diff --git a/Mesh_3/include/CGAL/Compact_mesh_cell_base_3.h b/Mesh_3/include/CGAL/Compact_mesh_cell_base_3.h
index 5a386ef5d99..47b47922f1f 100644
--- a/Mesh_3/include/CGAL/Compact_mesh_cell_base_3.h
+++ b/Mesh_3/include/CGAL/Compact_mesh_cell_base_3.h
@@ -242,9 +242,9 @@ public:
   typedef typename GT::Weighted_point_3 Weighted_point_3;
 
 
-  typedef Point*           Point_container;
-  typedef Point*           Point_iterator;
-  typedef const Point*     Point_const_iterator;
+  typedef Weighted_point_3*           Point_container;
+  typedef Weighted_point_3*           Point_iterator;
+  typedef const Weighted_point_3*     Point_const_iterator;
 
 public:
   void invalidate_circumcenter() const
diff --git a/Mesh_3/include/CGAL/Mesh_3/C3T3_helpers.h b/Mesh_3/include/CGAL/Mesh_3/C3T3_helpers.h
index e01cefea2d5..89d0c8b5dcc 100644
--- a/Mesh_3/include/CGAL/Mesh_3/C3T3_helpers.h
+++ b/Mesh_3/include/CGAL/Mesh_3/C3T3_helpers.h
@@ -387,6 +387,7 @@ class C3T3_helpers_base
 protected:
   typedef typename Tr::Geom_traits          Gt;
   typedef typename Tr::Bare_point           Bare_point;
+  typedef typename Tr::Weighted_point       Weighted_point;
   typedef typename Gt::FT                   FT;
   typedef typename Tr::Vertex_handle        Vertex_handle;
   typedef typename Tr::Cell_handle          Cell_handle;
@@ -402,7 +403,7 @@ protected:
 
 public:
   // Dummy locks/unlocks
-  bool try_lock_point(const Bare_point &, int = 0) const
+  bool try_lock_point(const Weighted_point &, int = 0) const
   {
     return true;
   }
@@ -412,7 +413,7 @@ public:
     return true;
   }
 
-  bool try_lock_point_no_spin(const Bare_point &, int = 0) const
+  bool try_lock_point_no_spin(const Weighted_point &, int = 0) const
   {
     return true;
   }
@@ -433,7 +434,7 @@ public:
   }
 
 
-  bool is_point_locked_by_this_thread(const Bare_point &) const
+  bool is_point_locked_by_this_thread(const Weighted_point &) const
   { return false; }
 
   bool is_cell_locked_by_this_thread(const Cell_handle &) const
@@ -457,7 +458,8 @@ class C3T3_helpers_base<Tr, Parallel_tag>
 {
 protected:
   typedef typename Tr::Geom_traits          Gt;
-  typedef typename Gt::Bare_point              Bare_point;
+  typedef typename Tr::Bare_point           Bare_point;
+  typedef typename Tr::Weighted_point       Weighted_point;
   typedef typename Tr::Vertex_handle        Vertex_handle;
   typedef typename Tr::Cell_handle          Cell_handle;
   typedef typename Tr::Facet                Facet;
@@ -475,7 +477,7 @@ public:
     return m_lock_ds;
   }*/
 
-  bool try_lock_point(const Bare_point &p, int lock_radius = 0) const
+  bool try_lock_point(const Weighted_point &p, int lock_radius = 0) const
   {
     if (m_lock_ds)
     {
@@ -493,7 +495,7 @@ public:
     return true;
   }
 
-  bool try_lock_point_no_spin(const Bare_point &p, int lock_radius = 0) const
+  bool try_lock_point_no_spin(const Weighted_point &p, int lock_radius = 0) const
   {
     if (m_lock_ds)
     {
@@ -537,7 +539,7 @@ public:
     return success;
   }
 
-  bool is_point_locked_by_this_thread(const Bare_point &p) const
+  bool is_point_locked_by_this_thread(const Weighted_point &p) const
   {
     bool locked = true;
     if (m_lock_ds)
@@ -634,11 +636,13 @@ class C3T3_helpers
 
   typedef typename Gt::Vector_3         Vector_3;
   typedef typename Gt::Bare_point          Bare_point;
+  typedef typename Gt::Weighted_point_3          Weighted_point;
   typedef typename Gt::Plane_3          Plane_3;
   typedef typename Gt::FT               FT;
   typedef typename Gt::Tetrahedron_3    Tetrahedron;
 
   typedef typename Gt::Construct_point_3    Construct_point_3;
+  typedef typename Gt::Construct_weighted_point_3    Construct_weighted_point_3;
 
   typedef typename Tr::Vertex_handle    Vertex_handle;
   typedef typename Tr::Cell_handle      Cell_handle;
@@ -715,6 +719,7 @@ public:
     , tr_(c3t3.triangulation())
     , domain_(domain)
     , wp2p_(tr_.geom_traits().construct_point_3_object())
+    , p2wp_(tr_.geom_traits().construct_weighted_point_3_object())
   { }
 
   /**
@@ -733,7 +738,7 @@ public:
    */
   template <typename SliverCriterion, typename OutputIterator>
   std::pair<bool,Vertex_handle>
-  update_mesh(const Bare_point& new_position,
+  update_mesh(const Weighted_point& new_position,
               const Vertex_handle& old_vertex,
               const SliverCriterion& criterion,
               OutputIterator modified_vertices,
@@ -747,7 +752,7 @@ public:
    */
   template <typename SliverCriterion, typename OutputIterator>
   std::pair<bool,Vertex_handle>
-  update_mesh_topo_change(const Bare_point& new_position,
+  update_mesh_topo_change(const Weighted_point& new_position,
                           const Vertex_handle& old_vertex,
                           const SliverCriterion& criterion,
                           OutputIterator modified_vertices,
@@ -760,7 +765,7 @@ public:
    * Insert into modified vertices the vertices which are impacted by to move.
    */
   template <typename OutputIterator>
-  Vertex_handle update_mesh(const Bare_point& new_position,
+  Vertex_handle update_mesh(const Weighted_point& new_position,
                             const Vertex_handle& old_vertex,
                             OutputIterator modified_vertices,
                             bool fill_modified_vertices = true);
@@ -769,7 +774,7 @@ public:
    * Updates mesh moving vertex \c old_vertex to \c new_position. Returns the
    * new vertex of the triangulation.
    */
-  Vertex_handle update_mesh(const Bare_point& new_position,
+  Vertex_handle update_mesh(const Weighted_point& new_position,
                             const Vertex_handle& old_vertex)
   {
     return update_mesh(new_position, old_vertex, Emptyset_iterator(), false);
@@ -821,11 +826,11 @@ public:
    * The second one (with the could_lock_zone param) is for the parallel version
    */
   Vertex_handle move_point(const Vertex_handle& old_vertex,
-                           const Bare_point& new_position,
+                           const Weighted_point& new_position,
                            Outdated_cell_set& outdated_cells_set,
                            Moving_vertices_set& moving_vertices) const;
   Vertex_handle move_point(const Vertex_handle& old_vertex,
-                           const Bare_point& new_position,
+                           const Weighted_point& new_position,
                            Outdated_cell_set& outdated_cells_set,
                            Moving_vertices_set& moving_vertices,
                            bool *could_lock_zone) const;
@@ -1631,7 +1636,7 @@ private:
    */
   template <typename SliverCriterion, typename OutputIterator>
   std::pair<bool,Vertex_handle>
-  update_mesh_no_topo_change(const Bare_point& new_position,
+  update_mesh_no_topo_change(const Weighted_point& new_position,
                              const Vertex_handle& old_vertex,
                              const SliverCriterion& criterion,
                              OutputIterator modified_vertices,
@@ -1644,13 +1649,13 @@ private:
   template < typename OutdatedCellsOutputIterator,
              typename DeletedCellsOutputIterator >
   Vertex_handle move_point(const Vertex_handle& old_vertex,
-                           const Bare_point& new_position,
+                           const Weighted_point& new_position,
                            OutdatedCellsOutputIterator outdated_cells,
                            DeletedCellsOutputIterator deleted_cells) const;
 
   Vertex_handle
   move_point_topo_change(const Vertex_handle& old_vertex,
-                         const Bare_point& new_position,
+                         const Weighted_point& new_position,
                          Outdated_cell_set& outdated_cells_set,
                          bool *could_lock_zone = NULL) const;
 
@@ -1658,22 +1663,22 @@ private:
              typename DeletedCellsOutputIterator >
   Vertex_handle
   move_point_topo_change(const Vertex_handle& old_vertex,
-                         const Bare_point& new_position,
+                         const Weighted_point& new_position,
                          OutdatedCellsOutputIterator outdated_cells,
                          DeletedCellsOutputIterator deleted_cells) const;
 
   Vertex_handle move_point_topo_change(const Vertex_handle& old_vertex,
-                                       const Bare_point& new_position) const;
+                                       const Weighted_point& new_position) const;
 
   template < typename OutdatedCellsOutputIterator >
   Vertex_handle
   move_point_no_topo_change(const Vertex_handle& old_vertex,
-                            const Bare_point& new_position,
+                            const Weighted_point& new_position,
                             OutdatedCellsOutputIterator outdated_cells) const;
 
   Vertex_handle
   move_point_no_topo_change(const Vertex_handle& old_vertex,
-                            const Bare_point& new_position) const;
+                            const Weighted_point& new_position) const;
 
   /**
    * Returns the least square plane from v, using adjacent surface points
@@ -1698,7 +1703,7 @@ private:
    */
   template<typename OutputIterator>
   Vertex_handle revert_move(const Vertex_handle& new_vertex,
-                            const Bare_point& old_point,
+                            const Weighted_point& old_point,
                             OutputIterator outdated_cells)
   {
     // Move vertex
@@ -1753,14 +1758,14 @@ private:
   template <typename OutputIterator>
   OutputIterator
   get_conflict_zone_topo_change(const Vertex_handle& vertex,
-                                const Bare_point& conflict_point,
+                                const Weighted_point& conflict_point,
                                 OutputIterator conflict_cells) const;
 
   template <typename CellsOutputIterator,
             typename FacetsOutputIterator>
   void
   get_conflict_zone_topo_change(const Vertex_handle& v,
-                                const Bare_point& conflict_point,
+                                const Weighted_point& conflict_point,
                                 CellsOutputIterator insertion_conflict_cells,
                                 FacetsOutputIterator insertion_conflict_boundary,
                                 CellsOutputIterator removal_conflict_cells,
@@ -1772,7 +1777,7 @@ private:
              typename DeletedCellsOutputIterator >
   Vertex_handle
   move_point_topo_change_conflict_zone_known(const Vertex_handle& old_vertex,
-                                             const Bare_point& new_position,
+                                             const Weighted_point& new_position,
                                              const Facet& insertion_boundary_facet,
                                              ConflictCellsInputIterator insertion_conflict_cells_begin,
                                              ConflictCellsInputIterator insertion_conflict_cells_end,
@@ -2398,6 +2403,7 @@ private:
   Tr& tr_;
   const MeshDomain& domain_;
   Construct_point_3 wp2p_;
+  Construct_weighted_point_3 p2wp_;
 }; // class C3T3_helpers
 
 
@@ -2405,7 +2411,7 @@ template <typename C3T3, typename MD>
 template <typename SliverCriterion, typename OutputIterator>
 std::pair<bool,typename C3T3_helpers<C3T3,MD>::Vertex_handle>
 C3T3_helpers<C3T3,MD>::
-update_mesh(const Bare_point& new_position,
+update_mesh(const Weighted_point& new_position,
             const Vertex_handle& old_vertex,
             const SliverCriterion& criterion,
             OutputIterator modified_vertices,
@@ -2444,7 +2450,7 @@ template <typename C3T3, typename MD>
 template <typename SliverCriterion, typename OutputIterator>
 std::pair<bool,typename C3T3_helpers<C3T3,MD>::Vertex_handle>
 C3T3_helpers<C3T3,MD>::
-update_mesh_no_topo_change(const Bare_point& new_position,
+update_mesh_no_topo_change(const Weighted_point& new_position,
                            const Vertex_handle& old_vertex,
                            const SliverCriterion& criterion,
                            OutputIterator modified_vertices,
@@ -2461,7 +2467,7 @@ update_mesh_no_topo_change(const Bare_point& new_position,
   // Get old values
   criterion.before_move(c3t3_cells(conflict_cells));
   // std::cerr << "old_sliver_value=" << old_sliver_value << std::endl;
-  Bare_point old_position = wp2p_(old_vertex->point());
+  Weighted_point old_position = old_vertex->point();
 
   // Move point
   reset_circumcenter_cache(conflict_cells);
@@ -2503,7 +2509,7 @@ template <typename C3T3, typename MD>
 template <typename SliverCriterion, typename OutputIterator>
 std::pair<bool,typename C3T3_helpers<C3T3,MD>::Vertex_handle>
 C3T3_helpers<C3T3,MD>::
-update_mesh_topo_change(const Bare_point& new_position,
+update_mesh_topo_change(const Weighted_point& new_position,
                         const Vertex_handle& old_vertex,
                         const SliverCriterion& criterion,
                         OutputIterator modified_vertices,
@@ -2545,7 +2551,7 @@ update_mesh_topo_change(const Bare_point& new_position,
 
   criterion.before_move(c3t3_cells(conflict_cells));
   // std::cerr << "old_sliver_value=" << old_sliver_value << std::endl;
-  Bare_point old_position = wp2p_(old_vertex->point());
+  Weighted_point old_position = old_vertex->point();
 
   // Keep old boundary
   Vertex_set old_incident_surface_vertices;
@@ -2621,7 +2627,7 @@ template <typename C3T3, typename MD>
 template <typename OutputIterator>
 typename C3T3_helpers<C3T3,MD>::Vertex_handle
 C3T3_helpers<C3T3,MD>::
-update_mesh(const Bare_point& new_position,
+update_mesh(const Weighted_point& new_position,
             const Vertex_handle& old_vertex,
             OutputIterator modified_vertices,
             bool fill_vertices)
@@ -2764,7 +2770,7 @@ rebuild_restricted_delaunay(OutdatedCells& outdated_cells,
       // Update moving vertices (it becomes new_vertex)
       moving_vertices.erase(*it);
 
-      Vertex_handle new_vertex = update_mesh(new_pos,*it);
+      Vertex_handle new_vertex = update_mesh(p2wp_(new_pos),*it);
       c3t3_.set_dimension(new_vertex,2);
 
       moving_vertices.insert(new_vertex);
@@ -2897,7 +2903,7 @@ template <typename OutdatedCellsOutputIterator,
 typename C3T3_helpers<C3T3,MD>::Vertex_handle
 C3T3_helpers<C3T3,MD>::
 move_point(const Vertex_handle& old_vertex,
-           const Bare_point& new_position,
+           const Weighted_point& new_position,
            OutdatedCellsOutputIterator outdated_cells,
            DeletedCellsOutputIterator deleted_cells) const
 {
@@ -2929,7 +2935,7 @@ template <typename C3T3, typename MD>
 typename C3T3_helpers<C3T3,MD>::Vertex_handle
 C3T3_helpers<C3T3,MD>::
 move_point(const Vertex_handle& old_vertex,
-           const Bare_point& new_position,
+           const Weighted_point& new_position,
            Outdated_cell_set& outdated_cells_set,
            Moving_vertices_set& moving_vertices) const
 {
@@ -2963,7 +2969,7 @@ template <typename C3T3, typename MD>
 typename C3T3_helpers<C3T3,MD>::Vertex_handle
 C3T3_helpers<C3T3,MD>::
 move_point(const Vertex_handle& old_vertex,
-           const Bare_point& new_position,
+           const Weighted_point& new_position,
            Outdated_cell_set& outdated_cells_set,
            Moving_vertices_set& moving_vertices,
            bool *could_lock_zone) const
@@ -3039,7 +3045,7 @@ template <typename C3T3, typename MD>
 typename C3T3_helpers<C3T3,MD>::Vertex_handle
 C3T3_helpers<C3T3,MD>::
 move_point_topo_change(const Vertex_handle& old_vertex,
-                       const Bare_point& new_position,
+                       const Weighted_point& new_position,
                        Outdated_cell_set& outdated_cells_set,
                        bool *could_lock_zone) const
 {
@@ -3098,7 +3104,7 @@ template <typename OutdatedCellsOutputIterator,
 typename C3T3_helpers<C3T3,MD>::Vertex_handle
 C3T3_helpers<C3T3,MD>::
 move_point_topo_change(const Vertex_handle& old_vertex,
-                       const Bare_point& new_position,
+                       const Weighted_point& new_position,
                        OutdatedCellsOutputIterator outdated_cells,
                        DeletedCellsOutputIterator deleted_cells) const
 {
@@ -3137,7 +3143,7 @@ typename C3T3_helpers<C3T3,MD>::Vertex_handle
 C3T3_helpers<C3T3,MD>::
 move_point_topo_change_conflict_zone_known(
     const Vertex_handle& old_vertex,
-    const Bare_point& new_position,
+    const Weighted_point& new_position,
     const Facet& insertion_boundary_facet,
     ConflictCellsInputIterator insertion_conflict_cells_begin,//ordered
     ConflictCellsInputIterator insertion_conflict_cells_end,
@@ -3148,7 +3154,7 @@ move_point_topo_change_conflict_zone_known(
                                              //o.w. deleted_cells will point to null pointer or so and crash
                                              const
 {
-  Bare_point old_position = wp2p_(old_vertex->point());
+  Weighted_point old_position = old_vertex->point();
   // make one set with conflict zone
   Cell_set conflict_zone;
   std::set_union(insertion_conflict_cells_begin, insertion_conflict_cells_end,
@@ -3203,7 +3209,7 @@ template <typename C3T3, typename MD>
 typename C3T3_helpers<C3T3,MD>::Vertex_handle
 C3T3_helpers<C3T3,MD>::
 move_point_topo_change(const Vertex_handle& old_vertex,
-                       const Bare_point& new_position) const
+                       const Weighted_point& new_position) const
 {
   // Insert new_vertex, remove old_vertex
   int dimension = c3t3_.in_dimension(old_vertex);
@@ -3230,7 +3236,7 @@ template <typename OutdatedCellsOutputIterator>
 typename C3T3_helpers<C3T3,MD>::Vertex_handle
 C3T3_helpers<C3T3,MD>::
 move_point_no_topo_change(const Vertex_handle& old_vertex,
-                          const Bare_point& new_position,
+                          const Weighted_point& new_position,
                           OutdatedCellsOutputIterator outdated_cells) const
 {
 
@@ -3246,7 +3252,7 @@ template <typename C3T3, typename MD>
 typename C3T3_helpers<C3T3,MD>::Vertex_handle
 C3T3_helpers<C3T3,MD>::
 move_point_no_topo_change(const Vertex_handle& old_vertex,
-                          const Bare_point& new_position) const
+                          const Weighted_point& new_position) const
 {
   // Change vertex position
   old_vertex->set_point(new_position);
@@ -3768,7 +3774,7 @@ template <typename CellsOutputIterator,
 void
 C3T3_helpers<C3T3,MD>::
 get_conflict_zone_topo_change(const Vertex_handle& v,
-                              const Bare_point& conflict_point,
+                              const Weighted_point& conflict_point,
                               CellsOutputIterator insertion_conflict_cells,
                               FacetsOutputIterator insertion_conflict_boundary,
                               CellsOutputIterator removal_conflict_cells,
@@ -3805,7 +3811,7 @@ template <typename OutputIterator>
 OutputIterator
 C3T3_helpers<C3T3,MD>::
 get_conflict_zone_topo_change(const Vertex_handle& vertex,
-                              const Bare_point& conflict_point,
+                              const Weighted_point& conflict_point,
                               OutputIterator conflict_cells) const
 {
   // Get triangulation_vertex incident cells
diff --git a/Mesh_3/include/CGAL/Mesh_3/Lloyd_move.h b/Mesh_3/include/CGAL/Mesh_3/Lloyd_move.h
index 202a20f9de4..ca195b4751b 100644
--- a/Mesh_3/include/CGAL/Mesh_3/Lloyd_move.h
+++ b/Mesh_3/include/CGAL/Mesh_3/Lloyd_move.h
@@ -112,7 +112,7 @@ public:
   
 private:
   /**
-   * Project_on_plane defines operator() to project Weighted_point object on plane.
+   * Project_on_plane defines operator() to project point object on plane.
    */
   struct Project_on_plane
   {
@@ -126,14 +126,14 @@ private:
   };
   
   /**
-   * To_2d defines operator() to transform Weighted_point into Point_2
+   * To_2d defines operator() to transform Bare_point into Point_2
    */
   struct To_2d
   {
     To_2d(const Aff_transformation_3& to_2d) : to_2d_(to_2d) {}
     
-    Point_2 operator()(const Weighted_point& p) const
-    { return Point_2(to_2d_.transform(p.point()).x(), to_2d_.transform(p.point()).y()); }
+    Point_2 operator()(const Bare_point& p) const
+    { return Point_2(to_2d_.transform(p).x(), to_2d_.transform(p).y()); }
     
   private:
     const Aff_transformation_3& to_2d_;
@@ -146,7 +146,7 @@ private:
   {
     To_3d(const Aff_transformation_3& to_3d) : to_3d_(to_3d) {}
     
-    Weighted_point operator()(const Point_2& p) const
+    Bare_point operator()(const Point_2& p) const
     { return to_3d_.transform((Bare_point(p.x(),p.y(),0))); }
     
   private:
@@ -221,16 +221,16 @@ private:
       }
       case 2: // centroid 
       {
-        const Weighted_point& a = points.front();
-        const Weighted_point& b = points.back();
+        const Bare_point& a = points.front();
+        const Bare_point& b = points.back();
         return centroid_segment_move(v,a,b,sizing_field);
         break;
       }
       case 3: // triangle centroid 
       {
-        const Weighted_point& a = points.at(0);
-        const Weighted_point& b = points.at(1);
-        const Weighted_point& c = points.at(2);
+        const Bare_point& a = points.at(0);
+        const Bare_point& b = points.at(1);
+        const Bare_point& c = points.at(2);
         return centroid_triangle_move(v,a,b,c,sizing_field);
         break;
       }
@@ -275,14 +275,15 @@ private:
    * Return move from \c v to centroid of segment [a,b]
    */
   Vector_3 centroid_segment_move(const Vertex_handle& v,
-                                 const Weighted_point& a,
-                                 const Weighted_point& b,
+                                 const Bare_point& a,
+                                 const Bare_point& b,
                                  const Sizing_field& sizing_field) const
   {
     typename Gt::Construct_vector_3 vector =
       Gt().construct_vector_3_object();
     
-    const Weighted_point& p = v->point();
+    typename Gt::Construct_point_3 wp2p = Gt().construct_point_3_object();
+    const Bare_point& p = wp2p(v->point());
     
     FT da = density_1d(a,v,sizing_field);
     FT db = density_1d(b,v,sizing_field);
@@ -295,15 +296,16 @@ private:
    * Return move from \c v to centroid of triangle [a,b,c]
    */
   Vector_3 centroid_triangle_move(const Vertex_handle& v,
-                                  const Weighted_point& a,
-                                  const Weighted_point& b,
-                                  const Weighted_point& c,
+                                  const Bare_point& a,
+                                  const Bare_point& b,
+                                  const Bare_point& c,
                                   const Sizing_field& sizing_field) const
   {
     typename Gt::Construct_vector_3 vector =
       Gt().construct_vector_3_object();
     
-    const Weighted_point& p = v->point();
+    typename Gt::Construct_point_3 wp2p = Gt().construct_point_3_object();
+    const Bare_point& p = wp2p(v->point());
     
     FT da = density_2d<true>(a,v,sizing_field);
     FT db = density_2d<false>(b,v,sizing_field);
@@ -353,7 +355,7 @@ private:
                            std::back_inserter(ch_2d));
     
     // Lift back convex hull to 3D 
-    std::vector<Weighted_point> polygon_3d;
+    std::vector<Bare_point> polygon_3d;
     polygon_3d.reserve(ch_2d.size());
     std::transform(ch_2d.begin(), ch_2d.end(),
                    std::back_inserter(polygon_3d), To_3d(to_3d));
@@ -389,13 +391,13 @@ private:
     const Weighted_point& vertex_position = v->point();
     
     // Use as reference point to triangulate
-    const Weighted_point& a = *first++;
-    const Weighted_point* b = &(*first++);
+    const Bare_point& a = *first++;
+    const Bare_point* b = &(*first++);
     
     // Treat first point (optimize density_2d call)
-    const Weighted_point& c = *first++;
+    const Bare_point& c = *first++;
     
-    Weighted_point triangle_centroid = centroid(a,*b,c);
+    Bare_point triangle_centroid = centroid(a,*b,c);
     FT density = density_2d<true>(triangle_centroid, v, sizing_field);
     
     FT sum_masses = density * area(a,*b,c);
@@ -406,9 +408,9 @@ private:
     // Next points
     while ( first != last )
     {
-      const Weighted_point& c = *first++;
+      const Bare_point& c = *first++;
 
-      Weighted_point triangle_centroid = centroid(a,*b,c);
+      Bare_point triangle_centroid = centroid(a,*b,c);
       FT density = density_2d<false>(triangle_centroid, v, sizing_field);
       FT mass = density * area(a,*b,c);
       
@@ -427,7 +429,7 @@ private:
    * Returns the transformation from reference_point to plane
    */
   Aff_transformation_3 compute_to_3d_transform(const Plane_3& plane,
-                                               const Weighted_point& reference_point) const
+                                               const Bare_point& reference_point) const
   {
     typename Gt::Construct_base_vector_3 base =
       Gt().construct_base_vector_3_object();
@@ -453,7 +455,7 @@ private:
    * returns density_1d
    */
   template <typename Sizing_field>
-  FT density_1d(const Weighted_point& p,
+  FT density_1d(const Bare_point& p,
                 const Vertex_handle& v,
                 const Sizing_field& sizing_field) const
   {
@@ -468,7 +470,7 @@ private:
    * returns density_2d
    */
   template <bool use_v, typename Sizing_field>
-  FT density_2d(const Weighted_point& p,
+  FT density_2d(const Bare_point& p,
                 const Vertex_handle& v,
                 const Sizing_field& sizing_field) const
   {
@@ -483,7 +485,7 @@ private:
    * returns density_3d
    */
   template <typename Sizing_field>
-  FT density_3d(const Weighted_point& p,
+  FT density_3d(const Bare_point& p,
                 const Cell_handle& cell,
                 const Sizing_field& sizing_field) const
   {
@@ -519,23 +521,26 @@ private:
     typename Gt::Construct_vector_3 vector =
       Gt().construct_vector_3_object();
     
+    typename Gt::Construct_point_3 wp2p =
+      Gt().construct_point_3_object();
+    
     Cell_circulator current_cell = tr.incident_cells(edge);
     Cell_circulator done = current_cell;
     
     // a & b are fixed points
-    const Weighted_point& a = v->point();
-    const Weighted_point b = tr.dual(current_cell++);
+    const Bare_point& a = wp2p(v->point());
+    const Bare_point b = tr.dual(current_cell++);
     CGAL_assertion(current_cell != done);
     
     // c & d are moving points
-    Weighted_point c = tr.dual(current_cell++);
+    Bare_point c = tr.dual(current_cell++);
     CGAL_assertion(current_cell != done);
     
     while ( current_cell != done )
     {
-      const Weighted_point d = tr.dual(current_cell++);
+      const Bare_point d = tr.dual(current_cell++);
       
-      Weighted_point tet_centroid = centroid(a,b,c,d);
+      Bare_point tet_centroid = centroid(a,b,c,d);
       
       // Compute mass
       FT density = density_3d(tet_centroid, current_cell, sizing_field);
diff --git a/Mesh_3/include/CGAL/Mesh_3/Mesh_global_optimizer.h b/Mesh_3/include/CGAL/Mesh_3/Mesh_global_optimizer.h
index 6bd2356271c..81d9935d38b 100644
--- a/Mesh_3/include/CGAL/Mesh_3/Mesh_global_optimizer.h
+++ b/Mesh_3/include/CGAL/Mesh_3/Mesh_global_optimizer.h
@@ -346,6 +346,7 @@ private:
 #endif // CGAL_LINKED_WITH_TBB
   {
     typename Gt::Construct_point_3 wp2p = Gt().construct_point_3_object();
+    typename Gt::Construct_weighted_point_3 p2wp = Gt().construct_weighted_point_3_object();
     // Get move for each moving vertex
     typename Moving_vertices_set::iterator vit = moving_vertices.begin();
     for ( ; vit != moving_vertices.end() ; )
@@ -359,7 +360,7 @@ private:
         Bare_point new_position = translate(wp2p(oldv->point()),move);
         FT size = (Sizing_field::is_vertex_update_needed ?
           sizing_field_(new_position, oldv) : 0);
-        moves.push_back(cpp11::make_tuple(oldv,new_position,size));
+        moves.push_back(cpp11::make_tuple(oldv,p2wp(new_position),size));
       }
       else // CGAL::NULL_VECTOR == move
       {
@@ -480,6 +481,7 @@ private:
     {
       Vector_3 move = m_mgo.compute_move(oldv);
       typename Gt::Construct_point_3 wp2p = Gt().construct_point_3_object();
+      typename Gt::Construct_weighted_point_3 p2wp = Gt().construct_weighted_point_3_object();
       if ( CGAL::NULL_VECTOR != move )
       {
         Bare_point new_position = m_translate(wp2p(oldv->point()), move);
@@ -489,7 +491,7 @@ private:
         //if( !Th().inside_protecting_balls(tr_, oldv, new_position))
         //note : this is not happening for Lloyd and ODT so it's commented
         //       maybe for a new global optimizer it should be de-commented
-        m_moves.push_back(cpp11::make_tuple(oldv, new_position, size));
+        m_moves.push_back(cpp11::make_tuple(oldv, p2wp(new_position), size));
       }
       else // CGAL::NULL_VECTOR == move
       {
@@ -574,7 +576,7 @@ private:
       for( size_t i = r.begin() ; i != r.end() ; ++i)
       {
         const Vertex_handle& v = cpp11::get<0>(m_moves[i]);
-        const Bare_point& new_position = wp2p(cpp11::get<1>(m_moves[i]));
+        const Weighted_point& new_position = cpp11::get<1>(m_moves[i]);
         // Get size at new position
         if ( MGO::Sizing_field::is_vertex_update_needed )
         {
@@ -937,7 +939,7 @@ update_mesh(const Moves_vector& moves,
       typename Gt::Construct_point_3 wp2p = Gt().construct_point_3_object();
 
       const Vertex_handle& v = cpp11::get<0>(*it);
-      const Bare_point& new_position = wp2p(cpp11::get<1>(*it));
+      const Weighted_point& new_position = cpp11::get<1>(*it);
       // Get size at new position
       if ( Sizing_field::is_vertex_update_needed )
       {
@@ -1002,7 +1004,7 @@ void
 Mesh_global_optimizer<C3T3,Md,Mf,V_>::
 fill_sizing_field()
 {
-  std::map<Weighted_point,FT> value_map;
+  std::map<Bare_point,FT> value_map;
 
 #ifdef CGAL_LINKED_WITH_TBB
   // Parallel
@@ -1032,7 +1034,8 @@ fill_sizing_field()
         vit != tr_.finite_vertices_end();
         ++vit)
     {
-      value_map.insert(std::make_pair(vit->point(),
+      typename Gt::Construct_point_3 wp2p = Gt().construct_point_3_object();
+      value_map.insert(std::make_pair(wp2p(vit->point()),
                                       average_circumradius_length(vit)));
     }
   }
diff --git a/Mesh_3/include/CGAL/Mesh_3/Mesh_sizing_field.h b/Mesh_3/include/CGAL/Mesh_3/Mesh_sizing_field.h
index 5f8bf8cadae..67ce3228ae1 100644
--- a/Mesh_3/include/CGAL/Mesh_3/Mesh_sizing_field.h
+++ b/Mesh_3/include/CGAL/Mesh_3/Mesh_sizing_field.h
@@ -96,7 +96,7 @@ class Mesh_sizing_field
 {
   // Types
   typedef typename Tr::Geom_traits   Gt;
-  typedef typename Tr::Weighted_point Weighted_point;
+  typedef typename Tr::Bare_point Bare_point;
   typedef typename Gt::FT            FT;
 
   typedef typename Tr::Vertex_handle      Vertex_handle;
@@ -115,43 +115,43 @@ public:
   /**
    * Fill sizing field, using size associated to point in \c value_map
    */
-  void fill(const std::map<Weighted_point, FT>& value_map);
+  void fill(const std::map<Bare_point, FT>& value_map);
 
   /**
    * Returns size at point \c p.
    */
-  FT operator()(const Weighted_point& p) const
+  FT operator()(const Bare_point& p) const
   { return this->operator()(p, this->get_last_cell()); }
 
   /**
    * Returns size at point \c p, using \c v to accelerate \c p location
    * in triangulation
    */
-  FT operator()(const Weighted_point& p, const Vertex_handle& v) const
+  FT operator()(const Bare_point& p, const Vertex_handle& v) const
   { return this->operator()(p,v->cell()); }
 
   /**
    * Returns size at point \c p.
    */
-  FT operator()(const Weighted_point& p, const Cell_handle& c) const;
+  FT operator()(const Bare_point& p, const Cell_handle& c) const;
 
   /**
    * Returns size at point \c p. Assumes that p is the centroid of c.
    */
-  FT operator()(const Weighted_point& p, const std::pair<Cell_handle,bool>& c) const;
+  FT operator()(const Bare_point& p, const std::pair<Cell_handle,bool>& c) const;
 
 private:
   /**
    * Returns size at point \c p, by interpolation into tetrahedron.
    */
-  FT interpolate_on_cell_vertices(const Weighted_point& p,
+  FT interpolate_on_cell_vertices(const Bare_point& p,
                                   const Cell_handle& cell) const;
 
   /**
    * Returns size at point \c p, by interpolation into facet (\c cell is assumed
    * to be an infinite cell).
    */
-  FT interpolate_on_facet_vertices(const Weighted_point& p,
+  FT interpolate_on_facet_vertices(const Bare_point& p,
                                    const Cell_handle& cell) const;
 
 private:
@@ -172,7 +172,7 @@ Mesh_sizing_field(Tr& tr)
 template <typename Tr, bool B>
 void
 Mesh_sizing_field<Tr,B>::
-fill(const std::map<Weighted_point, FT>& value_map)
+fill(const std::map<Bare_point, FT>& value_map)
 {
   typedef typename Tr::Finite_vertices_iterator  Fvi;
 
@@ -180,8 +180,8 @@ fill(const std::map<Weighted_point, FT>& value_map)
         vit != tr_.finite_vertices_end() ;
         ++ vit )
   {
-    typename std::map<Weighted_point, FT>::const_iterator find_result =
-      value_map.find(vit->point());
+    typename std::map<Bare_point, FT>::const_iterator find_result =
+      value_map.find(tr_.geom_traits().construct_point_3_object()(vit->point()));
 
     if ( find_result != value_map.end() )
     {
@@ -198,16 +198,16 @@ fill(const std::map<Weighted_point, FT>& value_map)
 template <typename Tr, bool B>
 typename Mesh_sizing_field<Tr,B>::FT
 Mesh_sizing_field<Tr,B>::
-operator()(const Weighted_point& p, const Cell_handle& c) const
+operator()(const Bare_point& p, const Cell_handle& c) const
 {
 #ifdef CGAL_MESH_3_SIZING_FIELD_INEXACT_LOCATE
   //use the inexact locate (much faster than locate) to get a hint
   //and then use locate to check whether p is really inside hint
   // if not, an exact locate will be performed
-  Cell_handle hint = tr_.inexact_locate(p,c);
-  const Cell_handle cell = tr_.locate(p, hint);
+  Cell_handle hint = tr_.inexact_locate(tr_.geom_traits().construct_weighted_point_3_object()(p),c);
+  const Cell_handle cell = tr_.locate(tr_.geom_traits().construct_weighted_point_3_object()(p), hint);
 #else
-  const Cell_handle cell = tr_.locate(p,c);
+  const Cell_handle cell = tr_.locate(tr_.geom_traits().construct_weighted_point_3_object()(p),c);
 #endif
   this->set_last_cell(cell);
 
@@ -221,7 +221,7 @@ operator()(const Weighted_point& p, const Cell_handle& c) const
 template <typename Tr, bool B>
 typename Mesh_sizing_field<Tr,B>::FT
 Mesh_sizing_field<Tr,B>::
-operator()(const Weighted_point&, const std::pair<Cell_handle,bool>& c) const
+operator()(const Bare_point&, const std::pair<Cell_handle,bool>& c) const
 {
   // Assumes that p is the centroid of c
   const Cell_handle& cell = c.first;
@@ -239,10 +239,10 @@ operator()(const Weighted_point&, const std::pair<Cell_handle,bool>& c) const
 template <typename Tr, bool B>
 typename Mesh_sizing_field<Tr,B>::FT
 Mesh_sizing_field<Tr,B>::
-interpolate_on_cell_vertices(const Weighted_point& p, const Cell_handle& cell) const
+interpolate_on_cell_vertices(const Bare_point& p, const Cell_handle& cell) const
 {
-  typename Gt::Compute_volume_3 volume =
-    Gt().compute_volume_3_object();
+  typename Gt::Compute_volume_3 volume = tr_.geom_traits().compute_volume_3_object();
+  typename Gt::Construct_point_3 wp2p = tr_.geom_traits().construct_point_3_object();
 
   // Interpolate value using tet vertices values
   const FT& va = cell->vertex(0)->meshing_info();
@@ -250,10 +250,10 @@ interpolate_on_cell_vertices(const Weighted_point& p, const Cell_handle& cell) c
   const FT& vc = cell->vertex(2)->meshing_info();
   const FT& vd = cell->vertex(3)->meshing_info();
 
-  const Weighted_point& a = cell->vertex(0)->point();
-  const Weighted_point& b = cell->vertex(1)->point();
-  const Weighted_point& c = cell->vertex(2)->point();
-  const Weighted_point& d = cell->vertex(3)->point();
+  const Bare_point& a = wp2p(cell->vertex(0)->point());
+  const Bare_point& b = wp2p(cell->vertex(1)->point());
+  const Bare_point& c = wp2p(cell->vertex(2)->point());
+  const Bare_point& d = wp2p(cell->vertex(3)->point());
 
   const FT abcp = CGAL::abs(volume(a,b,c,p));
   const FT abdp = CGAL::abs(volume(a,d,b,p));
@@ -272,11 +272,11 @@ interpolate_on_cell_vertices(const Weighted_point& p, const Cell_handle& cell) c
 template <typename Tr, bool B>
 typename Mesh_sizing_field<Tr,B>::FT
 Mesh_sizing_field<Tr,B>::
-interpolate_on_facet_vertices(const Weighted_point& p, const Cell_handle& cell) const
+interpolate_on_facet_vertices(const Bare_point& p, const Cell_handle& cell) const
 {
-  typename Gt::Compute_area_3 area =
-    Gt().compute_area_3_object();
+  typename Gt::Compute_area_3 area =  tr_.geom_traits().compute_area_3_object();
 
+  typename Gt::Construct_point_3 wp2p = tr_.geom_traits().construct_point_3_object();
   // Find infinite vertex and put it in k0
   int k0 = 0;
   int k1 = 1;
@@ -295,9 +295,9 @@ interpolate_on_facet_vertices(const Weighted_point& p, const Cell_handle& cell)
   const FT& vb = cell->vertex(k2)->meshing_info();
   const FT& vc = cell->vertex(k3)->meshing_info();
 
-  const Weighted_point& a = cell->vertex(k1)->point();
-  const Weighted_point& b = cell->vertex(k2)->point();
-  const Weighted_point& c = cell->vertex(k3)->point();
+  const Bare_point& a = wp2p(cell->vertex(k1)->point());
+  const Bare_point& b = wp2p(cell->vertex(k2)->point());
+  const Bare_point& c = wp2p(cell->vertex(k3)->point());
 
   const FT abp = area(a,b,p);
   const FT acp = area(a,c,p);
diff --git a/Mesh_3/include/CGAL/Mesh_3/Mesher_3.h b/Mesh_3/include/CGAL/Mesh_3/Mesher_3.h
index 6de90fb577f..c11aec4bf0f 100644
--- a/Mesh_3/include/CGAL/Mesh_3/Mesher_3.h
+++ b/Mesh_3/include/CGAL/Mesh_3/Mesher_3.h
@@ -555,7 +555,8 @@ initialize()
         float(tbb::task_scheduler_init::default_num_threads())
         * Concurrent_mesher_config::get().num_pseudo_infinite_vertices_per_core);
       for (int i = 0 ; i < NUM_PSEUDO_INFINITE_VERTICES ; ++i, ++random_point)
-        r_c3t3_.add_far_point(r_c3t3_.triangulation().geom_traits().construct_translated_point_3_object()(*random_point, center));
+        r_c3t3_.add_far_point(r_c3t3_.triangulation().geom_traits().construct_weighted_point_3_object()
+                              (r_c3t3_.triangulation().geom_traits().construct_translated_point_3_object()(*random_point, center)));
     
 #  ifdef CGAL_CONCURRENT_MESH_3_VERBOSE
       std::cerr << "done." << std::endl;
diff --git a/Mesh_3/include/CGAL/Mesh_3/Odt_move.h b/Mesh_3/include/CGAL/Mesh_3/Odt_move.h
index 7466d2b8726..4f5d34b900a 100644
--- a/Mesh_3/include/CGAL/Mesh_3/Odt_move.h
+++ b/Mesh_3/include/CGAL/Mesh_3/Odt_move.h
@@ -51,6 +51,7 @@ class Odt_move
   typedef typename Tr::Vertex_handle Vertex_handle;
   typedef typename Tr::Facet Facet;
   typedef typename Tr::Weighted_point Weighted_point;
+  typedef typename Tr::Bare_point Bare_point;
   
   typedef typename std::vector<Cell_handle>   Cell_vector;
   typedef typename std::vector<Vertex_handle> Vertex_vector;
@@ -133,7 +134,7 @@ private:
     typename Gt::Construct_centroid_3 centroid =
     Gt().construct_centroid_3_object();
     
-    Weighted_point c = centroid(tr.tetrahedron(cell));
+    Bare_point c = centroid(tr.tetrahedron(cell));
     FT s = sizing_field(c,std::make_pair(cell,true));
     CGAL_assertion(!is_zero(s));
 
diff --git a/Mesh_3/include/CGAL/Mesh_3/Refine_facets_manifold_base.h b/Mesh_3/include/CGAL/Mesh_3/Refine_facets_manifold_base.h
index 7783d2db858..8e913d7cb34 100644
--- a/Mesh_3/include/CGAL/Mesh_3/Refine_facets_manifold_base.h
+++ b/Mesh_3/include/CGAL/Mesh_3/Refine_facets_manifold_base.h
@@ -117,6 +117,7 @@ public:
   typedef Complex3InTriangulation3 C3t3;
   typedef MeshDomain Mesh_domain;
   typedef typename Tr::Point Point;
+  typedef typename Tr::Bare_point Bare_point;
   typedef typename Tr::Facet Facet;
   typedef typename Tr::Vertex_handle Vertex_handle;
 
@@ -125,6 +126,7 @@ public:
 protected:
   typedef typename Tr::Geom_traits GT;
   typedef typename GT::FT FT;
+  typedef typename GT::Construct_point_3 Construct_point_3;
   typedef typename Tr::Edge Edge;
   typedef typename Tr::Cell_handle Cell_handle;
 
@@ -179,13 +181,14 @@ private:
 
   FT compute_sq_distance_to_facet_center(const Facet& f,
                                          const Vertex_handle v) const {
-    const Point& fcenter = f.first->get_facet_surface_center(f.second);
-    const Point& vpoint = v->point();
+    Construct_point_3 wp2p = this->r_tr_.geom_traits().construct_point_3_object();
+    const Bare_point& fcenter = f.first->get_facet_surface_center(f.second);
+    const Bare_point& vpoint = wp2p(v->point());
 
     return
-      this->r_tr_.geom_traits().compute_squared_distance_3_object()
-      (fcenter.point(), vpoint.point())
-      - vpoint.weight();
+      this->r_tr_.geom_traits().compute_squared_distance_3_object()(fcenter,
+                                                                    vpoint)
+      - v->point().weight();
   }
 
   Facet
diff --git a/Mesh_3/include/CGAL/Mesh_3/Robust_weighted_circumcenter_filtered_traits_3.h b/Mesh_3/include/CGAL/Mesh_3/Robust_weighted_circumcenter_filtered_traits_3.h
index 84d7aa95f0f..b60f6ffbd8c 100644
--- a/Mesh_3/include/CGAL/Mesh_3/Robust_weighted_circumcenter_filtered_traits_3.h
+++ b/Mesh_3/include/CGAL/Mesh_3/Robust_weighted_circumcenter_filtered_traits_3.h
@@ -174,6 +174,7 @@ public:
   typedef Bare_point                                  result_type;
  
  typename Rt::Construct_point_3 wp2p = Rt().construct_point_3_object();
+ typename Rt::Construct_weighted_point_3 p2wp = Rt().construct_weighted_point_3_object();
   
   Bare_point operator() ( const Weighted_point & p,
                           const Weighted_point & q,
@@ -215,7 +216,7 @@ public:
           return res;
       } else {
       // Fast output
-      if ( power_side_of_oriented_power_sphere(p,q,r,s,res) == CGAL::ON_POSITIVE_SIDE )
+        if ( power_side_of_oriented_power_sphere(p,q,r,s,p2wp(res)) == CGAL::ON_POSITIVE_SIDE )
         return res;
     }
     }
diff --git a/Mesh_3/include/CGAL/Mesh_3/Sliver_perturber.h b/Mesh_3/include/CGAL/Mesh_3/Sliver_perturber.h
index e8e4e467908..863713bbffc 100644
--- a/Mesh_3/include/CGAL/Mesh_3/Sliver_perturber.h
+++ b/Mesh_3/include/CGAL/Mesh_3/Sliver_perturber.h
@@ -1312,7 +1312,7 @@ perturb_vertex( PVertex pv
   }
   
   Point_3 p = wp2p(pv.vertex()->point());
-  if (!helper_.try_lock_point_no_spin(p) || ! Gt().equal_3_object()(p,wp2p(pv.vertex()->point())))
+  if (!helper_.try_lock_point_no_spin(pv.vertex()->point()) || ! Gt().equal_3_object()(p,wp2p(pv.vertex()->point())))
   {
 #ifdef CGAL_CONCURRENT_MESH_3_PROFILING
     bcounter.increment_branch_2(); // THIS is an early withdrawal!
diff --git a/Mesh_3/include/CGAL/Mesh_3/Triangulation_helpers.h b/Mesh_3/include/CGAL/Mesh_3/Triangulation_helpers.h
index 7ffd8e0551a..43f42f8bc77 100644
--- a/Mesh_3/include/CGAL/Mesh_3/Triangulation_helpers.h
+++ b/Mesh_3/include/CGAL/Mesh_3/Triangulation_helpers.h
@@ -41,6 +41,7 @@ class Triangulation_helpers
 {
   typedef typename Tr::Geom_traits              Gt;
   typedef typename Tr::Bare_point               Bare_point;
+  typedef typename Tr::Weighted_point           Weighted_point;
   typedef typename Tr::Vertex_handle            Vertex_handle;
   typedef typename Tr::Cell_handle              Cell_handle;
   typedef std::vector<Cell_handle>              Cell_vector;
@@ -65,18 +66,18 @@ class Triangulation_helpers
   {
     /// When the requested will be about vh, the returned point will be p
     /// instead of vh->point()
-    Point_getter(const Vertex_handle &vh, const Bare_point&p)
+    Point_getter(const Vertex_handle &vh, const Weighted_point&p)
       : m_vh(vh), m_p(p)
     {}
 
-    const Bare_point& operator()(const Vertex_handle &vh) const
+    const Weighted_point& operator()(const Vertex_handle &vh) const
     {
-      return (vh == m_vh ? m_p : vh->point().point()); //  AF: Can this be NOT a weighted point?
+      return (vh == m_vh ? m_p : vh->point());
     }
 
   private:
     const Vertex_handle m_vh;
-    const Bare_point &m_p;
+    const Weighted_point &m_p;
   };
 
 public:
@@ -89,7 +90,7 @@ public:
    */
   void move_point(Tr& tr,
                   const Vertex_handle& v,
-                  const Bare_point& p) const;
+                  const Weighted_point& p) const;
 
   /**
    * Returns true if moving \c v to \c p makes no topological
@@ -97,21 +98,21 @@ public:
    */
   bool no_topological_change(const Tr& tr,
                              const Vertex_handle& v,
-                             const Bare_point& p,
+                             const Weighted_point& p,
                              Cell_vector& cells_tos) const;
   bool no_topological_change__without_set_point(
                              const Tr& tr,
                              const Vertex_handle& v,
-                             const Bare_point& p,
+                             const Weighted_point& p,
                              Cell_vector& cells_tos) const;
 
   bool no_topological_change(const Tr& tr,
                              const Vertex_handle& v,
-                             const Bare_point& p) const;
+                             const Weighted_point& p) const;
   bool no_topological_change__without_set_point(
                              const Tr& tr,
                              const Vertex_handle& v,
-                             const Bare_point& p) const;
+                             const Weighted_point& p) const;
 
 
   bool inside_protecting_balls(const Tr& tr,
@@ -137,7 +138,7 @@ void
 Triangulation_helpers<Tr>::
 move_point(Tr& tr,
            const Vertex_handle& v,
-           const Bare_point& p) const
+           const Weighted_point& p) const
 {
   if ( no_topological_change(tr, v, p) )
     v->set_point(p);
@@ -153,12 +154,12 @@ bool
 Triangulation_helpers<Tr>::
 no_topological_change(const Tr& tr,
                       const Vertex_handle& v0,
-                      const Bare_point& p,
+                      const Weighted_point& p,
                       Cell_vector& cells_tos) const
 {
   Tr::Geom_traits::Construct_point_3 wp2p = tr.geom_traits().construct_point_3_object();
   bool np = true;
-  Bare_point fp = wp2p(v0->point());
+  const Weighted_point& fp = v0->point();
   v0->set_point(p);
 
   if(!well_oriented(tr, cells_tos))
@@ -222,7 +223,7 @@ Triangulation_helpers<Tr>::
 no_topological_change__without_set_point(
   const Tr& tr,
   const Vertex_handle& v0,
-  const Bare_point& p,
+  const Weighted_point& p,
   Cell_vector& cells_tos) const
 {
   bool np = true;
@@ -311,7 +312,7 @@ bool
 Triangulation_helpers<Tr>::
 no_topological_change(const Tr& tr,
                       const Vertex_handle& v0,
-                      const Bare_point& p) const
+                      const Weighted_point& p) const
 {
   Cell_vector cells_tos;
   cells_tos.reserve(64);
@@ -325,7 +326,7 @@ Triangulation_helpers<Tr>::
 no_topological_change__without_set_point(
                       const Tr& tr,
                       const Vertex_handle& v0,
-                      const Bare_point& p) const
+                      const Weighted_point& p) const
 {
   Cell_vector cells_tos;
   cells_tos.reserve(64);
diff --git a/Mesh_3/include/CGAL/Mesh_3/vertex_perturbation.h b/Mesh_3/include/CGAL/Mesh_3/vertex_perturbation.h
index 3c1c5ad3582..2f82639ce12 100644
--- a/Mesh_3/include/CGAL/Mesh_3/vertex_perturbation.h
+++ b/Mesh_3/include/CGAL/Mesh_3/vertex_perturbation.h
@@ -445,6 +445,7 @@ protected:
     typedef typename C3T3::Triangulation::Geom_traits Gt;
     typedef typename Gt::FT       FT;
     typedef typename C3T3::Triangulation::Bare_point  Bare_point;
+    typedef typename C3T3::Triangulation::Weighted_point  Weighted_point;
     typedef Triangulation_helpers<typename C3T3::Triangulation> Th;
     
     typename Gt::Compute_squared_length_3 sq_length =
@@ -456,6 +457,9 @@ protected:
     typename Gt::Construct_point_3 wp2p =
       Gt().construct_point_3_object();
     
+    typename Gt::Construct_weighted_point_3 p2wp =
+      Gt().construct_weighted_point_3_object();
+    
     // create a helper
     typedef C3T3_helpers<C3T3,MeshDomain> C3T3_helpers;
     C3T3_helpers helper(c3t3, domain);
@@ -466,8 +470,8 @@ protected:
     FT sq_norm = this->compute_perturbation_sq_amplitude(v, c3t3, sq_step_size_);
     FT step_length = CGAL::sqrt(sq_norm/sq_length(gradient_vector));
     Bare_point new_loc = translate(wp2p(v->point()), step_length * gradient_vector);
-    
     Bare_point final_loc = new_loc;
+
     if ( c3t3.in_dimension(v) < 3 )
       final_loc = helper.project_on_surface(new_loc, v);
 
@@ -477,7 +481,7 @@ protected:
     if (could_lock_zone)
     {
       while(Th().no_topological_change__without_set_point(c3t3.triangulation(), 
-                                                          v, final_loc) 
+                                                          v, p2wp(final_loc)) 
             && (++i <= max_step_nb_) )
       {
         new_loc = translate(new_loc, step_length * gradient_vector);
@@ -490,7 +494,7 @@ protected:
     }
     else
     {
-      while( Th().no_topological_change(c3t3.triangulation(), v, final_loc) 
+      while( Th().no_topological_change(c3t3.triangulation(), v, p2wp(final_loc)) 
             && (++i <= max_step_nb_) )
       {
         new_loc = translate(new_loc, step_length * gradient_vector);
@@ -508,7 +512,7 @@ protected:
       return std::make_pair(false,v);
 
     // we know that there will be a combinatorial change
-    return helper.update_mesh_topo_change(final_loc,
+    return helper.update_mesh_topo_change(p2wp(final_loc),
                                           v,
                                           criterion,
                                           std::back_inserter(modified_vertices),
@@ -1255,9 +1259,12 @@ private:
     typename Gt::Construct_translated_point_3 translate =
       Gt().construct_translated_point_3_object();
     
+    typename Gt::Construct_weighted_point_3 p2wp =
+      Gt().construct_weighted_point_3_object();
+    
     modified_vertices.clear();
 
-    // Create an helper
+    // Create a helper
     typedef C3T3_helpers<C3T3,MeshDomain> C3T3_helpers;
     C3T3_helpers helper(c3t3, domain);
     
@@ -1291,7 +1298,7 @@ private:
       // try to move vertex
       std::vector<Vertex_handle> tmp_mod_vertices;
       std::pair<bool,Vertex_handle> update =
-        helper.update_mesh(new_location,
+        helper.update_mesh(p2wp(new_location),
                            moving_vertex,
                            criterion,
                            std::back_inserter(tmp_mod_vertices),
diff --git a/Triangulation_3/include/CGAL/Regular_traits_adaptor.h b/Triangulation_3/include/CGAL/Regular_traits_adaptor.h
index 190ecaca14b..880da748ee3 100644
--- a/Triangulation_3/include/CGAL/Regular_traits_adaptor.h
+++ b/Triangulation_3/include/CGAL/Regular_traits_adaptor.h
@@ -123,11 +123,46 @@ public:
     return f(p0, p1);
   }
 
+  typename cpp11::result_of< Functor(Point_3,Point_3, Point_3) >::type operator() (const Point_3& p0, const Point_3& p1, const Point_3& p2) const
+  {
+    return f(p0, p1, p2);
+  }
+
+  typename cpp11::result_of< Functor(Point_3,Point_3, Point_3,Point_3) >::type operator() (const Point_3& p0, const Point_3& p1, const Point_3& p2, const Point_3& p3) const
+  {
+    return f(p0, p1, p2, p3);
+  }
+
+ typename cpp11::result_of< Functor(Point_3,Origin) >::type operator() (const Point_3& p0, const Origin& o) const
+  {
+    return f(p0, o);
+  }
+
+  typename cpp11::result_of< Functor(Point_3,Origin) >::type operator() (const Origin& o, const Point_3& p0) const
+  {
+    return f(o, p0);
+  }
+
+  typename cpp11::result_of< Functor(Plane_3,Point_3) >::type operator() (const Plane_3& pl, const Point_3& p) const
+  {
+    return f(pl, p);
+  }
+
    typename cpp11::result_of< Functor(Point_3,Point_3) >::type  operator() (const Weighted_point_3& p0, const Weighted_point_3& p1) const
   {
     return f(cp(p0), cp(p1));
   }
 
+   typename cpp11::result_of< Functor(Point_3,Point_3) >::type  operator() (const Point_3& p0, const Weighted_point_3& p1) const
+  {
+    return f(p0, cp(p1));
+  }
+
+   typename cpp11::result_of< Functor(Point_3,Point_3) >::type  operator() (const Weighted_point_3& p0, const Point_3& p1) const
+  {
+    return f(cp(p0), p1);
+  }
+
   typename cpp11::result_of<Functor(Plane_3,Point_3)>::type operator() (const Plane_3& p0, const Weighted_point_3& p1) const
   {
     return f(p0, cp(p1));
diff --git a/Triangulation_3/include/CGAL/Regular_triangulation_3.h b/Triangulation_3/include/CGAL/Regular_triangulation_3.h
index 4d43e6838e4..2a69f76941b 100644
--- a/Triangulation_3/include/CGAL/Regular_triangulation_3.h
+++ b/Triangulation_3/include/CGAL/Regular_triangulation_3.h
@@ -550,6 +550,27 @@ namespace CGAL {
 
   public:
 
+
+    Cell_handle locate(const Bare_point& bp) const
+    {
+      return this->locate(geom_traits().construct_weighted_point_3_object()(bp));
+    }
+
+
+    Cell_handle locate(const Bare_point& bp, Locate_type & lt, int & li, int & lj,
+                         Cell_handle start = Cell_handle()) const
+    {
+      return this->locate(geom_traits().construct_weighted_point_3_object()(bp),
+                          lt, li, lj, start);
+    }
+
+
+    Cell_handle locate(const Bare_point& bp, Locate_type & lt, int & li, int & lj,
+                         Vertex_handle start) const
+    {
+      return this->locate(geom_traits().construct_weighted_point_3_object()(bp),
+                          lt, li, lj, start);
+    }
 #ifndef CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
     template < class InputIterator >
     std::ptrdiff_t
@@ -791,6 +812,11 @@ namespace CGAL {
     Vertex_handle insert(const Weighted_point & p, Locate_type lt,
       Cell_handle c, int li, int, bool *could_lock_zone = NULL);
 
+    Vertex_handle insert(const Bare_point & p)
+    {
+      return insert(geom_traits().construct_weighted_point_3_object()(p, typename Geom_traits::FT(0)));
+    }
+
     template <class CellIt>
     Vertex_handle
       insert_in_hole(const Weighted_point & p, CellIt cell_begin, CellIt cell_end,