diff --git a/Arrangement_on_surface_2/examples/Arrangement_on_surface_2/polylines.cpp b/Arrangement_on_surface_2/examples/Arrangement_on_surface_2/polylines.cpp
index b9a211852da..6475e747a81 100644
--- a/Arrangement_on_surface_2/examples/Arrangement_on_surface_2/polylines.cpp
+++ b/Arrangement_on_surface_2/examples/Arrangement_on_surface_2/polylines.cpp
@@ -41,6 +41,7 @@ int main ()
   points1[3] = Point_2 (4, 4);
   points1[4] = Point_2 (6, 0);
   Polyline_2            pi1 (&points1[0], &points1[5]);
+  //  insert (arr, pi1);
 
   /* Polyline's outline:
    *
@@ -67,6 +68,7 @@ int main ()
   points2.push_back (Point_2 (5, 3));
   points2.push_back (Point_2 (4, 2));
   Polyline_2            pi2 (points2.begin(), points2.end());
+  //  insert (arr, pi2);
 
   /* Polyline's outline:
    *         
@@ -83,6 +85,8 @@ int main ()
   points3[2] = Point_2 (3, 6);
   points3[3] = Point_2 (5, 2);
   Polyline_2            pi3 (points3.begin(), points3.end());
+  //  insert (arr, pi3);
+
 
   /*Polyline's outline:
    *
@@ -101,11 +105,20 @@ int main ()
   points4.push_back (Point_2(0,1));
   points4.push_back (Point_2(-1,0));
   Polyline_2 pi4 (points4.begin(),points4.end());
-  
-  //  insert (arr, pi1);
-  //  insert (arr, pi2);
-  //  insert (arr, pi3);
-  insert (arr, pi4);
+  //  insert (arr, pi4); 
+
+  /*Polyline's outline:
+   *
+   */
+  std::list<Point_2> points5;
+  points5.push_back (Point_2(-2,-2));
+  points5.push_back (Point_2(2,-2));
+  points5.push_back (Point_2(2,2));
+  points5.push_back (Point_2(-2,2));
+  points5.push_back (Point_2(-2,-2));
+  Polyline_2 pi5 (points5.begin(),points5.end());
+  insert (arr, pi5); 
+
   
   print_arrangement (arr);
   return 0;
diff --git a/Arrangement_on_surface_2/include/CGAL/Arr_polyline_traits_2.h b/Arrangement_on_surface_2/include/CGAL/Arr_polyline_traits_2.h
index 3b2bf3d7cf2..ba3f330f4a0 100644
--- a/Arrangement_on_surface_2/include/CGAL/Arr_polyline_traits_2.h
+++ b/Arrangement_on_surface_2/include/CGAL/Arr_polyline_traits_2.h
@@ -481,19 +481,25 @@ public:
         return oi;
       }
 
+      // Polyline contains at least 2 segments!
+
       typename Segment_traits_2::Compare_xy_2 comp_xy =
         seg_traits->compare_xy_2_object();
+      typename Segment_traits_2::Is_vertical_2 is_vertical =
+	seg_traits->is_vertical_2_object();
       Construct_x_monotone_curve_2 construct_x_monotone_curve = 
 	m_traits->construct_x_monotone_curve_2_object();
 
       const_seg_iterator it_start = start_seg;
       const_seg_iterator it_curr = start_seg;
 
-      for (; it_next != end_seg; ++it_next)
+      for (/*it_next was advanced earlier*/; it_next != end_seg; ++it_next)
       {
+	// TODO: Improve this test. Avoid double tests of geometrical elements.
         if ((comp_xy(max_v(*it_curr), min_v(*it_next)) != EQUAL) &&
             (comp_xy(min_v(*it_curr), max_v(*it_next)) != EQUAL) )
         {
+	  // Construct an x-monotone curve from the sub-range which was found
           *oi++ = make_object(construct_x_monotone_curve(it_start, it_next));
           it_start = it_next;
         }
@@ -531,6 +537,9 @@ public:
     void operator()(const X_monotone_curve_2& cv, const Point_2& p,
                     X_monotone_curve_2& c1, X_monotone_curve_2& c2) const
     {
+
+      // TODO: Should this be rewritten using iterators over the segments?
+
       typename Segment_traits_2::Construct_min_vertex_2 min_vertex =
         m_seg_traits->construct_min_vertex_2_object();
       typename Segment_traits_2::Construct_max_vertex_2 max_vertex =
@@ -954,6 +963,20 @@ public:
       return (X_monotone_curve_2(pts + 0, pts + 2));
     }
 
+    /*! Returns an x-monotone curve consists of one given segment.
+     * \param seg input segment
+     * \return A polyline with one segment, namely seg.
+     * TODO: Implement this overload
+     * TODO: Use this implementations in the Make_x_monotone_2, when there's
+     *       only one segment.
+     */
+    X_monotone_curve_2 operator()(const Segment_2 seg) const
+    {      
+      X_monotone_curve_2 res;
+      return res;
+    }
+
+
     template <typename InputIterator>
     X_monotone_curve_2 operator()(InputIterator begin, InputIterator end) const
     {