyperbolic_surface_triangulation_2<Traits>& Hyperbolic_surface_triangulation_2);
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/*!
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Reads the triangulation from a stream.
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The format of the input should be the same as the format of the output of the 'from_stream' method.
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The format of the input should be the same as the format of the output of the '<<' operator.
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*/
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template<class Traits> void operator>>(std::istream& s, Hyperbolic_surface_triangulation_2<Traits>& triangulation);
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void operator>>(std::istream& s, Hyperbolic_surface_triangulation_2<Traits>& triangulation);
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/// @}
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};
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} //namespace CGAL
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}; // namespace CGAL
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@ -1,30 +1,17 @@
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// Copyright (c) 2024 INRIA Nancy - Grand Est (France). LIGM Marne-la-Vallée (France)
|
||||
// All rights reserved.
|
||||
|
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namespace CGAL {
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namespace CGAL{
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/*!
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\ingroup PkgHyperbolicSurfaceTriangulation2TraitsClasses
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Traits class offered by \cgal as a default model for the traits concept `HyperbolicSurfacesTraits_2`.
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\tparam HyperbolicTraitsClass must be a model of `HyperbolicDelaunayTriangulationTraits_2`.
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|
||||
The default value for `HyperbolicTraitsClass` is `CGAL::Hyperbolic_Delaunay_triangulation_CK_traits_2<CGAL::Circular_kernel_2<CGAL::Cartesian<CGAL::Gmpq>,CGAL::Algebraic_kernel_for_circles_2_2<CGAL::Gmpq>>>`,
|
||||
which provides exact constructions and predicates over rational numbers.
|
||||
|
||||
\cgalModels{HyperbolicSurfacesTraits_2}
|
||||
*/
|
||||
template<class HyperbolicTraitsClass>
|
||||
class Hyperbolic_surfaces_traits_2 : public HyperbolicTraitsClass {
|
||||
public:
|
||||
/// \name Types
|
||||
/// @{
|
||||
typedef typename HyperbolicTraitsClass::FT FT;
|
||||
typedef typename HyperbolicTraitsClass::Hyperbolic_point_2 Hyperbolic_point_2;
|
||||
typedef Complex_without_sqrt<FT> Complex;
|
||||
/// @}
|
||||
};
|
||||
|
||||
|
||||
} //namespace CGAL
|
||||
}; // namespace CGAL
|
||||
|
|
|
|||
|
|
@ -5,9 +5,6 @@
|
|||
\ingroup PkgHyperbolicSurfaceTriangulation2Concepts
|
||||
\cgalConcept
|
||||
|
||||
The concept `HyperbolicSurfacesTraits_2` describes the set of requirements
|
||||
to be fulfilled by any class instantiating the template parameter of the main classes of the package.
|
||||
|
||||
\cgalRefines{HyperbolicDelaunayTriangulationTraits_2}
|
||||
|
||||
\cgalHasModelsBegin
|
||||
|
|
@ -19,8 +16,16 @@ public:
|
|||
/// \name Types
|
||||
/// @{
|
||||
/*!
|
||||
Represents a complex number over a field: must be a model of 'CGAL::ComplexWithoutSqrt'.
|
||||
Field type.
|
||||
*/
|
||||
typedef unspecified_type Complex;
|
||||
typedef unspecified_type FT;
|
||||
/*!
|
||||
Represents a point in the Poincaré disk model of the hyperbolic plane.
|
||||
*/
|
||||
typedef unspecified_type Hyperbolic_point_2;
|
||||
/*!
|
||||
Represents a complex number over a field: must be a model of ComplexWithoutSqrt.
|
||||
*/
|
||||
typedef unspecified_type Complex;
|
||||
/// @}
|
||||
};
|
||||
|
|
|
|||
|
|
@ -14,63 +14,68 @@ namespace CGAL {
|
|||
<img src="header.svg" style="max-width:80%; width:80%;"/>
|
||||
</center>
|
||||
|
||||
This package enables handling and building triangulations of closed orientable hyperbolic surfaces.
|
||||
Facilities are offered such as the Delaunay flip algorithm, and the construction of a portion of the lift of the triangulation in the Poincaré disk model of the hyperbolic plane.
|
||||
A triangulation of a surface can be generated from a convex fundamental domain of the surface. A method is offered that randomly generates such domains in genus two.
|
||||
This package enables building and handling triangulations of closed orientable hyperbolic surfaces.
|
||||
Functionalities are offered such as the Delaunay flip algorithm, and the construction of a portion of the lift of the triangulation in the Poincaré disk model of the hyperbolic plane.
|
||||
A triangulation of a surface can be generated from a convex fundamental domain of the surface. A method is offered that generates such domains in genus two.
|
||||
|
||||
|
||||
\section Section_Hyperbolic_Surface_Triangulations_Background Hyperbolic surfaces
|
||||
|
||||
\subsection Section_Hyperbolic_Surface_Triangulations_surfaces Hyperbolic surfaces
|
||||
|
||||
We assume some familiarity with basic notions from covering space theory, and from the theory of hyperbolic surfaces.
|
||||
The <b>Poincaré disk</b> \f$ \mathbb{D} \f$ is a model of the hyperbolic plane whose point set is the open unit disk of the complex plane \f$ \mathbb{C} \f$.
|
||||
This package is concerned with the closed (compact, and without boundary) and orientable hyperbolic surfaces \f$ S \f$.
|
||||
In this package, every hyperbolic surface \f$ S \f$ is closed (compact, and without boundary) and orientable : this is without further mention.
|
||||
The Poincaré disk \f$ \mathbb{D} \f$ is a universal covering space for \f$ S \f$, whose projection map \f$ \pi : \mathbb{D} \to S \f$ is a (local) isometry.
|
||||
The pre-image \f$ \pi^{-1}(x) \f$ of a point \f$ x \in S \f$ is infinite, its points are the <b>lifts</b> of \f$ x \f$.
|
||||
We usually denote by \f$ \widetilde x \f$ a lift of \f$ x \f$.
|
||||
Paths, and triangulations of \f$ S \f$ can also be lifted in \f$ \mathbb{D} \f$.
|
||||
|
||||
\subsection Section_Hyperbolic_Surface_Triangulations_domains Fundamental domains and triangulations
|
||||
|
||||
Let \f$ S \f$ be a closed orientable hyperbolic surface. For representing \f$ S \f$ on a computer, we cut \f$ S \f$ into "manageable" pieces.
|
||||
A graph \f$ G \f$ embedded in \f$ S \f$ is a <b>cellular decomposition</b> of \f$ S \f$ if every face (every connected component of \f$ S \setminus G \f$ ) has genus zero; It is in that sense that the faces of \f$ G \f$ are "manageable".
|
||||
In this document, it is implicit that if a graph \f$ G \f$ is embedded in a hyperbolic surface \f$ S \f$, then every edge of \f$ G \f$ is geodesic in \f$ S \f$.
|
||||
Let \f$ S \f$ be a hyperbolic surface. For representing \f$ S \f$ on a computer, we cut \f$ S \f$ into "manageable" pieces.
|
||||
A graph \f$ G \f$ embedded on \f$ S \f$ is a <b>cellular decomposition</b> of \f$ S \f$ if every face (every connected component of \f$ S \setminus G \f$ ) is a topological disk.
|
||||
In this document, every edge of a graph \f$ G \f$ embedded on \f$ S \f$ is a geodesic on \f$ S \f$.
|
||||
We consider two types of cellular decompositions of \f$ S \f$:
|
||||
|
||||
<ul>
|
||||
<li>We consider cellular decompositions \f$ G \f$ of \f$ S \f$ that have only one face.
|
||||
Cutting \f$ S \f$ open at the edges of \f$ G \f$ results in a hyperbolic polygon \f$ P \f$, that we call <b>fundamental domain</b> for \f$ S \f$.
|
||||
Cutting \f$ S \f$ open at the edges of \f$ G \f$ results in a hyperbolic polygon \f$ P \f$, which is a <b>fundamental domain</b> for \f$ S \f$.
|
||||
The edges of \f$ P \f$ are paired, so that every edge of \f$ G \f$ is cut into two edges that are paired in \f$ P \f$.
|
||||
Every closed orientable hyperbolic surface admits a fundamental domain \f$ P \f$ that is <b>convex</b>, in that the interior angles of \f$ P \f$ do not exceed \f$ \pi \f$.
|
||||
Every hyperbolic surface admits a fundamental domain \f$ P \f$ that is <b>convex</b>, in that the interior angles of \f$ P \f$ do not exceed \f$ \pi \f$.
|
||||
|
||||
<li>Also, we consider <b>triangulations</b> of \f$ S \f$.
|
||||
A cellular decomposition \f$ T \f$ of \f$ S \f$ is a triangulation if every face of \f$ T \f$ is a "triangle", in the sense that it admits three incidences with edges of \f$ T \f$.
|
||||
A cellular decomposition \f$ T \f$ of \f$ S \f$ is a triangulation if every face of \f$ T \f$ is a "triangle": it admits three incidences with edges of \f$ T \f$.
|
||||
Observe that this definition allows for triangulations with only one vertex.
|
||||
</ul>
|
||||
|
||||
A triangulation of \f$ S \f$ can be obtained from a convex fundamental domain \f$ P \f$ of \f$ S \f$ by triangulating the interior \f$ P \f$, and by gluing back the boundary edges that are paired in \f$ P \f$.
|
||||
The assumption that \f$ P \f$ is convex ensures that the interior of \f$ P \f$ can be triangulated naively by insertion of arbitrary arcs of \f$ P \f$.
|
||||
A triangulation of \f$ S \f$ can be obtained from a convex fundamental domain \f$ P \f$ of \f$ S \f$ by triangulating the interior of \f$ P \f$, and by gluing back the boundary edges that are paired in \f$ P \f$.
|
||||
The assumption that \f$ P \f$ is convex ensures that the interior of \f$ P \f$ can be triangulated naively by insertion of any maximal set of pairwise-disjoint arcs of \f$ P \f$.
|
||||
|
||||
\subsection Section_Hyperbolic_Surface_Triangulations_generation Generation of convex fundamental domains
|
||||
|
||||
We aim at generating convex fundamental domains in a randomized way.
|
||||
We represent every domain as a polygon in the Poincaré disk, given by the list of its vertices, and endowed with side pairings.
|
||||
In order to perform fast and exact computations with the domains generated, every vertex must be a complex number whose type supports fast and exact computations.
|
||||
Under this constraint, we do not know how to generate domains of surfaces of genus greater than two.
|
||||
In genus two, this package can generate a convex fundamental domain \f$ P \f$, whose side pairings are \f$ A B C D A B C D \f$, and whose eight vertices \f$ z_0, \dots, z_7 \f$ belong to \f$ \mathbb{Q} + i \mathbb{Q} \f$ (their real and imaginary parts are rational numbers).
|
||||
Here the side pairings and the vertices are in counter-clockwise order, the side between \f$ z_0 \f$ and \f$ z_1 \f$ is labeled \f$ A \f$, and the side between \f$ z_4 \f$ and \f$ z_5 \f$ is also labeled \f$ A \f$.
|
||||
Those octagons are symmetric in the sense that \f$ z_i = z_{i+4} \f$ for every \f$ i \f$, where indices are modulo eight.
|
||||
Such octagons are described in \cgalCite{aigon2005hyperbolic} (without the constraint that the vertices belong to \f$ \mathbb{Q} + i \mathbb{Q} \f$).
|
||||
The exact generation process can be found in \cgalCite{despre2022experimental}, together with a proof that the surfaces that can be generated in this way are dense in the space of surfaces genus two.
|
||||
This package can generate a convex fundamental domain \f$ P \f$ of a surface of genus two, with eight vertices \f$ z_0, \dots, z_7 \in \mathbb{C} \f$, whose side pairings are \f$ A B C D \overline{A} \overline{B} \overline{C} \overline{D} \f$.
|
||||
The vertices and the side pairings are in counter-clockwise order, the side between \f$ z_0 \f$ and \f$ z_1 \f$ is \f$ A \f$, and the side between \f$ z_4 \f$ and \f$ z_5 \f$ is \f$ \overline{A} \f$.
|
||||
Those octagons are symmetric, .i.e \f$ z_i = z_{i+4} \f$ for every \f$ i \f$, where indices are modulo eight.
|
||||
Such octagons are described in \cgalCite{aigon2005hyperbolic}.
|
||||
|
||||
<center>
|
||||
<img src="octagon.svg" style="max-width:20%; width:20%;"/>
|
||||
</center>
|
||||
|
||||
\section Subsection_Hyperbolic_Surface_Triangulations_Representation Representation
|
||||
|
||||
\subsection Subsection_Hyperbolic_Surface_Triangulations_Data_Structure Data structure
|
||||
\subsection Subsection_Hyperbolic_Surface_Triangulations_DS_Domains Data structure for domains
|
||||
|
||||
We represent domains by the list of their vertices in the Poincaré disk, and by the list of their sides pairings.
|
||||
We represent every domain as a polygon in the Poincaré disk, given by the list of its vertices, and by the list of its side pairings.
|
||||
Concerning the generation of domains, in order to perform fast and exact computations with the domains generated, every vertex must be a complex number whose type supports fast and exact computations.
|
||||
Under this constraint, it is not known how to generate domains of surfaces of genus greater than two.
|
||||
In genus two, this package generates domains whose vertices belong to \f$ \mathbb{Q} + i \mathbb{Q} \f$ (their real and imaginary parts are rational numbers).
|
||||
The exact generation process can be found in \cgalCite{despre2022experimental}, together with a proof that the surfaces that can be generated in this way are dense in the space of surfaces genus two.
|
||||
|
||||
Let \f$ T \f$ be a triangulation of a closed orientable hyperbolic surface.
|
||||
We represent \f$ T \f$ by an instance of CGAL::Combinatorial_map whose edges are attributed complex numbers.
|
||||
The complex number \f$ R_T(e) \in \mathbb{C} \f$ attributed to an edge \f$ e \f$ of \f$ T \f$ is the <b>cross ratio</b> of \f$ e \f$ in \f$ T \f$, defined as follows.
|
||||
\subsection Subsection_Hyperbolic_Surface_Triangulations_DS_Triangulations Data structure for triangulations
|
||||
|
||||
Let \f$ T \f$ be a triangulation of a hyperbolic surface.
|
||||
We represent \f$ T \f$ by an instance of CGAL::Combinatorial_map whose edges are decorated with complex numbers.
|
||||
The complex number \f$ R_T(e) \in \mathbb{C} \f$ decorating an edge \f$ e \f$ of \f$ T \f$ is the <b>cross ratio</b> of \f$ e \f$ in \f$ T \f$, defined as follows.
|
||||
Consider the lift \f$ \widetilde T \f$ of \f$ T \f$ in the Poincaré disk \f$ \mathbb{D} \f$.
|
||||
In \f$ \widetilde T \f$, let \f$ \widetilde e \f$ be a lift of \f$ e \f$.
|
||||
Orient \f$ \widetilde e \f$ arbitrarily, and let \f$ z_0 \in \mathbb{D} \f$ and \f$ z_2 \in \mathbb{D} \f$ be respectively the first and second vertices of \f$ \widetilde e \f$.
|
||||
|
|
@ -78,6 +83,11 @@ In \f$ \widetilde T \f$, consider the triangle on the right of \f$ \widetilde e
|
|||
Similarly, consider the triangle on the left of \f$ \widetilde e \f$, and let \f$ z_3 \in \mathbb{D} \f$ be the third vertex of this triangle.
|
||||
Then \f$ R_T(e) = (z_3-z_1)*(z_2-z_0) / ((z_3-z_0)*(z_2-z_1)) \f$.
|
||||
This definition does not depend on the choice of the lift \f$ \widetilde e \f$, nor on the orientation of \f$ \widetilde e \f$.
|
||||
See \cgalCite{despre2022experimental} for details.
|
||||
|
||||
<center>
|
||||
<img src="crossratio.svg" style="max-width:15%; width:15%;"/>
|
||||
</center>
|
||||
|
||||
While the triangulation \f$ T \f$ is unambiguously determined by the combinatorial map and its cross ratios, the internal representation of \f$ T \f$ can contain some additional data: the anchor.
|
||||
The anchor is used when building a portion of the lift of \f$ T \f$ in the Poincaré disk \f$ \mathbb{D} \f$.
|
||||
|
|
@ -86,11 +96,11 @@ It contains a lift \f$ t \f$ of a triangle of \f$ T \f$ in \f$ \mathbb{D} \f$: \
|
|||
|
||||
\subsection Subsection_Hyperbolic_Surface_Triangulations_Delaunay Delaunay flip algorithm
|
||||
|
||||
Let \f$ T \f$ be a triangulation of a closed orientable hyperbolic surface. An edge \f$ e \f$ of \f$ T \f$ satisfies the <b>Delaunay criterion</b> if the imaginary part of its cross ratio \f$R_T(e)\f$ is non-positive.
|
||||
(This definition has an equivalent "empty disk" formulation in the package \ref PkgHyperbolicTriangulation2).
|
||||
Let \f$ T \f$ be a triangulation of a hyperbolic surface. An edge \f$ e \f$ of \f$ T \f$ satisfies the <b>Delaunay criterion</b> if the imaginary part of its cross ratio \f$R_T(e)\f$ is non-positive.
|
||||
This definition is equivalent to the usual "empty disk" formulation.
|
||||
Then \f$ T \f$ is a <b>Delaunay triangulation</b> if every edge of \f$ T \f$ satisfies the Delaunay criterion.
|
||||
If an edge \f$e \f$ of \f$ T \f$ does not satisfy the Delaunay criterion, then the two triangles incident to \f$ e \f$ form a strictly convex quadrilateron, so \f$ e \f$ can be deleted from \f$ T \f$ and replaced by the other diagonal of the quadrilateron.
|
||||
This operation is called a <b>flip</b>.
|
||||
This operation is called a Delaunay <b>flip</b>.
|
||||
When a flip occurs, the cross ratios of the edges involved are modified via simple formulas.
|
||||
The <b>Delaunay flip algorithm</b> flips edges that do not satisfy the Delaunay criterion as long as possible, with no preference on the order of the flips.
|
||||
This algorithm terminates, and outputs a Delaunay triangulation of \f$ S \f$ \cgalCite{despre2020flipping}.
|
||||
|
|
@ -99,17 +109,16 @@ This algorithm terminates, and outputs a Delaunay triangulation of \f$ S \f$ \cg
|
|||
|
||||
|
||||
The package contains three main classes:
|
||||
- `CGAL::Hyperbolic_surface_triangulation_2` represents a triangulation of a closed orientable hyperbolic surface. Offers facilities such as the generation of the triangulation from a convex fundamental domain, the Delaunay flip algorithm, and the construction of a portion of the lift of the triangulation in the Poincaré disk.
|
||||
- `CGAL::Hyperbolic_surface_triangulation_2` represents a triangulation of a hyperbolic surface. It offers functionalities such as the generation of the triangulation from a convex fundamental domain, the Delaunay flip algorithm, and the construction of a portion of the lift of the triangulation in the Poincaré disk.
|
||||
|
||||
- `CGAL::Hyperbolic_fundamental_domain_2` represents a convex fundamental domain of a closed orientable hyperbolic surface.
|
||||
|
||||
- `CGAL::Hyperbolic_fundamental_domain_factory_2` is a factory class, whose only purpose is to randomly generate some convex fundamental domains of surfaces of genus two.
|
||||
- `CGAL::Hyperbolic_fundamental_domain_2` represents a convex fundamental domain of a hyperbolic surface.
|
||||
|
||||
- `CGAL::Hyperbolic_fundamental_domain_factory_2` is a factory class, whose purpose is to generate some convex fundamental domains of surfaces of genus two.
|
||||
|
||||
The package also contains the secondary class `CGAL::Hyperbolic_isometry_2` to deal with isometries in the Poincaré disk.
|
||||
|
||||
The concept `HyperbolicSurfacesTraits_2` is a refinement of `HyperbolicDelaunayTriangulationTraits_2`.
|
||||
This concept describes the requirements for the template parameter of the classes of the package.
|
||||
It describes the template parameter of most classes of the package.
|
||||
It is modeled by the class `CGAL::Hyperbolic_surfaces_traits_2`.
|
||||
|
||||
The concept `ComplexWithoutSqrt` describes a complex number type that does not use square root. It is modeled by the class `CGAL::Complex_without_sqrt`.
|
||||
|
|
@ -120,16 +129,16 @@ The concept `ComplexWithoutSqrt` describes a complex number type that does not u
|
|||
|
||||
\section Section_Hyperbolic_Surface_Triangulations_Example Example
|
||||
|
||||
The example below randomly generates a convex fundamental domain of a surface of genus two, triangulates the domain, applies the Delaunay flip algorithm to the resulting triangulation, and saves and prints the Delaunay triangulation.
|
||||
The example below generates a convex fundamental domain of a surface of genus two, triangulates the domain, applies the Delaunay flip algorithm to the resulting triangulation, and saves and prints the Delaunay triangulation.
|
||||
\cgalExample{Hyperbolic_surface_triangulation_2_Example/example.cpp}
|
||||
|
||||
|
||||
\section Section_Hyperbolic_Surface_Implementation_History Design and implementation history
|
||||
|
||||
This package implements the Delaunay flip algorithm described in the hyperbolic setting by Vincent Despré, Jean-Marc Schlenker, and Monique Teillaud in \cgalCite{despre2020flipping} (with a different data structure for representing triangulations).
|
||||
This package implements the Delaunay flip algorithm described in the hyperbolic setting by Vincent Despré, Jean-Marc Schlenker, and Monique Teillaud in \cgalCite{despre2020flipping} (with a different data structure for representing triangulations, see \cgalCite{despre2022experimental}).
|
||||
It also implements the generation of domains described by Vincent Despré, Loïc Dubois, Benedikt Kolbe, and Monique Teillaud in \cgalCite{despre2022experimental}, based on results of Aline Aigon-Dupuy, Peter Buser, Michel Cibils, Alfred F Künzle, and Frank Steiner \cgalCite{aigon2005hyperbolic}.
|
||||
The code and the documentation of the package were entirely written by Loïc Dubois, under regular discussions with Vincent Despré and Monique Teillaud.
|
||||
The authors aknowledge support from the ANR SoS.
|
||||
The code and the documentation of the package were written by Loïc Dubois, under regular discussions with Vincent Despré and Monique Teillaud.
|
||||
The authors aknowledge support from the grants <a href="https://sos.loria.fr/">SoS</a> and <a href="https://perso.math.u-pem.fr/sabourau.stephane/min-max/min-max.html">MIN-MAX</a> of the French National Research Agency ANR.
|
||||
|
||||
*/
|
||||
} /* namespace CGAL */
|
||||
|
|
|
|||
|
|
@ -19,7 +19,7 @@
|
|||
|
||||
\cgalPkgSummaryBegin
|
||||
\cgalPkgAuthors{Vincent Despré, Loïc Dubois, and Monique Teillaud}
|
||||
\cgalPkgDesc{This package enables handling and building triangulations of closed orientable hyperbolic surfaces. Facilities are offered such as the Delaunay flip algorithm, and the construction of a portion of the lift of the triangulation in the Poincaré disk model of the hyperbolic plane. Triangulations can be generated by triangulating a convex fundamental domain in the Poincaré disk model. A method is offered that generates such domains in genus two.}
|
||||
\cgalPkgDesc{This package enables building and handling triangulations of closed orientable hyperbolic surfaces. Functionalities are offered such as the Delaunay flip algorithm, and the construction of a portion of the lift of the triangulation in the Poincaré disk model of the hyperbolic plane. Triangulations can be generated by triangulating a convex fundamental domain in the Poincaré disk model. A method is offered that generates such domains in genus two.}
|
||||
\cgalPkgManuals{Chapter_Hyperbolic_Surface_Triangulations,PkgHyperbolicSurfaceTriangulation2Ref}
|
||||
\cgalPkgSummaryEnd
|
||||
|
||||
|
|
@ -39,20 +39,18 @@
|
|||
\cgalCRPSection{Concepts}
|
||||
|
||||
- `HyperbolicSurfacesTraits_2` describes the set of requirements to be fulfilled by any class instantiating the template parameter of most classes of the package.
|
||||
- `ComplexWithoutSqrt` describes a complex number type that does not use square root.
|
||||
|
||||
\cgalCRPSection{Classes}
|
||||
|
||||
- `CGAL::Hyperbolic_surface_triangulation_2` represents a triangulation of a closed orientable hyperbolic surface. Offers facilities such as the generation of the triangulation from a convex fundamental domain, the Delaunay flip algorithm, and the construction of a portion of the lift of the triangulation in the hyperbolic plane.
|
||||
- `CGAL::Hyperbolic_surface_triangulation_2` represents a triangulation of a closed orientable hyperbolic surface. It offers facilities such as the generation of the triangulation from a convex fundamental domain, the Delaunay flip algorithm, and the construction of a portion of the lift of the triangulation in the hyperbolic plane.
|
||||
|
||||
- `CGAL::Hyperbolic_fundamental_domain_2` represents a fundamental domain of a closed orientable hyperbolic surface.
|
||||
|
||||
- `CGAL::Hyperbolic_fundamental_domain_factory_2` is a factory class, whose only purpose is to randomly generate some convex domains of surfaces of genus two.
|
||||
- `CGAL::Hyperbolic_fundamental_domain_factory_2` is a factory class, whose only purpose is to generate some convex domains of surfaces of genus two.
|
||||
|
||||
- `CGAL::Hyperbolic_isometry_2` represents an isometry in the Poincaré disk model. Facilities are offered to compose isometries, and apply an isometry to a point.
|
||||
|
||||
- `CGAL::Complex_without_sqrt` represents a complex number.
|
||||
|
||||
A model for the concept `HyperbolicSurfacesTraits_2` is provided:
|
||||
- `CGAL::Hyperbolic_surfaces_traits_2` inherits from its template parameter HyperbolicTraitsClass. HyperbolicTraitsClass must be a model of `CGAL::HyperbolicDelaunayTriangulationTraits_2`.
|
||||
Models for `HyperbolicSurfacesTraits_2` and `ComplexWithoutSqrt` are provided : `CGAL::Hyperbolic_surfaces_traits_2` and `CGAL::Complex_without_sqrt`.
|
||||
|
||||
*/
|
||||
|
|
|
|||
|
|
@ -151,7 +151,7 @@ typename Traits::FT Hyperbolic_fundamental_domain_factory_2<Traits>::exact_numbe
|
|||
if (x< 0){
|
||||
return _FT(0)-exact_number_from_float(-x);
|
||||
}
|
||||
return _FT(int(x*_DENOMINATOR_FOR_GENERATION)%_DENOMINATOR_FOR_GENERATION, _DENOMINATOR_FOR_GENERATION);
|
||||
return _FT(int(x*_DENOMINATOR_FOR_GENERATION)%_DENOMINATOR_FOR_GENERATION)/_FT( _DENOMINATOR_FOR_GENERATION);
|
||||
}
|
||||
|
||||
template<class Traits>
|
||||
|
|
|
|||
Loading…
Reference in New Issue