diff --git a/Triangulation_3/include/CGAL/Triangulation_3.h b/Triangulation_3/include/CGAL/Triangulation_3.h
index 795aca2d6a0..a60b57f9676 100644
--- a/Triangulation_3/include/CGAL/Triangulation_3.h
+++ b/Triangulation_3/include/CGAL/Triangulation_3.h
@@ -1646,7 +1646,7 @@ private:
   Vertex_handle>::type Vertex_handle_unique_hash_map;
 
   Vertex_triple make_vertex_triple(const Facet& f) const;
-  void make_canonical(Vertex_triple& t) const;
+  void make_canonical_oriented_triple(Vertex_triple& t) const;
 
   template < class VertexRemover >
   VertexRemover& make_hole_2D(Vertex_handle v, std::list<Edge_2D>& hole,
@@ -4215,7 +4215,7 @@ make_vertex_triple(const Facet& f) const
 template < class Gt, class Tds, class Lds >
 void
 Triangulation_3<Gt,Tds,Lds>::
-make_canonical(Vertex_triple& t) const
+make_canonical_oriented_triple(Vertex_triple& t) const
 {
   int i = (t.first < t.second) ? 0 : 1;
   if(i==0)
@@ -4770,7 +4770,7 @@ make_hole_3D(Vertex_handle v,
     Cell_handle opp_cit = (*cit)->neighbor(indv);
     Facet f(opp_cit, opp_cit->index(*cit));
     Vertex_triple vt = make_vertex_triple(f);
-    make_canonical(vt);
+    make_canonical_oriented_triple(vt);
     outer_map[vt] = f;
     for(int i=0; i<4; i++)
     {
@@ -4797,7 +4797,7 @@ make_hole_3D(Vertex_handle v,
     Cell_handle opp_cit = (*cit)->neighbor(indv);
     Facet f(opp_cit, opp_cit->index(*cit));
     Vertex_triple vt = make_vertex_triple(f);
-    make_canonical(vt);
+    make_canonical_oriented_triple(vt);
     outer_map[vt] = f;
     for(int i=0; i<4; i++)
     {
@@ -4985,7 +4985,7 @@ remove_3D(Vertex_handle v, VertexRemover& remover)
         Facet f = std::pair<Cell_handle,int>(it,i);
         Vertex_triple vt_aux = make_vertex_triple(f);
         Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         inner_map[vt]= f;
       }
     }
@@ -5000,7 +5000,7 @@ remove_3D(Vertex_handle v, VertexRemover& remover)
         Facet f = std::pair<Cell_handle,int>(it,i);
         Vertex_triple vt_aux = make_vertex_triple(f);
         Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         inner_map[vt]= f;
       }
     }
@@ -5043,7 +5043,7 @@ remove_3D(Vertex_handle v, VertexRemover& remover)
       {
         Facet f = std::pair<Cell_handle,int>(new_ch,i);
         Vertex_triple vt = make_vertex_triple(f);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         std::swap(vt.second,vt.third);
 
         typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
@@ -5181,7 +5181,7 @@ remove_3D(Vertex_handle v, VertexRemover& remover,
         Facet f = std::pair<Cell_handle,int>(it,i);
         Vertex_triple vt_aux = make_vertex_triple(f);
         Vertex_triple vt(vmap[vt_aux.first],vmap[vt_aux.third],vmap[vt_aux.second]);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         inner_map[vt]= f;
       }
     }
@@ -5196,7 +5196,7 @@ remove_3D(Vertex_handle v, VertexRemover& remover,
         Facet f = std::pair<Cell_handle,int>(it,i);
         Vertex_triple vt_aux = make_vertex_triple(f);
         Vertex_triple vt(vmap[vt_aux.first],vmap[vt_aux.third],vmap[vt_aux.second]);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         inner_map[vt]= f;
       }
     }
@@ -5241,7 +5241,7 @@ remove_3D(Vertex_handle v, VertexRemover& remover,
       {
         Facet f = std::pair<Cell_handle,int>(new_ch,i);
         Vertex_triple vt = make_vertex_triple(f);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         std::swap(vt.second,vt.third);
 
         typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
@@ -5485,7 +5485,7 @@ remove_3D(Vertex_handle v, VertexRemover& remover, OutputItCells fit)
         Facet f = std::pair<Cell_handle,int>(it,i);
         Vertex_triple vt_aux = make_vertex_triple(f);
         Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         inner_map[vt] = f;
       }
     }
@@ -5499,7 +5499,7 @@ remove_3D(Vertex_handle v, VertexRemover& remover, OutputItCells fit)
         Facet f = std::pair<Cell_handle,int>(it,i);
         Vertex_triple vt_aux = make_vertex_triple(f);
         Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         inner_map[vt] = f;
       }
     }
@@ -5546,7 +5546,7 @@ remove_3D(Vertex_handle v, VertexRemover& remover, OutputItCells fit)
       {
         Facet f = std::pair<Cell_handle,int>(new_ch,i);
         Vertex_triple vt = make_vertex_triple(f);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         std::swap(vt.second, vt.third);
         typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
         if(oit2 == outer_map.end())
@@ -5873,7 +5873,7 @@ move_if_no_collision(Vertex_handle v, const Point& p,
         Facet f = std::pair<Cell_handle,int>(it,i);
         Vertex_triple vt_aux = make_vertex_triple(f);
         Vertex_triple vt(vmap[vt_aux.first],vmap[vt_aux.third],vmap[vt_aux.second]);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         inner_map[vt]= f;
       }
     }
@@ -5888,7 +5888,7 @@ move_if_no_collision(Vertex_handle v, const Point& p,
         Facet f = std::pair<Cell_handle,int>(it,i);
         Vertex_triple vt_aux = make_vertex_triple(f);
         Vertex_triple vt(vmap[vt_aux.first],vmap[vt_aux.third],vmap[vt_aux.second]);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         inner_map[vt]= f;
       }
     }
@@ -5934,7 +5934,7 @@ move_if_no_collision(Vertex_handle v, const Point& p,
       {
         Facet f = std::pair<Cell_handle,int>(new_ch,i);
         Vertex_triple vt = make_vertex_triple(f);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         std::swap(vt.second,vt.third);
         typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
         if(oit2 == outer_map.end())
@@ -6325,7 +6325,7 @@ move_if_no_collision_and_give_new_cells(Vertex_handle v, const Point& p,
         Facet f = std::pair<Cell_handle,int>(it,i);
         Vertex_triple vt_aux = make_vertex_triple(f);
         Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         inner_map[vt]= f;
       }
     }
@@ -6340,7 +6340,7 @@ move_if_no_collision_and_give_new_cells(Vertex_handle v, const Point& p,
         Facet f = std::pair<Cell_handle,int>(it,i);
         Vertex_triple vt_aux = make_vertex_triple(f);
         Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         inner_map[vt]= f;
       }
     }
@@ -6387,7 +6387,7 @@ move_if_no_collision_and_give_new_cells(Vertex_handle v, const Point& p,
       {
         Facet f = std::pair<Cell_handle, int>(new_ch, i);
         Vertex_triple vt = make_vertex_triple(f);
-        make_canonical(vt);
+        make_canonical_oriented_triple(vt);
         std::swap(vt.second,vt.third);
         typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
         if(oit2 == outer_map.end())
@@ -6519,7 +6519,7 @@ _make_big_hole_3D(Vertex_handle v,
 
       Facet f(opp_cit, opp_i);
       Vertex_triple vt = make_vertex_triple(f);
-      make_canonical(vt);
+      make_canonical_oriented_triple(vt);
       outer_map[vt] = f;
       v1->set_cell(opp_cit);
       v2->set_cell(opp_cit);
@@ -6664,7 +6664,7 @@ _remove_cluster_3D(InputIterator first, InputIterator beyond, VertexRemover& rem
           Facet f = std::pair<Cell_handle,int>(it,index);
           Vertex_triple vt_aux = this->make_vertex_triple(f);
           Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
-          this->make_canonical(vt);
+          this->make_canonical_oriented_triple(vt);
           inner_map[vt]= f;
         }
       }
@@ -6679,7 +6679,7 @@ _remove_cluster_3D(InputIterator first, InputIterator beyond, VertexRemover& rem
           Facet f = std::pair<Cell_handle,int>(it,index);
           Vertex_triple vt_aux = this->make_vertex_triple(f);
           Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
-          this->make_canonical(vt);
+          this->make_canonical_oriented_triple(vt);
           inner_map[vt]= f;
         }
       }
@@ -6728,7 +6728,7 @@ _remove_cluster_3D(InputIterator first, InputIterator beyond, VertexRemover& rem
         {
           Facet f = std::pair<Cell_handle,int>(new_ch,index);
           Vertex_triple vt = this->make_vertex_triple(f);
-          this->make_canonical(vt);
+          this->make_canonical_oriented_triple(vt);
           std::swap(vt.second,vt.third);
           typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
           if(oit2 == outer_map.end())