diff --git a/Triangulation_2/include/CGAL/Triangulation_hierarchy_2.h b/Triangulation_2/include/CGAL/Triangulation_hierarchy_2.h
index 29513248384..215a48dace2 100644
--- a/Triangulation_2/include/CGAL/Triangulation_hierarchy_2.h
+++ b/Triangulation_2/include/CGAL/Triangulation_hierarchy_2.h
@@ -40,17 +40,17 @@
 
 namespace CGAL {
 
-// parameterization of the  hierarchy
-//const float Triangulation_hierarchy_2__ratio    = 30.0;
-const int Triangulation_hierarchy_2__ratio      = 30;
-const int   Triangulation_hierarchy_2__minsize  = 20;
-const int   Triangulation_hierarchy_2__maxlevel = 5;
-// maximal number of points is 30^5 = 24 millions !
 
 template <class Tr_>
 class Triangulation_hierarchy_2
   : public Tr_
 {
+  // parameterization of the hierarchy
+  // maximal number of points is 30^5 = 24 millions !
+  enum Ratio : int { ratio = 30 };
+  enum { minsize = 20};
+  enum { maxlevel = 5};
+
 public:
   typedef Tr_                                  Tr_Base;
   typedef typename Tr_Base::Geom_traits        Geom_traits;
@@ -84,13 +84,13 @@ public:
  private:
   void init_hierarchy() {
     hierarchy[0] = this;
-    for(int i=1; i<Triangulation_hierarchy_2__maxlevel; ++i)
+    for(int i=1; i<maxlevel; ++i)
       hierarchy[i] = &hierarchy_triangulations[i-1];
   }
 
  // here is the stack of triangulations which form the hierarchy
-  std::array<Tr_Base,Triangulation_hierarchy_2__maxlevel-1> hierarchy_triangulations;
-  std::array<Tr_Base*,Triangulation_hierarchy_2__maxlevel> hierarchy;
+  std::array<Tr_Base,maxlevel-1> hierarchy_triangulations;
+  std::array<Tr_Base*,maxlevel> hierarchy;
   boost::rand48  random;
 
 public:
@@ -163,7 +163,7 @@ public:
       // hints[i] is the face of the previously inserted point in level i.
       // Thanks to spatial sort, they are better hints than what the hierarchy
       // would give us.
-      Face_handle hints[Triangulation_hierarchy_2__maxlevel];
+      Face_handle hints[maxlevel];
       for (typename std::vector<Point>::const_iterator p = points.begin(), end = points.end();
               p != end; ++p)
       {
@@ -260,7 +260,7 @@ private:
                       int& li,
                       Face_handle loc,
                       Face_handle
-                      pos[Triangulation_hierarchy_2__maxlevel]) const;
+                      pos[maxlevel]) const;
 
   int random_level();
 
@@ -291,7 +291,7 @@ Triangulation_hierarchy_2<Tr_>::
 Triangulation_hierarchy_2(const Geom_traits& traits)
   : Tr_Base(traits)
   , hierarchy_triangulations(
-           make_filled_array<Triangulation_hierarchy_2__maxlevel-1,
+           make_filled_array<maxlevel-1,
                              Tr_Base>(traits))
 {
   init_hierarchy();
@@ -325,7 +325,7 @@ Triangulation_hierarchy_2<Tr_>::
 copy_triangulation(const Triangulation_hierarchy_2<Tr_> &tr)
 {
   {
-    for(int i=0;i<Triangulation_hierarchy_2__maxlevel;++i)
+    for(int i=0;i<maxlevel;++i)
     hierarchy[i]->copy_triangulation(*tr.hierarchy[i]);
   }
 
@@ -343,7 +343,7 @@ copy_triangulation(const Triangulation_hierarchy_2<Tr_> &tr)
   add_hidden_vertices_into_map(Weighted_tag(), V);
 
   {
-    for(int i=1;i<Triangulation_hierarchy_2__maxlevel;++i) {
+    for(int i=1;i<maxlevel;++i) {
       for( Finite_vertices_iterator it=hierarchy[i]->finite_vertices_begin();
            it != hierarchy[i]->finite_vertices_end(); ++it) {
         // down pointer goes in original instead in copied triangulation
@@ -396,7 +396,7 @@ void
 Triangulation_hierarchy_2<Tr_>::
 clear()
 {
-  for(int i=0;i<Triangulation_hierarchy_2__maxlevel;++i)
+  for(int i=0;i<maxlevel;++i)
     if(hierarchy[i]) hierarchy[i]->clear();
 }
 
@@ -410,7 +410,7 @@ is_valid(bool verbose, int level) const
   int i;
   Finite_vertices_iterator it;
   //verify correctness of triangulation at all levels
-  for(i=0;i<Triangulation_hierarchy_2__maxlevel;++i) {
+  for(i=0;i<maxlevel;++i) {
     if(verbose) // pirnt  number of vertices at each level
       std::cout << "number_of_vertices "
                 <<  hierarchy[i]->number_of_vertices() << std::endl;
@@ -421,13 +421,13 @@ is_valid(bool verbose, int level) const
        it != hierarchy[0]->finite_vertices_end(); ++it)
     result = result && ( it->down() ==   Vertex_handle());
   //verify that other levels have down pointer and reciprocal link is fine
-  for(i=1;i<Triangulation_hierarchy_2__maxlevel;++i)
+  for(i=1;i<maxlevel;++i)
     for( it = hierarchy[i]->finite_vertices_begin();
          it != hierarchy[i]->finite_vertices_end(); ++it)
       result = result &&
                ( &*(it->down()->up())  ==  &*(it) );
   //verify that levels have up pointer and reciprocal link is fine
-  for(i=0;i<Triangulation_hierarchy_2__maxlevel-1;++i)
+  for(i=0;i<maxlevel-1;++i)
     for( it = hierarchy[i]->finite_vertices_begin();
          it != hierarchy[i]->finite_vertices_end(); ++it)
       result = result && ( it->up() ==  Vertex_handle() ||
@@ -445,7 +445,7 @@ insert(const Point &p, Face_handle loc)
   Locate_type lt;
   int i;
   // locate using hierarchy
-  Face_handle positions[Triangulation_hierarchy_2__maxlevel];
+  Face_handle positions[maxlevel];
   locate_in_all(p,lt,i,loc,positions);
   Vertex_handle vertex=hierarchy[0]->Tr_Base::insert(p,lt,positions[0],i);
   Vertex_handle previous=vertex;
@@ -480,7 +480,7 @@ insert(const Point& p,
     // locate using hierarchy
     Locate_type ltt;
     int lii;
-    Face_handle positions[Triangulation_hierarchy_2__maxlevel];
+    Face_handle positions[maxlevel];
     locate_in_all(p,ltt,lii,loc,positions);
     //insert in higher levels
     int level  = 1;
@@ -514,7 +514,7 @@ remove(Vertex_handle v )
   while(1){
     hierarchy[l++]->remove(v);
     if (u == Vertex_handle()) break;
-    if (l >= Triangulation_hierarchy_2__maxlevel) break;
+    if (l >= maxlevel) break;
     v=u; u=v->up();
   }
 }
@@ -531,7 +531,7 @@ remove_and_give_new_faces(Vertex_handle v, OutputItFaces fit)
     if(l==0) hierarchy[l++]->remove_and_give_new_faces(v, fit);
     else hierarchy[l++]->remove(v);
     if (u == Vertex_handle()) break;
-    if (l >= Triangulation_hierarchy_2__maxlevel) break;
+    if (l >= maxlevel) break;
     v=u; u=v->up();
   }
 }
@@ -565,16 +565,19 @@ template <class Tr_>
 typename Triangulation_hierarchy_2<Tr_>::Vertex_handle
 Triangulation_hierarchy_2<Tr_>::
 move_if_no_collision(Vertex_handle v, const Point &p) {
-  Vertex_handle u=v->up(), norm = v;
-  int l = 0 ;
-  while(1) {
-    Vertex_handle w = hierarchy[l++]->move_if_no_collision(v, p);
-    if(w != v) return w;
-    if (u == Vertex_handle()) break;
-    if (l >= Triangulation_hierarchy_2__maxlevel) break;
-    v=u; u=v->up();
+
+  CGAL_precondition(!this->is_infinite(v));
+  if(v->point() == p) return v;
+  Vertex_handle ans = hierarchy[0]->move_if_no_collision(v, p);
+  if(ans != v) return ans; // ans is an existing vertex at p and v was not changed
+  for (int l = 1; l < maxlevel; ++l) {
+    Vertex_handle u = v->up();
+    if (u == Vertex_handle())
+      break;
+    hierarchy[l]->move_if_no_collision(u, p);
+    v = u;
   }
-  return norm;
+  return ans;
 }
 
 template <class Tr_>
@@ -597,22 +600,18 @@ Triangulation_hierarchy_2<Tr_>::
 move_if_no_collision_and_give_new_faces(Vertex_handle v, const Point &p,
                                         OutputItFaces oif)
 {
-  Vertex_handle u=v->up(), norm = v;
-  int l = 0 ;
-  while(1){
-    Vertex_handle w;
-    if(l == 0)
-      w =
-        hierarchy[l++]->move_if_no_collision_and_give_new_faces(v, p, oif);
-    else w = hierarchy[l++]->move_if_no_collision(v, p);
-
-    if(w != v) return w;
-
-    if (u == Vertex_handle()) break;
-    if (l >= Triangulation_hierarchy_2__maxlevel) break;
-    v=u; u=v->up();
+  CGAL_precondition(!this->is_infinite(v));
+  if(v->point() == p) return v;
+  Vertex_handle ans = hierarchy[0]->move_if_no_collision_and_give_new_faces(v, p, oif);
+  if(ans != v) return ans; // ans is an existing vertex at p and v was not changed
+  for (int l = 1; l < maxlevel; ++l) {
+    Vertex_handle u = v->up();
+    if (u == Vertex_handle())
+      break;
+    hierarchy[l]->move_if_no_collision(u, p);
+    v = u;
   }
-  return norm;
+  return ans;
 }
 
 template <class Tr_>
@@ -627,7 +626,7 @@ Triangulation_hierarchy_2<Tr_>::insert_and_give_new_faces(const Point  &p,
   Locate_type lt;
   int i;
   // locate using hierarchy
-  Face_handle positions[Triangulation_hierarchy_2__maxlevel];
+  Face_handle positions[maxlevel];
   locate_in_all(p,lt,i,loc,positions);
   Vertex_handle vertex=
     hierarchy[0]->Tr_Base::insert_and_give_new_faces(p,lt,positions[0],i,oif);
@@ -666,7 +665,7 @@ insert_and_give_new_faces(const Point  &p,
     // locate using hierarchy
     Locate_type ltt;
     int lii;
-    Face_handle positions[Triangulation_hierarchy_2__maxlevel];
+    Face_handle positions[maxlevel];
     locate_in_all(p,ltt,lii,loc,positions);
     //insert in higher levels
     int level  = 1;
@@ -686,7 +685,7 @@ typename Triangulation_hierarchy_2<Tr_>::Face_handle
 Triangulation_hierarchy_2<Tr_>::
 locate(const Point& p, Locate_type& lt, int& li, Face_handle loc) const
 {
-  Face_handle positions[Triangulation_hierarchy_2__maxlevel];
+  Face_handle positions[maxlevel];
   locate_in_all(p,lt,li,loc,positions);
   return positions[0];
 }
@@ -708,11 +707,11 @@ locate_in_all(const Point& p,
               Locate_type& lt,
               int& li,
               Face_handle loc,
-              Face_handle pos[Triangulation_hierarchy_2__maxlevel]) const
+              Face_handle pos[maxlevel]) const
 {
   Face_handle position;
   Vertex_handle nearest;
-  int level  = Triangulation_hierarchy_2__maxlevel;
+  int level  = maxlevel;
   typename Geom_traits::Compare_distance_2
     closer = geom_traits().compare_distance_2_object();
 
@@ -721,7 +720,7 @@ locate_in_all(const Point& p,
 
   // find the highest level with enough vertices that is at the same time 2D
   while ( (hierarchy[--level]->number_of_vertices()
-           < static_cast<size_type> (Triangulation_hierarchy_2__minsize ))
+           < static_cast<size_type> (minsize ))
           || (hierarchy[level]->dimension()<2) ){
     if ( ! level) break;  // do not go below 0
   }
@@ -729,7 +728,7 @@ locate_in_all(const Point& p,
     level--;
   }
 
-  for (int i=level+1; i<Triangulation_hierarchy_2__maxlevel;++i) pos[i]=nullptr;
+  for (int i=level+1; i<maxlevel;++i) pos[i]=nullptr;
   while(level > 0) {
     pos[level]=position=hierarchy[level]->locate(p, position);
     // locate at that level from "position"
@@ -770,10 +769,10 @@ int
 Triangulation_hierarchy_2<Tr_>::
 random_level()
 {
-  boost::geometric_distribution<> proba(1.0/Triangulation_hierarchy_2__ratio);
+  boost::geometric_distribution<> proba(1.0/double(ratio));
   boost::variate_generator<boost::rand48&, boost::geometric_distribution<> > die(random, proba);
 
-  return (std::min)(die(), Triangulation_hierarchy_2__maxlevel)-1;
+  return (std::min)(die(), (int)maxlevel)-1;
 
 }
 
diff --git a/Triangulation_3/include/CGAL/Triangulation_hierarchy_3.h b/Triangulation_3/include/CGAL/Triangulation_hierarchy_3.h
index d15a5bdcfe5..bf73537b25d 100644
--- a/Triangulation_3/include/CGAL/Triangulation_hierarchy_3.h
+++ b/Triangulation_3/include/CGAL/Triangulation_hierarchy_3.h
@@ -62,7 +62,7 @@ class Triangulation_hierarchy_3
 {
   // parameterization of the hierarchy
   // maximal number of points is 30^5 = 24 millions !
-  enum { ratio = 30 };
+  enum Ratio : int { ratio = 30 };
   enum { minsize = 20};
   enum { maxlevel = 5};
 
@@ -504,26 +504,54 @@ is_valid(bool verbose, int level) const
   bool result = true;
 
   // verify correctness of triangulation at all levels
-  for(int i=0; i<maxlevel; ++i)
+  for(int i=0; i<maxlevel; ++i){
         result = result && hierarchy[i]->is_valid(verbose, level);
-
-  // verify that lower level has no down pointers
+        if(verbose && (! result)){
+          std::cerr << "triangulation at level " << i << " invalid" << std::endl;
+        }
+  }
+  // verify that lowest level has no down pointers
   for( Finite_vertices_iterator it = hierarchy[0]->finite_vertices_begin(),
-       end = hierarchy[0]->finite_vertices_end(); it != end; ++it)
+       end = hierarchy[0]->finite_vertices_end(); it != end; ++it){
     result = result && (it->down() == Vertex_handle());
+    if(verbose && (! result)){
+          std::cerr << "lowest level has a down pointer" << std::endl;
+    }
+  }
 
-  // verify that other levels has down pointer and reciprocal link is fine
+  // verify that other levels have down pointer and reciprocal link is fine
   for(int j=1; j<maxlevel; ++j)
     for( Finite_vertices_iterator it = hierarchy[j]->finite_vertices_begin(),
          end = hierarchy[j]->finite_vertices_end(); it != end; ++it)
-      result = result && &*(it) == &*(it->down()->up());
+    {
+      result = result && (it->down() != Vertex_handle());
+      if(verbose && (! result)){
+        std::cerr << "missing down pointer" << std::endl;
+      }
+      if(it->down() == Vertex_handle()){
+        return false;
+      }
+      result = result && Vertex_handle(it) == Vertex_handle(it->down()->up());
+      if(verbose && (! result)){
+          std::cerr << "wrong reciprocal link with down()" << std::endl;
+      }
+      result = result && Vertex_handle(it)->point() == Vertex_handle(it->down())->point();
+      if(verbose && (! result)){
+          std::cerr << "inconsistent vertex positions" << std::endl;
+      }
+    }
 
-  // verify that other levels has down pointer and reciprocal link is fine
+  // verify that all levels have up pointer and reciprocal link is fine
   for(int k=0; k<maxlevel-1; ++k)
     for( Finite_vertices_iterator it = hierarchy[k]->finite_vertices_begin(),
          end = hierarchy[k]->finite_vertices_end(); it != end; ++it)
+    {
       result = result && ( it->up() == Vertex_handle() ||
-                &*it == &*(it->up())->down() );
+                Vertex_handle(it) == Vertex_handle(it->up())->down() );
+      if(verbose && (! result)){
+          std::cerr << "wrong reciprocal link with up()" << std::endl;
+      }
+    }
 
   return result;
 }
@@ -718,14 +746,13 @@ move_if_no_collision(Vertex_handle v, const Point & p)
 {
   CGAL_precondition(!this->is_infinite(v));
   if(v->point() == p) return v;
-  Vertex_handle ans;
-  for (int l = 0; l < maxlevel; ++l) {
+  Vertex_handle ans = hierarchy[0]->move_if_no_collision(v, p);
+  if(ans != v) return ans; // ans is an existing vertex at p and v was not changed
+  for (int l = 1; l < maxlevel; ++l) {
     Vertex_handle u = v->up();
-    if(l) hierarchy[l]->move_if_no_collision(v, p);
-    else ans = hierarchy[l]->move_if_no_collision(v, p);
-    if(ans != v) return ans;
     if (u == Vertex_handle())
       break;
+    hierarchy[l]->move_if_no_collision(u, p);
     v = u;
   }
   return ans;
@@ -753,17 +780,15 @@ Triangulation_hierarchy_3<Tr>::
 move_if_no_collision_and_give_new_cells(
   Vertex_handle v, const Point & p, OutputItCells fit)
 {
-  CGAL_precondition(!is_infinite(v));
+  CGAL_precondition(!this->is_infinite(v));
   if(v->point() == p) return v;
-  Vertex_handle ans;
-  for (int l = 0; l < maxlevel; ++l) {
+  Vertex_handle ans = hierarchy[0]->move_if_no_collision_and_give_new_cells(v, p, fit);
+  if(ans != v) return ans; // ans is an existing vertex at p and v was not changed
+  for (int l = 1; l < maxlevel; ++l) {
     Vertex_handle u = v->up();
-    if(l) hierarchy[l]->move_if_no_collision(v, p);
-    else ans =
-           hierarchy[l]->move_if_no_collision_and_give_new_cells(v, p, fit);
-    if(ans != v) return ans;
     if (u == Vertex_handle())
       break;
+    hierarchy[l]->move_if_no_collision(u, p);
     v = u;
   }
   return ans;