diff --git a/Combinatorial_map/include/CGAL/Combinatorial_map_functionalities.h b/Combinatorial_map/include/CGAL/Combinatorial_map_functionalities.h
index 38a6f3d5884..ff1a8e6f506 100644
--- a/Combinatorial_map/include/CGAL/Combinatorial_map_functionalities.h
+++ b/Combinatorial_map/include/CGAL/Combinatorial_map_functionalities.h
@@ -1080,11 +1080,34 @@ namespace CGAL {
     }
 
     // Simplify the path by removing all spurs
+    // @return true iff at least one spur was removed
     bool remove_spurs(Path_on_surface<Map>& path) const
     {
+      /* TODO TEMPO POUR TEST */
+      Path_on_surface_with_rle<Map> prle(path);
+      prle.remove_spurs();
+      /* TODO FIN TEMPO POUR TEST */
+
       bool res=false;
       while(remove_spurs_one_step(path))
       { res=true; }
+
+      /* TODO TEMPO POUR TEST */
+      Path_on_surface<Map> p2(prle);
+      if (p2!=path)
+      {
+        std::cout<<"Error: p2!=path: "<<std::endl
+                <<"p2= "<<p2<<"; "<<std::endl
+               <<"path="<<path<<"; "<<std::endl
+              <<"prle="<<prle<<std::endl;
+        prle.display_pos_and_neg_turns();
+      }
+      else
+      {
+        std::cout<<"GOOD JOB for path "<<p2<<std::endl;
+      }
+      /* TODO FIN TEMPO POUR TEST */
+
       return res;
     }
 
@@ -1341,7 +1364,7 @@ namespace CGAL {
 
       bool modified=false;
       // std::cout<<"RS ";
-      remove_spurs_one_step(path);
+      remove_spurs(path); // TODO BEFORE remove_spurs_one_step(path);
 
       /* display(); display_pos_and_neg_turns();
       std::cout<<std::endl; */
@@ -1355,7 +1378,8 @@ namespace CGAL {
           /* std::cout<<"BF "; display(); display_pos_and_neg_turns();
           std::cout<<std::endl; */
 
-          modified=modified || remove_spurs_one_step(path);
+          modified=modified || remove_spurs(path); // TODO BEFORE remove_spurs_one_step(path);
+
 
           /* std::cout<<"RS "; display(); display_pos_and_neg_turns();
           std::cout<<std::endl; */
diff --git a/Combinatorial_map/include/CGAL/Path_on_surface.h b/Combinatorial_map/include/CGAL/Path_on_surface.h
index 9ca34d5e856..5657c447c47 100644
--- a/Combinatorial_map/include/CGAL/Path_on_surface.h
+++ b/Combinatorial_map/include/CGAL/Path_on_surface.h
@@ -355,9 +355,7 @@ public:
   {
     assert(is_valid());
     assert(i<m_path.size());
-
-    if (!is_closed() && i==length()-1)
-    { return 0; }
+    assert (is_closed() || i<length()-1);
 
     Dart_const_handle d1=get_ith_dart(i);
     Dart_const_handle d2=get_next_dart(i);
@@ -381,9 +379,7 @@ public:
   {
     assert(is_valid());
     assert(i<m_path.size());
-
-    if (!is_closed() && i==length()-1)
-    { return 0; }
+    assert (is_closed() || i<length()-1);
 
     Dart_const_handle d1=m_map.template beta<2>(get_ith_dart(i));
     Dart_const_handle d2=m_map.template beta<2>(get_next_dart(i));
@@ -402,7 +398,6 @@ public:
     return res;
   }
 
-
   std::vector<std::size_t> compute_positive_turns() const
   {
     std::vector<std::size_t> res;
@@ -410,17 +405,9 @@ public:
 
     std::size_t i;
     for (i=0; i<m_path.size()-1; ++i)
-    {
-      if (m_path[i]==m_map.template beta<2>(m_path[i+1]))
-      { res.push_back(0); }
-      else { res.push_back(next_positive_turn(i)); }
-    }
+    { res.push_back(next_positive_turn(i)); }
     if (is_closed())
-    {
-      if (m_path[i]==m_map.template beta<2>(m_path[0]))
-      { res.push_back(0); }
-      else { res.push_back(next_positive_turn(i)); }
-    }
+    { res.push_back(next_positive_turn(i)); }
     return res;
   }
 
@@ -431,17 +418,9 @@ public:
 
     std::size_t i;
     for (i=0; i<m_path.size()-1; ++i)
-    {
-      if (m_path[i]==m_map.template beta<2>(m_path[i+1]))
-      { res.push_back(0); }
-      else { res.push_back(next_negative_turn(i)); }
-    }
+    { res.push_back(next_negative_turn(i)); }
     if (is_closed())
-    {
-      if (m_path[i]==m_map.template beta<2>(m_path[0]))
-      { res.push_back(0); }
-      else { res.push_back(next_negative_turn(i)); }
-    }
+    { res.push_back(next_negative_turn(i)); }
     return res;
   }
 
diff --git a/Combinatorial_map/include/CGAL/Path_on_surface_with_rle.h b/Combinatorial_map/include/CGAL/Path_on_surface_with_rle.h
index e8c8392f57c..24ddc1e0d8c 100644
--- a/Combinatorial_map/include/CGAL/Path_on_surface_with_rle.h
+++ b/Combinatorial_map/include/CGAL/Path_on_surface_with_rle.h
@@ -25,6 +25,7 @@
 #include <utility>
 #include <iostream>
 #include <iterator>
+#include <vector>
 
 namespace CGAL {
 
@@ -40,6 +41,9 @@ public:
   typedef Map_ Map;
   typedef typename Map::Dart_handle Dart_handle;
   typedef typename Map::Dart_const_handle Dart_const_handle;
+  typedef std::list<std::pair<Dart_const_handle, int> > List_of_dart_length;
+  typedef typename List_of_dart_length::iterator List_iterator;
+  typedef typename List_of_dart_length::const_iterator List_const_iterator;
 
   typedef Path_on_surface_with_rle<Map> Self;
 
@@ -52,6 +56,8 @@ public:
                                                                 m_is_closed(apath.is_closed()),
                                                                 m_length(apath.length())
   {
+    if (apath.is_empty()) return;
+
     std::size_t i=0, j=0, starti=0, length=0;
     bool positive_flat=false;
     bool negative_flat=false;    
@@ -160,6 +166,276 @@ public:
     return true;
   }
 
+  void advance_iterator(List_iterator& it)
+  {
+    assert(it!=m_path.end());
+    it=std::next(it);
+    if (is_closed() && it==m_path.end())
+    { it=m_path.begin(); } // Here the path is closed, and it is the last element of the list
+  }
+
+  void advance_iterator(List_const_iterator& it) const
+  {
+    assert(it!=m_path.end());
+    it=std::next(it);
+    if (is_closed() && it==m_path.end())
+    { it=m_path.begin(); } // Here the path is closed, and it is the last element of the list
+  }
+
+  void retreat_iterator(List_iterator& it)
+  {
+    assert(it!=m_path.end());
+    assert(it!=m_path.begin() || is_closed());
+    if (is_closed() && it==m_path.begin())
+    { it=m_path.end(); }
+    it=std::prev(it);
+  }
+
+  void retreat_iterator(List_const_iterator& it) const
+  {
+    assert(it!=m_path.end());
+    assert(it!=m_path.begin() || is_closed());
+    if (is_closed() && it==m_path.begin())
+    { it=m_path.end(); }
+    it=std::prev(it);
+  }
+
+  List_iterator next_iterator(const List_iterator& it)
+  {
+    List_iterator res=it;
+    advance_iterator(res);
+    return res;
+  }
+
+  List_const_iterator next_iterator(const List_const_iterator& it) const
+  {
+    List_const_iterator res=it;
+    advance_iterator(res);
+    return res;
+  }
+
+  List_iterator prev_iterator(const List_iterator& it) const
+  {
+    List_iterator res=it;
+    retreat_iterator(res);
+    return res;
+ }
+
+  List_const_iterator prev_iterator(const List_const_iterator& it) const
+  {
+    List_const_iterator res=it;
+    retreat_iterator(res);
+    return res;
+ }
+
+  // @return true iff there is a dart after it
+  bool next_dart_exist(const List_const_iterator& it) const
+  {
+    assert(it!=m_path.end());
+    return is_closed() || std::next(it)!=m_path.end();
+  }
+
+  // Return true if it is the beginning of a spur.
+  bool is_spur(const List_const_iterator& it) const
+  {
+    assert(it!=m_path.end());
+    return it->second==0 &&
+        next_dart_exist(it) &&
+        m_map.template beta<2>(it->first)==next_iterator(it)->first;
+  }
+
+  // Remove the spur given by it; move it to the element before it
+  // (m_path.end() if the path becomes empty).
+  void remove_spur(List_iterator& it)
+  {
+    assert(is_spur(it));
+    it=m_path.erase(it); // Erase the first dart of the spur
+    if (is_closed() && it==m_path.end())
+    { it=m_path.begin(); }
+
+    if (it->second==0) // a flat part having only one dart
+    {
+      it=m_path.erase(it); // Erase the second dart of the spur
+      if (is_closed() && it==m_path.end())
+      { it=m_path.begin(); }
+    }
+    else
+    { // Here we need to reduce the length of the flat part
+      if (it->second>0)
+      {
+        --(it->second);
+        it->first=m_map.template beta<1, 2, 1>(it->first);
+      }
+      else
+      {
+        ++(it->second);
+        it->first=m_map.template beta<2, 0, 2, 0, 2>(it->first);
+      }
+    }
+
+    // Now move it to the element before the removed spur
+    // except if the path has become empty, or if it is not closed
+    // and we are in its first element.
+    if (m_path.empty())
+    {
+      assert(it==m_path.end());
+      m_is_closed=false;
+    }
+    else if (is_closed() || it!=m_path.begin())
+    {
+      retreat_iterator(it); // go to the previous element
+      // TODO we need to test if the previous flat part should be merged with the next one
+      if (compute_positive_turns(it, next(it))==2 || compute_negative_turns(it, next(it))==2 )
+      {
+
+      }
+    }
+  }
+
+  // Move it to the next spur after it. Go to m_path.end() if there is no
+  // spur in the path.
+  void move_to_next_spur(List_iterator& it)
+  {
+     assert(it!=m_path.end());
+    List_iterator itend=(is_closed()?it:m_path.end());
+    do
+    {
+      advance_iterator(it);
+      if (is_spur(it)) { return; }
+    }
+    while(it!=itend);
+    it=m_path.end(); // Here there is no spur in the whole path
+  }
+
+  // Simplify the path by removing all spurs
+  // @return true iff at least one spur was removed
+  bool remove_spurs()
+  {
+    bool res=false;
+    List_iterator it=m_path.begin();
+    while(it!=m_path.end())
+    {
+      if (is_spur(it)) { remove_spur(it); res=true; }
+      else { move_to_next_spur(it); }
+    }
+    return res;
+  }
+
+  /// @return the turn between dart it and next dart.
+  ///         (turn is position of the second edge in the cyclic ordering of
+  ///          edges starting from the first edge around the second extremity
+  ///          of the first dart)
+  std::size_t next_positive_turn(const List_const_iterator& it) const
+  {
+    assert(is_valid());
+    assert(it!=m_path.end());
+    assert(is_closed() || std::next(it)!=m_path.end());
+
+    Dart_const_handle d1=it->first;
+    if (it->second!=0)
+    {
+      if (it->second<0) { return -(it->second); }
+      else { return it->second; }
+    }
+
+    Dart_const_handle d2=next_iterator(it)->first;
+    assert(d1!=d2);
+
+    if (d2==m_map.template beta<2>(d1))
+    { return 0; }
+
+    std::size_t res=1;
+    while (m_map.template beta<1>(d1)!=d2)
+    {
+      ++res;
+      d1=m_map.template beta<1, 2>(d1);
+    }
+    // std::cout<<"next_positive_turn="<<res<<std::endl;
+    return res;
+  }
+
+  /// Same than next_positive_turn but turning in reverse orientation around vertex.
+  std::size_t next_negative_turn(const List_const_iterator& it) const
+  {
+    assert(is_valid());
+    assert(it!=m_path.end());
+    assert(is_closed() || std::next(it)!=m_path.end());
+
+    Dart_const_handle d1=m_map.template beta<2>(it->first);
+    if (it->second!=0)
+    {
+      if (it->second<0) { return -(it->second); }
+      else { return it->second; }
+    }
+
+    Dart_const_handle d2=m_map.template beta<2>(next_iterator(it)->first);
+    assert(d1!=d2);
+
+    if (d2==m_map.template beta<2>(d1))
+    { return 0; }
+
+    std::size_t res=1;
+    while (m_map.template beta<0>(d1)!=d2)
+    {
+      ++res;
+      d1=m_map.template beta<0, 2>(d1);
+    }
+    // std::cout<<"next_negative_turn="<<res<<std::endl;
+    return res;
+  }
+
+  std::vector<std::size_t> compute_positive_turns() const
+  {
+    std::vector<std::size_t> res;
+    if (is_empty()) return res;
+    for (auto it=m_path.begin(), itend=m_path.end(); it!=itend; ++it)
+    {
+      if (is_closed() || std::next(it)!=m_path.end())
+      { res.push_back(next_positive_turn(it)); }
+    }
+    return res;
+  }
+
+  std::vector<std::size_t> compute_negative_turns() const
+  {
+    std::vector<std::size_t> res;
+    if (is_empty()) return res;
+    for (auto it=m_path.begin(), itend=m_path.end(); it!=itend; ++it)
+    {
+      if (is_closed() || std::next(it)!=m_path.end())
+      { res.push_back(next_negative_turn(it)); }
+    }
+    return res;
+  }
+
+  std::vector<std::size_t> compute_turns(bool positive) const
+  { return (positive?compute_positive_turns():compute_negative_turns()); }
+
+  void display_positive_turns() const
+  {
+    std::cout<<"+(";
+    std::vector<std::size_t> res=compute_positive_turns();
+    for (std::size_t i=0; i<res.size(); ++i)
+    { std::cout<<res[i]<<(i<res.size()-1?" ":""); }
+    std::cout<<")";
+  }
+
+  void display_negative_turns() const
+  {
+    std::cout<<"-(";
+    std::vector<std::size_t> res=compute_negative_turns();
+    for (std::size_t i=0; i<res.size(); ++i)
+    { std::cout<<res[i]<<(i<res.size()-1?" ":""); }
+    std::cout<<")";
+  }
+
+  void display_pos_and_neg_turns() const
+  {
+    display_positive_turns();
+    std::cout<<"  ";
+    display_negative_turns();
+  }
+
   void display() const
   {
     for (auto it=m_path.begin(), itend=m_path.end(); it!=itend; ++it)
@@ -179,7 +455,7 @@ public:
 
 protected:
   const Map& m_map; // The underlying map
-  std::list<std::pair<Dart_const_handle, int> > m_path; // The sequence of turning darts, plus the length of the flat part after the dart (a flat part is a sequence of dart with positive turn == 2). If negative value k, -k is the length of the flat part, for negative turns (-2).
+  List_of_dart_length m_path; // The sequence of turning darts, plus the length of the flat part after the dart (a flat part is a sequence of dart with positive turn == 2). If negative value k, -k is the length of the flat part, for negative turns (-2).
   bool m_is_closed; // True iff the path is a cycle
   std::size_t m_length;
 };