// Copyright (c) 2009 INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
// You can redistribute it and/or modify it under the terms of the GNU
// General Public License as published by the Free Software Foundation,
// either version 3 of the License, or (at your option) any later version.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL: https://github.com/CGAL/cgal/blob/releases/CGAL-4.14.3/Mesh_3/include/CGAL/Labeled_mesh_domain_3.h $
// $Id: Labeled_mesh_domain_3.h e4aaf24 2019-06-18T15:50:15+02:00 Laurent Rineau
// SPDX-License-Identifier: GPL-3.0+
//
//
// Author(s)     : Stéphane Tayeb, Aymeric PELLE
//
//******************************************************************************
// File Description :
// class Labeled_mesh_domain_3. See class description.
//******************************************************************************

#ifndef CGAL_LABELED_MESH_DOMAIN_3_H
#define CGAL_LABELED_MESH_DOMAIN_3_H

#include <CGAL/license/Mesh_3.h>

#include <CGAL/disable_warnings.h>

#include <CGAL/Mesh_3/config.h>

#include <CGAL/Bbox_3.h>
#include <CGAL/point_generators_3.h>
#include <CGAL/boost/parameter.h>
#include <boost/shared_ptr.hpp>
#include <CGAL/tuple.h>
#include <CGAL/Origin.h>

#include <CGAL/result_of.h>
#include <CGAL/function.h>

#include <CGAL/internal/Mesh_3/Handle_IO_for_pair_of_int.h>
#include <CGAL/internal/Mesh_3/indices_management.h>

// support for `CGAL::Image_3`
#include <CGAL/Image_3.h>
#include <CGAL/Mesh_3/Image_to_labeled_function_wrapper.h>

// support for implicit functions
#include <CGAL/Implicit_to_labeling_function_wrapper.h>

#include <CGAL/boost/parameter.h>
#include <boost/parameter/preprocessor.hpp>
#include <boost/parameter/name.hpp>
#ifdef CGAL_MESH_3_VERBOSE
#  include <boost/format.hpp>
#endif
#include <boost/optional.hpp>

namespace CGAL {
namespace Mesh_3 {
namespace internal {

  struct Identity {
    template <typename T>
    const T& operator()(const T& x) { return x; }
  };

  template<typename T>
  struct Greater_than {
    typedef T argument_type;
    typedef bool result_type;
    Greater_than(const T& second) : second(second) {}
    bool operator()(const T& first) const {
      return std::greater<T>()(first, second);
    }
    T second;
  };

  struct Do_not_delete {
    template <typename T> void operator()(T*) const { }
  };

  /// Returns a box enclosing image \c im
  inline Bbox_3 compute_bounding_box(const Image_3& im)
  {
    return Bbox_3(-1+im.tx(),-1+im.ty(),-1+im.tz(),
                  double(im.xdim())*im.vx()+im.tx()+1,
                  double(im.ydim())*im.vy()+im.ty()+1,
                  double(im.zdim())*im.vz()+im.tz()+1);
  }

  template <typename Image_values_to_subdom_indices>
  struct Create_gray_image_values_to_subdomain_indices {
    typedef Image_values_to_subdom_indices type;
    template <typename FT>
    type operator()(Image_values_to_subdom_indices functor, const FT&) const {
      return functor;
    }
  };

  // specialization for `Null_functor`: create the default functor
  template <>
  struct Create_gray_image_values_to_subdomain_indices<Null_functor> {
    typedef Mesh_3::internal::Greater_than<double> type;
    template <typename FT>
    type operator()(Null_functor, const FT& iso_value) const {
      return type(iso_value);
    }
  };

  template <typename Image_values_to_subdom_indices>
  struct Create_labeled_image_values_to_subdomain_indices {
    typedef Image_values_to_subdom_indices type;
    type operator()(Image_values_to_subdom_indices functor) const {
      return functor;
    }
  };

  // specialization for `Null_functor`: create the default functor
  template <>
  struct Create_labeled_image_values_to_subdomain_indices<Null_functor> {
    typedef Identity type;
    type operator()(Null_functor) const {
      return type();
    }
  };

} // end namespace CGAL::Mesh_3::internal
} // end namespace CGAL::Mesh_3

struct Null_subdomain_index {
  template <typename T>
  bool operator()(const T& x) const { return 0 == x; }
};

template <typename Subdomain_index>
struct Construct_pair_from_subdomain_indices {
  typedef std::pair<Subdomain_index, Subdomain_index> result_type;

  result_type operator()(Subdomain_index a, Subdomain_index b) const {
    return result_type(a, b);
  }
}; // end class template Construct_pair_from_subdomain_indices

template <typename Geom_traits,
          typename Subdomain_index,
          typename Surface_patch_index_>
class Labeled_mesh_domain_3_impl_details
{
protected:
  typedef Surface_patch_index_ Surface_patch_index;
  typedef typename Geom_traits::Point_3 Point_3;
  typedef typename Geom_traits::Sphere_3 Sphere_3;
  typedef typename Geom_traits::Iso_cuboid_3 Iso_cuboid_3;
  typedef typename Geom_traits::FT FT;
  typedef boost::shared_ptr<CGAL::Random> CGAL_Random_share_ptr_t;
  /// Returns squared error bound from \c bbox and \c error
  FT squared_error_bound(const Iso_cuboid_3& bbox, const FT& error) const
  {
    typename Geom_traits::Compute_squared_distance_3 squared_distance =
                             Geom_traits().compute_squared_distance_3_object();
    return squared_distance((bbox.min)(), (bbox.max)())*error*error/4;
  }

  static Iso_cuboid_3 iso_cuboid(const Bbox_3& bbox)
  {
    const Point_3 p_min(bbox.xmin(), bbox.ymin(), bbox.zmin());
    const Point_3 p_max(bbox.xmax(), bbox.ymax(), bbox.zmax());

    return Iso_cuboid_3(p_min,p_max);
  }

  static Iso_cuboid_3 iso_cuboid(const typename Geom_traits::Sphere_3& sphere)
  {
    return iso_cuboid(sphere.bbox());
  }

  static Iso_cuboid_3 iso_cuboid(const typename Geom_traits::Iso_cuboid_3& c)
  {
    return c;
  }

  static Construct_pair_from_subdomain_indices<Subdomain_index>
  construct_pair_functor() {
    return Construct_pair_from_subdomain_indices<Subdomain_index>();
  }

  template <class ArgumentPack>
  Labeled_mesh_domain_3_impl_details(ArgumentPack const& args)
    : function_(args[parameters::function])
    , bbox_(iso_cuboid(args[parameters::bounding_object]))
    , cstr_s_p_index(args[parameters::construct_surface_patch_index |
                          construct_pair_functor()])
    , null(args[parameters::null_subdomain_index | Null_subdomain_index()])
    , p_rng_(args[parameters::p_rng|0] == 0 ?
             CGAL_Random_share_ptr_t(new CGAL::Random(0)) :
             CGAL_Random_share_ptr_t(args[parameters::p_rng|(CGAL::Random*)(0)],
                                     Mesh_3::internal::Do_not_delete()))
    , squared_error_bound_
      ( squared_error_bound(bbox_,
                            args[parameters::relative_error_bound|FT(1e-3)]))
  {
  }

  template <typename Function,
            typename Bounding_object,
            typename Null,
            typename Construct_surface_patch_index>
  Labeled_mesh_domain_3_impl_details(const Function& f,
                                     const Bounding_object& bounding,
                                     const FT& error_bound,
                                     Construct_surface_patch_index cstr_s_p_i,
                                     Null null,
                                     CGAL::Random* p_rng)
    : function_(f)
    , bbox_(iso_cuboid(bounding))
    , cstr_s_p_index(cstr_s_p_i)
    , null(null)
    , p_rng_(p_rng == 0 ?
             CGAL_Random_share_ptr_t(new CGAL::Random(0)) :
             CGAL_Random_share_ptr_t(p_rng, Mesh_3::internal::Do_not_delete()))
    , squared_error_bound_(squared_error_bound(bbox_,error_bound))
  {}

  /// The function which answers subdomain queries
  typedef CGAL::cpp11::function<Subdomain_index(const Point_3&)> Function;
  Function function_;
  /// The bounding box
  const Iso_cuboid_3 bbox_;

  typedef CGAL::cpp11::function<
    Surface_patch_index(Subdomain_index,
                        Subdomain_index)> Construct_surface_patch_index;
  Construct_surface_patch_index cstr_s_p_index;
  /// The functor that decides which sub-domain indices correspond to the
  /// outside of the domain.
  typedef CGAL::cpp11::function<bool(Subdomain_index)> Null;
  Null null;
  /// The random number generator used by Construct_initial_points
  CGAL_Random_share_ptr_t p_rng_;
  /// Error bound relative to sphere radius
  FT squared_error_bound_;
}; // Labeled_mesh_domain_3_impl_details

/**
 * \class Labeled_mesh_domain_3
 *
 * Function f must take his values into N.
 * Let p be a Point.
 *  - f(p)=0 means that p is outside domain.
 *  - f(p)=a, a!=0 means that p is inside subdomain a.
 *
 *  Any boundary facet is labelled <a,b>, with a<b, where a and b are the
 *  tags of it's incident subdomain.
 *  Thus, a boundary facet of the domain is labelled <0,b>, where b!=0.
 */
template<class BGT,
         class Subdomain_index_ = int,
         class Surface_patch_index_ = std::pair<Subdomain_index_,
                                                Subdomain_index_> >
class Labeled_mesh_domain_3 :
    protected Labeled_mesh_domain_3_impl_details<BGT,
                                                 Subdomain_index_,
                                                 Surface_patch_index_>
{
public:
  //-------------------------------------------------------
  // Index Types
  //-------------------------------------------------------
  /// Type of indexes for cells of the input complex
  typedef Subdomain_index_                  Subdomain_index;
  typedef boost::optional<Subdomain_index>  Subdomain;

  /// Type of indexes for cells of the input complex
  typedef Surface_patch_index_                  Surface_patch_index;
  typedef boost::optional<Surface_patch_index>  Surface_patch;

  /// Type of indexes to characterize the lowest dimensional face of the input
  /// complex on which a vertex lie
  typedef typename CGAL::Mesh_3::internal::
    Index_generator<Subdomain_index, Surface_patch_index>::Index Index;

private:
  typedef Labeled_mesh_domain_3_impl_details<BGT,
                                             Subdomain_index,
                                             Surface_patch_index
                                             > Impl_details;
  typedef typename Impl_details::Null     Null;
  typedef typename Impl_details::Construct_surface_patch_index
                                          Construct_surface_patch_index;
  typedef typename Impl_details::Function Function;

public:
  /// Geometric object types
  typedef typename BGT::Point_3    Point_3;
  typedef typename BGT::Segment_3  Segment_3;
  typedef typename BGT::Ray_3      Ray_3;
  typedef typename BGT::Line_3     Line_3;
  typedef typename BGT::Vector_3   Vector_3;
  typedef typename BGT::Sphere_3   Sphere_3;
  typedef CGAL::Bbox_3             Bbox_3;

protected:
  typedef typename BGT::Iso_cuboid_3 Iso_cuboid_3;

public:
  // Kernel_traits compatibility
  typedef BGT R;
  // access Function type from inherited class
  typedef Function Fct;

  typedef CGAL::cpp11::tuple<Point_3,Index,int> Intersection;


  typedef typename BGT::FT FT;
  typedef BGT Geom_traits;


  BOOST_PARAMETER_CONSTRUCTOR(Labeled_mesh_domain_3,
                              (Impl_details),
                              parameters::tag,
                              (required
                               (function_,(Function))
                               (bounding_object_,*)
                               )
                              (optional
                               (relative_error_bound_, (const FT&))
                               (p_rng_, (CGAL::Random*))
                               (null_subdomain_index_,(Null))
                               (construct_surface_patch_index_,
                                    (Construct_surface_patch_index))
                               )
                              )
  using Impl_details::construct_pair_functor;
  /**
   * Backward-compatibility constructors, with `null_subdomain_index` as
   * fourth parameter.
   * @{
   */
  Labeled_mesh_domain_3(const Function& f,
                        const Sphere_3& bounding_sphere,
                        const FT& error_bound = FT(1e-3),
                        Null null = Null_subdomain_index(),
                        CGAL::Random* p_rng = NULL)
    : Impl_details(f, bounding_sphere,
                   error_bound,
                   construct_pair_functor(),
                   null, p_rng) {}

  Labeled_mesh_domain_3(const Function& f,
                        const Bbox_3& bbox,
                        const FT& error_bound = FT(1e-3),
                        Null null = Null_subdomain_index(),
                        CGAL::Random* p_rng = NULL)
    : Impl_details(f, bbox,
                   error_bound,
                   construct_pair_functor(),
                   null, p_rng) {}

  Labeled_mesh_domain_3(const Function& f,
                        const Iso_cuboid_3& bbox,
                        const FT& error_bound = FT(1e-3),
                        Null null = Null_subdomain_index(),
                        CGAL::Random* p_rng = NULL)
    : Impl_details(f, bbox, error_bound,
                   construct_pair_functor(),
                   null, p_rng)
  {}
  /**
   * @}
   */

  /// Named constructors
  /// @{
#if defined(BOOST_MSVC)
#  pragma warning(push)
#  pragma warning(disable: 4003)
#endif  
  BOOST_PARAMETER_MEMBER_FUNCTION(
                                  (Labeled_mesh_domain_3),
                                  static create_gray_image_mesh_domain,
                                  parameters::tag,
                                  (required
                                   (image_, (const CGAL::Image_3&))
                                   )
                                  (optional
                                   (iso_value_, *, 0)
                                   (value_outside_, *, 0)
                                   (relative_error_bound_, (const FT&),
                                    FT(1e-3))
                                   (p_rng_, (CGAL::Random*), (CGAL::Random*)(0))
                                   (image_values_to_subdomain_indices_, *,
                                    Null_functor())
                                   (null_subdomain_index_,*,Null_functor())
                                   (construct_surface_patch_index_, *,
                                    Null_functor())
                                   )
                                  )
  {
    namespace p = CGAL::parameters;
    return Labeled_mesh_domain_3
      (create_gray_image_wrapper
       (image_,
        iso_value_,
        image_values_to_subdomain_indices_,
        value_outside_),
       Mesh_3::internal::compute_bounding_box(image_),
       p::relative_error_bound = relative_error_bound_,
       p::p_rng = p_rng_,
       p::null_subdomain_index =
         create_null_subdomain_index(null_subdomain_index_),
       p::construct_surface_patch_index =
         create_construct_surface_patch_index(construct_surface_patch_index_));
  }

  BOOST_PARAMETER_MEMBER_FUNCTION(
                                  (Labeled_mesh_domain_3),
                                  static create_labeled_image_mesh_domain,
                                  parameters::tag,
                                  (required
                                   (image_, (const CGAL::Image_3&))
                                   )
                                  (optional
                                   (relative_error_bound_, (const FT&),
                                    FT(1e-3))
                                   (value_outside_, *, 0)
                                   (p_rng_, (CGAL::Random*), (CGAL::Random*)(0))
                                   (image_values_to_subdomain_indices_, *,
                                    Null_functor())
                                   (null_subdomain_index_,*,Null_functor())
                                   (construct_surface_patch_index_, *,
                                    Null_functor())
                                   )
                                  )
  {
    namespace p = CGAL::parameters;
    return Labeled_mesh_domain_3
      (create_labeled_image_wrapper
       (image_,
        image_values_to_subdomain_indices_,
        value_outside_),
       Mesh_3::internal::compute_bounding_box(image_),
       p::relative_error_bound = relative_error_bound_,
       p::p_rng = p_rng_,
       p::null_subdomain_index =
         create_null_subdomain_index(null_subdomain_index_),
       p::construct_surface_patch_index =
         create_construct_surface_patch_index(construct_surface_patch_index_));
  }

  BOOST_PARAMETER_MEMBER_FUNCTION(
                                  (Labeled_mesh_domain_3),
                                  static create_implicit_mesh_domain,
                                  parameters::tag,
                                  (required
                                   (function_, *)
                                   (bounding_object_,*)
                                   )
                                  (optional
                                   (relative_error_bound_, (const FT&),
                                    FT(1e-3))
                                   (p_rng_, (CGAL::Random*), (CGAL::Random*)(0))
                                   (null_subdomain_index_,*,Null_functor())
                                   (construct_surface_patch_index_, *,
                                    Null_functor())
                                   )
                                  )
  {
    namespace p = CGAL::parameters;
    return Labeled_mesh_domain_3
      (make_implicit_to_labeling_function_wrapper<BGT>(function_),
       bounding_object_,
       p::relative_error_bound = relative_error_bound_,
       p::p_rng = p_rng_,
       p::null_subdomain_index =
         create_null_subdomain_index(null_subdomain_index_),
       p::construct_surface_patch_index =
         create_construct_surface_patch_index(construct_surface_patch_index_));
  }

#if defined(BOOST_MSVC)
#  pragma warning(pop)
#endif
  /// @}

  /**
   * Constructs  a set of \ccc{n} points on the surface, and output them to
   *  the output iterator \ccc{pts} whose value type is required to be
   *  \ccc{std::pair<Points_3, Index>}.
   */
  struct Construct_initial_points
  {
    Construct_initial_points(const Labeled_mesh_domain_3& domain)
      : r_domain_(domain) {}

    template<class OutputIterator>
    OutputIterator operator()(OutputIterator pts, const int n = 12) const;

  private:
    const Labeled_mesh_domain_3& r_domain_;
  };

  /// Returns Construct_initial_points object
  Construct_initial_points construct_initial_points_object() const
  {
    return Construct_initial_points(*this);
  }

  /**
   * Returns a bounding box of the domain
   */
  Bbox_3 bbox() const {
    return this->bbox_.bbox();
  }

  /**
   * Returns true if point~\ccc{p} is in the domain. If \ccc{p} is in the
   *  domain, the parameter index is set to the index of the subdomain
   *  including $p$. It is set to the default value otherwise.
   */
  struct Is_in_domain
  {
    Is_in_domain(const Labeled_mesh_domain_3& domain) : r_domain_(domain) {}

    Subdomain operator()(const Point_3& p) const
    {
      // null(f(p)) means p is outside the domain
      Subdomain_index index = (r_domain_.function_)(p);
      if ( r_domain_.null(index) )
        return Subdomain();
      else
        return Subdomain(index);
    }
  private:
    const Labeled_mesh_domain_3& r_domain_;
  };

  /// Returns Is_in_domain object
  Is_in_domain is_in_domain_object() const { return Is_in_domain(*this); }

  /**
   * Returns true is the element \ccc{type} intersect properly any of the
   * surface patches describing the either the domain boundary or some
   * subdomain boundary.
   * \ccc{Type} is either \ccc{Segment_3}, \ccc{Ray_3} or \ccc{Line_3}.
   * Parameter index is set to the index of the intersected surface patch
   * if \ccc{true} is returned and to the default \ccc{Surface_patch_index}
   * value otherwise.
   */
  struct Do_intersect_surface
  {
    Do_intersect_surface(const Labeled_mesh_domain_3& domain)
      : r_domain_(domain) {}

    Surface_patch operator()(const Segment_3& s) const
    {
      return this->operator()(s.source(), s.target());
    }

    Surface_patch operator()(const Ray_3& r) const
    {
      return clip_to_segment(r);
    }

    Surface_patch operator()(const Line_3& l) const
    {
      return clip_to_segment(l);
    }

  private:
    /// Returns true if points \c a & \c b do not belong to the same subdomain
    /// \c index is set to the surface index of subdomains f(a), f(b)
    Surface_patch operator()(const Point_3& a, const Point_3& b) const
    {
      // If f(a) != f(b), then [a,b] intersects some surface. Here we consider
      // [a,b] intersects surface_patch labelled <f(a),f(b)> (or <f(b),f(a)>).
      // It may be false, further rafinement will improve precision
      const Subdomain_index value_a = r_domain_.function_(a);
      const Subdomain_index value_b = r_domain_.function_(b);

      if ( value_a != value_b ) {
        if( r_domain_.null(value_a) && r_domain_.null(value_b) )
          return Surface_patch();
        else
          return Surface_patch(r_domain_.make_surface_index(value_a, value_b));
      }
      else
        return Surface_patch();
    }

    /**
     * Clips \c query to a segment \c s, and call operator()(s)
     */
    template<typename Query>
    Surface_patch clip_to_segment(const Query& query) const
    {
      typename cpp11::result_of<typename BGT::Intersect_3(Query, Iso_cuboid_3)>::type
        clipped = CGAL::intersection(query, r_domain_.bbox_);

      if(clipped)
#if CGAL_INTERSECTION_VERSION > 1
        if(const Segment_3* s = boost::get<Segment_3>(&*clipped))
          return this->operator()(*s);
#else
        if(const Segment_3* s = object_cast<Segment_3>(&clipped))
          return this->operator()(*s);
#endif

      return Surface_patch();
    }

  private:
    const Labeled_mesh_domain_3& r_domain_;
  };

  /// Returns Do_intersect_surface object
  Do_intersect_surface do_intersect_surface_object() const
  {
    return Do_intersect_surface(*this);
  }

  /**
   * Returns a point in the intersection of the primitive \ccc{type}
   * with some boundary surface.
   * \ccc{Type1} is either \ccc{Segment_3}, \ccc{Ray_3} or \ccc{Line_3}.
   * The integer \ccc{dimension} is set to the dimension of the lowest
   * dimensional face in the input complex containing the returned point, and
   * \ccc{index} is set to the index to be stored at a mesh vertex lying
   * on this face.
   */
  struct Construct_intersection
  {
    Construct_intersection(const Labeled_mesh_domain_3& domain)
      : r_domain_(domain) {}

    Intersection operator()(const Segment_3& s) const
    {
#ifndef CGAL_MESH_3_NO_LONGER_CALLS_DO_INTERSECT_3
      CGAL_precondition(r_domain_.do_intersect_surface_object()(s));
#endif // NOT CGAL_MESH_3_NO_LONGER_CALLS_DO_INTERSECT_3
      return this->operator()(s.source(),s.target());
    }

    Intersection operator()(const Ray_3& r) const
    {
      return clip_to_segment(r);
    }

    Intersection operator()(const Line_3& l) const
    {
      return clip_to_segment(l);
    }

  private:
    /**
     * Returns a point in the intersection of [a,b] with the surface
     * \c a must be the source point, and \c b the out point. It's important
     * because it drives bisection cuts.
     * Indeed, the returned point is the first intersection from \c [a,b]
     * with a subdomain surface.
     */
    Intersection operator()(const Point_3& a, const Point_3& b) const
    {
      // Functors
      typename BGT::Compute_squared_distance_3 squared_distance =
                                      BGT().compute_squared_distance_3_object();
      typename BGT::Construct_midpoint_3 midpoint =
                                      BGT().construct_midpoint_3_object();

      // Non const points
      Point_3 p1 = a;
      Point_3 p2 = b;
      Point_3 mid = midpoint(p1, p2);

      // Cannot be const: those values are modified below.
      Subdomain_index value_at_p1 = r_domain_.function_(p1);
      Subdomain_index value_at_p2 = r_domain_.function_(p2);
      Subdomain_index value_at_mid = r_domain_.function_(mid);

      // If both extremities are in the same subdomain,
      // there is no intersection.
      // This should not happen...
      if( value_at_p1 == value_at_p2 )
      {
        return Intersection();
      }
      if( r_domain_.null(value_at_p1) && r_domain_.null(value_at_p2) ) {
        return Intersection();
      }

      // Else lets find a point (by bisection)
      // Bisection ends when the point is near than error bound from surface
      while(true)
      {
        // If the two points are enough close, then we return midpoint
        if ( squared_distance(p1, p2) < r_domain_.squared_error_bound_ )
        {
          CGAL_assertion(value_at_p1 != value_at_p2 &&
             ! ( r_domain_.null(value_at_p1) && r_domain_.null(value_at_p2) ) );
          const Surface_patch_index sp_index =
            r_domain_.make_surface_index(value_at_p1, value_at_p2);
          const Index index = r_domain_.index_from_surface_patch_index(sp_index);
          return Intersection(mid, index, 2);
        }

        // Else we must go on
        // Here we consider that p1(a) is the source point. Thus, we keep p1 and
        // change p2 if f(p1)!=f(p2).
        // That allows us to find the first intersection from a of [a,b] with
        // a surface.
        if ( value_at_p1 != value_at_mid &&
             ! ( r_domain_.null(value_at_p1) && r_domain_.null(value_at_mid) ) )
        {
          p2 = mid;
          value_at_p2 = value_at_mid;
        }
        else
        {
          p1 = mid;
          value_at_p1 = value_at_mid;
        }

        mid = midpoint(p1, p2);
        value_at_mid = r_domain_.function_(mid);
      }
    }

    /// Clips \c query to a segment \c s, and call operator()(s)
    template<typename Query>
    Intersection clip_to_segment(const Query& query) const
    {
      typename cpp11::result_of<typename BGT::Intersect_3(Query, Iso_cuboid_3)>::type
        clipped = CGAL::intersection(query, r_domain_.bbox_);

      if(clipped)
#if CGAL_INTERSECTION_VERSION > 1
        if(const Segment_3* s = boost::get<Segment_3>(&*clipped))
          return this->operator()(*s);
#else
        if(const Segment_3* s = object_cast<Segment_3>(&clipped))
          return this->operator()(*s);
#endif

      return Intersection();
    }

  private:
    const Labeled_mesh_domain_3& r_domain_;
  };

  /// Returns Construct_intersection object
  Construct_intersection construct_intersection_object() const
  {
    return Construct_intersection(*this);
  }

  /**
   * Returns the index to be stored in a vertex lying on the surface identified
   * by \c index.
   */
  Index index_from_surface_patch_index(const Surface_patch_index& index) const
  { return Index(index); }

  /**
   * Returns the index to be stored in a vertex lying in the subdomain
   * identified by \c index.
   */
  Index index_from_subdomain_index(const Subdomain_index& index) const
  { return Index(index); }

  /**
   * Returns the \c Surface_patch_index of the surface patch
   * where lies a vertex with dimension 2 and index \c index.
   */
  Surface_patch_index surface_patch_index(const Index& index) const
  { return boost::get<Surface_patch_index>(index); }

  /**
   * Returns the index of the subdomain containing a vertex
   *  with dimension 3 and index \c index.
   */
  Subdomain_index subdomain_index(const Index& index) const
  { return boost::get<Subdomain_index>(index); }

  // -----------------------------------
  // Backward Compatibility
  // -----------------------------------
#ifndef CGAL_MESH_3_NO_DEPRECATED_SURFACE_INDEX
  typedef Surface_patch_index   Surface_index;

  Index index_from_surface_index(const Surface_index& index) const
  { return index_from_surface_patch_index(index); }

  Surface_index surface_index(const Index& index) const
  { return surface_patch_index(index); }
#endif // CGAL_MESH_3_NO_DEPRECATED_SURFACE_INDEX
  // -----------------------------------
  // End backward Compatibility
  // -----------------------------------

protected:
  /// Returns Surface_patch_index from \c i and \c j
  Surface_patch_index make_surface_index(const Subdomain_index i,
                                         const Subdomain_index j) const
  {
    if(i < j)
      return this->cstr_s_p_index(i, j);
    else
      return this->cstr_s_p_index(j, i);
  }

  /// Returns the bounding sphere of an Iso_cuboid_3
  Sphere_3 bounding_sphere(const Iso_cuboid_3& bbox) const
  {
    typename BGT::Construct_sphere_3 sphere = BGT().construct_sphere_3_object();
    return sphere((bbox.min)(), (bbox.max)());
  }

  template <typename Image_word_type,
            typename FT, typename FT2, typename Functor>
  static
  Function
  create_gray_image_wrapper_with_known_word_type
  (const CGAL::Image_3& image,
   const FT& iso_value,
   const Functor& image_values_to_subdomain_indices,
   const FT2& value_outside)
  {
    using Mesh_3::internal::Create_gray_image_values_to_subdomain_indices;
    typedef Create_gray_image_values_to_subdomain_indices<Functor> C_i_v_t_s_i;
    typedef typename C_i_v_t_s_i::type Image_values_to_subdomain_indices;
    Image_values_to_subdomain_indices transform_fct =
      C_i_v_t_s_i()(image_values_to_subdomain_indices, iso_value);

    typedef Mesh_3::Image_to_labeled_function_wrapper<Image_word_type,
                                                      double,
                                                      Subdomain_index,
                                                      false>           Wrapper;
    return Wrapper(image,
                   transform_fct,
                   transform_fct(value_outside));
  }

  template <typename FT, typename FT2, typename Functor>
  static
  Function
  create_gray_image_wrapper(const CGAL::Image_3& image,
                            const FT& iso_value,
                            const Functor& image_values_to_subdomain_indices,
                            const FT2& value_outside)
  {
    CGAL_IMAGE_IO_CASE(image.image(),
       return create_gray_image_wrapper_with_known_word_type<Word>
                       (image,
                        iso_value,
                        image_values_to_subdomain_indices,
                        value_outside);
                       );
    CGAL_error_msg("This place should never be reached, because it would mean "
                   "the image word type is a type that is not handled by "
                   "CGAL_ImageIO.");
    return Function();
  }

  template <typename Image_word_type,
            typename FT, typename Functor>
  static
  Function
  create_labeled_image_wrapper_with_known_word_type
  (const CGAL::Image_3& image,
   const Functor& image_values_to_subdomain_indices,
   const FT& value_outside)
  {
    using Mesh_3::internal::Create_labeled_image_values_to_subdomain_indices;
    typedef Create_labeled_image_values_to_subdomain_indices<Functor> C_i_v_t_s_i;
    typedef typename C_i_v_t_s_i::type Image_values_to_subdomain_indices;
    Image_values_to_subdomain_indices transform_fct =
      C_i_v_t_s_i()(image_values_to_subdomain_indices);

    typedef Mesh_3::Image_to_labeled_function_wrapper<Image_word_type,
                                                      int,
                                                      Subdomain_index> Wrapper;
    return Wrapper(image,
                   transform_fct,
                   transform_fct(value_outside));
  }

  template <typename FT, typename Functor>
  static
  Function
  create_labeled_image_wrapper(const CGAL::Image_3& image,
                               const Functor& image_values_to_subdomain_indices,
                               const FT& value_outside)
  {
    CGAL_IMAGE_IO_CASE(image.image(),
       return create_labeled_image_wrapper_with_known_word_type<Word>
                       (image,
                        image_values_to_subdomain_indices,
                        value_outside);
                       );
    CGAL_error_msg("This place should never be reached, because it would mean "
                   "the image word type is a type that is not handled by "
                   "CGAL_ImageIO.");
    return Function();
  }

  static
  Construct_surface_patch_index
  create_construct_surface_patch_index(const Null_functor&) {
    return Impl_details::construct_pair_functor();
  }

  template <typename Functor>
  static
  Construct_surface_patch_index
  create_construct_surface_patch_index(const Functor& functor) {
    return functor;
  }

  static Null create_null_subdomain_index(const Null_functor&) {
    return Null_subdomain_index();
  }

  template <typename Functor>
  static Null create_null_subdomain_index(const Functor& functor) {
    return functor;
  }

public:
  /// Returns bounding box
  const Iso_cuboid_3& bounding_box() const { return this->bbox_; }

};  // end class Labeled_mesh_domain_3

//-------------------------------------------------------
// Method implementation
//-------------------------------------------------------
template<class BGT, class Subdomain_index, class Surface_patch_index>
template<class OutputIterator>
OutputIterator
Labeled_mesh_domain_3<BGT, Subdomain_index, Surface_patch_index>::
Construct_initial_points::operator()(OutputIterator pts,
                                     const int nb_points) const
{
  // Create point_iterator on and in bounding_sphere
  typedef Random_points_on_sphere_3<Point_3> Random_points_on_sphere_3;
  typedef Random_points_in_sphere_3<Point_3> Random_points_in_sphere_3;

  const FT squared_radius = BGT().compute_squared_radius_3_object()(
      r_domain_.bounding_sphere(r_domain_.bbox_));

  const double radius = std::sqrt(CGAL::to_double(squared_radius));

  CGAL::Random& rng = *(r_domain_.p_rng_);
  Random_points_on_sphere_3 random_point_on_sphere(radius, rng);
  Random_points_in_sphere_3 random_point_in_sphere(radius, rng);

  // Get some functors
  typename BGT::Construct_segment_3 segment_3 =
                              BGT().construct_segment_3_object();
  typename BGT::Construct_vector_3 vector_3 =
                              BGT().construct_vector_3_object();
  typename BGT::Construct_translated_point_3 translate =
                              BGT().construct_translated_point_3_object();
  typename BGT::Construct_center_3 center = BGT().construct_center_3_object();

  // Get translation from origin to sphere center
  Point_3 center_pt = center(r_domain_.bounding_sphere(r_domain_.bbox_));
  const Vector_3 sphere_translation = vector_3(CGAL::ORIGIN, center_pt);

  // Create nb_point points
  int n = nb_points;
#ifdef CGAL_MESH_3_VERBOSE
  std::cerr << "construct initial points (nb_points: " << nb_points << ")\n";
#endif
  while ( 0 != n )
  {
    // Get a random segment
    const Point_3 random_point = translate(*random_point_on_sphere,
                                           sphere_translation);
    const Segment_3 random_seg = segment_3(center_pt, random_point);

    // Add the intersection to the output if it exists
    Surface_patch surface = r_domain_.do_intersect_surface_object()(random_seg);
    if ( surface )
    {
      const Point_3 intersect_pt = CGAL::cpp11::get<0>(
          r_domain_.construct_intersection_object()(random_seg));
      *pts++ = std::make_pair(intersect_pt,
                              r_domain_.index_from_surface_patch_index(*surface));
      --n;

#ifdef CGAL_MESH_3_VERBOSE
      std::cerr << boost::format("\r             \r"
                                 "%1%/%2% initial point(s) found...")
                   % (nb_points - n)
                   % nb_points;
#endif
    }
    else
    {
      // Get a new random point into sphere as center of object
      // It may be necessary if the center of the domain is empty, e.g. torus
      // In general case, it is good for input point dispersion
      ++random_point_in_sphere;
      center_pt = translate(*random_point_in_sphere, sphere_translation);
    }
    ++random_point_on_sphere;
  }

#ifdef CGAL_MESH_3_VERBOSE
  std::cerr << "\n";
#endif
  return pts;
}


}  // end namespace CGAL

#include <CGAL/enable_warnings.h>

#endif // CGAL_LABELED_MESH_DOMAIN_3_H
