diff --git a/Packages/Triangulation_2/include/CGAL/Triangulation_default_data_structure_2.h b/Packages/Triangulation_2/include/CGAL/Triangulation_default_data_structure_2.h
index f72b1aff151..2899fca20fb 100644
--- a/Packages/Triangulation_2/include/CGAL/Triangulation_default_data_structure_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Triangulation_default_data_structure_2.h
@@ -259,9 +259,38 @@ public:
 
   void insert_in_face(Vertex* v, Face* f)
     //insert in face
+    // vertex v will replace f->vertex(0)
   {
     CGAL_triangulation_precondition( v != NULL & f != NULL);
-    f->insert_in_face(v);
+    
+    Vertex* v0 = f->vertex(0);
+    Vertex* v2 = f->vertex(2);
+    Vertex* v1 = f->vertex(1);
+    
+    Face* n1 = f->neighbor(1);
+    Face* n2 = f->neighbor(2);
+    int i1,i2 ;
+    if (n1 != NULL) {i1= cw(n1->index(f->vertex(cw(1))));}
+    if (n2 != NULL) {i2 = cw(n2->index(f->vertex(cw(2))));}
+    
+    Face* f1 = new Face(v0, v, v2,
+			f, n1, NULL);
+    
+    Face* f2 = new Face(v0, v1, v,
+			f, NULL, n2);
+
+    f1->set_neighbor(2, f2);
+    f2->set_neighbor(1, f1);
+    if (n1 != NULL) {n1->set_neighbor(i1,f1);}
+    if (n2 != NULL) {n2->set_neighbor(i2,f2);}
+
+    f->set_vertex(0, v);
+    f->set_neighbor(1, f1);
+    f->set_neighbor(2, f2);
+
+    if( v0->face() == f  ) {  v0->set_face(f2); }
+    v->set_face(f);
+
     set_number_of_vertices(number_of_vertices() +1);
   }
 
@@ -281,9 +310,9 @@ public:
     int in;
     CGAL_triangulation_assertion( n->has_vertex(f->vertex(cw(i)), in));
     in = cw(in);
-    f->insert_in_face(v);
+    insert_in_face(v,f);
     flip(n,in); 
-    set_number_of_vertices(number_of_vertices() +1);
+    return;
   }
 
   // the following function insert in 1_dim triangulation
@@ -310,20 +339,46 @@ public:
   void remove_degree_3(Vertex* v, Face* f = NULL)
     // remove a vertex of degree 3
   {
-    CGAL_triangulation_assertion(v != NULL);
-    CGAL_triangulation_assertion(v != _infinite_vertex);
-    CGAL_triangulation_assertion(v->degree() == 3);
-
-    // this cannot happens because of third assertion;
-//     if (number_of_vertices()==1){
-//       set_number_of_vertices(0);
-//       set_finite_vertex(NULL);
-//       delete v;
-//       return;
-//     }
+    CGAL_triangulation_precondition(v != NULL);
+    CGAL_triangulation_precondition(v != _infinite_vertex);
+    CGAL_triangulation_precondition(v->degree() == 3);
 
     if (f == NULL) {f= v->face();}
     else { CGAL_triangulation_assertion( f->has_vertex(v));}
+      
+    int i = f->index(v);
+    Face* left = f->neighbor(cw(i));
+    Face* right = f->neighbor(ccw(i));
+    Face *ll, *rr;
+        
+    int li = left->index(this);
+    int ri = right->index(this);
+    Vertex* q = left->vertex(li);
+    CGAL_triangulation_assertion( left->vertex(li) == right->vertex(ri));
+    
+    ll = left->neighbor(cw(li));
+    if(ll != NULL) {
+      int lli = ll->index(left);
+      ll->set_neighbor(lli, f);
+    } 
+    f->set_neighbor(cw(i), ll);
+    if (f->vertex(ccw(i))->face() == left) f->vertex(ccw(i))->set_face(f);    
+        
+    rr = right->neighbor(ccw(ri));
+    if(rr != NULL) {
+      int rri = rr->index(right);
+      rr->set_neighbor(rri, f);
+    } 
+    f->set_neighbor(ccw(i), rr);
+    if (f->vertex(cw(i))->face() == right) f->vertex(cw(i))->set_face(f);  
+        
+    f->set_vertex(i, q);
+    if (q->face() == right || q->face() == left) {
+	   q->set_face(f);
+    }
+    delete right;
+    delete left;
+        
     // take care of _finite_vertex data member
     if (finite_vertex() == v){
       int i=f->index(v);
@@ -332,23 +387,8 @@ public:
       set_finite_vertex( vv);
     }
 
-// this cannot happens because of third assertion;
-//     if (number_of_vertices() == 2){
-//       Face* ff = f->neighbor(f->index(infinite_vertex()));
-//       delete f;
-//       delete ff;
-//       delete v;
-//       set_number_of_vertices(1);
-//     }
-//    else
-      {
-      f->remove(v); //f->remove returns true because degree 3 has been asserted
-      delete v;
-      //update number of vertices
-      // Vertex* vv= finite_vertex();
-      //Face* f = vv->face();
-      set_number_of_vertices( number_of_vertices() -1);
-    }
+    delete v;
+    set_number_of_vertices( number_of_vertices() -1);
   } 
 
   void remove_second(Vertex* v)
diff --git a/Packages/Triangulation_2/include/CGAL/Triangulation_ds_face_2.h b/Packages/Triangulation_2/include/CGAL/Triangulation_ds_face_2.h
index 6d96b60aae6..ee0ebd4b7ec 100644
--- a/Packages/Triangulation_2/include/CGAL/Triangulation_ds_face_2.h
+++ b/Packages/Triangulation_2/include/CGAL/Triangulation_ds_face_2.h
@@ -131,65 +131,61 @@ public:
 //    
 
   //Additionnal Operations
-  void insert_in_face(Vertex*& v)
-  {
-    CGAL_triangulation_precondition(v != NULL);
-    Vertex* v0 = vertex(0);
-    Vertex* v2 = vertex(2);
-    Vertex* v1 =vertex(1);
-    
-    Face* f1 = neighbor(1);
-    Face* f2 = neighbor(2);
-    int i1 = cw(f1->index(vertex(cw(1))));
-    int i2 = cw(f2->index(vertex(cw(2))));
-    
-    int ccwi, cwi;
-    if(ccwptr != NULL) {
-      ccwi = cw(ccwptr->index(v0));
-    }
-    if(cwptr != NULL) {
-      cwi  = ccw(cwptr->index(v0));
-    }
-    
-    Face* n1ptr = new Face(v0, v, v2,
-			   this, f1, NULL);
-    
-    Face* n2ptr = new Face(v0, v1, v,
-			   this, NULL, f2);
 
-    n1ptr->set_neighbor(2, n2ptr);
-    n2ptr->set_neighbor(1, n1ptr);
-    f1->set_neighbor(i1,n1ptr);
-    f2->set_neighbor(i2,n1ptr);
-
-    set_vertex(0, v);
-    set_neighbor(1, n1ptr);
-    set_neighbor(2, n2ptr);
-
-    if( v0->face ==this  ) {
-      v0->set_face(n2ptr);
-    }
-     v->set_face(this);
-   }
+  //the following function has been moved to the tds class
+//   void insert_in_face(Vertex*& v)
+//   {
+//     CGAL_triangulation_precondition(v != NULL);
+//     Vertex* v0 = vertex(0);
+//     Vertex* v2 = vertex(2);
+//     Vertex* v1 = vertex(1);
+//     
+//     Face* f1 = neighbor(1);
+//     Face* f2 = neighbor(2);
+//     int i1,i2 ;
+//     if (f1 != NULL) {i1= cw(f1->index(vertex(cw(1))));}
+//     if (f2 != NULL) {i2 = cw(f2->index(vertex(cw(2))));}
+//     
+//     Face* n1ptr = new Face(v0, v, v2,
+// 			   this, f1, NULL);
+//     
+//     Face* n2ptr = new Face(v0, v1, v,
+// 			   this, NULL, f2);
+// 
+//     n1ptr->set_neighbor(2, n2ptr);
+//     n2ptr->set_neighbor(1, n1ptr);
+//     if (f1 != NULL) {f1->set_neighbor(i1,n1ptr);}
+//     if (f2 != NULL) {f2->set_neighbor(i2,n2ptr);}
+// 
+//     set_vertex(0, v);
+//     set_neighbor(1, n1ptr);
+//     set_neighbor(2, n2ptr);
+// 
+//     if( v0->face() == this  ) {  v0->set_face(n2ptr); }
+//     v->set_face(this);
+//    }
     
-  void insert_on_edge(const Vertex* v, int i)
-  {
-    
-      CGAL_triangulation_precondition(v != NULL); 
-      Face* n = neighbor(i);
-
-      // The following seems natural, but it may fail if the faces
-      // this and n are neighbors on two edges (1-dim triangulation,
-      // with infinite faces
-      // int in = n->index(this);
-
-      int in;
-      CGAL_triangulation_assertion( n->has_vertex(vertex(cw(i),in)));
-      in = cw(in);
-      insert_in_face(v);
-      n->flip(in); 
-  }
+//the following function has been moved to the tds class
+//   void insert_on_edge(const Vertex* v, int i)
+//   {
+//     
+//       CGAL_triangulation_precondition(v != NULL); 
+//       Face* n = neighbor(i);
+// 
+//       // The following seems natural, but it may fail if the faces
+//       // this and n are neighbors on two edges (1-dim triangulation,
+//       // with infinite faces
+//       // int in = n->index(this);
+// 
+//       int in;
+//       CGAL_triangulation_assertion( n->has_vertex(vertex(cw(i),in)));
+//       in = cw(in);
+//       insert_in_face(v);
+//       n->flip(in); 
+//   }
+// 
 
+  //no longer necessary
 //   bool insert_outside(const Vertex* v, int i)
 //   {
 //     CGAL_triangulation_precondition(v != NULL);
@@ -210,140 +206,142 @@ public:
 //     }
     
 
-  bool remove(Vertex* v)
-  {
-    CGAL_triangulation_precondition(v != NULL);
-    CGAL_triangulation_precondition( has_vertex(v));
-     int i = index(v);
-    
-        Face* left, *right, *ll, *rr;
-    
-        left = neighbor(cw(i));
-        right = neighbor(ccw(i));
-    
-//         if(left == NULL || right == NULL) {
-//             if(left == NULL && right == NULL) {
-//                 Face* n = neighbor(i);
-//                 if(n != NULL) {
-//                     int ni = n->index(this);
-//                     n->set_neighbor(ni, NULL);
-//                     Vertex* q = vertex(cw(i));
-//                     if(q != NULL){
-//                         q->set_face(n);
-//                     }
-//                 } else {
-//                     cerr << "removal of boundary vertex failed as face has"
-//                          << "no neighbors" << endl;
-//                 }
-//                 handle().Delete();
-//                 v.Delete();
-//                 return true;
-//             } else {
-//                 cerr << "removal of boundary vertex with degree != 2 failed";
-//                 cerr << endl;
-//                 return false;
-//             }
+  // this function has been moved to the Tds class
+//   bool remove(Vertex* v)
+//   {
+//     CGAL_triangulation_precondition(v != NULL);
+//     CGAL_triangulation_precondition( has_vertex(v));
+//      int i = index(v);
+//     
+//         Face* left, *right, *ll, *rr;
+//     
+//         left = neighbor(cw(i));
+//         right = neighbor(ccw(i));
+//     
+// //         if(left == NULL || right == NULL) {
+// //             if(left == NULL && right == NULL) {
+// //                 Face* n = neighbor(i);
+// //                 if(n != NULL) {
+// //                     int ni = n->index(this);
+// //                     n->set_neighbor(ni, NULL);
+// //                     Vertex* q = vertex(cw(i));
+// //                     if(q != NULL){
+// //                         q->set_face(n);
+// //                     }
+// //                 } else {
+// //                     cerr << "removal of boundary vertex failed as face has"
+// //                          << "no neighbors" << endl;
+// //                 }
+// //                 handle().Delete();
+// //                 v.Delete();
+// //                 return true;
+// //             } else {
+// //                 cerr << "removal of boundary vertex with degree != 2 failed";
+// //                 cerr << endl;
+// //                 return false;
+// //             }
+// //         }
+// //     
+//         if(v->degree() != 3){
+//             cerr << "removal of internal vertex with degree != 3 failed";
+//             cerr << endl;
+//             return false;
 //         }
 //     
-        if(v->degree() != 3){
-            cerr << "removal of internal vertex with degree != 3 failed";
-            cerr << endl;
-            return false;
-        }
-    
-        int li = left->index(this);
-	int ri = right->index(this);
-        Vertex* q = left->vertex(li);
-	CGAL_triangulation_assertion( left->vertex(li) == right->vertex(ri));
-    
-        ll = left->neighbor(cw(li));
-        if(ll != NULL) {
-            int lli = ll->index(left);
-            ll->set_neighbor(lli, this);
-        } 
-        set_neighbor(cw(i), ll);
-	if (vertex(ccw(i))->face() == left) vertex(ccw(i))->set_face(this);    
-        
-    
-        
-        rr = right->neighbor(ccw(ri));
-        if(rr != NULL) {
-            int rri = rr->index(right);
-            rr->set_neighbor(rri, this);
-        } 
-        set_neighbor(ccw(i), rr);
-	if (vertex(cw(i))->face() == right) vertex(cw(i))->set_face(this);  
-        
-	set_vertex(i, q);
-	if (q->face() == right || q->face() == left) {
-	   q->set_face(this);
-	}
+//         int li = left->index(this);
+// 	int ri = right->index(this);
+//         Vertex* q = left->vertex(li);
+// 	CGAL_triangulation_assertion( left->vertex(li) == right->vertex(ri));
+//     
+//         ll = left->neighbor(cw(li));
+//         if(ll != NULL) {
+//             int lli = ll->index(left);
+//             ll->set_neighbor(lli, this);
+//         } 
+//         set_neighbor(cw(i), ll);
+// 	if (vertex(ccw(i))->face() == left) vertex(ccw(i))->set_face(this);    
+//         
+//     
+//         
+//         rr = right->neighbor(ccw(ri));
+//         if(rr != NULL) {
+//             int rri = rr->index(right);
+//             rr->set_neighbor(rri, this);
+//         } 
+//         set_neighbor(ccw(i), rr);
+// 	if (vertex(cw(i))->face() == right) vertex(cw(i))->set_face(this);  
+//         
+// 	set_vertex(i, q);
+// 	if (q->face() == right || q->face() == left) {
+// 	   q->set_face(this);
+// 	}
+// 
+// 	delete right;
+//     	delete left;
+//         
+//         delete v;
+//         return true;   
+//     }
+//     
 
-	delete right;
-    	delete left;
-        
-        delete v;
-        return true;   
-    }
-    
-  void flip(int i)
-  {
-    Face* n  = neighbor(i);
-    
-    Vertex* v_cw = vertex(cw(i));
-    Vertex*  v_ccw = vertex(ccw(i));
-
-    // we should not attempt to flip two faces which are adjacent on two edges
-    // This configuration happens in 1-dim triangulation
-
-    int ni;
-    CGAL_triangulation_assertion( n->has_vertex(v_cw,ni));
-    ni = cw(ni);
-    CGAL_triangulation_assertion( vertex(i) != n->vertex(ni));
-    CGAL_triangulation_assertion( this == n->neighbor(ni) );
-
-    // Old stuff
-    // The following seems natural, but it may fail if the faces
-    // this and n are neighbors on two edges (1-dim triangulation,
-    // with infinite faces
-    // int ni = n->index(this);
-    //
-    //  int ni = cw(n->index(v_cw));
-    //  CGAL_triangulation_assertion( this == n->neighbor(ni) );
-    
-    // bl == bottom left, tr == top right
-    Face* tr  = neighbor(ccw(i));
-    Face* bl = n->neighbor(ccw(ni));
-    int bli, tri;
-
-    // Old stuff which seems natural
-    // but makes problem if f and tr or n and bl are incident through two edges
-    // bli = bl->index(n);
-    // tri = tr->index(f);
-    bli = 3 - ( bl->index(n->vertex(ni)) + bl->index(n->vertex(cw(ni))) );
-    tri = 3 - (tr->index(vertex(i)) + tr->index(vertex(cw(i))));
-    
-    set_vertex(cw(i), n->vertex(ni));
-    n->set_vertex(cw(ni), vertex(i));
-    
-    // update the neighborhood relations
-    set_neighbor(i, bl);
-    bl->set_neighbor(bli, this);
-           
-    set_neighbor(ccw(i), n);
-    n->set_neighbor(ccw(ni), this);
-    
-    n->set_neighbor(ni, tr);
-    tr->set_neighbor(tri,n);    
-    
-    if(v_cw->face() == this) {
-      v_cw->set_face(n);
-    }
-    
-    if(v_ccw->face() == n) {
-      v_ccw->set_face(this);
-    }
-  }
+//   void flip(int i)
+//   {
+//     Face* n  = neighbor(i);
+//     
+//     Vertex* v_cw = vertex(cw(i));
+//     Vertex*  v_ccw = vertex(ccw(i));
+// 
+//     // we should not attempt to flip two faces which are adjacent on two edges
+//     // This configuration happens in 1-dim triangulation
+// 
+//     int ni;
+//     CGAL_triangulation_assertion( n->has_vertex(v_cw,ni));
+//     ni = cw(ni);
+//     CGAL_triangulation_assertion( vertex(i) != n->vertex(ni));
+//     CGAL_triangulation_assertion( this == n->neighbor(ni) );
+// 
+//     // Old stuff
+//     // The following seems natural, but it may fail if the faces
+//     // this and n are neighbors on two edges (1-dim triangulation,
+//     // with infinite faces
+//     // int ni = n->index(this);
+//     //
+//     //  int ni = cw(n->index(v_cw));
+//     //  CGAL_triangulation_assertion( this == n->neighbor(ni) );
+//     
+//     // bl == bottom left, tr == top right
+//     Face* tr  = neighbor(ccw(i));
+//     Face* bl = n->neighbor(ccw(ni));
+//     int bli, tri;
+// 
+//     // Old stuff which seems natural
+//     // but makes problem if f and tr or n and bl are incident through two edges
+//     // bli = bl->index(n);
+//     // tri = tr->index(f);
+//     bli = 3 - ( bl->index(n->vertex(ni)) + bl->index(n->vertex(cw(ni))) );
+//     tri = 3 - (tr->index(vertex(i)) + tr->index(vertex(cw(i))));
+//     
+//     set_vertex(cw(i), n->vertex(ni));
+//     n->set_vertex(cw(ni), vertex(i));
+//     
+//     // update the neighborhood relations
+//     set_neighbor(i, bl);
+//     bl->set_neighbor(bli, this);
+//            
+//     set_neighbor(ccw(i), n);
+//     n->set_neighbor(ccw(ni), this);
+//     
+//     n->set_neighbor(ni, tr);
+//     tr->set_neighbor(tri,n);    
+//     
+//     if(v_cw->face() == this) {
+//       v_cw->set_face(n);
+//     }
+//     
+//     if(v_ccw->face() == n) {
+//       v_ccw->set_face(this);
+//     }
+//   }
 
    bool is_valid(bool verbose = false, int level = 0) const
   {