diff --git a/Triangulation/include/CGAL/Delaunay_triangulation.h b/Triangulation/include/CGAL/Delaunay_triangulation.h
index 1c33c83c669..26881005bea 100644
--- a/Triangulation/include/CGAL/Delaunay_triangulation.h
+++ b/Triangulation/include/CGAL/Delaunay_triangulation.h
@@ -90,7 +90,7 @@ public:
     using Base::are_incident_full_cells_valid;
     using Base::coaffine_orientation_predicate;
     using Base::current_dimension;
-    using Base::compute_star;
+    using Base::star;
     using Base::incident_full_cells;
     using Base::geom_traits;
     using Base::get_visited;
diff --git a/Triangulation/include/CGAL/Triangulation.h b/Triangulation/include/CGAL/Triangulation.h
index c53f83fc880..2309aa24f53 100644
--- a/Triangulation/include/CGAL/Triangulation.h
+++ b/Triangulation/include/CGAL/Triangulation.h
@@ -395,9 +395,9 @@ public:
         return tds().incident_full_cells(v, out);
     }
     template< typename OutputIterator >
-    OutputIterator compute_star(const Face & f, OutputIterator out) const
+    OutputIterator star(const Face & f, OutputIterator out) const
     {
-        return tds().compute_star(f, out);
+        return tds().star(f, out);
     }
 
     template< typename OutputIterator >
diff --git a/Triangulation/include/CGAL/Triangulation_data_structure.h b/Triangulation/include/CGAL/Triangulation_data_structure.h
index 2ea53db2fc3..72a51240391 100644
--- a/Triangulation/include/CGAL/Triangulation_data_structure.h
+++ b/Triangulation/include/CGAL/Triangulation_data_structure.h
@@ -511,7 +511,7 @@ public:
     template< typename OutputIterator >
     OutputIterator incident_full_cells(Vertex_const_handle, OutputIterator) const; /* Concept */
     template< typename OutputIterator >
-    OutputIterator compute_star(const Face &, OutputIterator) const; /* Concept */
+    OutputIterator star(const Face &, OutputIterator) const; /* Concept */
 #ifndef CGAL_CFG_NO_CPP0X_DEFAULT_TEMPLATE_ARGUMENTS_FOR_FUNCTION_TEMPLATES 
     template< typename OutputIterator, typename Comparator = std::less<Vertex_const_handle> >
     OutputIterator incident_upper_faces(Vertex_const_handle v, const int d, OutputIterator out, Comparator cmp = Comparator())
@@ -581,7 +581,7 @@ template< class Dim, class Vb, class Sb >
 template< typename OutputIterator >
 OutputIterator
 Triangulation_data_structure<Dim, Vb, Sb>
-::compute_star(const Face & f, OutputIterator out) const /* Concept */
+::star(const Face & f, OutputIterator out) const /* Concept */
 {
     // CGAL_precondition_msg(is_full_cell(f.full_cell()), "the facet does not belong to the Triangulation");
     Star_traversal_predicate tp(*this, f);
@@ -730,7 +730,7 @@ Triangulation_data_structure<Dim, Vb, Sb>
     // compute the star of f
     simps.reserve(64);
     std::back_insert_iterator<std::vector<Full_cell_handle> > out(simps);
-    compute_star(f, out);
+    star(f, out);
     Vertex_handle v = insert_in_hole(simps.begin(), simps.end(), Facet(f.full_cell(), f.index(0)));
     for( int i = 0; i <= fd; ++i )
         delete_vertex(s.vertex(i));
diff --git a/Triangulation/include/CGAL/Triangulation_ds_full_cell.h b/Triangulation/include/CGAL/Triangulation_ds_full_cell.h
index af11fc66de1..53090ad825d 100644
--- a/Triangulation/include/CGAL/Triangulation_ds_full_cell.h
+++ b/Triangulation/include/CGAL/Triangulation_ds_full_cell.h
@@ -38,12 +38,12 @@ class Triangulation_ds_full_cell
 
 public:
     typedef typename TDS::Face                     Face;
-    typedef typename TDS::Vertex_handle            Vertex_handle;
-    typedef typename TDS::Full_cell_handle           Full_cell_handle;
+    typedef typename TDS::Vertex_handle            Vertex_handle; /* Concept */
     typedef typename TDS::Vertex_const_handle      Vertex_const_handle;
-    typedef typename TDS::Full_cell_const_handle     Full_cell_const_handle;
+    typedef typename TDS::Full_cell_handle         Full_cell_handle; /* Concept */
+    typedef typename TDS::Full_cell_const_handle   Full_cell_const_handle;
     template< typename TDS2 >
-    struct Rebind_TDS
+    struct Rebind_TDS /* Concept */
     {
         typedef Triangulation_ds_full_cell<TDS2, FullCellStoragePolicy> Other;
     };
@@ -66,8 +66,9 @@ private: // STORAGE
 
 public:
     typedef typename Vertex_handle_array::const_iterator    Vertex_handle_const_iterator;
+    typedef Vertex_handle_const_iterator    Vertex_handle_iterator; /* Concept */
 
-    Triangulation_ds_full_cell(const int dmax)
+    Triangulation_ds_full_cell(const int dmax) /* Concept */
     : flags_(0), combinatorics_(dmax)
     {
 		CGAL_assertion( dmax > 0 );
@@ -79,53 +80,53 @@ public:
         }
     }
 
-    Triangulation_ds_full_cell(const Triangulation_ds_full_cell & s)
+    Triangulation_ds_full_cell(const Triangulation_ds_full_cell & s) /* Concept */
     : flags_(s.flags_), combinatorics_(s.combinatorics_)
     {}
 
     ~Triangulation_ds_full_cell() {}
 
-    int ambient_dimension() const
+    int ambient_dimension() const /* Concept */
     {
         return vertices().size() - 1;
     }
 
-    Vertex_handle_const_iterator vertices_begin() const
+    Vertex_handle_const_iterator vertices_begin() const /* Concept */
     {
         return vertices().begin();
     }
 
-    Vertex_handle_const_iterator vertices_end() const
+    Vertex_handle_const_iterator vertices_end() const /* Concept */
     {
         return vertices().end();
     }
 
-    Vertex_handle vertex(const int i) const
+    Vertex_handle vertex(const int i) const /* Concept */
     {
         CGAL_precondition(0<=i && i<=ambient_dimension());
         return vertices()[i];
     }
 
-    Full_cell_handle neighbor(const int i) const
+    Full_cell_handle neighbor(const int i) const /* Concept */
     {
         CGAL_precondition(0<=i && i<=ambient_dimension());
         return neighbors()[i];
     }
 
-    int mirror_index(const int i) const
+    int mirror_index(const int i) const /* Concept */
     {
         CGAL_precondition(0<=i && i<=ambient_dimension());
         return combinatorics_.mirror_index(i);
     }
 
 	// Advanced...
-    Vertex_handle mirror_vertex(const int i, const int cur_dim) const
+    Vertex_handle mirror_vertex(const int i, const int cur_dim) const /* Concept */
     {
         CGAL_precondition(0<=i && i<=ambient_dimension());
         return combinatorics_.mirror_vertex(i, cur_dim);
     }
 
-    int index(Full_cell_const_handle s) const
+    int index(Full_cell_const_handle s) const /* Concept */
     {
         // WE ASSUME THE FULL CELL WE ARE LOOKING FOR INDEED EXISTS !
         CGAL_precondition(has_neighbor(s));
@@ -135,7 +136,7 @@ public:
         return index;
     }
 
-    int index(Vertex_const_handle v) const
+    int index(Vertex_const_handle v) const /* Concept */
     {
         // WE ASSUME THE VERTEX WE ARE LOOKING FOR INDEED EXISTS !
         CGAL_precondition(has_vertex(v));
@@ -145,31 +146,31 @@ public:
         return index;
     }
 
-	void set_vertex(const int i, Vertex_handle v)
+	void set_vertex(const int i, Vertex_handle v) /* Concept */
 	{
 		CGAL_precondition(0<=i && i<=ambient_dimension());
 		vertices()[i] = v;
 	}
 
-	void set_neighbor(const int i, Full_cell_handle s)
+	void set_neighbor(const int i, Full_cell_handle s) /* Concept */
 	{
 		CGAL_precondition(0<=i && i<=ambient_dimension());
 		neighbors()[i] = s;
 	}
 
-	void set_mirror_index(const int i, const int index)
+	void set_mirror_index(const int i, const int index) /* Concept */
 	{
 		CGAL_precondition(0<=i && i<=ambient_dimension());
 		combinatorics_.set_mirror_index(i, index);
 	}
 
-    bool has_vertex(Vertex_const_handle v) const
+    bool has_vertex(Vertex_const_handle v) const /* Concept */
     {
         int index;
         return has_vertex(v, index);
     }
 
-    bool has_vertex(Vertex_const_handle v, int & index) const
+    bool has_vertex(Vertex_const_handle v, int & index) const /* Concept */
     {
         const int d = ambient_dimension();
         index = 0;
@@ -178,13 +179,13 @@ public:
         return (index <= d);
     }
 
-    bool has_neighbor(Full_cell_const_handle s) const
+    bool has_neighbor(Full_cell_const_handle s) const /* Concept */
     {
         int index;
         return has_neighbor(s, index);
     }
 
-    bool has_neighbor(Full_cell_const_handle s, int & index) const
+    bool has_neighbor(Full_cell_const_handle s, int & index) const /* Concept */
     {
         const int d = ambient_dimension();
         index = 0;
@@ -193,21 +194,21 @@ public:
         return (index <= d);
     }
 
-    void swap_vertices(const int d1, const int d2)
+    void swap_vertices(const int d1, const int d2) /* Concept */
     {
         CGAL_precondition(0 <= d1 && d1<=ambient_dimension());
         CGAL_precondition(0 <= d2 && d2<=ambient_dimension());
         combinatorics_.swap_vertices(d1, d2);
     }
 
-    unsigned int get_flags() const { return flags_; }
+    unsigned int get_flags() const { return flags_; } /* Concept */
     // Don't forget that member variable flags_ is mutable...
-    void set_flags(unsigned int f) const { flags_ = f; }
+    void set_flags(unsigned int f) const { flags_ = f; } /* Concept */
 
     void*   for_compact_container() const { return combinatorics_.for_compact_container(); }
     void* & for_compact_container() { return combinatorics_.for_compact_container(); }
 
-    bool is_valid(bool verbose = true, int /* level */ = 0) const
+    bool is_valid(bool verbose = true, int /* level */ = 0) const /* Concept */
     {
         const int d = ambient_dimension();
         for( int i = 0; i <= d; ++i )
@@ -246,7 +247,7 @@ private:
 
 template < typename TDS, typename SSP >
 std::ostream &
-operator<<(std::ostream & O, const Triangulation_ds_full_cell<TDS,SSP> & s)
+operator<<(std::ostream & O, const Triangulation_ds_full_cell<TDS,SSP> & s) /* Concept */
 {
     /*if( is_ascii(O) )
     {
@@ -258,7 +259,7 @@ operator<<(std::ostream & O, const Triangulation_ds_full_cell<TDS,SSP> & s)
 
 template < typename TDS, typename SSP >
 std::istream &
-operator>>(std::istream & I, Triangulation_ds_full_cell<TDS,SSP> & s)
+operator>>(std::istream & I, Triangulation_ds_full_cell<TDS,SSP> & s) /* Concept */
 {
     /*if( is_ascii(I) )
     {}
diff --git a/Triangulation/include/CGAL/Triangulation_ds_vertex.h b/Triangulation/include/CGAL/Triangulation_ds_vertex.h
index 9010a67f6ad..6f29341ce4a 100644
--- a/Triangulation/include/CGAL/Triangulation_ds_vertex.h
+++ b/Triangulation/include/CGAL/Triangulation_ds_vertex.h
@@ -42,27 +42,27 @@ public:
     };
 
 protected: // DATA MEMBERS
-    Full_cell_handle full_cell_; // A handle to an adjacent full_cell
+    Full_cell_handle full_cell_; // A handle to an incident full_cell
 
 public:	
-    // Constructs a vertex with adjacent full_cell 's'
+    // Constructs a vertex with incident full_cell 's'
     Triangulation_ds_vertex(Full_cell_handle s) : full_cell_(s) /* Concept */
     {
         CGAL_assertion( Full_cell_handle() != s );
     }
-    // Constructs a vertex with no adjacent full_cell
+    // Constructs a vertex with no incident full_cell
     Triangulation_ds_vertex() : full_cell_() {} /* Concept */
 
     ~Triangulation_ds_vertex() {}
 
-    /// Set 's' as an adjacent full_cell
+    /// Set 's' as an incident full_cell
     void set_full_cell(Full_cell_handle s) /* Concept */
     {
 		CGAL_precondition( Full_cell_handle() != s );
         full_cell_ = s;
     }
 
-    /// Returns a full_cell adjacent to the vertex
+    /// Returns a full_cell incident to the vertex
     Full_cell_handle full_cell() const /* Concept */
     {
         return full_cell_;
@@ -72,7 +72,8 @@ public:
     {
         if( Full_cell_handle() == full_cell() )
         {
-            if( verbose ) CGAL_warning_msg(false, "vertex has no incident full cell.");
+            if( verbose )
+                CGAL_warning_msg(false, "vertex has no incident full cell.");
             return false;
         }
         return true;