// Copyright (c) 2014
// INRIA Saclay-Ile de France (France)
//
// This file is part of CGAL (www.cgal.org)
//
// $URL: https://github.com/CGAL/cgal/blob/v5.6.2/NewKernel_d/include/CGAL/NewKernel_d/function_objects_cartesian.h $
// $Id: function_objects_cartesian.h 9dd8235dc50 2022-08-10T23:48:47+02:00 Sébastien Loriot
// SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial
//
// Author(s)     : Marc Glisse

#ifndef CGAL_KERNEL_D_FUNCTION_OBJECTS_CARTESIAN_H
#define CGAL_KERNEL_D_FUNCTION_OBJECTS_CARTESIAN_H

#include <CGAL/NewKernel_d/utils.h>
#include <CGAL/Dimension.h>
#include <CGAL/Uncertain.h>
#include <CGAL/NewKernel_d/store_kernel.h>
#include <CGAL/type_traits/is_iterator.h>
#include <CGAL/iterator_from_indices.h>
#include <CGAL/number_utils.h>
#include <CGAL/Kernel/Return_base_tag.h>
#include <CGAL/transforming_iterator.h>
#include <CGAL/transforming_pair_iterator.h>
#include <CGAL/NewKernel_d/functor_tags.h>
#include <CGAL/NewKernel_d/functor_properties.h>
#include <CGAL/predicates/sign_of_determinant.h>
#include <functional>
#include <initializer_list>
#include <vector>

namespace CGAL {
namespace CartesianDKernelFunctors {
namespace internal {
template<class,int> struct Dimension_at_most { enum { value = false }; };
template<int a,int b> struct Dimension_at_most<Dimension_tag<a>,b> {
        enum { value = (a <= b) };
};
}

template<class R_,class D_=typename R_::Default_ambient_dimension,bool=internal::Dimension_at_most<D_,6>::value> struct Orientation_of_points : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Orientation_of_points)
        typedef R_ R;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_type<R, Orientation_tag>::type result_type;
        typedef typename R::LA::Square_matrix Matrix;

        template<class Iter>
        result_type operator()(Iter f, Iter e)const{
                typename Get_functor<R, Compute_point_cartesian_coordinate_tag>::type c(this->kernel());
                typename Get_functor<R, Point_dimension_tag>::type pd(this->kernel());
                Point const& p0=*f++;
                int d=pd(p0);
                Matrix m(d,d);
                // FIXME: this writes the vector coordinates in lines ? check all the other uses in this file, this may be wrong for some.
                for(int i=0;f!=e;++f,++i) {
                  Point const& p=*f;
                  for(int j=0;j<d;++j){
                    m(i,j)=c(p,j)-c(p0,j);
                    // should we cache the coordinates of p0 in case they are computed?
                  }
                }
                return R::LA::sign_of_determinant(std::move(m));
        }

        // Since the dimension is at least 2, there are at least 3 points and no ambiguity with iterators.
        // template <class...U,class=std::enable_if_t<std::is_same<Dimension_tag<sizeof...(U)-1>::value,typename R::Default_ambient_dimension>::value>>
        template <class...U,class=std::enable_if_t<(sizeof...(U)>=3)>>
        result_type operator()(U&&...u) const {
                return operator()({std::forward<U>(u)...});
        }

        template <class P>
        result_type operator()(std::initializer_list<P> l) const {
                return operator()(l.begin(),l.end());
        }
};

template<class R_,int d> struct Orientation_of_points<R_,Dimension_tag<d>,true> : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Orientation_of_points)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_type<R, Orientation_tag>::type result_type;
        template<class>struct Help;
        template<std::size_t...I>struct Help<std::index_sequence<I...> > {
                template<class C,class P,class T> result_type operator()(C const&c,P const&x,T&&t)const{
                        return sign_of_determinant<RT>(c(std::get<I/d>(t),I%d)-c(x,I%d)...);
                }
        };
        template<class P0,class...P> result_type operator()(P0 const&x,P&&...p)const{
                static_assert(d==sizeof...(P),"Wrong number of arguments");
                typename Get_functor<R, Compute_point_cartesian_coordinate_tag>::type c(this->kernel());
                return Help<std::make_index_sequence<d*d>>()(c,x,std::forward_as_tuple(std::forward<P>(p)...));
        }


        template<int N,class Iter,class...U> result_type help2(Dimension_tag<N>, Iter f, Iter const&e, U&&...u)const{
                auto const&p=*f;
                return help2(Dimension_tag<N-1>(),++f,e,std::forward<U>(u)...,p);
        }
        template<class Iter,class...U> result_type help2(Dimension_tag<0>, Iter CGAL_assertion_code(f), Iter const& CGAL_assertion_code(e), U&&...u)const{
                CGAL_assertion(f==e);
                return operator()(std::forward<U>(u)...);
        }
        template<class Iter>
        result_type operator()(Iter f, Iter e)const{
                return help2(Dimension_tag<d+1>(),f,e);
        }
};

template<class R_> struct Orientation_of_points<R_,Dimension_tag<1>,true> : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Orientation_of_points)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_type<R, Orientation_tag>::type result_type;
        result_type operator()(Point const&x, Point const&y) const {
                typename Get_functor<R, Compute_point_cartesian_coordinate_tag>::type c(this->kernel());
                // No sign_of_determinant(RT) :-(
                return CGAL::compare(c(y,0),c(x,0));
        }
        template<class Iter>
        result_type operator()(Iter f, Iter CGAL_assertion_code(e))const{
                Point const&x=*f;
                Point const&y=*++f;
                CGAL_assertion(++f==e);
                return operator()(x,y);
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Orientation_of_points_tag,(CartesianDKernelFunctors::Orientation_of_points<K>),(Point_tag),(Point_dimension_tag,Compute_point_cartesian_coordinate_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Orientation_of_vectors : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Orientation_of_vectors)
        typedef R_ R;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_type<R, Orientation_tag>::type result_type;
        typedef typename R::LA::Square_matrix Matrix;

        template<class Iter>
        result_type operator()(Iter f, Iter e)const{
                typename Get_functor<R, Compute_vector_cartesian_coordinate_tag>::type c(this->kernel());
                typename Get_functor<R, Vector_dimension_tag>::type vd(this->kernel());
                Vector const& v0=*f;
                int d=vd(v0);
                Matrix m(d,d);
                for(int j=0;j<d;++j){
                        m(0,j)=c(v0,j);
                }
                for(int i=1;++f!=e;++i) {
                        Vector const& v=*f;
                for(int j=0;j<d;++j){
                        m(i,j)=c(v,j);
                }
                }
                return R::LA::sign_of_determinant(std::move(m));
        }

        template <class...U,class=std::enable_if_t<(sizeof...(U)>=3)>>
        result_type operator()(U&&...u) const {
                return operator()({std::forward<U>(u)...});
        }

        template <class V>
        result_type operator()(std::initializer_list<V> l) const {
                return operator()(l.begin(),l.end());
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Orientation_of_vectors_tag,(CartesianDKernelFunctors::Orientation_of_vectors<K>),(Vector_tag),(Vector_dimension_tag,Compute_vector_cartesian_coordinate_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Linear_rank : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Linear_rank)
        typedef R_ R;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        // Computing a sensible Uncertain<int> is not worth it
        typedef int result_type;
        typedef typename R::LA::Dynamic_matrix Matrix;

        template<class Iter>
        result_type operator()(Iter f, Iter e)const{
                typename Get_functor<R, Compute_vector_cartesian_coordinate_tag>::type c(this->kernel());
                typename Get_functor<R, Vector_dimension_tag>::type vd(this->kernel());
                std::ptrdiff_t n=std::distance(f,e);
                if (n==0) return 0;
                Vector const& v0 = *f;
                int d=vd(v0);
                Matrix m(d,n);
                for(int j=0;j<d;++j){
                  m(j,0)=c(v0,j);
                }
                for(int i=1; ++f!=e; ++i){
                  Vector const& v = *f;
                  for(int j=0;j<d;++j){
                    m(j,i)=c(v,j);
                  }
                }
                return R::LA::rank(std::move(m));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Linear_rank_tag,(CartesianDKernelFunctors::Linear_rank<K>),(Vector_tag),(Vector_dimension_tag,Compute_vector_cartesian_coordinate_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Linearly_independent : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Linearly_independent)
        typedef R_ R;
        typedef typename Get_type<R, Bool_tag>::type result_type;

        template<class Iter>
        result_type operator()(Iter f, Iter e)const{
                typename Get_functor<R, Vector_dimension_tag>::type vd(this->kernel());
                std::ptrdiff_t n=std::distance(f,e);
                int d=vd(*f);
                if (n>d) return false;
                typename Get_functor<R, Linear_rank_tag>::type lr(this->kernel());
                return lr(f,e) == n;
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Linearly_independent_tag,(CartesianDKernelFunctors::Linearly_independent<K>),(Vector_tag),(Vector_dimension_tag,Linear_rank_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Contained_in_linear_hull : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Contained_in_linear_hull)
        typedef R_ R;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        // Computing a sensible Uncertain<bool> is not worth it
        typedef bool result_type;
        typedef typename R::LA::Dynamic_matrix Matrix;

        template<class Iter,class V>
        result_type operator()(Iter f, Iter e,V const&w)const{
                typename Get_functor<R, Compute_vector_cartesian_coordinate_tag>::type c(this->kernel());
                typename Get_functor<R, Vector_dimension_tag>::type vd(this->kernel());
                std::ptrdiff_t n=std::distance(f,e);
                if (n==0) return false;
                int d=vd(w);
                Matrix m(d,n+1);
                for(int i=0; f!=e; ++f,++i){
                  Vector const& v = *f;
                  for(int j=0;j<d;++j){
                    m(j,i)=c(v,j);
                  }
                }
                for(int j=0;j<d;++j){
                  m(j,n)=c(w,j);
                }
                int r1 = R::LA::rank(m);
                // FIXME: Don't use eigen directly, go through an interface in LA...
                m.conservativeResize(Eigen::NoChange, n);
                int r2 = R::LA::rank(std::move(m));
                return r1 == r2;
                // TODO: This is very very far from optimal...
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Contained_in_linear_hull_tag,(CartesianDKernelFunctors::Contained_in_linear_hull<K>),(Vector_tag),(Vector_dimension_tag,Compute_vector_cartesian_coordinate_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Affine_rank : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Affine_rank)
        typedef R_ R;
        typedef typename Get_type<R, Point_tag>::type Point;
        // Computing a sensible Uncertain<int> is not worth it
        typedef int result_type;
        typedef typename R::LA::Dynamic_matrix Matrix;

        template<class Iter>
        result_type operator()(Iter f, Iter e)const{
                typename Get_functor<R, Compute_point_cartesian_coordinate_tag>::type c(this->kernel());
                typename Get_functor<R, Point_dimension_tag>::type pd(this->kernel());
                int n=(int)std::distance(f,e);
                if (--n<=0) return n;
                Point const& p0 = *f;
                int d=pd(p0);
                Matrix m(d,n);
                for(int i=0; ++f!=e; ++i){
                  Point const& p = *f;
                  for(int j=0;j<d;++j){
                    m(j,i)=c(p,j)-c(p0,j);
                    // TODO: cache p0[j] in case it is computed?
                  }
                }
                return R::LA::rank(std::move(m));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Affine_rank_tag,(CartesianDKernelFunctors::Affine_rank<K>),(Point_tag),(Point_dimension_tag,Compute_point_cartesian_coordinate_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Affinely_independent : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Affinely_independent)
        typedef R_ R;
        typedef typename Get_type<R, Bool_tag>::type result_type;

        template<class Iter>
        result_type operator()(Iter f, Iter e)const{
                typename Get_functor<R, Point_dimension_tag>::type pd(this->kernel());
                std::ptrdiff_t n=std::distance(f,e);
                int d=pd(*f);
                if (--n>d) return false;
                typename Get_functor<R, Affine_rank_tag>::type ar(this->kernel());
                return ar(f,e) == n;
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Affinely_independent_tag,(CartesianDKernelFunctors::Affinely_independent<K>),(Point_tag),(Point_dimension_tag,Affine_rank_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Contained_in_simplex : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Contained_in_simplex)
        typedef R_ R;
        typedef typename Get_type<R, Point_tag>::type Point;
        // Computing a sensible Uncertain<*> is not worth it
        // typedef typename Get_type<R, Boolean_tag>::type result_type;
        typedef bool result_type;
        typedef typename Increment_dimension<typename R::Default_ambient_dimension>::type D1;
        typedef typename Increment_dimension<typename R::Max_ambient_dimension>::type D2;
        typedef typename R::LA::template Rebind_dimension<D1,D2>::Other LA;
        typedef typename LA::Dynamic_matrix Matrix;
        typedef typename LA::Dynamic_vector DynVec;
        typedef typename LA::Vector Vec;

        template<class Iter, class P>
        result_type operator()(Iter f, Iter e, P const&q)const{
                typename Get_functor<R, Compute_point_cartesian_coordinate_tag>::type c(this->kernel());
                typename Get_functor<R, Point_dimension_tag>::type pd(this->kernel());
                std::ptrdiff_t n=std::distance(f,e);
                if (n==0) return false;
                int d=pd(q);
                Matrix m(d+1,n);
                DynVec a(n);
                // FIXME: Should use the proper vector constructor (Iterator_and_last)
                Vec b(d+1);
                for(int j=0;j<d;++j) b[j]=c(q,j);
                b[d]=1;

                for(int i=0; f!=e; ++i,++f){
                  Point const& p = *f;
                  for(int j=0;j<d;++j){
                    m(j,i)=c(p,j);
                  }
                  m(d,i)=1;
                }
                // If the simplex has full dimension, there must be a solution, only the signs need to be checked.
                if (n == d+1)
                  LA::solve(a,std::move(m),std::move(b));
                else if (!LA::solve_and_check(a,std::move(m),std::move(b)))
                  return false;
                for(int i=0;i<n;++i){
                  if (a[i]<0) return false;
                }
                return true;
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Contained_in_simplex_tag,(CartesianDKernelFunctors::Contained_in_simplex<K>),(Point_tag),(Point_dimension_tag,Compute_point_cartesian_coordinate_tag));

namespace CartesianDKernelFunctors {
  namespace internal {
    template<class Ref_>
    struct Matrix_col_access {
      typedef Ref_ result_type;
      int col;
      Matrix_col_access(int r):col(r){}
      template<class Mat> Ref_ operator()(Mat const& m, std::ptrdiff_t row)const{
        return m(row,col);
      }
    };
  }
template<class R_> struct Linear_base : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Linear_base)
        typedef R_ R;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_type<R, FT_tag>::type FT;
        typedef void result_type;
        typedef typename R::LA::Dynamic_matrix Matrix;

        template<class Iter, class Oter>
        result_type operator()(Iter f, Iter e, Oter o)const{
                typename Get_functor<R, Compute_vector_cartesian_coordinate_tag>::type c(this->kernel());
                typename Get_functor<R, Vector_dimension_tag>::type vd(this->kernel());
                typename Get_functor<R, Construct_ttag<Vector_tag> >::type cv(this->kernel());
                std::ptrdiff_t n=std::distance(f,e);
                if (n==0) return;
                Vector const& v0 = *f;
                int d=vd(v0);
                Matrix m(d,n);
                for(int j=0;j<d;++j){
                  m(0,j)=c(v0,j);
                }
                for(int i=1; ++f!=e; ++i){
                  Vector const& v = *f;
                  for(int j=0;j<d;++j){
                    m(i,j)=c(v,j);
                  }
                }
                Matrix b = R::LA::basis(std::move(m));
                for(int i=0; i < R::LA::columns(b); ++i){
                  //*o++ = Vector(b.col(i));
                  typedef
                    decltype(std::declval<const Matrix>()(0,0))
                    Ref;
                  typedef Iterator_from_indices<Matrix, FT, Ref,
                          internal::Matrix_col_access<Ref> > IFI;
                  *o++ = cv(IFI(b,0,i),IFI(b,d,i));
                }
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Linear_base_tag,(CartesianDKernelFunctors::Linear_base<K>),(Vector_tag),(Vector_dimension_tag,Compute_vector_cartesian_coordinate_tag));

#if 0
namespace CartesianDKernelFunctors {
template<class R_,bool=std::is_same<typename R_::Point,typename R_::Vector>::value> struct Orientation : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Orientation)
        typedef R_ R;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_type<R, Orientation_tag>::type result_type;
        typedef typename Get_functor<R, Orientation_of_points_tag>::type OP;
        typedef typename Get_functor<R, Orientation_of_vectors_tag>::type OV;

        //FIXME!!!
        //when Point and Vector are distinct types, the dispatch should be made
        //in a way that doesn't instantiate a conversion from Point to Vector
        template<class Iter>
        result_type operator()(Iter const&f, Iter const& e)const{
                typename Get_functor<R, Point_dimension_tag>::type pd(this->kernel());
                typename std::iterator_traits<Iter>::difference_type d=std::distance(f,e);
                int dim=pd(*f); // BAD
                if(d==dim) return OV(this->kernel())(f,e);
                CGAL_assertion(d==dim+1);
                return OP(this->kernel())(f,e);
        }
        //TODO: version that takes objects directly instead of iterators
};

template<class R_> struct Orientation<R_,false> : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Orientation)
        typedef R_ R;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_type<R, Orientation_tag>::type result_type;
        typedef typename Get_functor<R, Orientation_of_points_tag>::type OP;
        typedef typename Get_functor<R, Orientation_of_vectors_tag>::type OV;
        typedef typename R::LA::Square_matrix Matrix;

        //FIXME!!!
        //when Point and Vector are distinct types, the dispatch should be made
        //in a way that doesn't instantiate a conversion from Point to Vector
        template<class Iter>
        std::enable_if_t<is_iterator_to<Iter,Point>::value,result_type>
        operator()(Iter const&f, Iter const& e)const{
                return OP(this->kernel())(f,e);
        }
        template<class Iter>
        std::enable_if_t<is_iterator_to<Iter,Vector>::value,result_type>
        operator()(Iter const&f, Iter const& e)const{
                return OV(this->kernel())(f,e);
        }
        //TODO: version that takes objects directly instead of iterators
};
}
#endif

namespace CartesianDKernelFunctors {
template<class R_> struct Power_side_of_power_sphere_raw : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Power_side_of_power_sphere_raw)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, FT_tag>::type FT;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_type<R, Oriented_side_tag>::type result_type;
        typedef typename Increment_dimension<typename R::Default_ambient_dimension>::type D1;
        typedef typename Increment_dimension<typename R::Max_ambient_dimension>::type D2;
        typedef typename R::LA::template Rebind_dimension<D1,D2>::Other LA;
        typedef typename LA::Square_matrix Matrix;

        template<class IterP, class IterW, class Pt, class Wt>
        result_type operator()(IterP f, IterP const& e, IterW fw, IterW const&/*ew*/, Pt const& p0, Wt const& w0) const {
          typedef typename Get_functor<R, Squared_distance_to_origin_tag>::type Sqdo;
          typename Get_functor<R, Compute_point_cartesian_coordinate_tag>::type c(this->kernel());
          typename Get_functor<R, Point_dimension_tag>::type pd(this->kernel());

          int d=pd(p0);
          Matrix m(d+1,d+1);
          if(CGAL::Is_stored<Sqdo>::value) {
            Sqdo sqdo(this->kernel());
            FT const& h0 = sqdo(p0) - w0;
            for(int i=0;f!=e;++f,++fw,++i) {
              Point const& p=*f;
              for(int j=0;j<d;++j){
                RT const& x=c(p,j);
                m(i,j)=x-c(p0,j);
              }
              m(i,d) = sqdo(p) - *fw - h0;
            }
          } else {
            for(int i=0;f!=e;++f,++fw,++i) {
              Point const& p=*f;
              m(i,d) = w0 - *fw;
              for(int j=0;j<d;++j){
                RT const& x=c(p,j);
                m(i,j)=x-c(p0,j);
                m(i,d)+=CGAL::square(m(i,j));
              }
            }
          }
          if(d%2)
            return -LA::sign_of_determinant(std::move(m));
          else
            return LA::sign_of_determinant(std::move(m));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Power_side_of_power_sphere_raw_tag,(CartesianDKernelFunctors::Power_side_of_power_sphere_raw<K>),(Point_tag),(Point_dimension_tag,Squared_distance_to_origin_tag,Compute_point_cartesian_coordinate_tag));

// TODO: make Side_of_oriented_sphere call Power_side_of_power_sphere_raw
namespace CartesianDKernelFunctors {
template<class R_> struct Side_of_oriented_sphere : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Side_of_oriented_sphere)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_type<R, Oriented_side_tag>::type result_type;
        typedef typename Increment_dimension<typename R::Default_ambient_dimension>::type D1;
        typedef typename Increment_dimension<typename R::Max_ambient_dimension>::type D2;
        typedef typename R::LA::template Rebind_dimension<D1,D2>::Other LA;
        typedef typename LA::Square_matrix Matrix;

        /* Undocumented, removed
        template<class Iter>
        result_type operator()(Iter f, Iter const& e)const{
          Point const& p0=*f++; // *--e ?
          return this->operator()(f,e,p0);
        }
        */

        template<class Iter>
        result_type operator()(Iter f, Iter const& e, Point const& p0) const {
          typedef typename Get_functor<R, Squared_distance_to_origin_tag>::type Sqdo;
          typename Get_functor<R, Compute_point_cartesian_coordinate_tag>::type c(this->kernel());
          typename Get_functor<R, Point_dimension_tag>::type pd(this->kernel());

          int d=pd(p0);
          Matrix m(d+1,d+1);
          if(CGAL::Is_stored<Sqdo>::value) {
            Sqdo sqdo(this->kernel());
            for(int i=0;f!=e;++f,++i) {
              Point const& p=*f;
              for(int j=0;j<d;++j){
                RT const& x=c(p,j);
                m(i,j)=x-c(p0,j);
              }
              m(i,d) = sqdo(p) - sqdo(p0);
            }
          } else {
            for(int i=0;f!=e;++f,++i) {
              Point const& p=*f;
              m(i,d) = 0;
              for(int j=0;j<d;++j){
                RT const& x=c(p,j);
                m(i,j)=x-c(p0,j);
                m(i,d)+=CGAL::square(m(i,j));
              }
            }
          }
          if(d%2)
            return -LA::sign_of_determinant(std::move(m));
          else
            return LA::sign_of_determinant(std::move(m));
        }

        template <class...U,class=std::enable_if_t<(sizeof...(U)>=4)>>
        result_type operator()(U&&...u) const {
                return operator()({std::forward<U>(u)...});
        }

        template <class P>
        result_type operator()(std::initializer_list<P> l) const {
                return operator()(l.begin(),l.end());
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Side_of_oriented_sphere_tag,(CartesianDKernelFunctors::Side_of_oriented_sphere<K>),(Point_tag),(Point_dimension_tag,Squared_distance_to_origin_tag,Compute_point_cartesian_coordinate_tag));

namespace CartesianDKernelFunctors {
template <class R_> struct Construct_circumcenter : Store_kernel<R_> {
  CGAL_FUNCTOR_INIT_STORE(Construct_circumcenter)
  typedef typename Get_type<R_, Point_tag>::type Point;
  typedef typename Get_type<R_, Vector_tag>::type Vector;
  typedef Point result_type;
  typedef typename Get_type<R_, FT_tag>::type FT;
  template <class Iter>
  result_type operator()(Iter f, Iter e)const{
    typename Get_functor<R_, Point_dimension_tag>::type pd(this->kernel());

    Point const& p0=*f;
    int d = pd(p0);
    int k = static_cast<int>(std::distance(f,e));
    // Sorted from fastest to slowest, whether the dimension is 2, 3 or 4. It may be worth checking at some point.
    CGAL_assume(k>=1);
    if(k==1) return p0;
    if(k==2){
      typename Get_functor<R_, Midpoint_tag>::type mid(this->kernel());
      return mid(p0, *++f);
    }
    // TODO: check for degenerate cases in all the following cases?
    if(k==3){
      // Same equations as in the general case, but solved by hand (Cramer)
      // (c-r).(p-r)=(p-r)²/2
      // (c-r).(q-r)=(q-r)²/2
      typename Get_functor<R_, Squared_length_tag>::type sl(this->kernel());
      typename Get_functor<R_, Scalar_product_tag>::type sp(this->kernel());
      typename Get_functor<R_, Scaled_vector_tag>::type sv(this->kernel());
      typename Get_functor<R_, Difference_of_points_tag>::type dp(this->kernel());
      typename Get_functor<R_, Translated_point_tag>::type tp(this->kernel());
      Iter f2=f;
      Point const& q=*++f2;
      Point const& r=*++f2;
      Vector u = dp(p0, r);
      Vector v = dp(q, r);
      FT uv = sp(u, v);
      FT u2 = sl(u);
      FT v2 = sl(v);
      FT den = 2 * (u2 * v2 - CGAL::square(uv));
      FT a = (u2 - uv) * v2 / den;
      FT b = (v2 - uv) * u2 / den;
      // Wasteful if we only want the radius
      return tp(tp(r, sv(u, a)), sv(v, b));
    }
    if (k == d+1)
    {
      // 2*(x-y).c == x^2-y^2
      typedef typename R_::LA LA;
      typedef typename LA::Square_matrix Matrix;
      typedef typename LA::Vector Vec;
      typedef typename LA::Construct_vector CVec;
      typename Get_functor<R_, Compute_point_cartesian_coordinate_tag>::type c(this->kernel());
      typename Get_functor<R_, Construct_ttag<Point_tag> >::type cp(this->kernel());
      typename Get_functor<R_, Squared_distance_to_origin_tag>::type sdo(this->kernel());
      FT const& n0 = sdo(p0);
      Matrix m(d,d);
      Vec b = typename CVec::Dimension()(d);
      // Write the point coordinates in lines.
      int i;
      for(i=0; ++f!=e; ++i) {
        Point const& p=*f;
        for(int j=0;j<d;++j) {
          m(i,j)=2*(c(p,j)-c(p0,j));
        }
        b[i] = sdo(p) - n0;
      }
      CGAL_assertion (i == d);
      //Vec res = typename CVec::Dimension()(d);;
      //LA::solve(res, std::move(m), std::move(b));
      // We already assume Eigen below...
      Vec res=m.partialPivLu().solve(b);
      return cp(d,LA::vector_begin(res),LA::vector_end(res));
    }

    {
      // The general case. ui=p(i+1)-p0, center-p0=c=sum ai*ui, c.ui=ui²/2, M*a=b with M symmetric
      typedef typename Increment_dimension<typename R_::Max_ambient_dimension>::type D2;
      typedef typename R_::LA::template Rebind_dimension<Dynamic_dimension_tag,D2>::Other LA;
      typedef typename LA::Square_matrix Matrix;
      typedef typename LA::Vector Vec;
      typedef typename LA::Construct_vector CVec;
      typename Get_functor<R_, Translated_point_tag>::type tp(this->kernel());
      typename Get_functor<R_, Scaled_vector_tag>::type sv(this->kernel());
      typename Get_functor<R_, Difference_of_points_tag>::type dp(this->kernel());
      typename Get_functor<R_, Scalar_product_tag>::type sp(this->kernel());
      Matrix m(k-1,k-1);
      Vec b = typename CVec::Dimension()(k-1);
      std::vector<Vector> vecs; vecs.reserve(k-1);
      while(++f!=e)
        vecs.emplace_back(dp(*f,p0));
      // Only need to fill the lower half
      for(int i=0;i<k-1;++i){
        for(int j=i;j<k-1;++j) {
          m(j,i)=sp(vecs[i],vecs[j]);
#if ! EIGEN_VERSION_AT_LEAST(3, 3, 5)
          m(i,j)=m(j,i);
#endif
        }
        b[i]=m(i,i)/2;
      }
      // Assumes Eigen...
#if EIGEN_VERSION_AT_LEAST(3, 3, 5)
      Vec res=m.ldlt().solve(b);
#else
      // Older versions of Eigen use 1/highest as tolerance,
      // which we have no way to set to 0 for exact types.
      // Use something slow but that should work.
      Vec res=m.fullPivLu().solve(b);
#endif
      Point center=p0;
      // Wasteful if we only want the radius
      for(int i=0;i<k-1;++i)
        center=tp(center,sv(vecs[i],res[i]));
      return center;
    }
  }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Construct_circumcenter_tag,(CartesianDKernelFunctors::Construct_circumcenter<K>),(Point_tag,Vector_tag),(Construct_ttag<Point_tag>,Compute_point_cartesian_coordinate_tag,Scalar_product_tag,Squared_distance_to_origin_tag,Point_dimension_tag,Translated_point_tag,Scaled_vector_tag,Difference_of_points_tag,Squared_length_tag));

namespace CartesianDKernelFunctors {
template <class R_> struct Squared_circumradius : Store_kernel<R_> {
  CGAL_FUNCTOR_INIT_STORE(Squared_circumradius)
  typedef typename Get_type<R_, FT_tag>::type result_type;
  template <class Iter>
  result_type operator()(Iter f, Iter e)const{
    typename Get_functor<R_, Construct_circumcenter_tag>::type cc(this->kernel());
    typename Get_functor<R_, Squared_distance_tag>::type sd(this->kernel());
    return sd(cc(f, e), *f);
  }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Squared_circumradius_tag,(CartesianDKernelFunctors::Squared_circumradius<K>),(Point_tag),(Construct_circumcenter_tag,Squared_distance_tag));

namespace CartesianDKernelFunctors {
// TODO: implement it directly, it should be at least as fast as Side_of_oriented_sphere.
template<class R_> struct Side_of_bounded_sphere : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Side_of_bounded_sphere)
        typedef R_ R;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_type<R, Bounded_side_tag>::type result_type;

        template<class Iter>
        result_type operator()(Iter f, Iter const& e) const {
          Point const& p0 = *f++; // *--e ?
          typename Get_functor<R, Point_dimension_tag>::type pd(this->kernel());
          //FIXME: Doesn't work for non-full dimension.
          CGAL_assertion (std::distance(f,e) == pd(p0)+1);
          return operator() (f, e, p0);
        }

        template<class Iter>
        result_type operator()(Iter const& f, Iter const& e, Point const& p0) const {
          typename Get_functor<R, Side_of_oriented_sphere_tag>::type sos (this->kernel());
          typename Get_functor<R, Orientation_of_points_tag>::type op (this->kernel());
          // enum_cast is not very generic, but since this function isn't supposed to remain like this...
          return enum_cast<Bounded_side> (sos (f, e, p0) * op (f, e));
        }

        template <class...U,class=std::enable_if_t<(sizeof...(U)>=4)>>
        result_type operator()(U&&...u) const {
                return operator()({std::forward<U>(u)...});
        }

        template <class P>
        result_type operator()(std::initializer_list<P> l) const {
                return operator()(l.begin(),l.end());
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Side_of_bounded_sphere_tag,(CartesianDKernelFunctors::Side_of_bounded_sphere<K>),(Point_tag),(Side_of_oriented_sphere_tag,Orientation_of_points_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Side_of_bounded_circumsphere : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Side_of_bounded_circumsphere)
        typedef typename Get_type<R_, Bounded_side_tag>::type result_type;

        template<class Iter, class P>
        result_type operator()(Iter f, Iter const& e, P const& p0) const {
          // TODO: Special case when the dimension is full.
          typename Get_functor<R_, Construct_circumcenter_tag>::type cc(this->kernel());
          typename Get_functor<R_, Compare_distance_tag>::type cd(this->kernel());

          return enum_cast<Bounded_side>(cd(cc(f, e), *f, p0));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Side_of_bounded_circumsphere_tag,(CartesianDKernelFunctors::Side_of_bounded_circumsphere<K>),(Point_tag),(Squared_distance_tag,Construct_circumcenter_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Point_to_vector : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Point_to_vector)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_functor<R, Construct_ttag<Vector_tag> >::type CV;
        typedef typename Get_functor<R, Construct_ttag<Point_cartesian_const_iterator_tag> >::type CI;
        typedef Vector result_type;
        typedef Point argument_type;
        result_type operator()(argument_type const&v)const{
                CI ci(this->kernel());
                return CV(this->kernel())(ci(v,Begin_tag()),ci(v,End_tag()));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Point_to_vector_tag,(CartesianDKernelFunctors::Point_to_vector<K>),(Point_tag,Vector_tag),(Construct_ttag<Vector_tag>, Construct_ttag<Point_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Vector_to_point : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Vector_to_point)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_functor<R, Construct_ttag<Point_tag> >::type CP;
        typedef typename Get_functor<R, Construct_ttag<Vector_cartesian_const_iterator_tag> >::type CI;
        typedef Point result_type;
        typedef Vector argument_type;
        result_type operator()(argument_type const&v)const{
                CI ci(this->kernel());
                return CP(this->kernel())(ci(v,Begin_tag()),ci(v,End_tag()));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Vector_to_point_tag,(CartesianDKernelFunctors::Vector_to_point<K>),(Point_tag,Vector_tag),(Construct_ttag<Point_tag>, Construct_ttag<Vector_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Opposite_vector : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Opposite_vector)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_functor<R, Construct_ttag<Vector_tag> >::type CV;
        typedef typename Get_functor<R, Construct_ttag<Vector_cartesian_const_iterator_tag> >::type CI;
        typedef Vector result_type;
        typedef Vector argument_type;
        result_type operator()(Vector const&v)const{
                CI ci(this->kernel());
                return CV(this->kernel())(make_transforming_iterator(ci(v,Begin_tag()),std::negate<RT>()),make_transforming_iterator(ci(v,End_tag()),std::negate<RT>()));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Opposite_vector_tag,(CartesianDKernelFunctors::Opposite_vector<K>),(Vector_tag),(Construct_ttag<Vector_tag>, Construct_ttag<Vector_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Scaled_vector : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Scaled_vector)
        typedef R_ R;
        typedef typename Get_type<R, FT_tag>::type FT;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_functor<R, Construct_ttag<Vector_tag> >::type CV;
        typedef typename Get_functor<R, Construct_ttag<Vector_cartesian_const_iterator_tag> >::type CI;
        typedef Vector result_type;
        typedef Vector first_argument_type;
        typedef FT second_argument_type;
        result_type operator()(Vector const&v,FT const& s)const{
                CI ci(this->kernel());
                return CV(this->kernel())(make_transforming_iterator(ci(v,Begin_tag()),Scale<FT>(s)),make_transforming_iterator(ci(v,End_tag()),Scale<FT>(s)));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Scaled_vector_tag,(CartesianDKernelFunctors::Scaled_vector<K>),(Vector_tag),(Construct_ttag<Vector_tag>, Construct_ttag<Vector_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Sum_of_vectors : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Sum_of_vectors)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_functor<R, Construct_ttag<Vector_tag> >::type CV;
        typedef typename Get_functor<R, Construct_ttag<Vector_cartesian_const_iterator_tag> >::type CI;
        typedef Vector result_type;
        typedef Vector first_argument_type;
        typedef Vector second_argument_type;
        result_type operator()(Vector const&a, Vector const&b)const{
                CI ci(this->kernel());
                return CV(this->kernel())(make_transforming_pair_iterator(ci(a,Begin_tag()),ci(b,Begin_tag()),std::plus<RT>()),make_transforming_pair_iterator(ci(a,End_tag()),ci(b,End_tag()),std::plus<RT>()));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Sum_of_vectors_tag,(CartesianDKernelFunctors::Sum_of_vectors<K>),(Vector_tag),(Construct_ttag<Vector_tag>, Construct_ttag<Vector_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Difference_of_vectors : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Difference_of_vectors)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_functor<R, Construct_ttag<Vector_tag> >::type CV;
        typedef typename Get_functor<R, Construct_ttag<Vector_cartesian_const_iterator_tag> >::type CI;
        typedef Vector result_type;
        typedef Vector first_argument_type;
        typedef Vector second_argument_type;
        result_type operator()(Vector const&a, Vector const&b)const{
                CI ci(this->kernel());
                return CV(this->kernel())(make_transforming_pair_iterator(ci(a,Begin_tag()),ci(b,Begin_tag()),std::minus<RT>()),make_transforming_pair_iterator(ci(a,End_tag()),ci(b,End_tag()),std::minus<RT>()));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Difference_of_vectors_tag,(CartesianDKernelFunctors::Difference_of_vectors<K>),(Vector_tag),(Construct_ttag<Vector_tag>, Construct_ttag<Vector_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Translated_point : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Translated_point)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_functor<R, Construct_ttag<Point_tag> >::type CP;
        typedef typename Get_functor<R, Construct_ttag<Vector_cartesian_const_iterator_tag> >::type CVI;
        typedef typename Get_functor<R, Construct_ttag<Point_cartesian_const_iterator_tag> >::type CPI;
        typedef Point  result_type;
        typedef Point  first_argument_type;
        typedef Vector second_argument_type;
        result_type operator()(Point const&a, Vector const&b)const{
                CVI cvi(this->kernel());
                CPI cpi(this->kernel());
                return CP(this->kernel())(make_transforming_pair_iterator(cpi(a,Begin_tag()),cvi(b,Begin_tag()),std::plus<RT>()),make_transforming_pair_iterator(cpi(a,End_tag()),cvi(b,End_tag()),std::plus<RT>()));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Translated_point_tag,(CartesianDKernelFunctors::Translated_point<K>),(Point_tag, Vector_tag),(Construct_ttag<Point_tag>, Construct_ttag<Vector_cartesian_const_iterator_tag>, Construct_ttag<Point_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Difference_of_points : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Difference_of_points)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_functor<R, Construct_ttag<Vector_tag> >::type CV;
        typedef typename Get_functor<R, Construct_ttag<Point_cartesian_const_iterator_tag> >::type CI;
        typedef Vector result_type;
        typedef Point first_argument_type;
        typedef Point second_argument_type;
        result_type operator()(Point const&a, Point const&b)const{
                CI ci(this->kernel());
                return CV(this->kernel())(make_transforming_pair_iterator(ci(a,Begin_tag()),ci(b,Begin_tag()),std::minus<RT>()),make_transforming_pair_iterator(ci(a,End_tag()),ci(b,End_tag()),std::minus<RT>()));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Difference_of_points_tag,(CartesianDKernelFunctors::Difference_of_points<K>),(Point_tag, Vector_tag),(Construct_ttag<Vector_tag>, Construct_ttag<Point_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Midpoint : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Midpoint)
        typedef R_ R;
        typedef typename Get_type<R, FT_tag>::type FT;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_functor<R, Construct_ttag<Point_tag> >::type CP;
        typedef typename Get_functor<R, Construct_ttag<Point_cartesian_const_iterator_tag> >::type CI;
        typedef Point result_type;
        typedef Point first_argument_type;
        typedef Point second_argument_type;
        // There is a division, but it will be cast to RT afterwards anyway, so maybe we could use RT.
        struct Average : CGAL::cpp98::binary_function<FT,RT,FT> {
                FT operator()(FT const&a, RT const&b)const{
                        return (a+b)/2;
                }
        };
        result_type operator()(Point const&a, Point const&b)const{
                CI ci(this->kernel());
                //Divide<FT,int> half(2);
                //return CP(this->kernel())(make_transforming_iterator(make_transforming_pair_iterator(ci.begin(a),ci.begin(b),std::plus<FT>()),half),make_transforming_iterator(make_transforming_pair_iterator(ci.end(a),ci.end(b),std::plus<FT>()),half));
                return CP(this->kernel())(make_transforming_pair_iterator(ci(a,Begin_tag()),ci(b,Begin_tag()),Average()),make_transforming_pair_iterator(ci(a,End_tag()),ci(b,End_tag()),Average()));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Midpoint_tag,(CartesianDKernelFunctors::Midpoint<K>),(Point_tag),(Construct_ttag<Point_tag>, Construct_ttag<Point_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Squared_length : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Squared_length)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Vector_tag>::type Vector;
        typedef typename Get_functor<R, Construct_ttag<Vector_cartesian_const_iterator_tag> >::type CI;
        typedef RT result_type;
        typedef Vector argument_type;
        result_type operator()(Vector const&a)const{
                CI ci(this->kernel());
                typename Algebraic_structure_traits<RT>::Square f;
                // TODO: avoid this RT(0)+...
                return std::accumulate(make_transforming_iterator(ci(a,Begin_tag()),f),make_transforming_iterator(ci(a,End_tag()),f),RT(0));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Squared_length_tag,(CartesianDKernelFunctors::Squared_length<K>),(Vector_tag),(Construct_ttag<Vector_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Squared_distance_to_origin : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Squared_distance_to_origin)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_functor<R, Construct_ttag<Point_cartesian_const_iterator_tag> >::type CI;
        typedef RT result_type;
        typedef Point argument_type;
        result_type operator()(Point const&a)const{
                CI ci(this->kernel());
                typename Algebraic_structure_traits<RT>::Square f;
                // TODO: avoid this RT(0)+...
                return std::accumulate(make_transforming_iterator(ci(a,Begin_tag()),f),make_transforming_iterator(ci(a,End_tag()),f),RT(0));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Squared_distance_to_origin_tag,(CartesianDKernelFunctors::Squared_distance_to_origin<K>),(Point_tag),(Construct_ttag<Point_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Squared_distance : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Squared_distance)
        typedef R_ R;
        typedef typename Get_type<R, RT_tag>::type RT;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_functor<R, Construct_ttag<Point_cartesian_const_iterator_tag> >::type CI;
        typedef RT result_type;
        typedef Point first_argument_type;
        typedef Point second_argument_type;
        struct Sq_diff : CGAL::cpp98::binary_function<RT,RT,RT> {
                RT operator()(RT const&a, RT const&b)const{
                        return CGAL::square(a-b);
                }
        };
        result_type operator()(Point const&a, Point const&b)const{
                CI ci(this->kernel());
                Sq_diff f;
                // TODO: avoid this RT(0)+...
                return std::accumulate(make_transforming_pair_iterator(ci(a,Begin_tag()),ci(b,Begin_tag()),f),make_transforming_pair_iterator(ci(a,End_tag()),ci(b,End_tag()),f),RT(0));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Squared_distance_tag,(CartesianDKernelFunctors::Squared_distance<K>),(Point_tag),(Construct_ttag<Point_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Scalar_product : private Store_kernel<R_> {
  CGAL_FUNCTOR_INIT_STORE(Scalar_product)
  typedef R_ R;
  typedef typename Get_type<R, RT_tag>::type RT;
  typedef typename Get_type<R, Vector_tag>::type Vector;
  typedef typename Get_functor<R, Construct_ttag<Vector_cartesian_const_iterator_tag> >::type CI;
  typedef RT result_type;
  typedef Vector first_argument_type;
  typedef Vector second_argument_type;
  result_type operator()(Vector const&a, Vector const&b)const{
    CI ci(this->kernel());
    std::multiplies<RT> f;
    // TODO: avoid this RT(0)+...
    return std::accumulate(
        make_transforming_pair_iterator(ci(a,Begin_tag()),ci(b,Begin_tag()),f),
        make_transforming_pair_iterator(ci(a,  End_tag()),ci(b,  End_tag()),f),
        RT(0));
  }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Scalar_product_tag,(CartesianDKernelFunctors::Scalar_product<K>),(Vector_tag),(Construct_ttag<Vector_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Compare_distance : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Compare_distance)
        typedef R_ R;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_functor<R, Squared_distance_tag>::type CSD;
        typedef typename Get_type<R, Comparison_result_tag>::type result_type;
        typedef Point first_argument_type;
        typedef Point second_argument_type;
        typedef Point third_argument_type; // why am I doing this already?
        typedef Point fourth_argument_type;
        result_type operator()(Point const&a, Point const&b, Point const&c)const{
                CSD csd(this->kernel());
                return CGAL_NTS compare(csd(a,b),csd(a,c));
        }
        result_type operator()(Point const&a, Point const&b, Point const&c, Point const&d)const{
                CSD csd(this->kernel());
                return CGAL_NTS compare(csd(a,b),csd(c,d));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Compare_distance_tag,(CartesianDKernelFunctors::Compare_distance<K>),(Point_tag),(Squared_distance_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Less_point_cartesian_coordinate : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Less_point_cartesian_coordinate)
        typedef R_ R;
        typedef typename Get_type<R, Bool_tag>::type result_type;
        typedef typename Get_functor<R, Compute_point_cartesian_coordinate_tag>::type Cc;
        // TODO: This is_exact thing should be reengineered.
        // the goal is to have a way to tell: don't filter this
        typedef typename CGAL::Uses_no_arithmetic<Cc> Uses_no_arithmetic;

        template<class V,class W,class I>
        result_type operator()(V const&a, W const&b, I i)const{
                Cc c(this->kernel());
                return c(a,i)<c(b,i);
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Less_point_cartesian_coordinate_tag,(CartesianDKernelFunctors::Less_point_cartesian_coordinate<K>),(),(Compute_point_cartesian_coordinate_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Compare_point_cartesian_coordinate : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Compare_point_cartesian_coordinate)
        typedef R_ R;
        typedef typename Get_type<R, Comparison_result_tag>::type result_type;
        typedef typename Get_functor<R, Compute_point_cartesian_coordinate_tag>::type Cc;
        // TODO: This is_exact thing should be reengineered.
        // the goal is to have a way to tell: don't filter this
        typedef typename CGAL::Uses_no_arithmetic<Cc> Uses_no_arithmetic;

        template<class V,class W,class I>
        result_type operator()(V const&a, W const&b, I i)const{
                Cc c(this->kernel());
                return CGAL_NTS compare(c(a,i),c(b,i));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Compare_point_cartesian_coordinate_tag,(CartesianDKernelFunctors::Compare_point_cartesian_coordinate<K>),(),(Compute_point_cartesian_coordinate_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Compare_lexicographically : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Compare_lexicographically)
        typedef R_ R;
        typedef typename Get_type<R, Comparison_result_tag>::type result_type;
        typedef typename Get_functor<R, Construct_ttag<Point_cartesian_const_iterator_tag> >::type CI;
        // TODO: This is_exact thing should be reengineered.
        // the goal is to have a way to tell: don't filter this
        typedef typename CGAL::Uses_no_arithmetic<CI> Uses_no_arithmetic;

        template<class V,class W>
        result_type operator()(V const&a, W const&b)const{
                CI c(this->kernel());


                auto a_begin=c(a,Begin_tag());
                auto b_begin=c(b,Begin_tag());
                auto a_end=c(a,End_tag());
                result_type res;
                // can't we do slightly better for Uncertain<*> ?
                // after res=...; if(is_uncertain(res))return indeterminate<result_type>();
                do res=CGAL_NTS compare(*a_begin++,*b_begin++);
                while(a_begin!=a_end && res==EQUAL);
                return res;
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Compare_lexicographically_tag,(CartesianDKernelFunctors::Compare_lexicographically<K>),(),(Construct_ttag<Point_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Less_lexicographically : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Less_lexicographically)
        typedef R_ R;
        typedef typename Get_type<R, Bool_tag>::type result_type;
        typedef typename Get_functor<R, Compare_lexicographically_tag>::type CL;
        typedef typename CGAL::Uses_no_arithmetic<CL> Uses_no_arithmetic;

        template <class V, class W>
        result_type operator() (V const&a, W const&b) const {
                CL c (this->kernel());
                return c(a,b) < 0;
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Less_lexicographically_tag,(CartesianDKernelFunctors::Less_lexicographically<K>),(),(Compare_lexicographically_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Less_or_equal_lexicographically : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Less_or_equal_lexicographically)
        typedef R_ R;
        typedef typename Get_type<R, Bool_tag>::type result_type;
        typedef typename Get_functor<R, Compare_lexicographically_tag>::type CL;
        typedef typename CGAL::Uses_no_arithmetic<CL> Uses_no_arithmetic;

        template <class V, class W>
        result_type operator() (V const&a, W const&b) const {
                CL c (this->kernel());
                return c(a,b) <= 0;
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Less_or_equal_lexicographically_tag,(CartesianDKernelFunctors::Less_or_equal_lexicographically<K>),(),(Compare_lexicographically_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Equal_points : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Equal_points)
        typedef R_ R;
        typedef typename Get_type<R, Bool_tag>::type result_type;
        typedef typename Get_functor<R, Construct_ttag<Point_cartesian_const_iterator_tag> >::type CI;
        // TODO: This is_exact thing should be reengineered.
        // the goal is to have a way to tell: don't filter this
        typedef typename CGAL::Uses_no_arithmetic<CI> Uses_no_arithmetic;

        template<class V,class W>
        result_type operator()(V const&a, W const&b)const{
                CI c(this->kernel());


                auto a_begin=c(a,Begin_tag());
                auto b_begin=c(b,Begin_tag());
                auto a_end=c(a,End_tag());

                result_type res = true;
                // Is using CGAL::possibly for Uncertain really an optimization?
                do res = res & (*a_begin++ == *b_begin++);
                while(a_begin!=a_end && possibly(res));
                return res;
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Equal_points_tag,(CartesianDKernelFunctors::Equal_points<K>),(),(Construct_ttag<Point_cartesian_const_iterator_tag>));

namespace CartesianDKernelFunctors {
template<class R_> struct Oriented_side : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Oriented_side)
        typedef R_ R;
        typedef typename Get_type<R, Oriented_side_tag>::type result_type;
        typedef typename Get_type<R, Point_tag>::type Point;
        typedef typename Get_type<R, Hyperplane_tag>::type Hyperplane;
        typedef typename Get_type<R, Sphere_tag>::type Sphere;
        typedef typename Get_functor<R, Value_at_tag>::type VA;
        typedef typename Get_functor<R, Hyperplane_translation_tag>::type HT;
        typedef typename Get_functor<R, Squared_distance_tag>::type SD;
        typedef typename Get_functor<R, Squared_radius_tag>::type SR;
        typedef typename Get_functor<R, Center_of_sphere_tag>::type CS;

        result_type operator()(Hyperplane const&h, Point const&p)const{
                HT ht(this->kernel());
                VA va(this->kernel());
                return CGAL::compare(va(h,p),ht(h));
        }
        result_type operator()(Sphere const&s, Point const&p)const{
                SD sd(this->kernel());
                SR sr(this->kernel());
                CS cs(this->kernel());
                return CGAL::compare(sd(cs(s),p),sr(s));
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Oriented_side_tag,(CartesianDKernelFunctors::Oriented_side<K>),(Point_tag,Sphere_tag,Hyperplane_tag),(Value_at_tag,Hyperplane_translation_tag,Squared_distance_tag,Squared_radius_tag,Center_of_sphere_tag));

namespace CartesianDKernelFunctors {
template<class R_> struct Has_on_positive_side : private Store_kernel<R_> {
        CGAL_FUNCTOR_INIT_STORE(Has_on_positive_side)
        typedef R_ R;
        typedef typename Get_type<R, Bool_tag>::type result_type;
        typedef typename Get_functor<R, Oriented_side_tag>::type OS;

        template <class Obj, class Pt>
        result_type operator()(Obj const&o, Pt const&p)const{
                OS os(this->kernel());
                return os(o,p) == ON_POSITIVE_SIDE;
        }
};
}

CGAL_KD_DEFAULT_FUNCTOR(Has_on_positive_side_tag,(CartesianDKernelFunctors::Has_on_positive_side<K>),(),(Oriented_side_tag));

}
#include <CGAL/NewKernel_d/Coaffine.h>
#endif // CGAL_KERNEL_D_FUNCTION_OBJECTS_CARTESIAN_H
