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// Copyright (c) 2024 // INRIA Nancy (France), and Université Gustave Eiffel Marne-la-Vallee (France). // All rights reserved. // // This file is part of CGAL (www.cgal.org) // // $URL: https://github.com/CGAL/cgal/blob/v6.1/Triangulation_on_hyperbolic_surface_2/include/CGAL/Hyperbolic_isometry_2.h $ // $Id: include/CGAL/Hyperbolic_isometry_2.h b26b07a1242 $ // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial // // Author(s) : Vincent Despré, Loïc Dubois, Marc Pouget, Monique Teillaud #ifndef CGAL_HYPERBOLIC_ISOMETRY_2_H #define CGAL_HYPERBOLIC_ISOMETRY_2_H #include #include namespace CGAL { /* Represents a hyperbolic isometry in the Poincare disk model the hyperbolic plane. The isometry f is stored as list (c0, c1, c2, c3) of 4 complex numbers, so that f(z) = (c0 z + c1) / (c2 z + c3) holds on every complex z in the open unit disk. */ template class Hyperbolic_isometry_2 { public: typedef Hyperbolic_isometry_2 Self; typedef typename Traits::FT FT; typedef typename Traits::Complex Complex_number; typedef typename Traits::Hyperbolic_point_2 Point; Hyperbolic_isometry_2(); Hyperbolic_isometry_2(const Complex_number& c0, const Complex_number& c1, const Complex_number& c2, const Complex_number& c3); void set_to_identity(); // Can be used to set the coefficients manually. Warning: the implementation does not check that the resulting moebius transform fixes the unit circle. void set_coefficients(const Complex_number& c0, const Complex_number& c1, const Complex_number& c2, const Complex_number& c3); void set_coefficient(int index, const Complex_number& coefficient); // returns the index-th coefficient const Complex_number& get_coefficient(int index) const; // evaluates the isometry at point Point evaluate(const Point& point) const; Point operator()(const Point& point) const; private: Complex_number coefficients_[4]; }; // returns the composition of two isometries. template Hyperbolic_isometry_2 operator*(const Hyperbolic_isometry_2& iso1, const Hyperbolic_isometry_2& iso2); // template std::ostream& operator<<(std::ostream& s, const Hyperbolic_isometry_2& isometry); // When inverse is 'false', returns the hyperbolic translation that maps -p to zero, and zero to p. // Otherwise, returns the hyperbolic translation that maps p to zero, and zero to -p. template Hyperbolic_isometry_2 hyperbolic_translation(const typename Traits::Hyperbolic_point_2& p, bool inverse = false); // When inverse is 'false', returns the hyperbolic rotation around zero that maps p to q. // Otherwise, returns the hyperbolic rotation around zero that maps q to p. template Hyperbolic_isometry_2 hyperbolic_rotation(const typename Traits::Hyperbolic_point_2& p, const typename Traits::Hyperbolic_point_2& q, bool inverse = false); // returns the hyperbolic isometry that maps p1 to q1 and p2 to q2 template Hyperbolic_isometry_2 isometry_pairing_the_sides(const typename Traits::Hyperbolic_point_2& p1, const typename Traits::Hyperbolic_point_2& p2, const typename Traits::Hyperbolic_point_2& q1, const typename Traits::Hyperbolic_point_2& q2); //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////////// template Hyperbolic_isometry_2:: Hyperbolic_isometry_2() { set_to_identity(); } template Hyperbolic_isometry_2:: Hyperbolic_isometry_2(const Complex_number& c0, const Complex_number& c1, const Complex_number& c2, const Complex_number& c3) { set_coefficients(c0,c1,c2,c3); } //////////////////////////////////////////////////////////////////////////////// template void Hyperbolic_isometry_2:: set_to_identity() { set_coefficients(Complex_number(FT(1)), Complex_number(FT(0)), Complex_number(FT(0)), Complex_number(FT(1))); } template void Hyperbolic_isometry_2:: set_coefficients(const Complex_number& c0, const Complex_number& c1, const Complex_number& c2, const Complex_number& c3) { set_coefficient(0, c0); set_coefficient(1, c1); set_coefficient(2, c2); set_coefficient(3, c3); } template void Hyperbolic_isometry_2:: set_coefficient(int index, const Complex_number& coefficient) { coefficients_[index] = coefficient; } //////////////////////////////////////////////////////////////////////////////// template const typename Traits::Complex& Hyperbolic_isometry_2:: get_coefficient(int index) const { return coefficients_[index]; } //////////////////////////////////////////////////////////////////////////////// template typename Traits::Hyperbolic_point_2 Hyperbolic_isometry_2:: evaluate(const Point& point) const { Complex_number z(point.x(), point.y()); Complex_number numerator_of_the_result = coefficients_[0] * z + coefficients_[1]; Complex_number denominator_of_the_result = coefficients_[2] * z + coefficients_[3]; Complex_number result = numerator_of_the_result / denominator_of_the_result; return Point(result.real(), result.imag()); } template typename Traits::Hyperbolic_point_2 Hyperbolic_isometry_2:: operator()(const Point& point) const { return evaluate(point); } //////////////////////////////////////////////////////////////////////////////// template Hyperbolic_isometry_2 operator*(const Hyperbolic_isometry_2& iso1, const Hyperbolic_isometry_2& iso2) { Hyperbolic_isometry_2 result; for (int i=0; i<2; i++) { for (int j=0; j<2; j++) { result.set_coefficient(2*i+j, iso1.get_coefficient(2*i) * iso2.get_coefficient(j) + iso1.get_coefficient(2*i+1) * iso2.get_coefficient(2+j)); } } return result; } //////////////////////////////////////////////////////////////////////////////// template Hyperbolic_isometry_2 hyperbolic_translation(const typename Traits::Hyperbolic_point_2& p, bool inverse) { typename Traits::Complex one (typename Traits::FT(1)); typename Traits::Complex z; if (inverse) { z = typename Traits::Complex(p.x(), p.y()); } else { z = - typename Traits::Complex(p.x(), p.y()); } Hyperbolic_isometry_2 result; result.set_coefficients(one, z, conj(z), one); return result; } template Hyperbolic_isometry_2 hyperbolic_rotation(const typename Traits::Hyperbolic_point_2& p, const typename Traits::Hyperbolic_point_2& q, bool inverse) { typename Traits::Complex zero (typename Traits::FT(0)); Hyperbolic_isometry_2 result; if (inverse) { result.set_coefficients(typename Traits::Complex(p.x(), p.y()), zero, zero, typename Traits::Complex(q.x(), q.y())); } else { result.set_coefficients(typename Traits::Complex(q.x(), q.y()), zero, zero, typename Traits::Complex(p.x(), p.y())); } return result; } template Hyperbolic_isometry_2 isometry_pairing_the_sides(const typename Traits::Hyperbolic_point_2& p1, const typename Traits::Hyperbolic_point_2& p2, const typename Traits::Hyperbolic_point_2& q1, const typename Traits::Hyperbolic_point_2& q2) { Hyperbolic_isometry_2 A,B,Binv,C; A = hyperbolic_translation(p1); B = hyperbolic_translation(q1); Binv = hyperbolic_translation(q1, true); C = hyperbolic_rotation(A.evaluate(p2), B.evaluate(q2)); return (Binv*C)*A; } } // namespace CGAL #endif // CGAL_HYPERBOLIC_ISOMETRY_2_H