diff --git a/Triangulation/doc/Triangulation/CGAL/Triangulation.h b/Triangulation/doc/Triangulation/CGAL/Triangulation.h
index ff09454570b..9c7bdbce752 100644
--- a/Triangulation/doc/Triangulation/CGAL/Triangulation.h
+++ b/Triangulation/doc/Triangulation/CGAL/Triangulation.h
@@ -4,10 +4,14 @@ namespace CGAL {
 /*!
 \ingroup PkgTriangulations
 
+\cgalModifBegin
 The class `Triangulation` is used to store and query the full cells and vertices of
-a triangulationin dimension \f$ d\f$. A special vertex, named
+a triangulationin dimension \f$ d\f$(see the 
+\ref Chapter_Triangulations "User Manual" for
+a definition of "triangulation"). A special vertex, named
 <I>infinite vertex</I>, is used to triangulate the outside of the convex
 hull of the points in so called <I>infinite cells</I>.
+\cgalModifEnd
 
 Parameters
 --------------
@@ -55,7 +59,7 @@ A point in Euclidean space.
 typedef TriangulationTraits::Point_d Point;
 
 /*!
-This indicates whether the dimension of the underlying space is static
+This indicates whether the maximal dimension is static
 (i.e.\ if the type of `Maximal_dimension` is `CGAL::Dimension_tag<int dim>`)
 or dynamic (i.e.\ if the type of `Maximal_dimension` is 
 `CGAL::Dynamic_dimension_tag`).
@@ -542,7 +546,7 @@ to the newly created full cells are output in the `out` output iterator.
 interior and not contain any vertex all of whose incident full cells are in
 \f$C\f$ . (This implies that `t.current_dimension()`\f$ \geq 2\f$ if
 \f$|C|>1\f$ .) The boundary of \f$C\f$ must be a triangulation of the sphere
-\f$ \mathcal S ^k-1\f$.
+\f$ \mathcal S ^d-1\f$.
 \cgalAdvancedEnd
 */
 template < typename ForwardIterator, typename OutputIterator >
diff --git a/Triangulation/doc/Triangulation/CGAL/Triangulation_data_structure.h b/Triangulation/doc/Triangulation/CGAL/Triangulation_data_structure.h
index a1ae58c3273..7183e779549 100644
--- a/Triangulation/doc/Triangulation/CGAL/Triangulation_data_structure.h
+++ b/Triangulation/doc/Triangulation/CGAL/Triangulation_data_structure.h
@@ -5,16 +5,16 @@ namespace CGAL {
 \ingroup PkgTriangulations
 
 This class is used for storing the combinatorial information of a triangulation 
-of dimension \f$ k\leq d\f$. 
+of dimension \f$ d\leq D\f$ (`D` is the maximal dimension).
 
 Parameters 
 -------------- 
 
 `Dimensionality` can be either <UL> 
 
-<LI>CGAL::`Dimension_tag<d>` for some integer `d`. This 
+<LI>CGAL::`Dimension_tag<D>` for some integer `D`. This 
 indicates that the triangulation data structure can store simplices (full cells) of dimension at most 
-`d`. The maximum dimension `d` is known by the compiler, which 
+`D`. The maximal dimension `D` is known by the compiler, which 
 triggers some optimizations. Or 
 
 <LI>CGAL::`Dynamic_dimension_tag`. In this case, the maximum 
@@ -78,16 +78,6 @@ bool is_valid(bool verbose = true) const;
 /// \name Types 
 /// @{
 
-/*! 
-\cgalAdvancedBegin
-A data member of type `Full_cell_data` is stored in every full cell
-(models of the concept `TriangulationDSFullCell`). It is used to mark
-some full cells, during modifications of the triangulation data
-structure.
-\cgalAdvancedEnd
-*/ 
-typedef Hidden_type Full_cell_data; 
-
 /// @} 
 
 /// \name Vertex insertion 
diff --git a/Triangulation/doc/Triangulation/Concepts/DelaunayTriangulationTraits.h b/Triangulation/doc/Triangulation/Concepts/DelaunayTriangulationTraits.h
index 330e180e2d9..e1f74172c59 100644
--- a/Triangulation/doc/Triangulation/Concepts/DelaunayTriangulationTraits.h
+++ b/Triangulation/doc/Triangulation/Concepts/DelaunayTriangulationTraits.h
@@ -36,9 +36,10 @@ defined by the points in range `[start,end)`.
 If the simplex is positively
 oriented, then the positive side of sphere corresponds geometrically
 to its bounded side.
-\pre `std::distance(start,end)=D+1`, where
-`Point_dimension_d(*it)` is \f$ D\f$ for all `it` in
-`[start,end)`. `Point_dimension_d(p)` is also \f$ D\f$.
+\cgalModifBegin
+\pre If `Dimension`=`CGAL::``Dimension_tag<D>`, 
+then `std::distance(start,end)=D+1`.
+\cgalModifEnd
 The points in range
 `[start,end)` must be affinely independent, i.e., the simplex must
 not be flat.
@@ -65,12 +66,9 @@ The points in range `[start,end)` and `p` are supposed to belong to the lower di
 whose orientation is given by `orient`.
 \pre `std::distance(start,end)=k+1` where \f$ k\f$ is the number of
 points used to construct `orient`.
-`Point_dimension_d(*it)` is \f$ D\f$ for all `it` in
-`[start,end)`. `Point_dimension_d(p)` is also \f$ D\f$.
 The points in range
 `[start,end)` must be affinely independent, i.e., the simplex must
-not be flat.
-
+not be flat. `p` must be in the flat generated by this simplex.
 */
 typedef Hidden_type In_flat_side_of_oriented_sphere_d;
 
diff --git a/Triangulation/doc/Triangulation/Concepts/FullCellData.h b/Triangulation/doc/Triangulation/Concepts/FullCellData.h
new file mode 100644
index 00000000000..b43239d84d5
--- /dev/null
+++ b/Triangulation/doc/Triangulation/Concepts/FullCellData.h
@@ -0,0 +1,40 @@
+
+/*!
+\ingroup PkgTriangulationsConcepts
+\cgalConcept
+
+\cgalModifBegin
+The concept `FullCellData` describes the requirements on the type which 
+is used to mark some full cells, during modifications of the triangulation data
+structure.
+\cgalModifEnd
+
+\sa `TriangulationDataStructure`
+\sa `TriangulationDSFullCell`
+*/
+
+class FullCellData 
+{
+public:
+/*!
+Clear all data.
+*/
+void clear();
+/*!
+Mark the full cell as visited.
+*/
+void mark_visited();
+/*!
+Mark the full cell as not visited.
+*/
+void clear_visited();
+/*!
+Returns `true` if the full cell is <b>not</b> marked as visited, `false` otherwise.
+*/
+bool is_clear()   const;
+/*!
+Returns `true` if the full cell is marked as visited, `false` otherwise.
+*/
+bool is_visited() const;
+
+}; /* end FullCellData */
diff --git a/Triangulation/doc/Triangulation/Concepts/TriangulationDSFace.h b/Triangulation/doc/Triangulation/Concepts/TriangulationDSFace.h
index 0f085388034..82b34129bcc 100644
--- a/Triangulation/doc/Triangulation/Concepts/TriangulationDSFace.h
+++ b/Triangulation/doc/Triangulation/Concepts/TriangulationDSFace.h
@@ -31,16 +31,23 @@ public:
 /// @{
 
 /*!
-Must be the same as the nested type
-`TriangulationDataStructure::Full_cell_handle` of the `TriangulationDataStructure` in which the `TriangulationDSFace` is
+\cgalModifBegin
+The `Triangulation_data_structure` in which the `TriangulationDSFace` is
 defined/used.
+Must be a model of the `TriangulationDataStructure` concept.
+\cgalModifEnd
+*/
+typedef Hidden_type Triangulation_data_structure;
+
+/*!
+Must be the same as the nested type
+`Triangulation_data_structure::Full_cell_handle`.
 */
 typedef Hidden_type Full_cell_handle;
 
 /*!
 Must be the same as the nested type
-`TriangulationDataStructure::Vertex_handle` of the `TriangulationDataStructure` in which the `TriangulationDSFace` is
-defined/used.
+`Triangulation_data_structure::Vertex_handle`.
 */
 typedef Hidden_type Vertex_handle;
 
diff --git a/Triangulation/doc/Triangulation/Concepts/TriangulationDSFullCell.h b/Triangulation/doc/Triangulation/Concepts/TriangulationDSFullCell.h
index a41f3f3fc74..40efd31add0 100644
--- a/Triangulation/doc/Triangulation/Concepts/TriangulationDSFullCell.h
+++ b/Triangulation/doc/Triangulation/Concepts/TriangulationDSFullCell.h
@@ -34,11 +34,19 @@ public:
 
 /// \name Types
 /// @{
+  
+/*!
+\cgalModifBegin
+The `Triangulation_data_structure` in which the `TriangulationDSFullCell` is
+defined/used.
+Must be a model of the `TriangulationDataStructure` concept.
+\cgalModifEnd
+*/
+typedef Hidden_type Triangulation_data_structure;
 
 /*!
 A handle to a vertex. It must be the same as the
-nested type `TriangulationDataStructure::Vertex_handle` of the `TriangulationDataStructure` in which the
-`TriangulationDSFullCell` is defined/used.
+nested type `TriangulationDataStructure::Vertex_handle`.
 */
 typedef Hidden_type Vertex_handle;
 
@@ -50,8 +58,7 @@ typedef Hidden_type Vertex_handle_iterator;
 
 /*!
 A handle to a full cell. It must be the same as the
-nested type `TriangulationDataStructure::Full_cell_handle` of the `TriangulationDataStructure` in which the
-`TriangulationDSFullCell` is defined/used.
+nested type `TriangulationDataStructure::Full_cell_handle`.
 */
 typedef Hidden_type Full_cell_handle;
 
@@ -299,19 +306,23 @@ void * for_compact_container() const;
 void * & for_compact_container();
 
 /*!
-Reads (possibly) non-combinatorial information about a full cell from the stream `is`
-into `c`.
-*/
-template<class TriangulationDataStructure> 
-std::istream& operator>>(std::istream & is, Triangulation_ds_full_cell<TriangulationDataStructure> & c);
-
-/*!
+\cgalModifBegin
 Writes (possibly) non-combinatorial information about full cell `c` to the stream
 `os`.
+\cgalModifEnd
 */
 template<class TriangulationDataStructure> 
 std::ostream& operator<<(std::ostream & os, const Triangulation_ds_full_cell<TriangulationDataStructure> & c);
 
+/*!
+\cgalModifBegin
+Reads from stream `is` the full cell information written 
+by `operator<<`.
+\cgalModifEnd
+*/
+template<class TriangulationDataStructure> 
+std::istream& operator>>(std::istream & is, Triangulation_ds_full_cell<TriangulationDataStructure> & c);
+
 /// @}
 
 }; /* end TriangulationDSFullCell */
diff --git a/Triangulation/doc/Triangulation/Concepts/TriangulationDSVertex.h b/Triangulation/doc/Triangulation/Concepts/TriangulationDSVertex.h
index 69d189f8644..44527e5b34b 100644
--- a/Triangulation/doc/Triangulation/Concepts/TriangulationDSVertex.h
+++ b/Triangulation/doc/Triangulation/Concepts/TriangulationDSVertex.h
@@ -29,11 +29,19 @@ public:
 
 /// \name Types
 /// @{
+  
+/*!
+\cgalModifBegin
+The `Triangulation_data_structure` in which the `TriangulationDSVertex` is
+defined/used.
+Must be a model of the `TriangulationDataStructure` concept.
+\cgalModifEnd
+*/
+typedef Hidden_type Triangulation_data_structure;
 
 /*!
 A handle to a cell. It must be the same as the
-nested type `TriangulationDataStructure::Full_cell_handle` of the `TriangulationDataStructure` in which the
-`TriangulationDSVertex` is defined/used.
+nested type `TriangulationDataStructure::Full_cell_handle`.
 */
 typedef Hidden_type Full_cell_handle;
 
@@ -121,19 +129,22 @@ void * for_compact_container() const;
 void * & for_compact_container();
 
 /*!
-Reads (possibly) non-combinatorial information about a vertex from the stream `is`
-into `v`.
-*/
-template<class TriangulationDataStructure> 
-std::istream& operator>>(std::istream & is, Triangulation_ds_vertex<TriangulationDataStructure> & v);
-
-/*!
+\cgalModifBegin
 Writes (possibly) non-combinatorial information about vertex `v` to the stream
 `os`.
+\cgalModifEnd
 */
 template<class TriangulationDataStructure> 
 std::ostream& operator<<(std::ostream & os, const Triangulation_ds_vertex<TriangulationDataStructure> & v);
 
+/*!
+\cgalModifBegin
+Reads from stream `is` the vertex information written by `operator<<`.
+\cgalModifEnd
+*/
+template<class TriangulationDataStructure> 
+std::istream& operator>>(std::istream & is, Triangulation_ds_vertex<TriangulationDataStructure> & v);
+
 /// @}
 
 }; /* end TriangulationDSVertex */
diff --git a/Triangulation/doc/Triangulation/Concepts/TriangulationDataStructure.h b/Triangulation/doc/Triangulation/Concepts/TriangulationDataStructure.h
index eacac8f6452..1582778bfd1 100644
--- a/Triangulation/doc/Triangulation/Concepts/TriangulationDataStructure.h
+++ b/Triangulation/doc/Triangulation/Concepts/TriangulationDataStructure.h
@@ -95,6 +95,12 @@ The full cell type. A model of the concept `TriangulationDSFullCell`.
 */
 typedef Hidden_type Full_cell;
 
+/*! 
+\cgalModifBegin
+A model of the concept `FullCellData`.
+\cgalModifEnd
+*/ 
+typedef Hidden_type Full_cell_data; 
 /*!
 
 The concept `TriangulationDataStructure` also defines a type for
diff --git a/Triangulation/doc/Triangulation/Concepts/TriangulationTraits.h b/Triangulation/doc/Triangulation/Concepts/TriangulationTraits.h
index 1cb5038e90b..582adb3d357 100644
--- a/Triangulation/doc/Triangulation/Concepts/TriangulationTraits.h
+++ b/Triangulation/doc/Triangulation/Concepts/TriangulationTraits.h
@@ -9,14 +9,11 @@ It brings the geometric ingredient to the
 definition of a triangulation, while the combinatorial ingredient is brought by 
 the second template parameter, `TriangulationDataStructure`. 
 
-Inserting a range of points in a triangulation is optimized using 
-spatial sorting, thus besides the requirements below, 
-a class provided as `TriangulationTraits` should also satisfy the concept 
-`SpatialSortingTraits_d`. 
-
-\cgalRefines ::SpatialSortingTraits_d If a range of points is inserted, the 
-traits must refine `SpatialSortingTraits_d`. This is not needed 
-if the points are inserted one by one. 
+\cgalModifBegin
+\cgalRefines `SpatialSortingTraits_d` If a range of points is inserted, the 
+traits must refine `SpatialSortingTraits_d` (this operation is optimized using 
+spatial sorting). This is not required if the points are inserted one by one.
+\cgalModifEnd
 
 \cgalHasModel `CGAL::Cartesian_d<FT, Dim, LA>`
 \cgalHasModel `CGAL::Epick_d<Dim>` (recommended)
@@ -31,13 +28,13 @@ public:
 /// \name Types 
 /// @{
 
-/*! 
-A type representing the dimension of the underlying space. it can be static 
-(`Maximal_dimension`=`CGAL::``Dimension_tag<int dim>`) or 
-dynamic (`Maximal_dimension`=`CGAL::``Dynamic_dimension_tag`). 
-This dimension must match the dimension of the predicate 
-`Orientation_d` but not necessarily the one of `Point_d`. 
-
+/*!
+\cgalModifBegin
+A type representing the dimension of the `Orientation_d` predicate 
+(but not necessarily the one of `Point_d`). 
+Tt can be static (`Dimension`=`CGAL::``Dimension_tag<int dim>`) or 
+dynamic (`Dimension`=`CGAL::``Dynamic_dimension_tag`).
+\cgalModifEnd
 */ 
 typedef Hidden_type Dimension; 
 
@@ -46,24 +43,16 @@ A type representing a point in Euclidean space.
 */ 
 typedef Hidden_type Point_d; 
 
-/*! 
-Functor returning the dimension of a `Point_d`. 
-Must provide 
-`int operator()(Point_d p)` returning the dimension of \f$ p\f$. 
-
-*/ 
-typedef Hidden_type Point_dimension_d; 
-
 /*! 
 A predicate object that must provide the 
 templated operator 
-`template<typename ForwardIterator> Orientation operator()(ForwardIterator start, ForwardIterator end)`. 
-The operator returns 
-`CGAL::POSITIVE`, `CGAL::NEGATIVE` or `CGAL::COPLANAR` depending on 
-the orientation of the simplex defined by the points in the range `[start, end)`. 
-\pre `std::distance(start,end)=D+1`, where 
-`Point_dimension_d(*it)` is \f$ D\f$ for all `it` in `[start,end)`. 
-
+`template<typename ForwardIterator> Orientation operator()(ForwardIterator start, ForwardIterator end)`.
+\cgalModifBegin
+The operator returns the orientation of the simplex defined by the points 
+in the range `[start, end)`; the value can be 
+`CGAL::POSITIVE`, `CGAL::NEGATIVE` or `CGAL::COPLANAR`.
+\pre If `Dimension`=`CGAL::``Dimension_tag<D>`, then `std::distance(start,end)=D+1`.
+\cgalModifEnd
 */ 
 typedef Hidden_type Orientation_d; 
 
@@ -73,11 +62,13 @@ the templated operator
 `template<typename ForwardIterator> bool operator()(ForwardIterator start, ForwardIterator end, const Point_d & p)`. 
 The operator returns `true` if and only if point `p` is 
 contained in the affine space spanned by the points in the range `[start, end)`. That affine space is also called the <I>affine hull</I> of the points 
-in the range. 
-\pre The \f$ k\f$ points in the range 
+in the range.
+\cgalModifBegin
+\pre If `Dimension`=`CGAL::``Dimension_tag<D>`, 
+then `std::distance(start,end)=D+1`.
+The points in the range 
 must be affinely independent. 
-`Point_dimension_d(*it)` is \f$ D\f$ for all `it` in 
-`[start,end)`, for some \f$ D\f$. 
+\cgalModifEnd
 \f$ 2\leq k\leq D\f$. 
 
 */ 
@@ -112,11 +103,13 @@ The flat spanned by the points in
 the range `R=[start, end)` can be oriented in two different ways, 
 the operator 
 returns an object that allow to orient that flat so that `R=[start, end)` 
-defines a positive simplex. 
-\pre The \f$ k\f$ points in the range 
-must be affinely independent. 
-`Point_dimension_d(*it)` is \f$ D\f$ for all `it` in `R` for 
-some \f$ D\f$. 
+defines a positive simplex.
+\cgalModifBegin
+\pre If `Dimension`=`CGAL::``Dimension_tag<D>`, 
+then `std::distance(start,end)=D+1`.
+The points in range
+`[start,end)` must be affinely independent.
+\cgalModifEnd
 \f$ 2\leq k\leq D\f$. 
 
 */ 
@@ -124,8 +117,6 @@ typedef Hidden_type Construct_flat_orientation_d;
 
 /*! 
 
-CJTODO: update this?
-
 A predicate object that must provide the 
 templated operator 
 `template<typename ForwardIterator> Orientation operator()(Flat_orientation_d orient,ForwardIterator start, ForwardIterator end)`. 
@@ -136,11 +127,7 @@ the orientation of the simplex defined by the points in the range `[start, end)`
 The points are supposed to belong to the lower dimensional flat 
 whose orientation is given by `orient`. 
 \pre `std::distance(start,end)=k` where \f$ k\f$ is the number of 
-points 
-used to construct `orient`. 
-`Point_dimension_d(*it)` is \f$ D\f$ for all `it` in 
-`[start,end)` where \f$ D\f$ is the dimension of the points used to 
-construct `orient`. 
+points used to construct `orient`. 
 \f$ 2\leq k\leq D\f$. 
 
 */ 
diff --git a/Triangulation/doc/Triangulation/PackageDescription.txt b/Triangulation/doc/Triangulation/PackageDescription.txt
index 6d88f229dc8..db27dfa2679 100644
--- a/Triangulation/doc/Triangulation/PackageDescription.txt
+++ b/Triangulation/doc/Triangulation/PackageDescription.txt
@@ -73,6 +73,8 @@ following concepts:
 
 `TriangulationDSFace`
 
+`FullCellData`
+
 ## Triangulations ##
 
 `TriangulationTraits`
@@ -89,26 +91,26 @@ The latter two concepts are also abbreviated respectively as `TrVertex` and `TrF
 
 ## Triangulation data structure ##
 
-\ref ::CGAL::Triangulation_data_structure<Dimensionality, TriangulationDSVertex, TriangulationDSFullCell>
+- `CGAL::Triangulation_data_structure<Dimensionality, TriangulationDSVertex, TriangulationDSFullCell>`
 
-\ref ::CGAL::Triangulation_ds_vertex<TriangulationDataStructure>
+- `CGAL::Triangulation_ds_vertex<TriangulationDataStructure>`
 
-\ref ::CGAL::Triangulation_ds_full_cell<TriangulationDataStructure, TDSFullCellStoragePolicy>
+- `CGAL::Triangulation_ds_full_cell<TriangulationDataStructure, TDSFullCellStoragePolicy>`
 
-\ref ::CGAL::Triangulation_face<TriangulationDataStructure>
+- `CGAL::Triangulation_face<TriangulationDataStructure>`
 
 ## (Geometric) triangulations ##
 
-\ref ::CGAL::Triangulation<TriangulationTraits, TriangulationDataStructure>
+- `CGAL::Triangulation<TriangulationTraits, TriangulationDataStructure>`
 
-\ref ::CGAL::Delaunay_triangulation<DelaunayTriangulationTraits, TriangulationDataStructure>
+- `CGAL::Delaunay_triangulation<DelaunayTriangulationTraits, TriangulationDataStructure>`
 
-\ref ::CGAL::Triangulation_vertex<TriangulationTraits, Data, TriangulationDSVertex>
+- `CGAL::Triangulation_vertex<TriangulationTraits, Data, TriangulationDSVertex>`
 
-\ref ::CGAL::Triangulation_full_cell<TriangulationTraits, Data, TriangulationDSFullCell>
+- `CGAL::Triangulation_full_cell<TriangulationTraits, Data, TriangulationDSFullCell>`
 
 ## Enums ##
 
-\ref ::CGAL::Triangulation::Locate_type
+- `CGAL::Triangulation::Locate_type`
 */
 
diff --git a/Triangulation/include/CGAL/Triangulation_ds_full_cell.h b/Triangulation/include/CGAL/Triangulation_ds_full_cell.h
index 6a116a9d71c..8221bacb620 100644
--- a/Triangulation/include/CGAL/Triangulation_ds_full_cell.h
+++ b/Triangulation/include/CGAL/Triangulation_ds_full_cell.h
@@ -36,6 +36,7 @@ class Triangulation_ds_full_cell
     typedef typename TDS::Maximal_dimension        Maximal_dimension;
 
 public:
+    typedef typename TDS                           Triangulation_data_structure;
     typedef typename TDS::Face                     Face;
     typedef typename TDS::Vertex_handle            Vertex_handle; /* Concept */
     typedef typename TDS::Vertex_const_handle      Vertex_const_handle;
@@ -287,6 +288,7 @@ class Triangulation_ds_full_cell<void, StoragePolicy>
 {
 public:
     typedef internal::Triangulation::Dummy_TDS  TDS;
+    typedef typename TDS                        Triangulation_data_structure;
     typedef TDS::Vertex_handle                  Vertex_handle;
     typedef TDS::Vertex_const_handle            Vertex_const_handle;
     typedef TDS::Full_cell_handle               Full_cell_handle;
diff --git a/Triangulation/include/CGAL/Triangulation_ds_vertex.h b/Triangulation/include/CGAL/Triangulation_ds_vertex.h
index ad3fb76dda3..3bd78745ff7 100644
--- a/Triangulation/include/CGAL/Triangulation_ds_vertex.h
+++ b/Triangulation/include/CGAL/Triangulation_ds_vertex.h
@@ -33,6 +33,7 @@ class Triangulation_ds_vertex
     typedef Triangulation_ds_vertex<TDS>    Self;
 
 public:
+    typedef typename TDS                    Triangulation_data_structure;
     typedef typename TDS::Full_cell_handle  Full_cell_handle; /* Concept */
 
     template <typename TDS2>
@@ -137,9 +138,9 @@ operator<<(std::ostream & os, const Triangulation_ds_vertex<TDS> & v) /* Concept
 template<>
 class Triangulation_ds_vertex<void>
 {
-    typedef internal::Triangulation::Dummy_TDS  Triangulation_ds;
 public:
-    typedef Triangulation_ds::Full_cell_handle  Full_cell_handle; /* Concept */
+    typedef internal::Triangulation::Dummy_TDS Triangulation_data_structure;
+    typedef Triangulation_data_structure::Full_cell_handle Full_cell_handle; /* Concept */
     template <typename TDS2>
     struct Rebind_TDS /* Concept */
     {
diff --git a/Triangulation/include/CGAL/Triangulation_face.h b/Triangulation/include/CGAL/Triangulation_face.h
index 1bbd77919da..d8809e1b7b3 100644
--- a/Triangulation/include/CGAL/Triangulation_face.h
+++ b/Triangulation/include/CGAL/Triangulation_face.h
@@ -28,6 +28,7 @@ class Triangulation_face
 {
     typedef typename internal::Dimen_plus_one<typename TDS::Maximal_dimension>::type Dimen_plus;
 public:
+    typedef typename TDS                            Triangulation_data_structure;
     typedef typename TDS::Full_cell_handle          Full_cell_handle; /* Concept */
     typedef typename TDS::Vertex_handle             Vertex_handle; /* Concept */
     typedef internal::S_or_D_array<int, Dimen_plus> Indices;