diff --git a/Packages/Envelope_3/include/CGAL/Envelope_overlay_functor.h b/Packages/Envelope_3/include/CGAL/Envelope_overlay_functor.h
index feec8b01a71..556ac46351e 100644
--- a/Packages/Envelope_3/include/CGAL/Envelope_overlay_functor.h
+++ b/Packages/Envelope_3/include/CGAL/Envelope_overlay_functor.h
@@ -50,6 +50,21 @@ public:
 protected:
 
   typedef typename Minimization_diagram_2::Dcel::Dcel_data_iterator Envelope_data_iterator;
+  typedef std::pair<Vertex_handle, Face_handle>                  Vertex_face_pair;
+  struct Less_vertex_face_pair
+  {
+    bool operator() (const Vertex_face_pair& vf1,
+                     const Vertex_face_pair& vf2) const
+    {
+      Vertex_handle v1 = vf1.first, v2 = vf2.first;
+      Face_handle f1 = vf1.second, f2= vf2.second;
+      return (&*v1 < &*v2 ||
+              (&*v1 == &*v2 && &*f1 < &*f2));
+    }
+  };
+  typedef std::map<Vertex_face_pair, Halfedge_handle,
+                   Less_vertex_face_pair>                        Boundary_cache;
+
 public:
 
   Envelope_overlay_functor(Minimization_diagram_2& md1,
@@ -58,6 +73,11 @@ public:
     : m_1(md1), m_2(md2), m_result(result)
   {
   }
+
+  ~Envelope_overlay_functor()
+  {
+    traversed_vertices.clear();
+  }
   
   void create_face (Face_const_handle1 f1, Face_const_handle2 f2, Res_face_handle res_f)
   {
@@ -244,7 +264,7 @@ public:
     res_h->set_aux_source(1, m_2.non_const_handle(f2));
 
     res_h->twin()->set_aux_source(0, m_1.non_const_handle(h1->twin()));
-    res_h->twin()->set_aux_source(1, m_2.non_const_handle(f2));
+    res_h->twin()->set_aux_source(1, m_2.non_const_handle(f2));       
 
     res_h->set_is_equal_aux_data_in_face(0, h1->get_is_equal_data_in_face());
     res_h->set_is_equal_aux_data_in_face(1, true);
@@ -288,92 +308,10 @@ public:
       CGAL_assertion(false);
 
     // update is_equal/has_equal data in target for the second source map
-    Vertex_handle vh2;
-    Halfedge_handle hh2;
-    Face_handle fh2;
-    // update target
-    Object trg_src2 = res_h->target()->get_aux_source(1);
-    if (assign(vh2, trg_src2))
-      // todo
-    {
-      if (vh2->is_isolated())
-        copy_halfedge_target_info_from_vertex_face_info(vh2, res_h, 1);
-      else
-      {
-        // we have a vertex vh2 on the boundary of the face f2
-        bool is_equal = vh2->is_equal_data(f2->begin_data(), f2->end_data());
-        bool has_equal = vh2->has_equal_data(f2->begin_data(), f2->end_data());
-
-        res_h->set_is_equal_aux_data_in_target(1, is_equal);
-        res_h->set_has_equal_aux_data_in_target(1, has_equal);
-      }
-    } 
-    else if (assign(hh2, trg_src2))
-      // we should find the halfedge (hh2 or hh2->twin()) that points to face
-      // f2, and check the in_face flags there
-    {
-      CGAL_assertion(hh2->face() == m_2.non_const_handle(f2) ||
-                     hh2->twin()->face() == m_2.non_const_handle(f2));
-      Face_handle f = m_2.non_const_handle(f2);
-      if (hh2->face() == f)
-        copy_halfedge_target_info_from_halfedge_face_info(hh2, res_h, 1);
-      else
-        copy_halfedge_target_info_from_halfedge_face_info(hh2->twin(), res_h, 1);
-    }
-    else
-    {
-      CGAL_assertion(assign(fh2, trg_src2));
-      assign(fh2, trg_src2);
-      // the edge and target are in the same face, so we set halfedge-target
-      // equal information to true, and has equal acoording to face data
-      res_h->set_is_equal_aux_data_in_target(1, true);
-      res_h->set_has_equal_aux_data_in_target(1, !fh2->has_no_data());
-//      set_halfedge_target_info(res_h, 1, true);
-    }
-    
-    // update source
-    Object src_src2 = res_h->source()->get_aux_source(1);
-    if (assign(vh2, src_src2))
-      // todo
-    {
-      // we have a vertex vh2 on the boundary of the face f2
-      // or an isolated vertex
-      if (vh2->is_isolated())
-        copy_halfedge_target_info_from_vertex_face_info(vh2, res_h->twin(), 1);
-      else
-      {
-        bool is_equal = vh2->is_equal_data(f2->begin_data(), f2->end_data());
-        bool has_equal = vh2->has_equal_data(f2->begin_data(), f2->end_data());
-
-        res_h->twin()->set_is_equal_aux_data_in_target(1, is_equal);
-        res_h->twin()->set_has_equal_aux_data_in_target(1, has_equal);
-      }
-    }
-    else if (assign(hh2, src_src2))
-      // we should find the halfedge (hh2 or hh2->twin()) that points to face
-      // f2, and check the in_face flags there
-    {
-      CGAL_assertion(hh2->face() == m_2.non_const_handle(f2) ||
-                     hh2->twin()->face() == m_2.non_const_handle(f2));
-      Face_handle f = m_2.non_const_handle(f2);
-      if (hh2->face() == f)
-        copy_halfedge_target_info_from_halfedge_face_info(hh2, res_h->twin(), 1);
-      else
-        copy_halfedge_target_info_from_halfedge_face_info(hh2->twin(), res_h->twin(), 1);
-    }
-
-    else
-    {
-      CGAL_assertion(assign(fh2, src_src2));
-      assign(fh2, src_src2);
-      // the edge and soucre are in the same face, so we set halfedge-target
-      // information to true
-      res_h->twin()->set_is_equal_aux_data_in_target(1, true);
-      res_h->twin()->set_has_equal_aux_data_in_target(1, !fh2->has_no_data());
-
-//      set_halfedge_target_info(res_h->twin(), 1, true);
-    }
-
+    update_halfedge_flags_in_face(res_h, m_2.non_const_handle(f2), 1);
+   
+    // update is_equal/has_equal data in source for the second source map
+    update_halfedge_flags_in_face(res_h->twin(), m_2.non_const_handle(f2), 1);
   }                           
 
 
@@ -387,6 +325,7 @@ public:
     res_h->twin()->set_aux_source(0, m_1.non_const_handle(f1));
     res_h->twin()->set_aux_source(1, m_2.non_const_handle(h2->twin()));
 
+    // update halfedge-face flags of the new halfedge
     res_h->set_is_equal_aux_data_in_face(0, true);
     res_h->set_is_equal_aux_data_in_face(1, h2->get_is_equal_data_in_face());
     res_h->set_has_equal_aux_data_in_face(0, !f1->has_no_data());
@@ -429,90 +368,9 @@ public:
       CGAL_assertion(false);
 
     // update is_equal/has_equal data in target for the first source map
-    Vertex_handle vh1;
-    Halfedge_handle hh1;
-    Face_handle fh1;
-    // update target
-    Object trg_src1 = res_h->target()->get_aux_source(0);
-    if (assign(vh1, trg_src1))
-      // todo
-    {
-      if (vh1->is_isolated())
-        copy_halfedge_target_info_from_vertex_face_info(vh1, res_h, 0);
-      else
-      {
-        // we have a vertex vh1 on the boundary of the face f1
-        bool is_equal = vh1->is_equal_data(f1->begin_data(), f1->end_data());
-        bool has_equal = vh1->has_equal_data(f1->begin_data(), f1->end_data());
-
-        res_h->set_is_equal_aux_data_in_target(0, is_equal);
-        res_h->set_has_equal_aux_data_in_target(0, has_equal);
-      }
-    }
-    else if (assign(hh1, trg_src1))
-      // we should find the halfedge (hh1 or hh1>twin()) that points to face
-      // f1, and check the in_face flags there
-    {
-      CGAL_assertion(hh1->face() == m_1.non_const_handle(f1) ||
-                     hh1->twin()->face() == m_1.non_const_handle(f1));
-      Face_handle f = m_1.non_const_handle(f1);
-      if (hh1->face() == f)
-        copy_halfedge_target_info_from_halfedge_face_info(hh1, res_h, 0);
-      else
-        copy_halfedge_target_info_from_halfedge_face_info(hh1->twin(), res_h, 0);
-    }
-    else
-    {
-      CGAL_assertion(assign(fh1, trg_src1));
-      assign(fh1, trg_src1);
-      // the edge and target are in the same face, so we set halfedge-target
-      // information to true
-      res_h->set_is_equal_aux_data_in_target(0, true);
-      res_h->set_has_equal_aux_data_in_target(0, !fh1->has_no_data());
-//      set_halfedge_target_info(res_h, 0, true);
-    }
-
-    // update source
-    Object src_src1 = res_h->source()->get_aux_source(0);
-    if (assign(vh1, src_src1))
-      // todo
-    {
-      if (vh1->is_isolated())
-        copy_halfedge_target_info_from_vertex_face_info(vh1, res_h->twin(), 0);
-      else
-      {
-        // we have a vertex vh1 on the boundary of the face f1
-        bool is_equal = vh1->is_equal_data(f1->begin_data(), f1->end_data());
-        bool has_equal = vh1->has_equal_data(f1->begin_data(), f1->end_data());
-
-        res_h->twin()->set_is_equal_aux_data_in_target(0, is_equal);
-        res_h->twin()->set_has_equal_aux_data_in_target(0, has_equal);
-      }
-    }
-    else if (assign(hh1, src_src1))
-      // we should find the halfedge (hh1 or hh1->twin()) that points to face
-      // f1, and check the in_face flags there
-    {
-      CGAL_assertion(hh1->face() == m_1.non_const_handle(f1) ||
-                     hh1->twin()->face() == m_1.non_const_handle(f1));
-      Face_handle f = m_1.non_const_handle(f1);
-      if (hh1->face() == f)
-        copy_halfedge_target_info_from_halfedge_face_info(hh1, res_h->twin(), 0);
-      else
-        copy_halfedge_target_info_from_halfedge_face_info(hh1->twin(), res_h->twin(), 0);
-    }
-
-    else
-    {
-      CGAL_assertion(assign(fh1, src_src1));
-      assign(fh1, src_src1);
-      // the edge and soucre are in the same face, so we set halfedge-target
-      // information to true
-      res_h->twin()->set_is_equal_aux_data_in_target(0, true);
-      res_h->twin()->set_has_equal_aux_data_in_target(0, !fh1->has_no_data());
-
-//      set_halfedge_target_info(res_h->twin(), 0, true);
-    }
+    update_halfedge_flags_in_face(res_h, m_1.non_const_handle(f1), 0);
+    // update is_equal/has_equal data in source for the first source map
+    update_halfedge_flags_in_face(res_h->twin(), m_1.non_const_handle(f1), 0);
   }
 
 protected:
@@ -545,10 +403,125 @@ protected:
     to->set_is_equal_aux_data_in_target(id, from->get_is_equal_data_in_face());
     to->set_has_equal_aux_data_in_target(id, from->get_has_equal_data_in_face());
   }
+
+  // find a halfedge that v is its target and f is its face
+  Halfedge_handle find_halfedge_by_vertex_and_face(Vertex_handle v, Face_handle f)
+  {
+    // should always invoke this method when v is on the boundary of f
+    
+    // for the complexity of the total algorithm, we only loop over
+    // the halfedges of each vertex once and cache the triples of
+    // vertex-face-halfedge for future such questions
+    Vertex_face_pair query(v, f);
+    typename Boundary_cache::iterator iter = traversed_vertices.find(query);
+    Halfedge_handle result;
+    if (iter == traversed_vertices.end())
+    {
+      // first time to check this vertex - traverse all its halfedges
+      // and update the map
+      typename Minimization_diagram_2::Halfedge_around_vertex_circulator vc =
+        v->incident_halfedges(),
+        vc_begin = vc;
+      do {
+        Halfedge_handle hh = vc;
+        // update the map
+        traversed_vertices[Vertex_face_pair(v, hh->face())] = hh;
+        // check for reult
+        if (hh->face() == f)
+          result = hh;
+        ++vc;
+      } while (vc != vc_begin);
+    }
+    else
+    {
+      // take it from the map
+      result = iter->second;
+    }
+    CGAL_assertion(result != Halfedge_handle());
+    return result;
+  }
+
+  // update halfedge-target flags of new_h that is created inside face in_face
+  // id is the source diagram where in_face comes from
+  // (i.e. the id of the aux information to update)
+  void update_halfedge_flags_in_face(Halfedge_handle new_h, Face_handle in_face, unsigned int id)
+  {
+      // update is_equal/has_equal data in target for the first source map
+    Vertex_handle vh;
+    Halfedge_handle hh;
+    Face_handle fh;
+    // update target
+    Object trg_src = new_h->target()->get_aux_source(id);
+    if (assign(vh, trg_src))
+      // todo
+    {
+      if (vh->is_isolated())
+        copy_halfedge_target_info_from_vertex_face_info(vh, new_h, id);
+      else
+      {
+        // we have a vertex vh on the boundary of the face in_face
+        // todo: get rid of this calculations: (using unknown value for has_equal flag)
+        CGAL_assertion_code(
+          bool calc_is_equal = vh->is_equal_data(in_face->begin_data(), in_face->end_data());
+        )
+        bool calc_has_equal = vh->has_equal_data(in_face->begin_data(), in_face->end_data());
+
+        // find the halfedge with target vh on the boundary of in_face
+        Halfedge_handle h_of_vh_and_in_face =
+               find_halfedge_by_vertex_and_face(vh, in_face);
+        // is_equal relationship is easy:
+        bool is_equal = h_of_vh_and_in_face->get_is_equal_data_in_face() &&
+                        h_of_vh_and_in_face->get_is_equal_data_in_target();
+        CGAL_assertion(is_equal == calc_is_equal);
+
+        // has_equal relationship is problematic in one case:
+        bool has_equal;
+        if (h_of_vh_and_in_face->get_has_equal_data_in_face() == true)
+          has_equal = h_of_vh_and_in_face->get_has_equal_data_in_target();
+        else
+          if (h_of_vh_and_in_face->get_has_equal_data_in_target() == false)
+            has_equal = false;
+          else if (h_of_vh_and_in_face->get_is_equal_data_in_target() == true)
+            has_equal = false;
+          else
+          {
+            CGAL_assertion_code(std::cout << "happen" << std::endl;)
+            has_equal = calc_has_equal;
+          }
+        CGAL_assertion(has_equal == calc_has_equal);
+
+        new_h->set_is_equal_aux_data_in_target(id, is_equal);
+        new_h->set_has_equal_aux_data_in_target(id, calc_has_equal);
+      }
+    }
+    else if (assign(hh, trg_src))
+      // we should find the halfedge (hh or hh>twin()) that points to face
+      // in_face, and check the halfedge-face flags there
+    {
+      CGAL_assertion(hh->face() == in_face || hh->twin()->face() == in_face);
+      if (hh->face() == in_face)
+        copy_halfedge_target_info_from_halfedge_face_info(hh, new_h, id);
+      else
+        copy_halfedge_target_info_from_halfedge_face_info(hh->twin(), new_h, id);
+    }
+    else
+    {
+      CGAL_assertion_code(bool b =)
+      assign(fh, trg_src);
+      CGAL_assertion(b);
+      // the edge and target are in the same face, so we set halfedge-target
+      // is_equal information to true, and has_equal information according to
+      // the face data
+      CGAL_assertion(fh == in_face);
+      new_h->set_is_equal_aux_data_in_target(id, true);
+      new_h->set_has_equal_aux_data_in_target(id, !fh->has_no_data());
+    }
+  }
   
   Minimization_diagram_2& m_1;
   Minimization_diagram_2& m_2;
-  Minimization_diagram_2& m_result;  
+  Minimization_diagram_2& m_result;
+  Boundary_cache          traversed_vertices; 
 };
 
 CGAL_END_NAMESPACE