// Copyright (c) 2006,2007,2008,2009,2010,2011 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/Arrangement_on_surface_2/include/CGAL/Arr_algebraic_segment_traits_2.h $
// $Id: include/CGAL/Arr_algebraic_segment_traits_2.h b26b07a1242 $
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s): Michael Kerber    <mkerber@mpi-inf.mpg.de>
//
// ============================================================================

#ifndef CGAL_ARR_ALGEBRAIC_SEGMENT_TRAITS_H
#define CGAL_ARR_ALGEBRAIC_SEGMENT_TRAITS_H

#include <CGAL/license/Arrangement_on_surface_2.h>


#include <CGAL/disable_warnings.h>

#include <CGAL/config.h>

#include <CGAL/Algebraic_kernel_d/flags.h>
#include <CGAL/Algebraic_kernel_d_1.h>
#include <CGAL/Algebraic_kernel_d/Algebraic_curve_kernel_2.h>

#include <CGAL/Curved_kernel_via_analysis_2/Curved_kernel_via_analysis_2_impl.h>

#include <optional>
#include <boost/none.hpp>

namespace CGAL {

template <class Coefficient_>
class Arr_algebraic_segment_traits_2 {

public:
    enum Site_of_point {
        POINT_IN_INTERIOR = 0,
        MIN_ENDPOINT = -1,
        MAX_ENDPOINT = 1 };

    // Template argument

    typedef Coefficient_ Coefficient;

    // 'derivation'
    typedef CGAL::Algebraic_kernel_d_1< Coefficient > Algebraic_kernel_d_1;

    typedef CGAL::Algebraic_curve_kernel_2< Algebraic_kernel_d_1 >
        Algebraic_kernel_d_2;

    typedef CGAL::Curved_kernel_via_analysis_2< Algebraic_kernel_d_2 > CKvA_2;

    typedef  CGAL::Arr_algebraic_segment_traits_2<Coefficient> Self;

    // constructs default
    Arr_algebraic_segment_traits_2 () {}


    // constructs copy
    Arr_algebraic_segment_traits_2 (const  Self& /* s */) { /* No state...*/}

    // assigns
    const Self& operator= (const Self& s) { return s; }

    // public types

    typedef typename Algebraic_kernel_d_2::Algebraic_real_1 Algebraic_real_1;
    typedef typename Algebraic_kernel_d_2::Bound Bound;

    typedef typename Algebraic_kernel_d_2::Polynomial_2 Polynomial_2;

    /// public types for AosTraits_2

    typedef typename Algebraic_kernel_d_2::Curve_analysis_2 Curve_2;

    typedef typename CKvA_2::Arc_2 X_monotone_curve_2;

    typedef typename CKvA_2::Point_2 Point_2;


    typedef typename CKvA_2::Has_left_category Has_left_category;
    typedef typename CKvA_2::Has_merge_category Has_merge_category;
    typedef typename CKvA_2::Has_do_intersect_category
      Has_do_intersect_category;

    typedef typename CKvA_2::Left_side_category Left_side_category;
    typedef typename CKvA_2::Bottom_side_category Bottom_side_category;
    typedef typename CKvA_2::Top_side_category Top_side_category;
    typedef typename CKvA_2::Right_side_category Right_side_category;

    typedef typename CKvA_2::Multiplicity Multiplicity;

    typedef typename CKvA_2::Compare_x_2 Compare_x_2;
    Compare_x_2 compare_x_2_object() const {
        return CKvA_2::instance().compare_x_2_object();
    }

    typedef typename CKvA_2::Compare_xy_2 Compare_xy_2;
    Compare_xy_2 compare_xy_2_object() const {
        return CKvA_2::instance().compare_xy_2_object();
    }

    typedef typename CKvA_2::Compare_endpoints_xy_2 Compare_endpoints_xy_2;
    Compare_endpoints_xy_2 compare_endpoints_xy_2_object() const {
        return CKvA_2::instance().compare_endpoints_xy_2_object();
    }

    typedef typename CKvA_2::Equal_2 Equal_2;
    Equal_2 equal_2_object() const {
        return CKvA_2::instance().equal_2_object();
    }

    typedef typename CKvA_2::Parameter_space_in_y_2 Parameter_space_in_y_2;
    Parameter_space_in_y_2 parameter_space_in_y_2_object() const {
        return CKvA_2::instance().parameter_space_in_y_2_object();
    }

    typedef typename CKvA_2::Compare_y_near_boundary_2
       Compare_y_near_boundary_2;
    Compare_y_near_boundary_2 compare_y_near_boundary_2_object() const {
        return CKvA_2::instance().compare_y_near_boundary_2_object();
    }

    typedef typename CKvA_2::Parameter_space_in_x_2 Parameter_space_in_x_2;
    Parameter_space_in_x_2 parameter_space_in_x_2_object() const {
        return CKvA_2::instance().parameter_space_in_x_2_object();
    }

    typedef typename CKvA_2::Compare_x_on_boundary_2 Compare_x_on_boundary_2;
    Compare_x_on_boundary_2 compare_x_on_boundary_2_object() const {
        return CKvA_2::instance().compare_x_on_boundary_2_object();
    }

    typedef typename CKvA_2::Compare_x_near_boundary_2 Compare_x_near_boundary_2;
    Compare_x_near_boundary_2 compare_x_near_boundary_2_object() const {
        return CKvA_2::instance().compare_x_near_boundary_2_object();
    }

    typedef typename CKvA_2::Construct_min_vertex_2 Construct_min_vertex_2;
    Construct_min_vertex_2 construct_min_vertex_2_object() const {
        return CKvA_2::instance().construct_min_vertex_2_object();
    }

    typedef typename CKvA_2::Construct_max_vertex_2 Construct_max_vertex_2;
    Construct_max_vertex_2 construct_max_vertex_2_object() const {
        return CKvA_2::instance().construct_max_vertex_2_object();
    }

    typedef typename CKvA_2::Construct_opposite_2 Construct_opposite_2;
    Construct_opposite_2 construct_opposite_2_object() const {
        return CKvA_2::instance().construct_opposite_2_object();
    }

    typedef typename CKvA_2::Is_vertical_2 Is_vertical_2;
    Is_vertical_2 is_vertical_2_object() const {
        return CKvA_2::instance().is_vertical_2_object();
    }

    typedef typename CKvA_2::Compare_y_at_x_2 Compare_y_at_x_2;
    Compare_y_at_x_2 compare_y_at_x_2_object() const {
        return CKvA_2::instance().compare_y_at_x_2_object();
    }

    typedef typename CKvA_2::Compare_y_at_x_left_2 Compare_y_at_x_left_2;
    Compare_y_at_x_left_2 compare_y_at_x_left_2_object() const {
        return CKvA_2::instance().compare_y_at_x_left_2_object();
    }

    typedef typename CKvA_2::Compare_y_at_x_right_2 Compare_y_at_x_right_2;
    Compare_y_at_x_right_2 compare_y_at_x_right_2_object() const {
        return CKvA_2::instance().compare_y_at_x_right_2_object();
    }

    typedef typename CKvA_2::Intersect_2 Intersect_2;
    Intersect_2 intersect_2_object() const {
        return CKvA_2::instance().intersect_2_object();
    }

    typedef typename CKvA_2::Split_2 Split_2;
    Split_2 split_2_object() const {
        return CKvA_2::instance().split_2_object();
    }

    typedef typename CKvA_2::Are_mergeable_2 Are_mergeable_2;
    Are_mergeable_2 are_mergeable_2_object() const {
        return CKvA_2::instance().are_mergeable_2_object();
    }

    typedef typename CKvA_2::Merge_2 Merge_2;
    Merge_2 merge_2_object() const {
        return CKvA_2::instance().merge_2_object();
    }

    typedef typename CKvA_2::Make_x_monotone_2 Make_x_monotone_2;
    Make_x_monotone_2 make_x_monotone_2_object() const {
        return Make_x_monotone_2(&CKvA_2::instance());
    }

    typedef typename CKvA_2::Is_on_2 Is_on_2;

    Is_on_2 is_on_2_object() const {
        return Is_on_2(&CKvA_2::instance());
    }

    typedef typename CKvA_2::Construct_point_2 Construct_point_2;

    Construct_point_2 construct_point_2_object() const {
        return Construct_point_2(&CKvA_2::instance());
    }


    class Construct_x_monotone_segment_2 : public
    CGAL::internal::Curved_kernel_via_analysis_2_Functors::
    Curved_kernel_via_analysis_2_functor_base< CKvA_2 > {

    public:

        typedef CGAL::internal::Curved_kernel_via_analysis_2_Functors::
        Curved_kernel_via_analysis_2_functor_base< CKvA_2 > Base;

        Construct_x_monotone_segment_2(CKvA_2* kernel) : Base(kernel) {}


    private:

        // helper function, returning the number of incident branches
        // of a point wrt to a curve
        std::pair<std::pair<int,int>, bool>
            branch_numbers_and_vertical(Curve_2 cv, Point_2 p) const {

            Equal_2 equal = this->_ckva()->equal_2_object();

            Algebraic_real_1 x = p.x();
            int no;
            bool is_event;
            cv.x_to_index(x,no,is_event);
            if(! is_event) {
                return std::make_pair(std::make_pair(1,1),false);
            }
            typename Curve_2::Status_line_1 status_line
                = cv.status_line_at_event(no);
            bool vertical = status_line.covers_line();
            for(int i = 0; i < status_line.number_of_events(); i++) {
                Point_2 curr_point(x,cv,i);
                if(equal(p,curr_point)) {
                    return std::make_pair(
                            status_line.number_of_incident_branches(i),
                            vertical);
                }
            }
            return std::make_pair(std::make_pair(0,0),vertical);
        }

        // abbreviation for convenience
        bool is_one_one(Curve_2 cv, Point_2 p) const {

            std::pair<std::pair<int,int>,bool> branches
                = branch_numbers_and_vertical(cv,p);
            return branches.first.first==1 && branches.first.second==1
                && !branches.second;

        }

      template <typename OutputIterator> OutputIterator
      x_monotone_segment(Curve_2 cv,
                         Point_2 p,
                         std::optional<Point_2> start,
                         std::optional<Point_2> end,
                         OutputIterator out) const
      {
        typedef std::variant<Point_2, X_monotone_curve_2>
                                                        Make_x_monotone_result;
        //CGAL_assertion(is_one_one(cv,p));

        std::list<X_monotone_curve_2> segs;

        Is_on_2 on_arc = this->_ckva()->is_on_2_object();
        auto left = this->_ckva()->construct_min_vertex_2_object();
        auto right = this->_ckva()->construct_max_vertex_2_object();
        Equal_2 equal = this->_ckva()->equal_2_object();

        std::vector<Make_x_monotone_result> arcs;
        this->_ckva()->make_x_monotone_2_object()(cv, std::back_inserter(arcs));
        auto it = arcs.begin();
        auto helper = it;
        const auto* it_seg_p = std::get_if<X_monotone_curve_2>(&(*it));
        while (it != arcs.end()) {
          if ( on_arc(p, *it_seg_p) ) break;
          it++;
          it_seg_p = std::get_if<X_monotone_curve_2>(&(*it));
        }

        bool left_on_arc = start && on_arc(start.value(), *it_seg_p);
        bool right_on_arc = end && on_arc(end.value(), *it_seg_p);

        if ( left_on_arc && right_on_arc ) {
          segs.push_back(it_seg_p->trim(start.value(),end.value()));
        }
        if (left_on_arc && (!right_on_arc)) {
          if (!it_seg_p->is_finite(CGAL::ARR_MAX_END) ||
             !equal(start.value(),right(*it_seg_p))) {
            if (it_seg_p->is_finite(CGAL::ARR_MIN_END) &&
                equal(start.value(),left(*it_seg_p)))
            {
              segs.push_back(*it_seg_p);
            }
            else {
              X_monotone_curve_2 split1,split2;
              it_seg_p->split(start.value(),split1,split2);
              segs.push_back(split2);
            }
          }
        }
        if ((!left_on_arc) && right_on_arc) {
          if (!it_seg_p->is_finite(CGAL::ARR_MIN_END) ||
              ! equal(left(*it_seg_p), end.value()))
          {
            if (it_seg_p->is_finite(CGAL::ARR_MAX_END) &&
                equal(end.value(), right(*it_seg_p)))
            {
              segs.push_back(*it_seg_p);
            }
            else {
              X_monotone_curve_2 split1,split2;
              it_seg_p->split(end.value(), split1, split2);
              segs.push_back(split1);
            }
          }
        }
        if ( (!left_on_arc) && (!right_on_arc)) {
          segs.push_back(*it_seg_p);
        }
        helper = it; // for later usage

        if (! left_on_arc) {

          Point_2 point_it;
          while (true) {
            if (it_seg_p->is_finite(CGAL::ARR_MIN_END) &&
               is_one_one(cv, left(*it_seg_p) ) ) {
              point_it = left(*it_seg_p);
            } else {
              CGAL_assertion(! start);
              break;
            }
            CGAL_assertion(it != arcs.begin());
            it--;
            it_seg_p = std::get_if<X_monotone_curve_2>(&(*it));
            while (! on_arc(point_it, *it_seg_p)) {
              CGAL_assertion(it != arcs.begin());
              it--;
              it_seg_p = std::get_if<X_monotone_curve_2>(&(*it));
            }
            if (start && on_arc(start.value(),*it_seg_p)) {
              segs.push_front(it_seg_p->trim(start.value(), right(*it_seg_p)));
              break;
            }
            else {
              segs.push_front(*it_seg_p);
            }
          }
        }
        if (! right_on_arc) {
          it = helper; // reset
          it_seg_p = std::get_if<X_monotone_curve_2>(&(*it));
          Point_2 point_it;
          while (true) {
            if (it_seg_p->is_finite(CGAL::ARR_MAX_END) &&
                is_one_one(cv,right(*it_seg_p) ) )
            {
              point_it=right(*it_seg_p);
            } else {
              CGAL_assertion(! end);
              break;
            }
            it++;
            CGAL_assertion(it != arcs.end());
            it_seg_p = std::get_if<X_monotone_curve_2>(&(*it));
            while(! on_arc(point_it, *it_seg_p)) {
              it++;
              CGAL_assertion(it != arcs.end());
              it_seg_p = std::get_if<X_monotone_curve_2>(&(*it));
            }
            if(end && on_arc(end.value(),*it_seg_p)) {
              segs.push_back(it_seg_p->trim(left(*it_seg_p),end.value()));
              break;
            }
            else {
              segs.push_back(*it_seg_p);
            }
          }
        }

        std::copy(segs.begin(), segs.end(), out);
        return out;

      }

    public:

        template<typename OutputIterator> OutputIterator
            operator() (Curve_2 cv,
                        Point_2 p,
                        Site_of_point site_of_p,
                        OutputIterator out) const {
            if(site_of_p==POINT_IN_INTERIOR) {
                return x_monotone_segment(cv,p,std::nullopt, std::nullopt,out);
            } else if(site_of_p==MIN_ENDPOINT) {
                return x_monotone_segment(cv,
                                       p,
                                       std::optional<Point_2>(p),
                                       std::nullopt,
                                       out);
            }
            CGAL_assertion(site_of_p==MAX_ENDPOINT);
            return x_monotone_segment(cv,
                                   p,
                                   std::nullopt,
                                   std::optional<Point_2>(p),
                                   out);
        }


        template<typename OutputIterator> OutputIterator
            operator() (Curve_2 cv,
                        Point_2 end_left,
                        Point_2 end_right,
                        OutputIterator out) const {



            CGAL_assertion((branch_numbers_and_vertical(cv,end_left).
                            first.second==1 &&
                            !branch_numbers_and_vertical(cv,end_left).second)||
                           (branch_numbers_and_vertical(cv,end_right).
                            first.first==1 &&
                            !branch_numbers_and_vertical(cv,end_right).second)
                           ||
                           (branch_numbers_and_vertical(cv,end_left).
                            first.second==0 &&
                            branch_numbers_and_vertical(cv,end_left).second) ||
                           (branch_numbers_and_vertical(cv,end_right).
                            first.first==0 &&
                            branch_numbers_and_vertical(cv,end_right).second));
            CGAL_assertion_code(
                    if((branch_numbers_and_vertical(cv,end_left).
                        first.second==0 &&
                        branch_numbers_and_vertical(cv,end_left).second) ||
                       (branch_numbers_and_vertical(cv,end_right).
                        first.first==0 &&
                        branch_numbers_and_vertical(cv,end_right).second)) {
                        Compare_x_2 equal_x
                            = this->_ckva()->compare_x_2_object(); )
                    CGAL_assertion(equal_x(end_left,end_right)==CGAL::EQUAL);
            CGAL_assertion_code(});


            if( (branch_numbers_and_vertical(cv,end_left).
                 first.second==0 &&
                 branch_numbers_and_vertical(cv,end_left).second) ||
                (branch_numbers_and_vertical(cv,end_left).
                 first.second==1 &&
                 !branch_numbers_and_vertical(cv,end_left).second)) {

                return x_monotone_segment
                    (cv,
                     end_left,
                     std::optional<Point_2>(end_left),
                     std::optional<Point_2>(end_right),
                     out);
            } else {
                return x_monotone_segment
                    (cv,
                     end_right,
                     std::optional<Point_2>(end_left),
                     std::optional<Point_2>(end_right),
                     out);
            }
        }

        template<typename OutputIterator> OutputIterator
          operator() (Point_2 p, Point_2 q, OutputIterator out) {
          bool same_x=(this->_ckva()->compare_x_2_object()(p,q)==CGAL::EQUAL);
          if(same_x) {
            Algebraic_real_1 x = p.x();
            Polynomial_2 f = Polynomial_2(x.polynomial());
            Curve_2 cv
              = this->_ckva()->kernel().construct_curve_2_object() (f);
            *out++=typename CKvA_2::Arc_2(p,q,cv);
            return out;
          }
          Algebraic_real_1 px
            =this->_ckva()->kernel().compute_x_2_object()(p.xy()),
            py=this->_ckva()->kernel().compute_y_2_object()(p.xy()),
            qx=this->_ckva()->kernel().compute_x_2_object()(q.xy()),
            qy=this->_ckva()->kernel().compute_y_2_object()(q.xy());
          bool p_rat=px.is_rational() && py.is_rational();
          bool q_rat=qx.is_rational() && qy.is_rational();
          if(p_rat && q_rat) {
            typedef typename Algebraic_real_1::Rational Rational;
            Rational pxr=px.rational(), pyr=py.rational(),
              qxr=qx.rational(), qyr=qy.rational();
            typedef CGAL::Fraction_traits<Rational> FT;
            typename FT::Decompose decompose;
            typedef typename FT::Numerator_type Numerator;
            typedef typename FT::Denominator_type Denominator;
            Rational term_at_y=qxr-pxr, term_at_x=-qyr+pyr,
              term_at_1=pxr*qyr-pyr*qxr;
            Denominator denom_curr;
            Numerator term_at_y_int, term_at_x_int, term_at_1_int;
            decompose(term_at_y, term_at_y_int, denom_curr);
            term_at_x=term_at_x*denom_curr;
            term_at_1=term_at_1*denom_curr;
            decompose(term_at_x,term_at_x_int,denom_curr);
            term_at_y_int=term_at_y_int*denom_curr;
            term_at_1=term_at_1*denom_curr;
            decompose(term_at_1,term_at_1_int,denom_curr);
            term_at_y_int=term_at_y_int*denom_curr;
            term_at_x_int=term_at_x_int*denom_curr;
            typedef typename CGAL::Polynomial_traits_d<Polynomial_2>
              ::Coefficient_type Polynomial_1;
            Polynomial_2 pol(Polynomial_1(term_at_1_int,term_at_x_int),
                             Polynomial_1(term_at_y_int));
            Curve_2 curve=this->_ckva()->kernel().construct_curve_2_object()
              (pol);

            CGAL_assertion(this->_ckva()->is_on_2_object()(p,curve));
            CGAL_assertion(this->_ckva()->is_on_2_object()(q,curve));
            return this->operator()(curve,p,q,out);
          }
          CGAL_precondition(same_x || (p_rat && q_rat));
          return out;
        }
    };

    Construct_x_monotone_segment_2 construct_x_monotone_segment_2_object()
        const {
        return Construct_x_monotone_segment_2(&CKvA_2::instance());
    }

/*

    class Construct_vertical_segment_2 : public
    CGAL::internal::Curved_kernel_via_analysis_2_Functors::
    Curved_kernel_via_analysis_2_functor_base< CKvA_2 > {

    public:

        typedef CGAL::internal::Curved_kernel_via_analysis_2_Functors::
        Curved_kernel_via_analysis_2_functor_base< CKvA_2 > Base;

        Construct_vertical_segment_2(CKvA_2* kernel) : Base(kernel) {}


    public:

        X_monotone_curve_2 operator() (Point_2 p, Point_2 q) {
            Algebraic_real_1 x = p.x();
            Polynomial_2 f = Polynomial_2(x.polynomial());
            Curve_2 cv
                = this->_ckva()->kernel().construct_curve_2_object() (f);
            return typename CKvA_2::Arc_2(p,q,cv);
        }


        template<typename OutputIterator> OutputIterator
            operator() (Point_2 p,
                        Site_of_point site_of_p) const {
            Algebraic_real_1 x = p.x();
            Polynomial_2 f = Polynomial_2(x.polynomial());
            Curve_2 cv = this->_ckva().kernel().construct_curve_2_object() (f);

            if(site_of_p==POINT_IN_INTERIOR) {
                return typename CKvA_2::Arc_2(x,cv);
            } else if(site_of_p==MIN_ENDPOINT) {
                return typename CKvA_2::Arc_2(p,ARR_MIN_END,cv);
            }
            CGAL_assertion(site_of_p==MAX_ENDPOINT);
            return typename CKvA_2::Arc_2(p,ARR_MAX_END,cv);

        }
    };

    Construct_vertical_segment_2 construct_vertical_segment_2_object() const {
        return Construct_vertical_segment_2(&CKvA_2::instance());
    }

*/

    class Construct_curve_2 : public
    CGAL::internal::Curved_kernel_via_analysis_2_Functors::
    Curved_kernel_via_analysis_2_functor_base< CKvA_2 > {

    public:

        typedef CGAL::internal::Curved_kernel_via_analysis_2_Functors::
        Curved_kernel_via_analysis_2_functor_base< CKvA_2 > Base;

        Construct_curve_2(CKvA_2* kernel) : Base(kernel) {}

        Curve_2 operator() (Polynomial_2 p) const {
            return this->_ckva()->kernel().construct_curve_2_object() (p);
        }
    };

    Construct_curve_2 construct_curve_2_object() const {
        return Construct_curve_2(&CKvA_2::instance());
    }

  /*

  // additional functionality (for not introducing a "general" arc)

  class Connect_points_2 {


    template< class OutputIterator >
    OutputIterator operator() (Curve_2, Point_2, Point_2, ...,
                               OutputIterator) {


      while (false || true) {
        *oi++ = X_monotone_curve_2(...);
      }

      return oi;
    }

    template< class OutputIterator >
    OutputIterator operator() (Curve_2, Point_2, Point_2, int, ...,
                               OutputIterator) {


      while (false || true) {
        *oi++ = X_monotone_curve_2(...);
      }

      return oi;
    }

  };

  */

};

} // namespace CGAL

#include <CGAL/enable_warnings.h>

#endif

