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// Copyright (c) 2006-2009 Max-Planck-Institute Saarbruecken (Germany). // All rights reserved. // // This file is part of CGAL (www.cgal.org) // // $URL: https://github.com/CGAL/cgal/blob/v6.1/Algebraic_kernel_d/include/CGAL/Algebraic_kernel_d/Xy_coordinate_2.h $ // $Id: include/CGAL/Algebraic_kernel_d/Xy_coordinate_2.h b26b07a1242 $ // SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial // // // Author(s) : Eric Berberich // Pavel Emeliyanenko // // ============================================================================ #ifndef CGAL_ALGEBRAIC_CURVE_KERNEL_XY_COORDINATE_2_H #define CGAL_ALGEBRAIC_CURVE_KERNEL_XY_COORDINATE_2_H #include #include #include #include namespace CGAL { namespace internal { template < class AlgebraicCurveKernel_2, class Rep_, class HandlePolicy_ , class Allocator_> //::boost::fast_pool_allocator > class Xy_coordinate_2; template < class AlgebraicCurveKernel_2 > class Xy_coordinate_2_rep { public: // this first template argument typedef AlgebraicCurveKernel_2 Algebraic_curve_kernel_2; // myself typedef Xy_coordinate_2_rep Self; typedef typename Algebraic_curve_kernel_2::Curve_analysis_2 Curve_analysis_2; typedef typename Algebraic_curve_kernel_2::Algebraic_real_1 Algebraic_real_1; typedef CGAL::Bbox_2 Bbox_2; typedef CGAL::Handle_with_policy Xy_coordinate_2_inst; // constructors public: // default constructor () Xy_coordinate_2_rep() : _m_arcno(-1) { } // standard constructor Xy_coordinate_2_rep(const Algebraic_real_1& x, const Curve_analysis_2& curve, int arcno) : _m_kernel(curve.kernel()),_m_x(x), _m_curve(curve), _m_arcno(arcno) { } // data const Algebraic_curve_kernel_2* _m_kernel; // x-coordinate Algebraic_real_1 _m_x; // supporting curve mutable Curve_analysis_2 _m_curve; // arc number on curve mutable int _m_arcno; // y-coordinate mutable std::optional< Algebraic_real_1 > _m_y; //! A bounding box for the given point mutable std::optional< std::pair > _m_bbox_2_pair; }; //! \brief class \c Xy_coordinate_2 represents a single root of a system of //! two polynomial equations in two variables that are models //! \c AlgebraicCurveKernel_2::Polynomial_2 //! //! \c Xy_coordinate_2 coordinate is represented by an \c Algebraic_real_1, //! a supporting curve and an arcno and is valid only for finite solutions, //! i.e., it cannot represent points at infinity template , class HandlePolicy_= CGAL::Handle_policy_union, class Allocator_ = CGAL_ALLOCATOR(Rep_) > class Xy_coordinate_2 : public ::CGAL::Handle_with_policy { public: //! \name public typedefs //!@{ //! this instance's first template parameter typedef AlgebraicCurveKernel_2 Algebraic_curve_kernel_2; //! this instance's second template parameter typedef Rep_ Rep; //! this instance's third template parameter typedef HandlePolicy_ Handle_policy; //! this instance's fourth template parameter typedef Allocator_ Allocator; //! this instance itself typedef Xy_coordinate_2 Self; //! an instance of AlgebraicKernel_1 typedef typename Algebraic_curve_kernel_2::Algebraic_kernel_d_1 Algebraic_kernel_d_1; typedef typename Algebraic_curve_kernel_2::Polynomial_1 Polynomial_1; typedef CGAL::Polynomial_traits_d Polynomial_traits_1; typedef typename Algebraic_curve_kernel_2::Polynomial_2 Polynomial_2; typedef CGAL::Polynomial_traits_d Polynomial_traits_2; //! type of (explicit) x- and y-coordinates typedef typename Algebraic_curve_kernel_2::Algebraic_real_1 Algebraic_real_1; //! Coefficient type typedef typename Algebraic_curve_kernel_2::Coefficient Coefficient; //! type of curve pair analysis typedef typename Algebraic_curve_kernel_2::Curve_pair_analysis_2 Curve_pair_analysis_2; //! type of curve analysis typedef typename Algebraic_curve_kernel_2::Curve_analysis_2 Curve_analysis_2; //! the handle superclass typedef ::CGAL::Handle_with_policy Base; //! type for approximation boundaries typedef typename Algebraic_curve_kernel_2::Bound Bound; //! type for bound intervals typedef boost::numeric::interval Bound_interval; //! Type for the bounding box typedef typename Rep::Bbox_2 Bbox_2; //!@} private: //! \name private methods //!@{ /*!\brief * Simplifies the representation of two points whose supporting curves * share a common part. */ bool _simplify(const Xy_coordinate_2& p, const Xy_coordinate_2& q) const { std::vector parts_of_f, parts_of_g, common; if(kernel()->decompose_2_object()(p.curve(), q.curve(), std::back_inserter(parts_of_f), std::back_inserter(parts_of_g), std::back_inserter(common))) { CGAL_assertion((parts_of_f.size() == 1 || parts_of_g.size() == 1) && common.size() == 1); if(parts_of_f.size() == 1) { p.simplify_by(kernel()->construct_curve_pair_2_object()( parts_of_f[0], common[0])); } if(parts_of_g.size() == 1) { q.simplify_by(kernel()->construct_curve_pair_2_object()( parts_of_g[0], common[0])); } return true; } return false; } //!@} public: //!\name Constructors //!@{ /*!\brief * default constructor * * A default-constructed point supports no operation other than * having \c CGAL::degree(curve()) return \c -1. */ Xy_coordinate_2() : Base(Rep()) { } /*!\brief * copy constructor */ #ifdef DOXYGEN_RUNNING Xy_coordinate_2(const Self& p) : Base(static_cast(p)) { } #endif /*!\brief * Point at \c x, on \c curve with \c arcno. Finite points on vertical arcs * are also constructed in this way */ Xy_coordinate_2(const Algebraic_real_1& x, const Curve_analysis_2& curve, int arcno) : Base(Rep(x, curve, arcno)) { CGAL_precondition(arcno >= 0); CGAL_precondition_code( typename Curve_analysis_2::Status_line_1 v = curve.status_line_for_x(x); ); CGAL_precondition(arcno >= 0 && arcno < v.number_of_events()); } /*!\brief * constructs a point from a given representation */ Xy_coordinate_2(Rep rep) : Base(rep) { } //!@} public: //!\name Access functions //!@{ /*!\brief * x-coordinate of the point */ const Algebraic_real_1& x() const { return this->ptr()->_m_x; } /*! * \brief y-coordinate of this point * * Note: In general, this method results in an extremely large polynomial * for the y-coordinate. It is recommended to use it carefully, * and using get_approximation_y() instead whenever approximations suffice. */ Algebraic_real_1 y() const { typedef std::vector< Algebraic_real_1 > Roots; // EBEB 2012-07-05 deactivated map for y-roots for not being used // typedef typename Curve_analysis_2::Status_line_1 Key; // EBEB 2012-07-05 deactivated map for y-roots for not being used // typedef Roots Data; // EBEB 2012-07-05 deactivated map for y-roots for not being used // typedef std::map< Key, Data, CGAL::Handle_id_less_than< Key > > // Y_root_map; // EBEB 2012-07-05 deactivated map for y-roots for not being used //static Y_root_map y_root_map; if (!this->ptr()->_m_y) { Polynomial_2 f = curve().primitive_polynomial_2(); // This will be the defining polynomial of y Polynomial_1 y_pol; // Filter: If we know that the point is critical, we can use // the resultant of f and f_y with respect to x as polynomial bool point_is_certainly_critical = false; typename Curve_analysis_2::Status_line_1 line = curve().status_line_at_exact_x(x()); // EBEB 2012-07-05 deactivated map for y-roots for not being used //typename Y_root_map::iterator yit = // y_root_map.find(line); // TODO: Cache resultant computation // exacus-related code shouldn't be used here //curve().x_to_index(x(),i,is_event); if (line.is_event()) { //typename Internal_curve_2::Event1_info ev_info = // curve().event_info(i); typename Curve_analysis_2::Status_line_1::Arc_pair ipair = line.number_of_incident_branches(arcno()); if (ipair.first != 1 || ipair.second != 1) { point_is_certainly_critical = true; y_pol = CGAL::make_square_free( CGAL::resultant (typename Polynomial_traits_2::Swap() (f,0,1), typename Polynomial_traits_2::Swap() (CGAL::differentiate(f),0,1)) ); // BUGFIX: y_pol might be zero: if(y_pol.is_zero()) { // force re-computation with bigger resultant point_is_certainly_critical=false; } } } if (!point_is_certainly_critical) { Polynomial_2 r(x().polynomial()); y_pol = CGAL::make_square_free( CGAL::resultant (typename Polynomial_traits_2::Swap() (f,0,1), typename Polynomial_traits_2::Swap() (r,0,1)) ); } typename Algebraic_kernel_d_1::Solve_1 real_roots; Roots y_roots; real_roots(y_pol, std::back_inserter(y_roots), false ); long prec = 16; typename Algebraic_curve_kernel_2::Approximate_absolute_y_2 approx_y=kernel()->approximate_absolute_y_2_object(); std::pair y_pair = approx_y(*this,prec); Bound_interval y_iv(y_pair.first,y_pair.second); typedef typename std::vector::const_iterator Iterator; std::list< Iterator > candidates; for (Iterator it = y_roots.begin(); it != y_roots.end(); it++) { Bound_interval it_interval(it->low(), it->high()); if (boost::numeric::overlap(it_interval, y_iv)) { candidates.push_back(it); } } CGAL_assertion(!candidates.empty()); while (candidates.size() > 1) { prec*=2; y_pair = approx_y(*this,prec); y_iv = Bound_interval(y_pair.first,y_pair.second); for (typename std::list< Iterator >::iterator dit, cit = candidates.begin(); cit != candidates.end(); ) { bool remove = false; Bound_interval cit_interval((*cit)->low(), (*cit)->high()); if (!boost::numeric::overlap(cit_interval, y_iv)) { dit = cit; remove = true; } cit++; if (remove) { candidates.erase(dit); } } } CGAL_assertion(static_cast< int >(candidates.size()) == 1); this->ptr()->_m_y = Algebraic_real_1( (*candidates.begin())->polynomial(), (*candidates.begin())->low(), (*candidates.begin())->high() ); } CGAL_postcondition(bool(this->ptr()->_m_y)); return *this->ptr()->_m_y; } /*!\brief * supporting curve of the point */ Curve_analysis_2 curve() const { return this->ptr()->_m_curve; } /*!\brief * arc number of point * */ int arcno() const { return this->ptr()->_m_arcno; } //!@} public: //!\name comparison predicates //!@{ /*!\brief * compares x-coordinates of \c *this with \c q * * do we need this method or one should use Algebraic_curve_kernel_2 * directly ? */ CGAL::Comparison_result compare_x(const Self& q) const { if(this->is_identical(q)) { return CGAL::EQUAL; } return kernel()->compare_1_object()(this->x(), q.x()); } /*!\brief * compares \c *this with \c q lexicographically */ CGAL::Comparison_result compare_xy(const Self& q, bool equal_x = false) const { if(this->is_identical(q)) return CGAL::EQUAL; CGAL::Comparison_result res = (equal_x ? CGAL::EQUAL : compare_x(q)); if(res == CGAL::EQUAL) { res = _compare_y_at_x(q); } return res; } //! equality bool operator == (const Self& q) const {return q.compare_xy(*this)== 0;} //! inequality bool operator != (const Self& q) const {return q.compare_xy(*this)!= 0;} //! less than in (x,y) lexicographic order bool operator < (const Self& q) const {return q.compare_xy(*this)> 0;} //! less-equal in (x,y) lexicographic order bool operator <= (const Self& q) const {return q.compare_xy(*this)>= 0;} //! greater than in (x,y) lexicographic order bool operator > (const Self& q) const {return q.compare_xy(*this)< 0;} //! greater-equal in (x,y) lexicographic order bool operator >= (const Self& q) const {return q.compare_xy(*this)<= 0;} //!@} //!@{ //! \name const Algebraic_curve_kernel_2* kernel() const { return this->ptr()->_m_kernel; } private: /*!\brief * compares y-coordinates for covertical points \c *this and \c q * * \pre x() == q.x() */ CGAL::Comparison_result _compare_y_at_x(const Self& q) const { CGAL_precondition(this->compare_x(q) == CGAL::EQUAL); Curve_analysis_2 f = curve(), g = q.curve(); if(f.is_identical(g)) return CGAL::sign(arcno() - q.arcno()); if(Self::_simplify(*this, q)) // restart since supporting curves might be equal now return _compare_y_at_x(q); Curve_pair_analysis_2 cpa_2 = kernel()->construct_curve_pair_2_object()(f, g); typename Curve_pair_analysis_2::Status_line_1 vline = cpa_2.status_line_for_x(x()); return CGAL::sign(vline.event_of_curve(arcno(), f) - vline.event_of_curve(q.arcno(), g)); } //!@} public: //!\name Reconstructing functions //!@{ /*!\brief * Simplifies the representation of a point. * * Given a decomposition of the point's supporting \c curve() into * a pair of two curves \c pair, this function searches this point * in the curve pair and resets the curve and the arcno to this * found arc. It can happen, that both curves of the pair fit this * condition (intersection of the two curves at this point), then it * chooses the simpler one (less total degree). * * \pre pair must be a decomposition of curve() */ void simplify_by(const Curve_pair_analysis_2& cpa_2) const { CGAL_precondition_code( Polynomial_2 mult = cpa_2.curve_analysis(0).polynomial_2() * cpa_2.curve_analysis(1).polynomial_2(); ); // common parts CGAL_precondition(CGAL::resultant(mult, curve().polynomial_2()).is_zero()); // full parts CGAL_precondition(CGAL::degree(mult) == CGAL::degree(curve().polynomial_2())); CGAL_precondition(CGAL::total_degree(mult) == CGAL::total_degree(curve().polynomial_2())); typename Curve_pair_analysis_2::Status_line_1 cpv_line = cpa_2.status_line_for_x(x()); // # of arcs must match CGAL_precondition_code( typename Curve_analysis_2::Status_line_1 cv_line = curve().status_line_for_x(x()); ); CGAL_precondition(cpv_line.number_of_events() == cv_line.number_of_events()); bool cid = false; std::pair p = cpv_line.curves_at_event(arcno()); if(p.first != -1 && p.second != -1) { // both curves involved: choose simpler one // Remark: In this case, a vertical line in the curves can be // ignored, since it has not been considered when constructing // the point from the composed curved (also including this vertical // line). Therefore, the old arc number is also valid in the curve // pair. Polynomial_2 ff = cpa_2.curve_analysis(0).polynomial_2(), gg = cpa_2.curve_analysis(1).polynomial_2(); if(total_degree(ff) > total_degree(gg)) cid = true; } else cid = (p.first == -1); // overwrite data this->ptr()->_m_curve = cpa_2.curve_analysis(cid); this->ptr()->_m_arcno = (cid == 0 ? p.first : p.second); } //! befriending output iterator // friend std::ostream& operator << <>(std::ostream& os, const Self& pt); //!@} public: //! Returns whether the x-coordinate equals zero bool is_x_zero() const { return CGAL::is_zero(this->ptr()->_m_x); } //! Returns whether the y-coordinate equals zero bool is_y_zero() const { CGAL::Sign lower_sign = CGAL::sign(this->lower_bound_y()), upper_sign = CGAL::sign(this->upper_bound_y()); if( lower_sign == CGAL::ZERO ||upper_sign == CGAL::ZERO) { if(lower_sign==upper_sign) { //both zero return true; } else { // one zero, one not...isol interval is OPEN return false; } } else if( lower_sign==upper_sign) { // zero not in isol interval return false; } else { // zero in interval, need to check Polynomial_1 constant_pol = CGAL::get_coefficient(curve().primitive_polynomial_2(),0); bool zero_is_root_of_local_pol = kernel()->is_zero_at_1_object()(constant_pol,this->ptr()->_m_x); // Since we know that y_iv is an _isolating_ interval, // we can immediately return return zero_is_root_of_local_pol; } } // returns a double approximation of the point std::pair to_double() const { typedef typename CGAL::Get_arithmetic_kernel::Arithmetic_kernel AT; typedef typename AT::Bigfloat_interval BFI; long old_prec = get_precision(BFI()); set_precision (BFI(), 53); // rely on double conversion of the x-type double double_x = CGAL::to_double(this->ptr()->_m_x); double double_y; if (this->lower_bound_y()==this->upper_bound_y()) { double_y = CGAL::to_double(convert_to_bfi(this->lower_bound_y())); } else if(is_y_zero()) { double_y = 0.; } else { while(CGAL::sign(this->lower_bound_y()) != CGAL::sign(this->upper_bound_y()) ) { this->refine_y(); } long final_prec = set_precision(BFI(),get_precision(BFI())+4); BFI bfi = CGAL::hull(convert_to_bfi(this->lower_bound_y()), convert_to_bfi(this->upper_bound_y())); while( !singleton(bfi) && get_significant_bits(bfi) < final_prec ){ this->refine_y(); bfi = CGAL::hull( convert_to_bfi(this->lower_bound_y()), convert_to_bfi(this->upper_bound_y())); } double_y = CGAL::to_double((CGAL::lower(bfi)+ CGAL::upper(bfi)) / 2); } set_precision(BFI(),old_prec); return std::make_pair(double_x, double_y); } public: void refine_y() const { this->curve().status_line_at_exact_x(this->x()).refine(this->arcno()); } Bound lower_bound_y() const { return this->curve().status_line_at_exact_x(this->x()). lower_bound(this->arcno()); } Bound upper_bound_y() const { return this->curve().status_line_at_exact_x(this->x()). upper_bound(this->arcno()); } #if CGAL_AK_ENABLE_DEPRECATED_INTERFACE void refine_x() const { this->x().refine(); } Bound lower_bound_x() const { return this->x().low(); } Bound upper_bound_x() const { return this->x().high(); } #endif // friend function to provide a fast hashing friend std::size_t hash_value(const Self& x) { return static_cast(x.id()); } }; // class Xy_coordinate_2 template < class AlgebraicCurveKernel_2, class Rep> std::ostream& operator<< (std::ostream& os, const Xy_coordinate_2& pt) { switch (::CGAL::IO::get_mode(os)) { case ::CGAL::IO::PRETTY: { os << "[x-coord: " << CGAL::to_double(pt.x()) << "; curve: " << pt.curve().polynomial_2() << "; arcno: " << pt.arcno() << "]\n"; break; } case ::CGAL::IO::BINARY: std::cerr << "BINARY format not yet implemented" << std::endl; break; default: // ASCII os << "Algebraic_real_xca_2("; os << pt.x(); os << ","; os << pt.curve(); os << ","; os << pt.arcno(); os << ")"; } return os; } template < class AlgebraicCurveKernel_2, class Rep_ > std::istream& operator >> ( std::istream& is, Xy_coordinate_2< AlgebraicCurveKernel_2, Rep_>& pt) { CGAL_precondition(CGAL::IO::is_ascii(is)); // this instance's first template argument typedef AlgebraicCurveKernel_2 Algebraic_curve_kernel_2; // this instance's second template argument typedef Rep_ Rep; // myself typedef Xy_coordinate_2< Algebraic_curve_kernel_2, Rep > Xy_coordinate_2; typedef typename Algebraic_curve_kernel_2::Algebraic_real_1 Algebraic_real_1; typedef typename Algebraic_curve_kernel_2::Curve_analysis_2 Curve_analysis_2; // x-coordinate Algebraic_real_1 x; // supporting curve Curve_analysis_2 curve; // arc number on curve int arcno; // read "Algebraic_real_xca_2(" swallow(is, 'A'); swallow(is, 'l'); swallow(is, 'g'); swallow(is, 'e'); swallow(is, 'b'); swallow(is, 'r'); swallow(is, 'a'); swallow(is, 'i'); swallow(is, 'c'); swallow(is, '_'); swallow(is, 'r'); swallow(is, 'e'); swallow(is, 'a'); swallow(is, 'l'); swallow(is, '_'); swallow(is, 'x'); swallow(is, 'c'); swallow(is, 'a'); swallow(is, '_'); swallow(is, '2'); swallow(is, '('); // read values is >> x; swallow(is, ','); is >> curve; swallow(is, ','); is >> arcno; // read the ")" swallow(is, ')'); pt = Xy_coordinate_2(x, curve, arcno); return is; } } // namespace internal } //namespace CGAL #endif // CGAL_ALGEBRAIC_CURVE_KERNEL_XY_COORDINATE_2_H