diff --git a/Packages/Cartesian_kernel/changes.txt b/Packages/Cartesian_kernel/changes.txt
index 35eff841ade..93b51240bc1 100644
--- a/Packages/Cartesian_kernel/changes.txt
+++ b/Packages/Cartesian_kernel/changes.txt
@@ -1,3 +1,6 @@
+Version 6.51 (22 January 2002)
+- Classes and funtion return type only refer to R::...
+
 Version 6.50 (9 January 2002)
 - Update compare_y_at_x.
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian.h
index 5cd0d3ae659..c0a93178203 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian.h
@@ -231,14 +231,11 @@ struct Cartesian
     typedef CGAL::Handle_for<CGAL::Twotuple<FT> >	Vector_handle_2;
     typedef CGAL::Handle_for<CGAL::Twotuple<FT> >	Direction_handle_2;
     typedef CGAL::Handle_for<CGAL::Threetuple<FT> >	Line_handle_2;
-    typedef CGAL::Handle_for<CGAL::Twotuple<CGAL_TYPENAME_MSVC_NULL
-	                     Kernel_base::Point_2> >    Ray_handle_2;
-    typedef CGAL::Handle_for<CGAL::Twotuple<CGAL_TYPENAME_MSVC_NULL
-	                     Kernel_base::Point_2> >    Segment_handle_2;
-    typedef CGAL::Handle_for<triple<Point_2_base, FT, Orientation> >
+    typedef CGAL::Handle_for<CGAL::Twotuple<Point_2> >  Ray_handle_2;
+    typedef CGAL::Handle_for<CGAL::Twotuple<Point_2> >  Segment_handle_2;
+    typedef CGAL::Handle_for<triple<Point_2, FT, Orientation> >
 	                                                Circle_handle_2;
-    typedef CGAL::Handle_for<CGAL::Threetuple<CGAL_TYPENAME_MSVC_NULL
-	                     Kernel_base::Point_2> >    Triangle_handle_2;
+    typedef CGAL::Handle_for<CGAL::Threetuple<Point_2> >    Triangle_handle_2;
     typedef CGAL::Handle_for<CGAL::Twotuple<Point_2> >  Iso_rectangle_handle_2;
     typedef CGAL::Handle_for_virtual< Aff_transformation_rep_baseC2<Self> >
 			                           Aff_transformation_handle_2;
@@ -246,21 +243,15 @@ struct Cartesian
     typedef CGAL::Handle_for<CGAL::Threetuple<FT> >	Point_handle_3;
     typedef CGAL::Handle_for<CGAL::Threetuple<FT> >	Vector_handle_3;
     typedef CGAL::Handle_for<CGAL::Threetuple<FT> >	Direction_handle_3;
-    typedef CGAL::Handle_for<std::pair<Point_3_base, Direction_3_base> >
+    typedef CGAL::Handle_for<std::pair<Point_3, Direction_3> >
 	                                                Line_handle_3;
     typedef CGAL::Handle_for<CGAL::Fourtuple<FT> >	Plane_handle_3;
-    typedef CGAL::Handle_for<CGAL::Twotuple<CGAL_TYPENAME_MSVC_NULL
-	                     Kernel_base::Point_3> >    Ray_handle_3;
-    typedef CGAL::Handle_for<CGAL::Twotuple<CGAL_TYPENAME_MSVC_NULL
-	                     Kernel_base::Point_3> >    Segment_handle_3;
-    typedef CGAL::Handle_for<triple<Point_3_base, FT, Orientation> >
-	                                                Sphere_handle_3;
-    typedef CGAL::Handle_for<CGAL::Threetuple<CGAL_TYPENAME_MSVC_NULL
-	                     Kernel_base::Point_3> >    Triangle_handle_3;
-    typedef CGAL::Handle_for<CGAL::Fourtuple<CGAL_TYPENAME_MSVC_NULL
-	                     Kernel_base::Point_3> >    Tetrahedron_handle_3;
-    typedef CGAL::Handle_for<CGAL::Twotuple<CGAL_TYPENAME_MSVC_NULL
-	                     Kernel_base::Point_3> >    Iso_cuboid_handle_3;
+    typedef CGAL::Handle_for<CGAL::Twotuple<Point_3> >  Ray_handle_3;
+    typedef CGAL::Handle_for<CGAL::Twotuple<Point_3> >  Segment_handle_3;
+    typedef CGAL::Handle_for<triple<Point_3, FT,Orientation> > Sphere_handle_3;
+    typedef CGAL::Handle_for<CGAL::Threetuple<Point_3> > Triangle_handle_3;
+    typedef CGAL::Handle_for<CGAL::Fourtuple<Point_3> > Tetrahedron_handle_3;
+    typedef CGAL::Handle_for<CGAL::Twotuple<Point_3> >  Iso_cuboid_handle_3;
     typedef CGAL::Handle_for_virtual< Aff_transformation_rep_baseC3<Self> >
 			                           Aff_transformation_handle_3;
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_2.h
index dcf0d786521..0930b6e2d44 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_2.h
@@ -49,10 +49,11 @@ class Aff_transformationC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   typedef typename R_::FT                   FT;
   typedef Aff_transformation_rep_baseC2<R_> Aff_t_base;
 
-  typedef typename R_::Kernel_base::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Vector_2             Vector_2;
-  typedef typename R_::Kernel_base::Direction_2          Direction_2;
-  typedef typename R_::Kernel_base::Line_2               Line_2;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Vector_2             Vector_2;
+  typedef typename R_::Direction_2          Direction_2;
+  typedef typename R_::Line_2               Line_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
 public:
   typedef R_                                R;
@@ -153,7 +154,7 @@ public:
   operator()(const Line_2 &l) const
   { return transform(l); }
 
-  Aff_transformationC2 inverse() const { return Ptr()->inverse(); }
+  Aff_transformation_2 inverse() const { return Ptr()->inverse(); }
 
   bool is_even() const { return Ptr()->is_even(); }
   bool is_odd() const { return ! (Ptr()->is_even()); }
@@ -163,7 +164,7 @@ public:
   FT m(int i, int j) const { return cartesian(i,j); }
   FT hm(int i, int j) const { return cartesian(i,j); }
 
-  Aff_transformationC2 operator*(const Aff_transformationC2 &t) const
+  Aff_transformation_2 operator*(const Aff_transformationC2 &t) const
   {
     return (*Ptr()) * (*t.Ptr());
   }
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_3.h
index 42e3108dc34..b704a709c34 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_3.h
@@ -48,15 +48,16 @@ class Aff_transformationC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   friend class PlaneC3<R_ CGAL_CTAG>; // FIXME: why ?
 #endif
 
+  typedef typename R_::FT                   FT;
+  typedef Aff_transformation_rep_baseC3<R_> Aff_t_base;
+
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 public:
   typedef R_                               R;
-  typedef typename R::FT                   FT;
-  typedef Aff_transformation_rep_baseC3<R> Aff_t_base;
-
-  typedef typename R::Kernel_base::Point_3              Point_3;
-  typedef typename R::Kernel_base::Vector_3             Vector_3;
-  typedef typename R::Kernel_base::Direction_3          Direction_3;
-  typedef typename R::Kernel_base::Plane_3              Plane_3;
 
   Aff_transformationC3()
   {
@@ -152,7 +153,7 @@ public:
   operator()(const Plane_3& p) const
   { return transform(p); } // FIXME : not compiled by the test-suite !
 
-  Aff_transformationC3 inverse() const { return Ptr()->inverse(); }
+  Aff_transformation_3 inverse() const { return Ptr()->inverse(); }
   
   bool is_even() const { return Ptr()->is_even(); }
   bool is_odd() const { return  ! (Ptr()->is_even()); }
@@ -162,11 +163,11 @@ public:
   FT m(int i, int j) const { return cartesian(i,j); }
   FT hm(int i, int j) const { return cartesian(i,j); }
 
-  Aff_transformationC3 operator*(const Aff_transformationC3 &t) const
+  Aff_transformation_3 operator*(const Aff_transformationC3 &t) const
   { return (*Ptr()) * (*t.Ptr()); }
 
 protected:
-  Aff_transformationC3        transpose() const { return Ptr()->transpose(); }
+  Aff_transformation_3        transpose() const { return Ptr()->transpose(); }
 };
 
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_rep_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_rep_2.h
index 31f374560c1..74a2e4d3802 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_rep_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_rep_2.h
@@ -31,12 +31,11 @@ class Aff_transformation_rep_baseC2
   : public Ref_counted_virtual
 {
 public:
-  typedef typename R::FT                        FT;
-
-  typedef typename R::Kernel_base::Point_2              Point_2;
-  typedef typename R::Kernel_base::Vector_2             Vector_2;
-  typedef typename R::Kernel_base::Direction_2          Direction_2;
-  typedef typename R::Kernel_base::Aff_transformation_2 Aff_transformation_2;
+  typedef typename R::FT                   FT;
+  typedef typename R::Point_2              Point_2;
+  typedef typename R::Vector_2             Vector_2;
+  typedef typename R::Direction_2          Direction_2;
+  typedef typename R::Aff_transformation_2 Aff_transformation_2;
 
   virtual ~Aff_transformation_rep_baseC2() {}
 
@@ -166,7 +165,7 @@ friend class Scaling_repC2<R>;
   }
 
 private:
-    FT   t11, t12, t13; // FIXME : use an array instead ?
+    FT   t11, t12, t13;
     FT   t21, t22, t23;
 };
 
@@ -178,7 +177,7 @@ template < class R >
 CGAL_KERNEL_LARGE_INLINE
 typename Aff_transformation_repC2<R>::Aff_transformation_2
 Aff_transformation_repC2<R>::
-inverse() const // FIXME : construction
+inverse() const
 {
   FT det = FT(1) / (t11 * t22 - t12 * t21);
   return Aff_transformation_2(
@@ -199,7 +198,7 @@ template < class R >
 CGAL_KERNEL_LARGE_INLINE
 typename Aff_transformation_repC2<R>::Aff_transformation_2
 Aff_transformation_repC2<R>::
-compose(const Aff_transformation_repC2<R> &t) const // FIXME : construction
+compose(const Aff_transformation_repC2<R> &t) const
 {
   return Aff_transformation_2(t.t11*t11 + t.t12*t21,
                               t.t11*t12 + t.t12*t22,
@@ -213,7 +212,7 @@ template < class R >
 CGAL_KERNEL_LARGE_INLINE
 typename Aff_transformation_repC2<R>::Aff_transformation_2
 Aff_transformation_repC2<R>::
-compose(const Translation_repC2<R> &t) const // FIXME : construction
+compose(const Translation_repC2<R> &t) const
 {
   return Aff_transformation_2(t11,
                               t12,
@@ -227,7 +226,7 @@ template < class R >
 CGAL_KERNEL_LARGE_INLINE
 typename Aff_transformation_repC2<R>::Aff_transformation_2
 Aff_transformation_repC2<R>::
-compose(const Rotation_repC2<R> &t) const // FIXME : construction
+compose(const Rotation_repC2<R> &t) const
 {
   return Aff_transformation_2(t.cosinus_*t11 - t.sinus_*t21,
                               t.cosinus_*t12 - t.sinus_*t22,
@@ -241,7 +240,7 @@ template < class R >
 CGAL_KERNEL_LARGE_INLINE
 typename Aff_transformation_repC2<R>::Aff_transformation_2
 Aff_transformation_repC2<R>::
-compose(const Scaling_repC2<R> &t) const // FIXME : construction
+compose(const Scaling_repC2<R> &t) const
 {
    return Aff_transformation_2(t.scalefactor_ * t11,
                                t.scalefactor_ * t12,
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_rep_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_rep_3.h
index f310f664168..ffa8c5ab569 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_rep_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Aff_transformation_rep_3.h
@@ -31,12 +31,11 @@ class Aff_transformation_rep_baseC3
   : public Ref_counted_virtual
 {
 public:
-  typedef typename R::FT                        FT;
-
-  typedef typename R::Kernel_base::Point_3              Point_3;
-  typedef typename R::Kernel_base::Vector_3             Vector_3;
-  typedef typename R::Kernel_base::Direction_3          Direction_3;
-  typedef typename R::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R::FT                   FT;
+  typedef typename R::Point_3              Point_3;
+  typedef typename R::Vector_3             Vector_3;
+  typedef typename R::Direction_3          Direction_3;
+  typedef typename R::Aff_transformation_3 Aff_transformation_3;
 
   virtual ~Aff_transformation_rep_baseC3(){}
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Circle_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Circle_2.h
index d25065ea60f..657fa95affb 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Circle_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Circle_2.h
@@ -30,15 +30,14 @@ template <class R_ >
 class CircleC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Circle_handle_2
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Circle_2             Circle_2;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Circle_handle_2           base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Vector_2             Vector_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
-
 public:
   typedef R_                                     R;
 
@@ -68,9 +67,7 @@ public:
 
     if (p != q) {
       Point_2 center = midpoint(p, q);
-      FT      squared_radius = squared_distance(p, center);
-
-      initialize_with(rep(center, squared_radius, orient));
+      initialize_with(rep(center, squared_distance(p, center), orient));
     } else
       initialize_with(rep(p, FT(0), orient));
   }
@@ -81,9 +78,7 @@ public:
     CGAL_kernel_precondition( orient != COLLINEAR);
 
     Point_2 center = circumcenter(p, q, r);
-    FT      squared_radius = squared_distance(p, center);
-
-    initialize_with(rep(center, squared_radius, orient));
+    initialize_with(rep(center, squared_distance(p, center), orient));
   }
 
   bool           operator==(const CircleC2 &s) const;
@@ -104,9 +99,9 @@ public:
    return Ptr()->third;
   }
 
-  CircleC2           opposite() const;
+  Circle_2           opposite() const;
 
-  CircleC2           orthogonal_transform(const Aff_transformation_2 &t) const;
+  Circle_2           orthogonal_transform(const Aff_transformation_2 &t) const;
 
   Oriented_side  oriented_side(const Point_2 &p) const;
   Bounded_side   bounded_side(const Point_2 &p) const;
@@ -161,7 +156,7 @@ CGAL_KERNEL_INLINE
 Bounded_side
 CircleC2<R CGAL_CTAG>::
 bounded_side(const typename CircleC2<R CGAL_CTAG>::Point_2 &p) const
-{ // FIXME : predicate
+{
   return Bounded_side(CGAL_NTS compare(squared_radius(),
                                        squared_distance(center(),p)));
 }
@@ -171,8 +166,7 @@ inline
 bool
 CircleC2<R CGAL_CTAG>::
 has_on_boundary(const typename CircleC2<R CGAL_CTAG>::Point_2 &p) const
-{ // FIXME: predicate
-  // return squared_distance(center(), p) == squared_radius();
+{
   return bounded_side(p) == ON_BOUNDARY;
 }
 
@@ -182,8 +176,6 @@ bool
 CircleC2<R CGAL_CTAG>::
 has_on_bounded_side(const typename CircleC2<R CGAL_CTAG>::Point_2 &p) const
 {
-    // FIXME: predicate
-  // return squared_distance(center(),p) < squared_radius();
   return bounded_side(p) == ON_BOUNDED_SIDE;
 }
 
@@ -193,8 +185,6 @@ bool
 CircleC2<R CGAL_CTAG>::
 has_on_unbounded_side(const typename CircleC2<R CGAL_CTAG>::Point_2 &p) const
 {
-    // FIXME: predicate
-  // return squared_distance(center(),p) > squared_radius();
   return bounded_side(p) == ON_UNBOUNDED_SIDE;
 }
 
@@ -225,13 +215,12 @@ inline
 bool
 CircleC2<R CGAL_CTAG>::is_degenerate() const
 {
-    // FIXME: predicate
   return CGAL_NTS is_zero(squared_radius());
 }
 
 template < class R >
 inline
-CircleC2<R CGAL_CTAG>
+typename CircleC2<R CGAL_CTAG>::Circle_2
 CircleC2<R CGAL_CTAG>::opposite() const
 {
   return CircleC2<R CGAL_CTAG>(center(),
@@ -254,11 +243,11 @@ CircleC2<R CGAL_CTAG>::bbox() const // FIXME : to_interval()
 
 template < class R >
 CGAL_KERNEL_INLINE
-CircleC2<R CGAL_CTAG>
+typename CircleC2<R CGAL_CTAG>::Circle_2
 CircleC2<R CGAL_CTAG>::orthogonal_transform
   (const typename CircleC2<R CGAL_CTAG>::Aff_transformation_2 &t) const
-{ // FIXME : construction
-  Vector_2 vec(FT(1), FT(0) );   // unit vector
+{
+  typename R::Vector_2 vec(FT(1), FT(0) );   // unit vector
   vec = vec.transform(t);             // transformed
   FT sq_scale = vec.squared_length();       // squared scaling factor
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Data_accessor_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Data_accessor_2.h
index e52278005c0..8829b28e6f7 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Data_accessor_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Data_accessor_2.h
@@ -31,7 +31,7 @@ class Data_accessorC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
 public:
     // Min_ellipse_2 wants FT public...
     typedef typename R_::FT                   FT;
-    typedef typename R_::Kernel_base::Point_2 Point;
+    typedef typename R_::Point_2              Point;
 
     typedef R_                           R;
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Direction_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Direction_2.h
index 0ada32638ef..7a36614c45b 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Direction_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Direction_2.h
@@ -28,18 +28,18 @@ template < class R_ >
 class DirectionC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Direction_handle_2
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Vector_2             Vector_2;
+  typedef typename R_::Line_2               Line_2;
+  typedef typename R_::Ray_2                Ray_2;
+  typedef typename R_::Segment_2            Segment_2;
+  typedef typename R_::Direction_2          Direction_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Direction_handle_2        base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Vector_2             Vector_2;
-  typedef typename R_::Kernel_base::Line_2               Line_2;
-  typedef typename R_::Kernel_base::Ray_2                Ray_2;
-  typedef typename R_::Kernel_base::Segment_2            Segment_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
-
 public:
   typedef R_                                     R;
 
@@ -71,14 +71,15 @@ public:
 	                            const DirectionC2 &d2) const;
   
   Vector_2 to_vector() const;
+  Vector_2 vector() const { return to_vector(); }
 
-  DirectionC2 perpendicular(const Orientation &o) const;
-  DirectionC2 transform(const Aff_transformation_2 &t) const
+  Direction_2 perpendicular(const Orientation &o) const;
+  Direction_2 transform(const Aff_transformation_2 &t) const
   {
     return t.transform(*this);
   }
 
-  DirectionC2 operator-() const;
+  Direction_2 operator-() const;
 
   const FT & delta(int i) const;
   const FT & dx() const
@@ -168,14 +169,14 @@ inline
 typename DirectionC2<R CGAL_CTAG>::Vector_2
 DirectionC2<R CGAL_CTAG>::to_vector() const
 {
-  return Vector_2(*this);
+  return Vector_2(dx(), dy());
 }
 
 template < class R >
 CGAL_KERNEL_MEDIUM_INLINE
-DirectionC2<R CGAL_CTAG>
+typename DirectionC2<R CGAL_CTAG>::Direction_2
 DirectionC2<R CGAL_CTAG>::perpendicular(const Orientation &o) const
-{ // FIXME : construction
+{
   CGAL_kernel_precondition(o != COLLINEAR);
   if (o == COUNTERCLOCKWISE)
     return DirectionC2<R CGAL_CTAG>(-dy(), dx());
@@ -185,9 +186,9 @@ DirectionC2<R CGAL_CTAG>::perpendicular(const Orientation &o) const
 
 template < class R >
 inline
-DirectionC2<R CGAL_CTAG>
+typename DirectionC2<R CGAL_CTAG>::Direction_2
 DirectionC2<R CGAL_CTAG>::operator-() const
-{ // FIXME : construction
+{
   return DirectionC2<R CGAL_CTAG>(-dx(), -dy());
 }
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h
index c450688dc6a..9db46eed66c 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h
@@ -29,16 +29,16 @@ class DirectionC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Direction_handle_3
 {
   typedef typename R_::FT                   FT;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Line_3               Line_3;
+  typedef typename R_::Ray_3                Ray_3;
+  typedef typename R_::Segment_3            Segment_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Direction_handle_3   base;
   typedef typename base::element_type       rep;
 
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Line_3               Line_3;
-  typedef typename R_::Kernel_base::Ray_3                Ray_3;
-  typedef typename R_::Kernel_base::Segment_3            Segment_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                R;
 
@@ -64,12 +64,14 @@ public:
   bool           operator!=(const DirectionC3 &d) const;
 
   Vector_3       to_vector() const;
-  DirectionC3           transform(const Aff_transformation_3 &t) const
+  Vector_3       vector() const { return to_vector(); }
+
+  Direction_3    transform(const Aff_transformation_3 &t) const
   {
     return t.transform(*this);
   }
 
-  DirectionC3           operator-() const;
+  Direction_3    operator-() const;
 
   const FT & delta(int i) const;
   const FT & dx() const
@@ -130,14 +132,14 @@ inline
 typename DirectionC3<R CGAL_CTAG>::Vector_3
 DirectionC3<R CGAL_CTAG>::to_vector() const
 {
-  return Vector_3(*this);
+  return Vector_3(dx(), dy(), dz());
 }
 
 template < class R >
 inline
-DirectionC3<R CGAL_CTAG> 
+typename DirectionC3<R CGAL_CTAG>::Direction_3
 DirectionC3<R CGAL_CTAG>::operator-() const
-{ // FIXME : construction
+{
   return DirectionC3<R>(-dx(), -dy(), -dz());
 }
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Iso_cuboid_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Iso_cuboid_3.h
index 8739a05d8f7..0514bba4923 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Iso_cuboid_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Iso_cuboid_3.h
@@ -31,13 +31,13 @@ class Iso_cuboidC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Iso_cuboid_handle_3
 {
   typedef typename R_::FT                   FT;
+  typedef typename R_::Iso_cuboid_3         Iso_cuboid_3;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Iso_cuboid_handle_3  base;
   typedef typename base::element_type       rep;
 
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                R;
 
@@ -45,7 +45,7 @@ public:
     : base(rep()) {}
 
   Iso_cuboidC3(const Point_3 &p, const Point_3 &q)
-  { // FIXME : construction
+  {
     FT minx, maxx, miny, maxy, minz, maxz;
     if (p.x() < q.x()) { minx = p.x(); maxx = q.x(); }
     else               { minx = q.x(); maxx = p.x(); }
@@ -92,7 +92,7 @@ public:
   Point_3 vertex(int i) const;
   Point_3 operator[](int i) const;
 
-  Iso_cuboidC3 transform(const Aff_transformation_3 &t) const
+  Iso_cuboid_3 transform(const Aff_transformation_3 &t) const
   {
     return Iso_cuboidC3(t.transform(min()), t.transform(max()));
   }
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Iso_rectangle_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Iso_rectangle_2.h
index 03a4c859a56..ee173dc8320 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Iso_rectangle_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Iso_rectangle_2.h
@@ -28,14 +28,14 @@ template <class R_>
 class Iso_rectangleC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Iso_rectangle_handle_2
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Iso_rectangle_2      Iso_rectangle_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Iso_rectangle_handle_2    base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
-
 public:
   typedef R_                                     R;
 
@@ -43,7 +43,7 @@ public:
     : base(rep()) {}
 
   Iso_rectangleC2(const Point_2 &p, const Point_2 &q)
-  { // FIXME : construction
+  {
     FT minx, maxx, miny, maxy;
     if (p.x() < q.x()) { minx = p.x(); maxx = q.x(); }
     else               { minx = q.x(); maxx = p.x(); }
@@ -85,7 +85,7 @@ public:
   Point_2 vertex(int i) const;
   Point_2 operator[](int i) const;
 
-  Iso_rectangleC2 transform(const Aff_transformation_2 &t) const
+  Iso_rectangle_2 transform(const Aff_transformation_2 &t) const
   {
     // FIXME : We need a precondition like this!!!
     // CGAL_kernel_precondition(t.is_axis_preserving());
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Line_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Line_2.h
index 52920438ad7..f5c49c0dd11 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Line_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Line_2.h
@@ -28,17 +28,17 @@ template < class R_ >
 class LineC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Line_handle_2
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Direction_2          Direction_2;
+  typedef typename R_::Ray_2                Ray_2;
+  typedef typename R_::Segment_2            Segment_2;
+  typedef typename R_::Line_2               Line_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Line_handle_2             base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Direction_2          Direction_2;
-  typedef typename R_::Kernel_base::Ray_2                Ray_2;
-  typedef typename R_::Kernel_base::Segment_2            Segment_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
-
 public:
   typedef R_                                     R;
 
@@ -79,8 +79,8 @@ public:
   FT              x_at_y(const FT &y) const;
   FT              y_at_x(const FT &x) const;
 
-  LineC2            perpendicular(const Point_2 &p) const;
-  LineC2            opposite() const;
+  Line_2          perpendicular(const Point_2 &p) const;
+  Line_2          opposite() const;
   Point_2         point(int i) const;
 
   Point_2         point() const;
@@ -98,7 +98,7 @@ public:
   bool            is_vertical() const;
   bool            is_degenerate() const;
 
-  LineC2            transform(const Aff_transformation_2 &t) const
+  Line_2          transform(const Aff_transformation_2 &t) const
   {
     return LineC2<R CGAL_CTAG>(t.transform(point(0)),
                                t.transform(direction()));
@@ -165,7 +165,7 @@ LineC2<R CGAL_CTAG>::y_at_x(const typename LineC2<R CGAL_CTAG>::FT &x) const
 
 template < class R >
 inline
-LineC2<R CGAL_CTAG>
+typename LineC2<R CGAL_CTAG>::Line_2
 LineC2<R CGAL_CTAG>::
 perpendicular(const typename LineC2<R CGAL_CTAG>::Point_2 &p) const
 {
@@ -174,9 +174,9 @@ perpendicular(const typename LineC2<R CGAL_CTAG>::Point_2 &p) const
 
 template < class R >
 inline
-LineC2<R CGAL_CTAG>
+typename LineC2<R CGAL_CTAG>::Line_2
 LineC2<R CGAL_CTAG>::opposite() const
-{ // FIXME : construction
+{
   return LineC2<R CGAL_CTAG>( -a(), -b(), -c() );
 }
 
@@ -209,7 +209,7 @@ template < class R >
 inline
 typename LineC2<R CGAL_CTAG>::Direction_2
 LineC2<R CGAL_CTAG>::direction() const
-{ // FIXME : construction
+{
   return Direction_2( b(), -a() );
 }
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Line_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Line_3.h
index 9e478f14081..f327a1e4af5 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Line_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Line_3.h
@@ -28,32 +28,32 @@ template < class R_ >
 class LineC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Line_handle_3
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Ray_3                Ray_3;
+  typedef typename R_::Line_3               Line_3;
+  typedef typename R_::Segment_3            Segment_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Line_handle_3             base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Plane_3              Plane_3;
-  typedef typename R_::Kernel_base::Ray_3                Ray_3;
-  typedef typename R_::Kernel_base::Segment_3            Segment_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                     R;
 
   LineC3()
     : base(rep()) {}
 
-  LineC3(const Point_3 &p, const Point_3 &q) // FIXME : construction
+  LineC3(const Point_3 &p, const Point_3 &q)
     : base(rep(p, (q-p).direction())) {}
 
-  LineC3(const Segment_3 &s) // FIXME : construction
+  LineC3(const Segment_3 &s)
     : base(rep(s.start(), (s.end() - s.start()).direction())) {}
 
-  LineC3(const Ray_3 &r) // FIXME : construction
+  LineC3(const Ray_3 &r)
     : base(rep(r.start(), (r.point(1) - r.start()).direction())) {}
 
   LineC3(const Point_3 &p, const Direction_3 &d)
@@ -63,7 +63,7 @@ public:
   bool        operator!=(const LineC3 &l) const;
 
   Plane_3     perpendicular_plane(const Point_3 &p) const;
-  LineC3        opposite() const;
+  Line_3      opposite() const;
 
   const Point_3 &     point() const
   {
@@ -81,7 +81,7 @@ public:
   bool        has_on(const Point_3 &p) const;
   bool        is_degenerate() const;
 
-  LineC3        transform(const Aff_transformation_3 &t) const
+  Line_3        transform(const Aff_transformation_3 &t) const
   {
     return LineC3<R>(t.transform(point()), t.transform(direction()));
   }
@@ -128,7 +128,7 @@ perpendicular_plane(const typename LineC3<R CGAL_CTAG>::Point_3 &p) const
 
 template < class R >
 inline
-LineC3<R CGAL_CTAG>
+typename LineC3<R CGAL_CTAG>::Line_3
 LineC3<R CGAL_CTAG>::opposite() const
 {
   return LineC3<R CGAL_CTAG>(point(), -direction());
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Plane_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Plane_3.h
index 4732743ac53..9105d79344d 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Plane_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Plane_3.h
@@ -28,20 +28,20 @@ template <class R_>
 class PlaneC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Plane_handle_3
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Line_3               Line_3;
+  typedef typename R_::Ray_3                Ray_3;
+  typedef typename R_::Segment_3            Segment_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Plane_handle_3            base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Line_3               Line_3;
-  typedef typename R_::Kernel_base::Ray_3                Ray_3;
-  typedef typename R_::Kernel_base::Segment_3            Segment_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                     R;
 
@@ -92,7 +92,7 @@ public:
   }
 
   Line_3       perpendicular_line(const Point_3 &p) const;
-  PlaneC3         opposite() const;
+  Plane_3      opposite() const;
 
   Point_3      point() const;
   Point_3      projection(const Point_3 &p) const;
@@ -106,7 +106,7 @@ public:
   Point_2      to_2d(const Point_3 &p) const;
   Point_3      to_3d(const Point_2 &p) const;
 
-  PlaneC3         transform(const Aff_transformation_3 &t) const
+  Plane_3      transform(const Aff_transformation_3 &t) const
   {
     if (t.is_even())
       return PlaneC3<R>(t.transform(point()),
@@ -200,7 +200,7 @@ PlaneC3<R CGAL_CTAG>::orthogonal_direction() const
 template < class R >
 typename PlaneC3<R CGAL_CTAG>::Vector_3
 PlaneC3<R CGAL_CTAG>::base1() const
-{ // FIXME : construction
+{
   if ( CGAL_NTS is_zero(a()) )  // parallel to x-axis
       return Vector_3(FT(1), FT(0), FT(0));
 
@@ -216,7 +216,7 @@ PlaneC3<R CGAL_CTAG>::base1() const
 template < class R >
 typename PlaneC3<R CGAL_CTAG>::Vector_3
 PlaneC3<R CGAL_CTAG>::base2() const
-{ // FIXME : construction
+{
   return cross_product(orthogonal_vector(), base1());
 }
 
@@ -224,7 +224,7 @@ template < class R >
 typename PlaneC3<R CGAL_CTAG>::Point_3
 PlaneC3<R CGAL_CTAG>::
 to_plane_basis(const typename PlaneC3<R CGAL_CTAG>::Point_3 &p) const
-{ // FIXME : construction
+{
   FT alpha, beta, gamma;
 
   solve(base1(), base2(), orthogonal_vector(), p - point(),
@@ -237,7 +237,7 @@ template < class R >
 typename PlaneC3<R CGAL_CTAG>::Point_2
 PlaneC3<R CGAL_CTAG>::
 to_2d(const typename PlaneC3<R CGAL_CTAG>::Point_3 &p) const
-{ // FIXME : construction
+{
   FT alpha, beta, gamma;
 
   solve(base1(), base2(), orthogonal_vector(), p - point(),
@@ -251,7 +251,7 @@ inline
 typename PlaneC3<R CGAL_CTAG>::Point_3
 PlaneC3<R CGAL_CTAG>::
 to_3d(const typename PlaneC3<R CGAL_CTAG>::Point_2 &p) const
-{ // FIXME : construction
+{
   return point() + p.x() * base1() + p.y() * base2();
 }
 
@@ -266,9 +266,9 @@ perpendicular_line(const typename PlaneC3<R CGAL_CTAG>::Point_3 &p) const
 
 template < class R >
 inline
-PlaneC3<R CGAL_CTAG>
+typename PlaneC3<R CGAL_CTAG>::Plane_3
 PlaneC3<R CGAL_CTAG>::opposite() const
-{ // FIXME : construction
+{
   return PlaneC3<R CGAL_CTAG>(-a(), -b(), -c(), -d());
 }
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Point_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Point_2.h
index fa94b3f764a..1bb31e5dc5c 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Point_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Point_2.h
@@ -31,14 +31,14 @@ template < class R_ >
 class PointC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Point_handle_2
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Vector_2             Vector_2;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Point_handle_2		 base;
   typedef typename base::element_type	         rep;
 
-  typedef typename R_::Kernel_base::Vector_2             Vector_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
-
 public:
   typedef R_                                     R;
 
@@ -109,7 +109,7 @@ public:
 
   Bbox_2 bbox() const;
 
-  PointC2 transform(const Aff_transformation_2 &t) const
+  Point_2 transform(const Aff_transformation_2 &t) const
   {
     return t.transform(*this);
   }
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Point_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Point_3.h
index cd4c1f888fd..fddbd601795 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Point_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Point_3.h
@@ -32,13 +32,12 @@ class PointC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Point_handle_3
 {
   typedef typename R_::FT                   FT;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Point_handle_3       base;
   typedef typename base::element_type       rep;
 
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                R;
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Ray_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Ray_2.h
index 8ee018d1216..4fc5584f464 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Ray_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Ray_2.h
@@ -28,16 +28,16 @@ template < class R_ >
 class RayC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Ray_handle_2
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Direction_2          Direction_2;
+  typedef typename R_::Line_2               Line_2;
+  typedef typename R_::Ray_2                Ray_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Ray_handle_2              base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Direction_2          Direction_2;
-  typedef typename R_::Kernel_base::Line_2               Line_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
-
 public:
   typedef R_                                     R;
 
@@ -66,9 +66,9 @@ public:
 
   Direction_2 direction() const;
   Line_2      supporting_line() const;
-  RayC2        opposite() const;
+  Ray_2        opposite() const;
 
-  RayC2        transform(const Aff_transformation_2 &t) const
+  Ray_2        transform(const Aff_transformation_2 &t) const
   {
     return RayC2<R>(t.transform(source()), t.transform(second_point()));
   }
@@ -139,7 +139,7 @@ RayC2<R CGAL_CTAG>::supporting_line() const
 
 template < class R >
 inline
-RayC2<R CGAL_CTAG>
+typename RayC2<R CGAL_CTAG>::Ray_2
 RayC2<R CGAL_CTAG>::opposite() const
 {
   return RayC2<R CGAL_CTAG>( source(), - direction() );
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Ray_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Ray_3.h
index ab5f89e9695..ea1f6b0e07f 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Ray_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Ray_3.h
@@ -28,16 +28,16 @@ template < class R_ >
 class RayC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Ray_handle_3
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Line_3               Line_3;
+  typedef typename R_::Ray_3                Ray_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Ray_handle_3              base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Line_3               Line_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                     R;
 
@@ -66,9 +66,9 @@ public:
 
   Direction_3 direction() const;
   Line_3      supporting_line() const;
-  RayC3        opposite() const;
+  Ray_3       opposite() const;
 
-  RayC3        transform(const Aff_transformation_3 &t) const
+  Ray_3       transform(const Aff_transformation_3 &t) const
   {
     return RayC3<R>(t.transform(source()), t.transform(second_point()));
   }
@@ -137,7 +137,7 @@ RayC3<R CGAL_CTAG>::supporting_line() const
 
 template < class R >
 inline
-RayC3<R CGAL_CTAG>
+typename RayC3<R CGAL_CTAG>::Ray_3
 RayC3<R CGAL_CTAG>::opposite() const
 {
   return RayC3<R CGAL_CTAG>( source(), - direction() );
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Segment_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Segment_2.h
index b879618a327..1ef626c9089 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Segment_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Segment_2.h
@@ -30,17 +30,16 @@ template < class R_ >
 class SegmentC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Segment_handle_2
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Direction_2          Direction_2;
+  typedef typename R_::Line_2               Line_2;
+  typedef typename R_::Segment_2            Segment_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Segment_handle_2          base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Vector_2             Vector_2;
-  typedef typename R_::Kernel_base::Direction_2          Direction_2;
-  typedef typename R_::Kernel_base::Line_2               Line_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
-
 public:
   typedef R_                                     R;
 
@@ -80,8 +79,8 @@ public:
 
   Direction_2 direction() const;
   Line_2      supporting_line() const;
-  SegmentC2        opposite() const;
-  SegmentC2        transform(const Aff_transformation_2 &t) const
+  Segment_2        opposite() const;
+  Segment_2        transform(const Aff_transformation_2 &t) const
   {
     return SegmentC2<R>(t.transform(source()), t.transform(target()));
   }
@@ -194,7 +193,7 @@ SegmentC2<R CGAL_CTAG>::supporting_line() const
 
 template < class R >
 inline
-SegmentC2<R CGAL_CTAG>
+typename SegmentC2<R CGAL_CTAG>::Segment_2
 SegmentC2<R CGAL_CTAG>::opposite() const
 {
   return SegmentC2<R CGAL_CTAG>(target(), source());
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Segment_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Segment_3.h
index 7fe3f983fe1..dea4cdca125 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Segment_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Segment_3.h
@@ -28,16 +28,16 @@ template < class R_ >
 class SegmentC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Segment_handle_3
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Line_3               Line_3;
+  typedef typename R_::Segment_3            Segment_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Segment_handle_3          base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Line_3               Line_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                     R;
 
@@ -75,8 +75,8 @@ public:
 
   Direction_3 direction() const;
   Line_3      supporting_line() const;
-  SegmentC3        opposite() const;
-  SegmentC3        transform(const Aff_transformation_3 &t) const
+  Segment_3   opposite() const;
+  Segment_3   transform(const Aff_transformation_3 &t) const
   {
     return SegmentC3<R>(t.transform(source()), t.transform(target()));
   }
@@ -189,7 +189,7 @@ SegmentC3<R CGAL_CTAG>::supporting_line() const
 
 template < class R >
 inline
-SegmentC3<R CGAL_CTAG>
+typename SegmentC3<R CGAL_CTAG>::Segment_3
 SegmentC3<R CGAL_CTAG>::opposite() const
 {
   return SegmentC3<R CGAL_CTAG>(target(), source());
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h
index e81fda9938a..ef094d5a7e9 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h
@@ -28,15 +28,15 @@ template <class R_>
 class SphereC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Sphere_handle_3
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Sphere_3             Sphere_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Sphere_handle_3           base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                     R;
 
@@ -113,7 +113,7 @@ public:
       return Ptr()->third;
   }
 
-  SphereC3 orthogonal_transform(const Aff_transformation_3 &t) const
+  Sphere_3 orthogonal_transform(const Aff_transformation_3 &t) const
   {
     // FIXME: precond: t.is_orthogonal() (*UNDEFINED*)
     Vector_3 vec(FT(1), FT(0));               // unit vector
@@ -130,7 +130,7 @@ public:
   bool is_degenerate() const;
 
   // Returns a circle with opposite orientation
-  SphereC3 opposite() const;
+  Sphere_3 opposite() const;
 
   Oriented_side  oriented_side(const Point_3 &p) const;
   //! precond: ! x.is_degenerate() (when available)
@@ -268,8 +268,8 @@ is_degenerate() const
 
 template < class R >
 inline
-SphereC3<R CGAL_CTAG> SphereC3<R CGAL_CTAG>::
-opposite() const
+typename SphereC3<R CGAL_CTAG>::Sphere_3
+SphereC3<R CGAL_CTAG>::opposite() const
 {
   return SphereC3<R CGAL_CTAG>(center(), squared_radius(),
                                CGAL::opposite(orientation()) );
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h
index d06b89c9653..2a814cf1fcd 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h
@@ -32,15 +32,15 @@ template <class R_>
 class TetrahedronC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Tetrahedron_handle_3
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Tetrahedron_3        Tetrahedron_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Tetrahedron_handle_3      base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Plane_3              Plane_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                     R;
 
@@ -59,7 +59,7 @@ public:
 
   Bbox_3     bbox() const;
 
-  TetrahedronC3       transform(const Aff_transformation_3 &t) const
+  Tetrahedron_3       transform(const Aff_transformation_3 &t) const
   {
     return TetrahedronC3<R>(t.transform(vertex(0)),
                 t.transform(vertex(1)),
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Triangle_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Triangle_2.h
index b15de566e95..1059b7b9e25 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Triangle_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Triangle_2.h
@@ -30,15 +30,15 @@ template <class R_>
 class TriangleC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Triangle_handle_2
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Vector_2             Vector_2;
+  typedef typename R_::Triangle_2           Triangle_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Triangle_handle_2         base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Vector_2             Vector_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
-
 public:
   typedef R_                                    R;
 
@@ -54,8 +54,8 @@ public:
   const Point_2 & vertex(int i) const;
   const Point_2 & operator[](int i) const;
 
-  TriangleC2           opposite() const;
-  TriangleC2           transform(const Aff_transformation_2 &t) const
+  Triangle_2           opposite() const;
+  Triangle_2           transform(const Aff_transformation_2 &t) const
   {
     return TriangleC2<R>(t.transform(vertex(0)),
                 t.transform(vertex(1)),
@@ -258,9 +258,8 @@ TriangleC2<R CGAL_CTAG>::bbox() const
 
 template < class R >
 inline
-TriangleC2<R CGAL_CTAG>
-TriangleC2<R CGAL_CTAG>::
-opposite() const
+typename TriangleC2<R CGAL_CTAG>::Triangle_2
+TriangleC2<R CGAL_CTAG>::opposite() const
 {
   return TriangleC2<R CGAL_CTAG>(vertex(0), vertex(2), vertex(1));
 }
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Triangle_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Triangle_3.h
index 9108762df1d..2dca0078b9b 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Triangle_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Triangle_3.h
@@ -28,16 +28,16 @@ template <class R_>
 class TriangleC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Triangle_handle_3
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Triangle_3           Triangle_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Triangle_handle_3         base;
   typedef typename base::element_type            rep;
 
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Plane_3              Plane_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                     R;
 
@@ -52,7 +52,7 @@ public:
 
   Plane_3    supporting_plane() const;
 
-  TriangleC3       transform(const Aff_transformation_3 &t) const
+  Triangle_3 transform(const Aff_transformation_3 &t) const
   {
     return TriangleC3<R>(t.transform(vertex(0)),
                 t.transform(vertex(1)),
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Vector_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Vector_2.h
index e44db0dc76d..0c711331662 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Vector_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Vector_2.h
@@ -28,15 +28,15 @@ template < class R_ >
 class VectorC2 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Vector_handle_2
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Direction_2          Direction_2;
+  typedef typename R_::Vector_2             Vector_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Vector_handle_2		 base;
   typedef typename base::element_type	         rep;
 
-  typedef typename R_::Kernel_base::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Direction_2          Direction_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
-
 public:
   typedef R_                                     R;
 
@@ -102,16 +102,16 @@ public:
       return 2;
   }
 
-  VectorC2 operator+(const VectorC2 &w) const;
-  VectorC2 operator-(const VectorC2 &w) const;
-  VectorC2 operator-() const;
+  Vector_2 operator+(const VectorC2 &w) const;
+  Vector_2 operator-(const VectorC2 &w) const;
+  Vector_2 operator-() const;
   FT operator*(const VectorC2 &w) const;
   FT squared_length() const;
-  VectorC2 operator/(const FT &c) const;
+  Vector_2 operator/(const FT &c) const;
   Direction_2 direction() const;
 
-  VectorC2 perpendicular(const Orientation &o) const;
-  VectorC2 transform(const Aff_transformation_2 &t) const
+  Vector_2 perpendicular(const Orientation &o) const;
+  Vector_2 transform(const Aff_transformation_2 &t) const
   {
     return t.transform(*this);
   }
@@ -182,7 +182,7 @@ VectorC2<R CGAL_CTAG>::homogeneous(int i) const
 
 template < class R >
 CGAL_KERNEL_INLINE
-VectorC2<R CGAL_CTAG>
+typename VectorC2<R CGAL_CTAG>::Vector_2
 VectorC2<R CGAL_CTAG>::operator+(const VectorC2<R CGAL_CTAG> &w) const
 {
   return VectorC2<R CGAL_CTAG>(x() + w.x(), y() + w.y());
@@ -190,7 +190,7 @@ VectorC2<R CGAL_CTAG>::operator+(const VectorC2<R CGAL_CTAG> &w) const
 
 template < class R >
 CGAL_KERNEL_INLINE
-VectorC2<R CGAL_CTAG>
+typename VectorC2<R CGAL_CTAG>::Vector_2
 VectorC2<R CGAL_CTAG>::operator-(const VectorC2<R CGAL_CTAG> &w) const
 {
   return VectorC2<R CGAL_CTAG>(x() - w.x(), y() - w.y());
@@ -198,7 +198,7 @@ VectorC2<R CGAL_CTAG>::operator-(const VectorC2<R CGAL_CTAG> &w) const
 
 template < class R >
 CGAL_KERNEL_INLINE
-VectorC2<R CGAL_CTAG>
+typename VectorC2<R CGAL_CTAG>::Vector_2
 VectorC2<R CGAL_CTAG>::operator-() const
 {
   return VectorC2<R CGAL_CTAG>(-x(), -y());
@@ -222,7 +222,7 @@ VectorC2<R CGAL_CTAG>::squared_length() const
 
 template < class R >
 CGAL_KERNEL_INLINE
-VectorC2<R CGAL_CTAG>
+typename VectorC2<R CGAL_CTAG>::Vector_2
 VectorC2<R CGAL_CTAG>::
 operator/(const typename VectorC2<R CGAL_CTAG>::FT &c) const
 {
@@ -239,7 +239,7 @@ VectorC2<R CGAL_CTAG>::direction() const
 
 template < class R >
 CGAL_KERNEL_MEDIUM_INLINE
-VectorC2<R CGAL_CTAG>
+typename VectorC2<R CGAL_CTAG>::Vector_2
 VectorC2<R CGAL_CTAG>::perpendicular(const Orientation &o) const
 {
   CGAL_kernel_precondition( o != COLLINEAR );
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Vector_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Vector_3.h
index cf3f20b1cb4..753a35a4c0b 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/Vector_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/Vector_3.h
@@ -28,15 +28,15 @@ template < class R_ >
 class VectorC3 CGAL_ADVANCED_KERNEL_PARTIAL_SPEC
   : public R_::Vector_handle_3
 {
-  typedef typename R_::FT                        FT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Vector_handle_3	   	 base;
   typedef typename base::element_type   	 rep;
 
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
-
 public:
   typedef R_                                     R;
 
@@ -111,14 +111,14 @@ public:
       return 3;
   }
 
-  VectorC3 operator+(const VectorC3 &w) const;
-  VectorC3 operator-(const VectorC3 &w) const;
-  VectorC3 operator-() const;
-  VectorC3 operator/(const FT &c) const;
+  Vector_3 operator+(const VectorC3 &w) const;
+  Vector_3 operator-(const VectorC3 &w) const;
+  Vector_3 operator-() const;
+  Vector_3 operator/(const FT &c) const;
   FT operator*(const VectorC3 &w) const;
   FT squared_length() const;
   Direction_3 direction() const;
-  VectorC3 transform(const Aff_transformation_3 &t) const
+  Vector_3 transform(const Aff_transformation_3 &t) const
   {
     return t.transform(*this);
   }
@@ -192,7 +192,7 @@ VectorC3<R CGAL_CTAG>::homogeneous(int i) const
 
 template < class R >
 inline
-VectorC3<R CGAL_CTAG>
+typename VectorC3<R CGAL_CTAG>::Vector_3
 VectorC3<R CGAL_CTAG>::
 operator+(const VectorC3<R CGAL_CTAG> &w) const
 {
@@ -201,7 +201,7 @@ operator+(const VectorC3<R CGAL_CTAG> &w) const
 
 template < class R >
 inline
-VectorC3<R CGAL_CTAG>
+typename VectorC3<R CGAL_CTAG>::Vector_3
 VectorC3<R CGAL_CTAG>::operator-(const VectorC3<R CGAL_CTAG> &w) const
 {
   return VectorC3<R CGAL_CTAG>(x() - w.x(), y() - w.y(), z() - w.z());
@@ -209,10 +209,10 @@ VectorC3<R CGAL_CTAG>::operator-(const VectorC3<R CGAL_CTAG> &w) const
 
 template < class R >
 inline
-VectorC3<R CGAL_CTAG>
+typename VectorC3<R CGAL_CTAG>::Vector_3
 VectorC3<R CGAL_CTAG>::operator-() const
 {
-  return VectorC3<R CGAL_CTAG>(-x(), -y(), -z());
+  return Vector_3(-x(), -y(), -z());
 }
 
 template < class R >
@@ -233,7 +233,7 @@ VectorC3<R CGAL_CTAG>::squared_length() const
 
 template < class R >
 inline
-VectorC3<R CGAL_CTAG>
+typename VectorC3<R CGAL_CTAG>::Vector_3
 VectorC3<R CGAL_CTAG>::
 operator/(const typename VectorC3<R CGAL_CTAG>::FT &c) const
 {
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/global_operators_2.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/global_operators_2.h
index b986cc30241..ae0e63e6387 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/global_operators_2.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/global_operators_2.h
@@ -26,23 +26,23 @@ CGAL_BEGIN_NAMESPACE
 
 template < class R >
 inline
-PointC2<R CGAL_CTAG>
+typename R::Point_2
 operator+(const PointC2<R CGAL_CTAG> &p, const VectorC2<R CGAL_CTAG> &v)
-{ // FIXME : construction
+{
   return PointC2<R CGAL_CTAG>(p.x() + v.x(), p.y() + v.y());
 }
 
 template < class R >
 inline
-PointC2<R CGAL_CTAG>
+typename R::Point_2
 operator-(const PointC2<R CGAL_CTAG> &p, const VectorC2<R CGAL_CTAG> &v)
-{ // FIXME : construction
+{
   return PointC2<R CGAL_CTAG>(p.x() - v.x(), p.y() - v.y());
 }
 
 template < class R >
 inline
-PointC2<R CGAL_CTAG>
+typename R::Point_2
 operator+(const Origin &, const VectorC2<R CGAL_CTAG> &v)
 {
   return PointC2<R CGAL_CTAG>(v);
@@ -50,23 +50,23 @@ operator+(const Origin &, const VectorC2<R CGAL_CTAG> &v)
 
 template < class R >
 inline
-PointC2<R CGAL_CTAG>
+typename R::Point_2
 operator-(const Origin &, const VectorC2<R CGAL_CTAG> &v)
-{ // FIXME : construction
+{
   return PointC2<R CGAL_CTAG>(-v);
 }
 
 template < class R >
 inline
-VectorC2<R CGAL_CTAG>
+typename R::Vector_2
 operator-(const PointC2<R CGAL_CTAG> &p, const PointC2<R CGAL_CTAG> &q)
-{ // FIXME : construction
+{
   return VectorC2<R CGAL_CTAG>(p.x() - q.x(), p.y() - q.y());
 }
 
 template < class R >
 inline
-VectorC2<R CGAL_CTAG>
+typename R::Vector_2
 operator-(const PointC2<R CGAL_CTAG> &p, const Origin &)
 {
   return VectorC2<R CGAL_CTAG>(p);
@@ -74,32 +74,25 @@ operator-(const PointC2<R CGAL_CTAG> &p, const Origin &)
 
 template < class R >
 inline
-VectorC2<R CGAL_CTAG>
+typename R::Vector_2
 operator-(const Origin &, const PointC2<R CGAL_CTAG> &p)
-{ // FIXME : construction
+{
   return VectorC2<R CGAL_CTAG>(-p.x(), -p.y());
 }
 
-#ifdef __SUNPRO_CC
-template < class FT, class R >
-CGAL_KERNEL_INLINE
-VectorC2<R CGAL_CTAG>
-operator*(const FT &c, const VectorC2<R CGAL_CTAG> &w)
-#else
 template < class R >
 CGAL_KERNEL_INLINE
-VectorC2<R CGAL_CTAG>
+typename R::Vector_2
 operator*(const typename R::FT &c, const VectorC2<R CGAL_CTAG> &w)
-#endif
-{ // FIXME : construction
+{
    return VectorC2<R CGAL_CTAG>(c * w.x(), c * w.y());
 }
 
 template < class R >
 CGAL_KERNEL_INLINE
-VectorC2<R CGAL_CTAG>
+typename R::Vector_2
 operator*(const VectorC2<R CGAL_CTAG> &w, const typename R::FT &c)
-{ // FIXME : construction
+{
    return VectorC2<R CGAL_CTAG>(c * w.x(), c * w.y());
 }
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/global_operators_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/global_operators_3.h
index 3236a420279..6413b5e1a2a 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/global_operators_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/global_operators_3.h
@@ -26,23 +26,23 @@ CGAL_BEGIN_NAMESPACE
 
 template < class R >
 inline
-PointC3<R CGAL_CTAG>
+typename R::Point_3
 operator+(const PointC3<R CGAL_CTAG> &p, const VectorC3<R CGAL_CTAG> &v)
-{ // FIXME : construction
+{
   return PointC3<R CGAL_CTAG>(p.x() + v.x(), p.y() + v.y(), p.z() + v.z());
 }
 
 template < class R >
 inline
-PointC3<R CGAL_CTAG>
+typename R::Point_3
 operator-(const PointC3<R CGAL_CTAG> &p, const VectorC3<R CGAL_CTAG> &v)
-{ // FIXME : construction
+{
   return PointC3<R CGAL_CTAG>(p.x() - v.x(), p.y() - v.y(), p.z() - v.z());
 }
 
 template < class R >
 inline
-PointC3<R CGAL_CTAG>
+typename R::Point_3
 operator+(const Origin &, const VectorC3<R CGAL_CTAG> &v)
 {
   return PointC3<R CGAL_CTAG>(v);
@@ -50,7 +50,7 @@ operator+(const Origin &, const VectorC3<R CGAL_CTAG> &v)
 
 template < class R >
 inline
-PointC3<R CGAL_CTAG>
+typename R::Point_3
 operator-(const Origin &, const VectorC3<R CGAL_CTAG> &v)
 {
   return PointC3<R CGAL_CTAG>(-v);
@@ -58,15 +58,15 @@ operator-(const Origin &, const VectorC3<R CGAL_CTAG> &v)
 
 template < class R >
 inline
-VectorC3<R CGAL_CTAG>
+typename R::Vector_3
 operator-(const PointC3<R CGAL_CTAG> &p, const PointC3<R CGAL_CTAG> &q)
-{ // FIXME : construction
+{
   return VectorC3<R CGAL_CTAG>(p.x() - q.x(), p.y() - q.y(), p.z() - q.z());
 }
 
 template < class R >
 inline
-VectorC3<R CGAL_CTAG>
+typename R::Vector_3
 operator-(const PointC3<R CGAL_CTAG> &p, const Origin &)
 {
   return VectorC3<R CGAL_CTAG>(p);
@@ -74,32 +74,25 @@ operator-(const PointC3<R CGAL_CTAG> &p, const Origin &)
 
 template < class R >
 inline
-VectorC3<R CGAL_CTAG>
+typename R::Vector_3
 operator-(const Origin &, const PointC3<R CGAL_CTAG> &p)
-{ // FIXME : construction
+{
   return VectorC3<R CGAL_CTAG>(-p.x(), -p.y(), -p.z());
 }
 
-#ifdef __SUNPRO_CC
-template < class FT, class R >
-CGAL_KERNEL_INLINE
-VectorC3<R CGAL_CTAG>
-operator*(const FT &c, const VectorC3<R CGAL_CTAG> &w)
-#else
 template < class R >
 CGAL_KERNEL_INLINE
-VectorC3<R CGAL_CTAG>
+typename R::Vector_3
 operator*(const typename R::FT &c, const VectorC3<R CGAL_CTAG> &w)
-#endif
-{ // FIXME : construction
+{
    return VectorC3<R CGAL_CTAG>(c * w.x(), c * w.y(), c * w.z());
 }
 
 template < class R >
 CGAL_KERNEL_INLINE
-VectorC3<R CGAL_CTAG>
+typename R::Vector_3
 operator*(const VectorC3<R CGAL_CTAG> &w, const typename R::FT &c)
-{ // FIXME : construction
+{
    return VectorC3<R CGAL_CTAG>(c * w.x(), c * w.y(), c * w.z());
 }
 
diff --git a/Packages/Cartesian_kernel/include/CGAL/Cartesian/vector_constructions_3.h b/Packages/Cartesian_kernel/include/CGAL/Cartesian/vector_constructions_3.h
index 0b09c638e9c..489fd8f876f 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Cartesian/vector_constructions_3.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Cartesian/vector_constructions_3.h
@@ -28,7 +28,7 @@
 CGAL_BEGIN_NAMESPACE
 
 template < class R >
-VectorC3<R CGAL_CTAG>
+typename VectorC3<R CGAL_CTAG>::Vector_3
 cross_product(const VectorC3<R CGAL_CTAG> &v,
               const VectorC3<R CGAL_CTAG> &w)
 {
diff --git a/Packages/Cartesian_kernel/include/CGAL/Simple_cartesian.h b/Packages/Cartesian_kernel/include/CGAL/Simple_cartesian.h
index 80e7effb927..5b8927863a6 100644
--- a/Packages/Cartesian_kernel/include/CGAL/Simple_cartesian.h
+++ b/Packages/Cartesian_kernel/include/CGAL/Simple_cartesian.h
@@ -241,16 +241,13 @@ struct Simple_cartesian
                                                  	Direction_handle_2;
     typedef CGAL::Simple_Handle_for<CGAL::Threetuple<FT> >
                                                         Line_handle_2;
-    typedef CGAL::Simple_Handle_for<CGAL::Twotuple<
-	                CGAL_TYPENAME_MSVC_NULL Kernel_base::Point_2> >
+    typedef CGAL::Simple_Handle_for<CGAL::Twotuple<Point_2> >
                                                        	Ray_handle_2;
-    typedef CGAL::Simple_Handle_for<CGAL::Twotuple<
-	                    CGAL_TYPENAME_MSVC_NULL Kernel_base::Point_2> >
+    typedef CGAL::Simple_Handle_for<CGAL::Twotuple<Point_2> >
                                                       	Segment_handle_2;
-    typedef CGAL::Simple_Handle_for<triple<Point_2_base, FT, Orientation> >
+    typedef CGAL::Simple_Handle_for<triple<Point_2, FT, Orientation> >
                                                       	Circle_handle_2;
-    typedef CGAL::Simple_Handle_for<CGAL::Threetuple<
-	                     CGAL_TYPENAME_MSVC_NULL Kernel_base::Point_2> >
+    typedef CGAL::Simple_Handle_for<CGAL::Threetuple<Point_2> >
                                                        	Triangle_handle_2;
     typedef CGAL::Simple_Handle_for<CGAL::Twotuple<Point_2> >
                                                  	Iso_rectangle_handle_2;
@@ -263,26 +260,21 @@ struct Simple_cartesian
                                                    	Vector_handle_3;
     typedef CGAL::Simple_Handle_for<CGAL::Threetuple<FT> >
                                                   	Direction_handle_3;
-    typedef CGAL::Simple_Handle_for<std::pair<Point_3_base, Direction_3_base> >
+    typedef CGAL::Simple_Handle_for<std::pair<Point_3, Direction_3> >
                                                      	Line_handle_3;
     typedef CGAL::Simple_Handle_for<CGAL::Fourtuple<FT> >
                                                    	Plane_handle_3;
-    typedef CGAL::Simple_Handle_for<CGAL::Twotuple<
-	                CGAL_TYPENAME_MSVC_NULL Kernel_base::Point_3> >
+    typedef CGAL::Simple_Handle_for<CGAL::Twotuple<Point_3> >
                                                   	Ray_handle_3;
-    typedef CGAL::Simple_Handle_for<CGAL::Twotuple<
-	                    CGAL_TYPENAME_MSVC_NULL Kernel_base::Point_3> >
+    typedef CGAL::Simple_Handle_for<CGAL::Twotuple<Point_3> >
                                                    	Segment_handle_3;
-    typedef CGAL::Simple_Handle_for<triple<Point_3_base, FT, Orientation> >
+    typedef CGAL::Simple_Handle_for<triple<Point_3, FT, Orientation> >
                                                     	Sphere_handle_3;
-    typedef CGAL::Simple_Handle_for<CGAL::Threetuple<
-	                     CGAL_TYPENAME_MSVC_NULL Kernel_base::Point_3> >
+    typedef CGAL::Simple_Handle_for<CGAL::Threetuple<Point_3> >
                                                   	Triangle_handle_3;
-    typedef CGAL::Simple_Handle_for<CGAL::Fourtuple<
-	                        CGAL_TYPENAME_MSVC_NULL Kernel_base::Point_3> >
+    typedef CGAL::Simple_Handle_for<CGAL::Fourtuple<Point_3> >
                                                    	Tetrahedron_handle_3;
-    typedef CGAL::Simple_Handle_for<CGAL::Twotuple<
-	                       CGAL_TYPENAME_MSVC_NULL Kernel_base::Point_3> >
+    typedef CGAL::Simple_Handle_for<CGAL::Twotuple<Point_3> >
                                                 	Iso_cuboid_handle_3;
     typedef CGAL::Handle_for_virtual< Aff_transformation_rep_baseC3<Self> >
 			                           Aff_transformation_handle_3;
diff --git a/Packages/H2/changes.txt b/Packages/H2/changes.txt
index db79acb0307..f77a8bdc076 100644
--- a/Packages/H2/changes.txt
+++ b/Packages/H2/changes.txt
@@ -1,3 +1,6 @@
+2.59   (22 January 2002)
+- Classes and funtion return type only refer to R::...
+
 2.58   (9 January 2002)
 - just resubmit...
 
diff --git a/Packages/H2/include/CGAL/Homogeneous/Aff_transformationH2.h b/Packages/H2/include/CGAL/Homogeneous/Aff_transformationH2.h
index eb659a69c23..3334735ede6 100644
--- a/Packages/H2/include/CGAL/Homogeneous/Aff_transformationH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/Aff_transformationH2.h
@@ -45,9 +45,9 @@ class Aff_transformation_rep_baseH2 : public Ref_counted_virtual
   public:
     typedef typename R::RT RT;
     typedef typename R::FT FT;
-    typedef typename R::Kernel_base::Point_2      Point_2;
-    typedef typename R::Kernel_base::Vector_2     Vector_2;
-    typedef typename R::Kernel_base::Direction_2  Direction_2;
+    typedef typename R::Point_2      Point_2;
+    typedef typename R::Vector_2     Vector_2;
+    typedef typename R::Direction_2  Direction_2;
 
     virtual             ~Aff_transformation_rep_baseH2(){}
 
@@ -73,9 +73,9 @@ class Aff_transformation_repH2 : public Aff_transformation_rep_baseH2<R>
   public:
     typedef typename R::RT RT;
     typedef typename R::FT FT;
-    typedef typename R::Kernel_base::Point_2      Point_2;
-    typedef typename R::Kernel_base::Vector_2     Vector_2;
-    typedef typename R::Kernel_base::Direction_2  Direction_2;
+    typedef typename R::Point_2      Point_2;
+    typedef typename R::Vector_2     Vector_2;
+    typedef typename R::Direction_2  Direction_2;
 
     Aff_transformation_repH2()
     {}
@@ -109,20 +109,12 @@ class Aff_transformation_repH2 : public Aff_transformation_rep_baseH2<R>
     virtual  Direction_2
              transform(const Direction_2& dir) const
              {
-               const RT RT0(0);
-               if ( g > RT0 )
-               {
+               if ( g > RT(0) )
                    return Direction_2( a * dir.x() + b * dir.y(),
                                        d * dir.x() + e * dir.y() );
-               }
-               else if ( g < RT0 )
-               {
+               else
                    return - Direction_2(a * dir.x() + b * dir.y(),
                                         d * dir.x() + e * dir.y() );
-               }
-               return Direction_2( a * dir.x() + b * dir.y(),
-                                   d * dir.x() + e * dir.y(),
-                                   g );
              }
 
     virtual  Aff_transformationH2<R>
@@ -173,9 +165,9 @@ class Identity_repH2 : public Aff_transformation_rep_baseH2<R>
   public:
     typedef typename R::RT         RT;
     typedef typename R::FT         FT;
-    typedef typename R::Kernel_base::Point_2      Point_2;
-    typedef typename R::Kernel_base::Vector_2     Vector_2;
-    typedef typename R::Kernel_base::Direction_2  Direction_2;
+    typedef typename R::Point_2      Point_2;
+    typedef typename R::Vector_2     Vector_2;
+    typedef typename R::Direction_2  Direction_2;
 
              Identity_repH2()
              {}
@@ -228,9 +220,9 @@ class Translation_repH2 : public Aff_transformation_rep_baseH2<R>
   public:
     typedef typename R::RT RT;
     typedef typename R::FT FT;
-    typedef typename R::Kernel_base::Point_2      Point_2;
-    typedef typename R::Kernel_base::Vector_2     Vector_2;
-    typedef typename R::Kernel_base::Direction_2  Direction_2;
+    typedef typename R::Point_2      Point_2;
+    typedef typename R::Vector_2     Vector_2;
+    typedef typename R::Direction_2  Direction_2;
 
              Translation_repH2()
              {}
@@ -283,9 +275,9 @@ class Rotation_repH2 : public Aff_transformation_rep_baseH2<R>
   public:
     typedef typename R::RT RT;
     typedef typename R::FT FT;
-    typedef typename R::Kernel_base::Point_2      Point_2;
-    typedef typename R::Kernel_base::Vector_2     Vector_2;
-    typedef typename R::Kernel_base::Direction_2  Direction_2;
+    typedef typename R::Point_2      Point_2;
+    typedef typename R::Vector_2     Vector_2;
+    typedef typename R::Direction_2  Direction_2;
 
              Rotation_repH2()
              {
@@ -359,9 +351,9 @@ class Scaling_repH2 : public Aff_transformation_rep_baseH2<R>
   public:
     typedef typename R::RT RT;
     typedef typename R::FT FT;
-    typedef typename R::Kernel_base::Point_2      Point_2;
-    typedef typename R::Kernel_base::Vector_2     Vector_2;
-    typedef typename R::Kernel_base::Direction_2  Direction_2;
+    typedef typename R::Point_2      Point_2;
+    typedef typename R::Vector_2     Vector_2;
+    typedef typename R::Direction_2  Direction_2;
 
              Scaling_repH2()
              {}
@@ -430,10 +422,10 @@ class Reflection_repH2 : public Aff_transformation_rep_baseH2<R>
   public:
     typedef typename R::RT RT;
     typedef typename R::FT FT;
-    typedef typename R::Kernel_base::Point_2      Point_2;
-    typedef typename R::Kernel_base::Vector_2     Vector_2;
-    typedef typename R::Kernel_base::Direction_2  Direction_2;
-    typedef typename R::Kernel_base::Line_2       Line_2;
+    typedef typename R::Point_2      Point_2;
+    typedef typename R::Vector_2     Vector_2;
+    typedef typename R::Direction_2  Direction_2;
+    typedef typename R::Line_2       Line_2;
 
              Reflection_repH2(const Line_2& l_) : l(l_) {}
 
@@ -506,10 +498,10 @@ class Aff_transformationH2
 {
   typedef typename R_::FT                        FT;
   typedef typename R_::RT                        RT;
-  typedef typename R_::Kernel_base::Point_2      Point_2;
-  typedef typename R_::Kernel_base::Vector_2     Vector_2;
-  typedef typename R_::Kernel_base::Direction_2  Direction_2;
-  typedef typename R_::Kernel_base::Line_2       Line_2;
+  typedef typename R_::Point_2      Point_2;
+  typedef typename R_::Vector_2     Vector_2;
+  typedef typename R_::Direction_2  Direction_2;
+  typedef typename R_::Line_2       Line_2;
 
 public:
   typedef R_                                    R;
diff --git a/Packages/H2/include/CGAL/Homogeneous/CircleH2.h b/Packages/H2/include/CGAL/Homogeneous/CircleH2.h
index 1b3d4947b1f..eb1ff3f2b3c 100644
--- a/Packages/H2/include/CGAL/Homogeneous/CircleH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/CircleH2.h
@@ -31,12 +31,10 @@ template <class R_>
 class CircleH2
   : public R_::Circle_handle_2
 {
-    typedef typename R_::FT                         FT;
-    typedef typename R_::RT                         RT;
-    typedef typename R_::Kernel_base::Point_2       Point_2;
-    typedef typename R_::Kernel_base::Vector_2      Vector_2;
-    typedef typename R_::Kernel_base::Aff_transformation_2 
-                                                    Aff_transformation_2;
+    typedef typename R_::FT                   FT;
+    typedef typename R_::RT                   RT;
+    typedef typename R_::Point_2              Point_2;
+    typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
     typedef typename R_::Circle_handle_2            Circle_handle_2_;
     typedef typename Circle_handle_2_::element_type Circle_ref_2;
@@ -290,7 +288,7 @@ CircleH2<R>
 CircleH2<R>::
 orthogonal_transform(const typename CircleH2<R>::Aff_transformation_2& t) const
 {
-  Vector_2 vec( RT(1), RT(0) );   // unit vector
+  typename R::Vector_2 vec( RT(1), RT(0) );   // unit vector
   vec = vec.transform(t);                     // transformed
   FT  sq_scale = FT( vec*vec );               // squared scaling factor
 
diff --git a/Packages/H2/include/CGAL/Homogeneous/Data_accessorH2.h b/Packages/H2/include/CGAL/Homogeneous/Data_accessorH2.h
index 03ab47b69ea..4dc299f2ab8 100644
--- a/Packages/H2/include/CGAL/Homogeneous/Data_accessorH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/Data_accessorH2.h
@@ -32,7 +32,7 @@ class Data_accessorH2
 public:
     typedef typename R::FT FT;
     typedef typename R::RT RT;
-    typedef typename R::Kernel_base::Point_2      Point_2;
+    typedef typename R::Point_2      Point_2;
 
     RT  get_hx( const Point_2 & p) const { return( p.hx()); }
     RT  get_hy( const Point_2 & p) const { return( p.hy()); }
diff --git a/Packages/H2/include/CGAL/Homogeneous/DirectionH2.h b/Packages/H2/include/CGAL/Homogeneous/DirectionH2.h
index 1921775264b..016fc36b50d 100644
--- a/Packages/H2/include/CGAL/Homogeneous/DirectionH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/DirectionH2.h
@@ -29,14 +29,14 @@ template < class R_ >
 class DirectionH2
   : public R_::Direction_handle_2
 {
-  typedef typename R_::FT                        FT;
-  typedef typename R_::RT                        RT;
-  typedef typename R_::Kernel_base::Point_2      Point_2;
-  typedef typename R_::Kernel_base::Vector_2     Vector_2;
-  typedef typename R_::Kernel_base::Line_2       Line_2;
-  typedef typename R_::Kernel_base::Ray_2        Ray_2;
-  typedef typename R_::Kernel_base::Segment_2    Segment_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Vector_2             Vector_2;
+  typedef typename R_::Line_2               Line_2;
+  typedef typename R_::Ray_2                Ray_2;
+  typedef typename R_::Segment_2            Segment_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Direction_handle_2        Direction_handle_2_;
   typedef typename Direction_handle_2_::element_type  Direction_ref_2;
@@ -68,6 +68,8 @@ public:
    DirectionH2(const RT& x, const RT& y)
       : Direction_handle_2_ ( Direction_ref_2( x, y, RT(1) )) {}
 
+   // TODO Not documented : should not exist , not used.
+   // we should also change Threetuple<RT> -> Twotuple<RT>
    DirectionH2(const RT& x, const RT& y, const RT& w )
    {
      if (w > RT(0)   )
@@ -88,6 +90,7 @@ public:
     DirectionH2<R> operator-() const;
 
     Vector_2       to_vector() const;
+    Vector_2       vector() const { return to_vector(); }
 
     const RT & x() const { return Ptr()->e0; };
     const RT & y() const { return Ptr()->e1; };
@@ -215,7 +218,7 @@ template <class R >
 CGAL_KERNEL_INLINE
 typename DirectionH2<R>::Vector_2
 DirectionH2<R>::to_vector() const
-{ return Vector_2(*this); }
+{ return Vector_2(dx(), dy(), typename R::RT(1)); }
 
 
 #ifndef CGAL_NO_OSTREAM_INSERT_DIRECTIONH2
diff --git a/Packages/H2/include/CGAL/Homogeneous/Iso_rectangleH2.h b/Packages/H2/include/CGAL/Homogeneous/Iso_rectangleH2.h
index a3d4afd7b8c..314678dc884 100644
--- a/Packages/H2/include/CGAL/Homogeneous/Iso_rectangleH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/Iso_rectangleH2.h
@@ -29,10 +29,10 @@ template <class R_>
 class Iso_rectangleH2
   : public R_::Iso_rectangle_handle_2
 {
-  typedef typename R_::FT                                FT;
-  typedef typename R_::RT                                RT;
-  typedef typename R_::Point_2                           Point_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Iso_rectangle_handle_2         Iso_rectangle_handle_2_;
   typedef typename Iso_rectangle_handle_2_::element_type  Iso_rectangle_ref_2;
diff --git a/Packages/H2/include/CGAL/Homogeneous/LineH2.h b/Packages/H2/include/CGAL/Homogeneous/LineH2.h
index 8c7c6053c63..16c830e0c8c 100644
--- a/Packages/H2/include/CGAL/Homogeneous/LineH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/LineH2.h
@@ -30,15 +30,14 @@ template < class R_ >
 class LineH2
   : public R_::Line_handle_2
 {
-    typedef typename R_::FT                                FT;
-    typedef typename R_::RT                                RT;
-    typedef typename R_::Kernel_base::Point_2              Point_2;
-    typedef typename R_::Kernel_base::Vector_2             Vector_2;
-    typedef typename R_::Kernel_base::Direction_2          Direction_2;
-    typedef typename R_::Kernel_base::Segment_2            Segment_2;
-    typedef typename R_::Kernel_base::Ray_2                Ray_2;
-    typedef typename R_::Kernel_base::Aff_transformation_2 
-                                                          Aff_transformation_2;
+    typedef typename R_::FT                   FT;
+    typedef typename R_::RT                   RT;
+    typedef typename R_::Point_2              Point_2;
+    typedef typename R_::Vector_2             Vector_2;
+    typedef typename R_::Direction_2          Direction_2;
+    typedef typename R_::Segment_2            Segment_2;
+    typedef typename R_::Ray_2                Ray_2;
+    typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
     typedef typename R_::Line_handle_2            Line_handle_2_;
     typedef typename Line_handle_2_::element_type Line_ref_2;
@@ -141,7 +140,7 @@ CGAL_KERNEL_INLINE
 LineH2<R>::LineH2(const typename LineH2<R>::Point_2& p,
 		  const typename LineH2<R>::Direction_2& d)
 {
-  Point_2 q = p + Vector_2(d);
+  Point_2 q = p + d.to_vector();
   initialize_with( Line_ref_2 (
             p.hy()*q.hw() - p.hw()*q.hy(),
             p.hw()*q.hx() - p.hx()*q.hw(),
diff --git a/Packages/H2/include/CGAL/Homogeneous/PointH2.h b/Packages/H2/include/CGAL/Homogeneous/PointH2.h
index 6a370f67921..1aa67fa156f 100644
--- a/Packages/H2/include/CGAL/Homogeneous/PointH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/PointH2.h
@@ -32,11 +32,11 @@ template < class R_ >
 class PointH2
   : public R_::Point_handle_2
 {
-  typedef typename R_::FT                        FT;
-  typedef typename R_::RT                        RT;
-  typedef typename R_::Kernel_base::Vector_2     Vector_2;
-  typedef typename R_::Kernel_base::Direction_2  Direction_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::Vector_2             Vector_2;
+  typedef typename R_::Direction_2          Direction_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Point_handle_2            Point_handle_2_;
   typedef typename Point_handle_2_::element_type Point_ref_2;
diff --git a/Packages/H2/include/CGAL/Homogeneous/RayH2.h b/Packages/H2/include/CGAL/Homogeneous/RayH2.h
index bb1d74aa833..1e272e6e9a1 100644
--- a/Packages/H2/include/CGAL/Homogeneous/RayH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/RayH2.h
@@ -30,14 +30,13 @@ template < class R_ >
 class RayH2
   : public R_::Ray_handle_2
 {
-    typedef typename R_::FT                                FT;
-    typedef typename R_::RT                                RT;
-    typedef typename R_::Kernel_base::Point_2              Point_2;
-    typedef typename R_::Kernel_base::Direction_2          Direction_2;
-    typedef typename R_::Kernel_base::Line_2               Line_2;
-    typedef typename R_::Kernel_base::Vector_2             Vector_2;
-    typedef typename R_::Kernel_base::Aff_transformation_2 
-                                                          Aff_transformation_2;
+    typedef typename R_::FT                   FT;
+    typedef typename R_::RT                   RT;
+    typedef typename R_::Point_2              Point_2;
+    typedef typename R_::Direction_2          Direction_2;
+    typedef typename R_::Line_2               Line_2;
+    typedef typename R_::Vector_2             Vector_2;
+    typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
     typedef typename R_::Ray_handle_2                      Ray_handle_2_;
     typedef typename Ray_handle_2_::element_type           Ray_ref_2;
diff --git a/Packages/H2/include/CGAL/Homogeneous/SegmentH2.h b/Packages/H2/include/CGAL/Homogeneous/SegmentH2.h
index 7077d93fcdf..696f18dff55 100644
--- a/Packages/H2/include/CGAL/Homogeneous/SegmentH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/SegmentH2.h
@@ -31,12 +31,12 @@ template < class R_ >
 class SegmentH2
   : public R_::Segment_handle_2
 {
-  typedef typename R_::FT                                FT;
-  typedef typename R_::RT                                RT;
-  typedef typename R_::Kernel_base::Point_2              Point_2;
-  typedef typename R_::Kernel_base::Line_2               Line_2;
-  typedef typename R_::Kernel_base::Direction_2          Direction_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Line_2               Line_2;
+  typedef typename R_::Direction_2          Direction_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Segment_handle_2            Segment_handle_2_;
   typedef typename Segment_handle_2_::element_type Segment_ref_2;
diff --git a/Packages/H2/include/CGAL/Homogeneous/TriangleH2.h b/Packages/H2/include/CGAL/Homogeneous/TriangleH2.h
index 93991982efc..9368f631dbb 100644
--- a/Packages/H2/include/CGAL/Homogeneous/TriangleH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/TriangleH2.h
@@ -30,12 +30,11 @@ template <class R_>
 class TriangleH2
   : public R_::Triangle_handle_2
 {
-    typedef typename R_::FT                                FT;
-    typedef typename R_::RT                                RT;
-    typedef typename R_::Kernel_base::Point_2              Point_2;
-    typedef typename R_::Kernel_base::Vector_2             Vector_2;
-    typedef typename R_::Kernel_base::Aff_transformation_2 
-                                                          Aff_transformation_2;
+    typedef typename R_::FT                   FT;
+    typedef typename R_::RT                   RT;
+    typedef typename R_::Point_2              Point_2;
+    typedef typename R_::Vector_2             Vector_2;
+    typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
     typedef typename R_::Triangle_handle_2            Triangle_handle_2_;
     typedef typename Triangle_handle_2_::element_type Triangle_ref_2;
diff --git a/Packages/H2/include/CGAL/Homogeneous/VectorH2.h b/Packages/H2/include/CGAL/Homogeneous/VectorH2.h
index 92c1cd44d1c..7e3d0981f25 100644
--- a/Packages/H2/include/CGAL/Homogeneous/VectorH2.h
+++ b/Packages/H2/include/CGAL/Homogeneous/VectorH2.h
@@ -32,11 +32,12 @@ template < class R_ >
 class VectorH2
   : public R_::Vector_handle_2
 {
-  typedef typename R_::FT                        FT;
-  typedef typename R_::RT                        RT;
-  typedef typename R_::Kernel_base::Point_2      Point_2;
-  typedef typename R_::Kernel_base::Direction_2  Direction_2;
-  typedef typename R_::Kernel_base::Aff_transformation_2 Aff_transformation_2;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::Point_2              Point_2;
+  typedef typename R_::Direction_2          Direction_2;
+  typedef typename R_::Vector_2             Vector_2;
+  typedef typename R_::Aff_transformation_2 Aff_transformation_2;
 
   typedef typename R_::Vector_handle_2             Vector_handle_2_;
   typedef typename Vector_handle_2_::element_type  Vector_ref_2;
@@ -47,9 +48,6 @@ public:
    VectorH2()
       : Vector_handle_2_ ( Vector_ref_2()) {}
 
-   VectorH2(const VectorH2<R>& v)
-      : Vector_handle_2_ (v) {}
-
    VectorH2(const Point_2& a, const Point_2& b)
       : Vector_handle_2_ (b-a) {}
 
@@ -85,18 +83,17 @@ public:
 
    int     dimension() const;
    Direction_2 direction() const;
-   VectorH2<R> transform(const Aff_transformation_2& t ) const;
-   VectorH2<R> perpendicular(const Orientation& o ) const;
+   Vector_2 transform(const Aff_transformation_2& t ) const;
+   Vector_2 perpendicular(const Orientation& o ) const;
 
    FT      operator*( const VectorH2<R>& v) const;
-   VectorH2<R> operator-() const;
-   VectorH2<R> opposite() const;
+   Vector_2 operator-() const;
+   Vector_2 opposite() const;
 
 // undocumented:
    VectorH2(const Direction_2 & dir)
       : Vector_handle_2_ ( dir) {}
 
-protected:
   VectorH2(const Point_2 & p)
      : Vector_handle_2_ ( p) {}
 };
@@ -176,19 +173,19 @@ VectorH2<R>::direction() const
 
 template < class R >
 inline
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 VectorH2<R>::operator-() const
 { return VectorH2<R>(- hx(), - hy(), hw() ); }
 
 template < class R >
 inline
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 VectorH2<R>::opposite() const
 { return VectorH2<R>(- hx(), - hy(), hw() ); }
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 operator+(const VectorH2<R>& u, const VectorH2<R>& v)
 {
   return VectorH2<R>( u.hx()*v.hw() + v.hx()*u.hw(),
@@ -198,7 +195,7 @@ operator+(const VectorH2<R>& u, const VectorH2<R>& v)
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 operator-(const VectorH2<R>& u, const VectorH2<R>& v)
 {
   return VectorH2<R>( u.hx()*v.hw() - v.hx()*u.hw(),
@@ -218,27 +215,34 @@ VectorH2<R>::operator*(const VectorH2<R>& v) const
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH2<R>
+typename R::Vector_2
 operator/(const VectorH2<R>& v, const typename R::RT& f)
 { return VectorH2<R>( v.hx(), v.hy(), v.hw()*f ); }
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH2<R>
+typename VectorH2<R>::Vector_2
+operator/(const VectorH2<R>& v, const typename R::FT& f)
+{ return VectorH2<R>( v.hx()*f.denominator(), v.hy()*f.denominator(),
+	              v.hw()*f.numerator() ); }
+
+template <class R>
+CGAL_KERNEL_INLINE
+typename R::Vector_2
 operator*(const VectorH2<R>& v, const typename R::RT& f)
 { return VectorH2<R>( v.hx()*f, v.hy()*f, v.hw() ); }
 
-#ifdef __SUNPRO_CC
-template <class RT, class R>
-CGAL_KERNEL_INLINE
-VectorH2<R>
-operator*(const RT& f, const VectorH2<R>& v)
-#else
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH2<R>
+typename VectorH2<R>::Vector_2
+operator*(const VectorH2<R>& v, const typename R::FT& f)
+{ return VectorH2<R>( v.hx()*f.numerator(), v.hy()*f.numerator(),
+	              v.hw()*f.denominator() ); }
+
+template <class R>
+CGAL_KERNEL_INLINE
+typename VectorH2<R>::Vector_2
 operator*(const typename R::RT& f, const VectorH2<R>& v)
-#endif
 { return VectorH2<R>( v.hx()*f, v.hy()*f, v.hw() ); }
 
 template <class R>
@@ -249,50 +253,49 @@ origin_plus_vector(const VectorH2<R>& v)
 
 template <class R>
 inline
-PointH2<R>
+typename R::Point_2
 operator+(const Origin&, const VectorH2<R>& v)
 { return origin_plus_vector( v ); }
 
 template <class R>
 inline
-PointH2<R>
+typename R::Point_2
 origin_minus_vector(const VectorH2<R>& v)
 { return PointH2<R>( v.opposite() ); }
 
 template <class R>
 inline
-PointH2<R>
+typename R::Point_2
 operator-(const Origin&, const VectorH2<R>& v)
 { return origin_minus_vector( v ); }
 
 template <class R>
 inline
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 point_minus_origin(const PointH2<R>& p)
 { return VectorH2<R>( p ); }
 
 template <class R>
 inline
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 operator-(const PointH2<R>& p, const Origin&)
 { return point_minus_origin( p ); }
 
 template <class R>
 inline
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 origin_minus_point(const PointH2<R>& p)
 { return  VectorH2<R>( p ).opposite(); }
 
 template <class R>
 inline
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 operator-(const Origin&, const PointH2<R>& p)
 { return  origin_minus_point( p ); }
 
-
 template <class R>
 CGAL_KERNEL_INLINE
-PointH2<R>
+typename R::Point_2
 operator+(const PointH2<R>& p, const VectorH2<R>& v)
 {
   return PointH2<R>( p.hx()*v.hw() + v.hx()*p.hw(),
@@ -302,7 +305,7 @@ operator+(const PointH2<R>& p, const VectorH2<R>& v)
 
 template <class R>
 CGAL_KERNEL_INLINE
-PointH2<R>
+typename R::Point_2
 operator-(const PointH2<R>& p, const VectorH2<R>& v)
 {
   return PointH2<R>( p.hx()*v.hw() - v.hx()*p.hw(),
@@ -312,7 +315,7 @@ operator-(const PointH2<R>& p, const VectorH2<R>& v)
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 operator-(const PointH2<R>& p, const PointH2<R>& q)
 {
   return VectorH2<R>( p.hx()*q.hw() - q.hx()*p.hw(),
@@ -326,7 +329,7 @@ operator-(const PointH2<R>& p, const PointH2<R>& q)
 
 template < class R >
 CGAL_KERNEL_INLINE
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 VectorH2<R>::perpendicular(const Orientation& o) const
 {
   CGAL_kernel_precondition(o != COLLINEAR);
@@ -342,7 +345,7 @@ VectorH2<R>::perpendicular(const Orientation& o) const
 
 template < class R >
 inline
-VectorH2<R>
+typename VectorH2<R>::Vector_2
 VectorH2<R>::
 transform(const typename VectorH2<R>::Aff_transformation_2& t) const
 { return t.transform(*this); }
diff --git a/Packages/H2/include/CGAL/Homogeneous/homogeneous_rep.h b/Packages/H2/include/CGAL/Homogeneous/homogeneous_rep.h
index 6fb6f4db0fa..855aacde0ac 100644
--- a/Packages/H2/include/CGAL/Homogeneous/homogeneous_rep.h
+++ b/Packages/H2/include/CGAL/Homogeneous/homogeneous_rep.h
@@ -157,14 +157,14 @@ class Homogeneous
     typedef CGAL::Handle_for< Threetuple<RT> >       Point_handle_2;
     typedef CGAL::Handle_for< Threetuple<RT> >       Vector_handle_2;
     typedef CGAL::Handle_for< Threetuple<RT> >       Direction_handle_2;
-    typedef CGAL::Handle_for< Twotuple<Point_2_base> > Ray_handle_2;
-    typedef CGAL::Handle_for< Threetuple<Point_2_base> >  Triangle_handle_2;
-    typedef CGAL::Handle_for< triple<Point_2_base, FT, Orientation> >
+    typedef CGAL::Handle_for< Twotuple<Point_2> >    Ray_handle_2;
+    typedef CGAL::Handle_for< Threetuple<Point_2> >  Triangle_handle_2;
+    typedef CGAL::Handle_for< triple<Point_2, FT, Orientation> >
 	                                             Circle_handle_2;
     typedef CGAL::Handle_for< Twotuple<Point_2> >
 	                                             Iso_rectangle_handle_2;
     typedef CGAL::Handle_for< Threetuple<RT> >       Line_handle_2;
-    typedef CGAL::Handle_for< Twotuple<Point_2_base> > Segment_handle_2;
+    typedef CGAL::Handle_for< Twotuple<Point_2> > Segment_handle_2;
     typedef CGAL::Handle_for_virtual< Aff_transformation_rep_baseH2<R> >
 			                           Aff_transformation_handle_2;
 
@@ -172,16 +172,14 @@ class Homogeneous
     typedef CGAL::Handle_for< Fourtuple<RT> >        Vector_handle_3;
     typedef CGAL::Handle_for< Fourtuple<RT> >        Direction_handle_3;
     typedef CGAL::Handle_for< Fourtuple<RT> >        Plane_handle_3;
-    typedef CGAL::Handle_for< std::pair<Point_3_base, Direction_3_base> >
-	Ray_handle_3;
-    typedef CGAL::Handle_for< std::pair<Point_3_base, Direction_3_base> >
-	Line_handle_3;
-    typedef CGAL::Handle_for< triple<Point_3_base, FT, Orientation> >
+    typedef CGAL::Handle_for< std::pair<Point_3, Direction_3> > Ray_handle_3;
+    typedef CGAL::Handle_for< std::pair<Point_3, Direction_3> > Line_handle_3;
+    typedef CGAL::Handle_for< triple<Point_3, FT, Orientation> >
 	                                             Sphere_handle_3;
-    typedef CGAL::Handle_for< Fourtuple<Point_3_base> >   Tetrahedron_handle_3;
-    typedef CGAL::Handle_for< Twotuple<Point_3_base> >  Segment_handle_3;
-    typedef CGAL::Handle_for< Threetuple<Point_3_base> >  Triangle_handle_3;
-    typedef CGAL::Handle_for< Twotuple<Point_3_base> > Iso_cuboid_handle_3;
+    typedef CGAL::Handle_for< Fourtuple<Point_3> >   Tetrahedron_handle_3;
+    typedef CGAL::Handle_for< Twotuple<Point_3> >    Segment_handle_3;
+    typedef CGAL::Handle_for< Threetuple<Point_3> >  Triangle_handle_3;
+    typedef CGAL::Handle_for< Twotuple<Point_3> >    Iso_cuboid_handle_3;
     typedef CGAL::Handle_for_virtual< Aff_transformation_rep_baseH3<R> >
 			                           Aff_transformation_handle_3;
 
diff --git a/Packages/H2/include/CGAL/Homogeneous/simple_homogeneous_rep.h b/Packages/H2/include/CGAL/Homogeneous/simple_homogeneous_rep.h
index 748a7c74ffc..be94552fb6e 100644
--- a/Packages/H2/include/CGAL/Homogeneous/simple_homogeneous_rep.h
+++ b/Packages/H2/include/CGAL/Homogeneous/simple_homogeneous_rep.h
@@ -157,15 +157,14 @@ class Simple_homogeneous
     typedef CGAL::Simple_Handle_for< Threetuple<RT> > Point_handle_2;
     typedef CGAL::Simple_Handle_for< Threetuple<RT> > Vector_handle_2;
     typedef CGAL::Simple_Handle_for< Threetuple<RT> > Direction_handle_2;
-    typedef CGAL::Simple_Handle_for< Twotuple<Point_2_base> > Ray_handle_2;
-    typedef CGAL::Simple_Handle_for< Threetuple<Point_2_base> >
-	Triangle_handle_2;
-    typedef CGAL::Simple_Handle_for< triple<Point_2_base, FT, Orientation> >
+    typedef CGAL::Simple_Handle_for< Twotuple<Point_2> > Ray_handle_2;
+    typedef CGAL::Simple_Handle_for< Threetuple<Point_2> > Triangle_handle_2;
+    typedef CGAL::Simple_Handle_for< triple<Point_2, FT, Orientation> >
 	                                              Circle_handle_2;
     typedef CGAL::Simple_Handle_for< Twotuple<Point_2> >
 	                                              Iso_rectangle_handle_2;
     typedef CGAL::Simple_Handle_for< Threetuple<RT> > Line_handle_2;
-    typedef CGAL::Simple_Handle_for< Twotuple<Point_2_base> > Segment_handle_2;
+    typedef CGAL::Simple_Handle_for< Twotuple<Point_2> > Segment_handle_2;
     typedef CGAL::Handle_for_virtual< Aff_transformation_rep_baseH2<R> >
 			                        Aff_transformation_handle_2;
 
@@ -173,19 +172,16 @@ class Simple_homogeneous
     typedef CGAL::Simple_Handle_for< Fourtuple<RT> > Vector_handle_3;
     typedef CGAL::Simple_Handle_for< Fourtuple<RT> > Direction_handle_3;
     typedef CGAL::Simple_Handle_for< Fourtuple<RT> > Plane_handle_3;
-    typedef CGAL::Simple_Handle_for<std::pair<Point_3_base, Direction_3_base> >
+    typedef CGAL::Simple_Handle_for<std::pair<Point_3, Direction_3> >
 	                                              Ray_handle_3;
-    typedef CGAL::Simple_Handle_for<std::pair<Point_3_base, Direction_3_base> >
+    typedef CGAL::Simple_Handle_for<std::pair<Point_3, Direction_3> >
 	                                              Line_handle_3;
-    typedef CGAL::Simple_Handle_for< triple<Point_3_base, FT, Orientation> >
+    typedef CGAL::Simple_Handle_for< triple<Point_3, FT, Orientation> >
 	                                              Sphere_handle_3;
-    typedef CGAL::Simple_Handle_for< Fourtuple<Point_3_base> >
-	Tetrahedron_handle_3;
-    typedef CGAL::Simple_Handle_for< Twotuple<Point_3_base> > Segment_handle_3;
-    typedef CGAL::Simple_Handle_for< Threetuple<Point_3_base> >
-	Triangle_handle_3;
-    typedef CGAL::Simple_Handle_for< Twotuple<Point_3_base> >
-	Iso_cuboid_handle_3;
+    typedef CGAL::Simple_Handle_for< Fourtuple<Point_3> > Tetrahedron_handle_3;
+    typedef CGAL::Simple_Handle_for< Twotuple<Point_3> > Segment_handle_3;
+    typedef CGAL::Simple_Handle_for< Threetuple<Point_3> > Triangle_handle_3;
+    typedef CGAL::Simple_Handle_for< Twotuple<Point_3> > Iso_cuboid_handle_3;
     typedef CGAL::Handle_for_virtual< Aff_transformation_rep_baseH3<R> >
 			                        Aff_transformation_handle_3;
 
diff --git a/Packages/H3/changes.txt b/Packages/H3/changes.txt
index 0870d43c30b..cb96c92d9a3 100644
--- a/Packages/H3/changes.txt
+++ b/Packages/H3/changes.txt
@@ -1,3 +1,6 @@
+2.44   (22 January 2002)
+- Classes and funtion return type only refer to R::...
+
 2.43   (4 January 2002)
 - Replace the small helper *rep classes by std::pair and triple.
 - Tetrahedron_3.orientation() is not cached anymore.
diff --git a/Packages/H3/include/CGAL/Homogeneous/Aff_transformationH3.h b/Packages/H3/include/CGAL/Homogeneous/Aff_transformationH3.h
index 242938c0ea5..610224b54da 100644
--- a/Packages/H3/include/CGAL/Homogeneous/Aff_transformationH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/Aff_transformationH3.h
@@ -48,11 +48,11 @@ public:
   typedef R_                       R;
   typedef typename R::FT           FT;
   typedef typename R::RT           RT;
-  typedef typename R::Kernel_base::Point_3        Point_3;
-  typedef typename R::Kernel_base::Vector_3       Vector_3;
-  typedef typename R::Kernel_base::Direction_3    Direction_3;
-  typedef typename R::Kernel_base::Plane_3        Plane_3;
-  typedef typename R::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R::Point_3        Point_3;
+  typedef typename R::Vector_3       Vector_3;
+  typedef typename R::Direction_3    Direction_3;
+  typedef typename R::Plane_3        Plane_3;
+  typedef typename R::Aff_transformation_3 Aff_transformation_3;
 
   virtual  ~Aff_transformation_rep_baseH3(){}
 
@@ -68,10 +68,10 @@ public:
   virtual  Plane_3
            transform(const Plane_3&) const = 0;
 
-  virtual  Aff_transformationH3<R>
+  virtual  Aff_transformation_3
            inverse() const = 0;
 
-  virtual  Aff_transformationH3<R>
+  virtual  Aff_transformation_3
            transpose() const = 0;
 
   virtual  Aff_transformation_repH3<R>
@@ -92,11 +92,11 @@ class Aff_transformation_repH3 : public Aff_transformation_rep_baseH3<R_>
 {
   typedef typename R_::FT           FT;
   typedef typename R_::RT           RT;
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Plane_3              Plane_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
 public:
   typedef R_                       R;
@@ -165,11 +165,11 @@ class Identity_repH3 : public Aff_transformation_rep_baseH3<R_>
 {
   typedef typename R_::RT    RT;
   typedef typename R_::FT    FT;
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Plane_3              Plane_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
 public:
   typedef R_                R;
@@ -225,11 +225,11 @@ class Translation_repH3 : public Aff_transformation_rep_baseH3<R_>
 {
   typedef typename R_::FT       FT;
   typedef typename R_::RT       RT;
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Plane_3              Plane_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
 public:
   typedef R_                    R;
@@ -278,12 +278,13 @@ private:
 template < class R_ >
 class Aff_transformationH3 : public R_::Aff_transformation_handle_3
 {
-  typedef typename R_::RT    RT;
-  typedef typename R_::FT    FT;
-  typedef typename R_::Kernel_base::Point_3        Point_3;
-  typedef typename R_::Kernel_base::Vector_3       Vector_3;
-  typedef typename R_::Kernel_base::Direction_3    Direction_3;
-  typedef typename R_::Kernel_base::Plane_3        Plane_3;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
 public:
   typedef R_                R;
@@ -323,7 +324,7 @@ public:
   Plane_3
   transform(const Plane_3& pl) const;
 
-  Aff_transformationH3<R>
+  Aff_transformation_3
   inverse()   const;
 
   Aff_transformationH3<R>
@@ -399,17 +400,14 @@ typename Aff_transformation_repH3<R>::Direction_3
 Aff_transformation_repH3<R>::
 transform(const typename Aff_transformation_repH3<R>::Direction_3& d) const
 {
-  Vector_3 v( d.to_vector() );
-  return Direction_3(t00 * v.hx() + t01 * v.hy() + t02 * v.hz(),
-                     t10 * v.hx() + t11 * v.hy() + t12 * v.hz(),
-                     t20 * v.hx() + t21 * v.hy() + t22 * v.hz(),
-                     t33 * v.hw() );
-
-/*
-  return Direction_3( t00 * d.hx() + t01 * d.hy() + t02 * d.hz(),
-                      t10 * d.hx() + t11 * d.hy() + t12 * d.hz(),
-                      t20 * d.hx() + t21 * d.hy() + t22 * d.hz() );
-*/
+    if (t33 > RT(0))
+        return Direction_3(t00 * d.hx() + t01 * d.hy() + t02 * d.hz(),
+                           t10 * d.hx() + t11 * d.hy() + t12 * d.hz(),
+                           t20 * d.hx() + t21 * d.hy() + t22 * d.hz());
+    else
+        return - Direction_3(t00 * d.hx() + t01 * d.hy() + t02 * d.hz(),
+                             t10 * d.hx() + t11 * d.hy() + t12 * d.hz(),
+                             t20 * d.hx() + t21 * d.hy() + t22 * d.hz());
 }
 
 template < class R >
@@ -840,7 +838,7 @@ transform(const typename Aff_transformationH3<R>::Plane_3& pl) const
 
 template < class R >
 inline
-Aff_transformationH3<R>
+typename Aff_transformationH3<R>::Aff_transformation_3
 Aff_transformationH3<R>::inverse() const
 { return Ptr()->inverse(); }
 
diff --git a/Packages/H3/include/CGAL/Homogeneous/DirectionH3.h b/Packages/H3/include/CGAL/Homogeneous/DirectionH3.h
index abd204c8caf..f497890e6bd 100644
--- a/Packages/H3/include/CGAL/Homogeneous/DirectionH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/DirectionH3.h
@@ -29,15 +29,14 @@ template < class R_ >
 class DirectionH3
   : public R_::Direction_handle_3
 {
-   typedef typename R_::RT     RT;
-   typedef typename R_::FT     FT;
-
-   typedef typename R_::Kernel_base::Point_3              Point_3;
-   typedef typename R_::Kernel_base::Vector_3             Vector_3;
-   typedef typename R_::Kernel_base::Segment_3            Segment_3;
-   typedef typename R_::Kernel_base::Line_3               Line_3;
-   typedef typename R_::Kernel_base::Ray_3                Ray_3;
-   typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+   typedef typename R_::RT                   RT;
+   typedef typename R_::FT                   FT;
+   typedef typename R_::Point_3              Point_3;
+   typedef typename R_::Vector_3             Vector_3;
+   typedef typename R_::Segment_3            Segment_3;
+   typedef typename R_::Line_3               Line_3;
+   typedef typename R_::Ray_3                Ray_3;
+   typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
    typedef typename R_::Direction_handle_3            Direction_handle_3_;
    typedef typename Direction_handle_3_::element_type Direction_ref_3;
@@ -63,6 +62,7 @@ class DirectionH3
   DirectionH3(const Segment_3 & s )
     : Direction_handle_3_(s.direction()) {}
 
+  // the fourth argument is not documented.  Should go away ?
   DirectionH3(const RT& x, const RT& y,
               const RT& z, const RT& w = RT(1) )
   {
@@ -83,6 +83,7 @@ class DirectionH3
   bool  operator!=( const DirectionH3<R>& d) const;
 
   Vector_3    to_vector() const;
+  Vector_3    vector() const { return to_vector(); }
 
   const RT & dx() const { return Ptr()->e0; }
   const RT & dy() const { return Ptr()->e1; }
@@ -150,7 +151,7 @@ template <class R >
 inline
 typename DirectionH3<R>::Vector_3
 DirectionH3<R>::to_vector() const
-{ return Vector_3(*this); }
+{ return Vector_3(dx(), dy(), dz(), RT(1)); }
 
 template <class R>
 CGAL_KERNEL_INLINE
diff --git a/Packages/H3/include/CGAL/Homogeneous/Iso_cuboidH3.h b/Packages/H3/include/CGAL/Homogeneous/Iso_cuboidH3.h
index 3b4e0dff0f1..dbde2d82b24 100644
--- a/Packages/H3/include/CGAL/Homogeneous/Iso_cuboidH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/Iso_cuboidH3.h
@@ -32,10 +32,10 @@ template <class R_>
 class Iso_cuboidH3
   : public R_::Iso_cuboid_handle_3
 {
-  typedef typename R_::RT    RT;
-  typedef typename R_::FT    FT;
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
 public:
   typedef R_                 R;
diff --git a/Packages/H3/include/CGAL/Homogeneous/LineH3.h b/Packages/H3/include/CGAL/Homogeneous/LineH3.h
index d6d0bd33667..6b262aa6867 100644
--- a/Packages/H3/include/CGAL/Homogeneous/LineH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/LineH3.h
@@ -30,15 +30,15 @@ template < class R_ >
 class LineH3
   : public R_::Line_handle_3
 {
-  typedef typename R_::RT    RT;
-  typedef typename R_::FT    FT;
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Segment_3            Segment_3;
-  typedef typename R_::Kernel_base::Ray_3                Ray_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Plane_3              Plane_3;
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Segment_3            Segment_3;
+  typedef typename R_::Ray_3                Ray_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Line_handle_3             Line_handle_3_;
   typedef typename Line_handle_3_::element_type  Line_ref_3;
diff --git a/Packages/H3/include/CGAL/Homogeneous/PlaneH3.h b/Packages/H3/include/CGAL/Homogeneous/PlaneH3.h
index 7033cafa474..3f722bac992 100644
--- a/Packages/H3/include/CGAL/Homogeneous/PlaneH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/PlaneH3.h
@@ -30,16 +30,17 @@ template < class R_ >
 class PlaneH3
   : public R_::Plane_handle_3
 {
-   typedef typename R_::RT     RT;
-   typedef typename R_::FT     FT;
-   typedef typename R_::Kernel_base::Point_2              Point_2;
-   typedef typename R_::Kernel_base::Point_3              Point_3;
-   typedef typename R_::Kernel_base::Vector_3             Vector_3;
-   typedef typename R_::Kernel_base::Line_3               Line_3;
-   typedef typename R_::Kernel_base::Segment_3            Segment_3;
-   typedef typename R_::Kernel_base::Ray_3                Ray_3;
-   typedef typename R_::Kernel_base::Direction_3          Direction_3;
-   typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+   typedef typename R_::RT                   RT;
+   typedef typename R_::FT                   FT;
+   typedef typename R_::Point_2              Point_2;
+   typedef typename R_::Point_3              Point_3;
+   typedef typename R_::Vector_3             Vector_3;
+   typedef typename R_::Line_3               Line_3;
+   typedef typename R_::Segment_3            Segment_3;
+   typedef typename R_::Ray_3                Ray_3;
+   typedef typename R_::Direction_3          Direction_3;
+   typedef typename R_::Plane_3              Plane_3;
+   typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
    typedef typename R_::Plane_handle_3            Plane_handle_3_;
    typedef typename Plane_handle_3_::element_type Plane_ref_3;
@@ -50,9 +51,7 @@ class PlaneH3
     PlaneH3()
       : Plane_handle_3_(Plane_ref_3()) {}
 
-    PlaneH3(const Point_3&,
-            const Point_3&,
-            const Point_3& );
+    PlaneH3(const Point_3&, const Point_3&, const Point_3& );
     PlaneH3(const RT& a, const RT& b,
             const RT& c, const RT& d );
     PlaneH3(const Point_3&, const Ray_3& );
@@ -74,7 +73,7 @@ class PlaneH3
     bool       operator!=( const PlaneH3<R>& ) const;
 
     Line_3  perpendicular_line(const Point_3& ) const;
-    PlaneH3<R> opposite() const;  // plane with opposite orientation
+    Plane_3 opposite() const;  // plane with opposite orientation
     Point_3 projection(const Point_3& ) const;
 
     Point_3 point() const;     // same point on the plane
@@ -95,7 +94,7 @@ class PlaneH3
 
     bool           is_degenerate() const;
 
-    PlaneH3<R> transform(const Aff_transformation_3& ) const;
+    Plane_3 transform(const Aff_transformation_3& ) const;
 
     Aff_transformation_3 transform_to_2d() const;
     Point_2   to_2d(const Point_3& )  const;
@@ -184,20 +183,20 @@ PlaneH3<R>::operator!=(const PlaneH3<R>& l) const
 template < class R >
 CGAL_KERNEL_INLINE
 PlaneH3<R>::PlaneH3(const typename PlaneH3<R>::Point_3& p,
-                        const typename PlaneH3<R>::Point_3& q,
-                        const typename PlaneH3<R>::Point_3& r)
+                    const typename PlaneH3<R>::Point_3& q,
+                    const typename PlaneH3<R>::Point_3& r)
 { new_rep(p,q,r); }
 
 template < class R >
 CGAL_KERNEL_INLINE
 PlaneH3<R>::PlaneH3(const RT& a, const RT& b,
-                        const RT& c, const RT& d)
+                    const RT& c, const RT& d)
 { new_rep(a,b,c,d); }
 
 template < class R >
 CGAL_KERNEL_INLINE
 PlaneH3<R>::PlaneH3(const typename PlaneH3<R>::Point_3& p ,
-                        const typename PlaneH3<R>::Line_3&  l)
+                    const typename PlaneH3<R>::Line_3&  l)
 { new_rep(p, l.point(0), l.point(1) ); }
 
 template < class R >
@@ -292,7 +291,7 @@ PlaneH3<R>::perpendicular_line(const typename PlaneH3<R>::Point_3& p) const
 
 template < class R >
 CGAL_KERNEL_INLINE
-PlaneH3<R>
+typename R::Plane_3
 PlaneH3<R>::opposite() const
 { return PlaneH3<R>(-a(), -b(), -c(), -d() ); }
 
@@ -388,7 +387,7 @@ PlaneH3<R>::orthogonal_vector() const
 { return Vector_3(a(), b(), c() ); }
 
 template < class R >
-PlaneH3<R>
+typename R::Plane_3
 PlaneH3<R>::transform(const typename PlaneH3<R>::Aff_transformation_3& t) const
 {
  return t.transform(*this);
diff --git a/Packages/H3/include/CGAL/Homogeneous/PointH3.h b/Packages/H3/include/CGAL/Homogeneous/PointH3.h
index 5bd2ad048ae..462a3b24d0b 100644
--- a/Packages/H3/include/CGAL/Homogeneous/PointH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/PointH3.h
@@ -33,11 +33,11 @@ template < class R_ >
 class PointH3
   : public R_::Point_handle_3
 {
-   typedef typename R_::RT     RT;
-   typedef typename R_::FT     FT;
-   typedef typename R_::Kernel_base::Vector_3             Vector_3;
-   typedef typename R_::Kernel_base::Direction_3          Direction_3;
-   typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+   typedef typename R_::RT                   RT;
+   typedef typename R_::FT                   FT;
+   typedef typename R_::Vector_3             Vector_3;
+   typedef typename R_::Direction_3          Direction_3;
+   typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
    typedef typename R_::Point_handle_3            Point_handle_3_;
    typedef typename Point_handle_3_::element_type Point_ref_3;
diff --git a/Packages/H3/include/CGAL/Homogeneous/RayH3.h b/Packages/H3/include/CGAL/Homogeneous/RayH3.h
index 8a478dbb0e3..3d1b111a360 100644
--- a/Packages/H3/include/CGAL/Homogeneous/RayH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/RayH3.h
@@ -30,12 +30,12 @@ template < class R_ >
 class RayH3
   : public R_::Ray_handle_3
 {
-   typedef typename R_::RT    RT;
-   typedef typename R_::FT    FT;
-   typedef typename R_::Kernel_base::Point_3              Point_3;
-   typedef typename R_::Kernel_base::Line_3               Line_3;
-   typedef typename R_::Kernel_base::Direction_3          Direction_3;
-   typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+   typedef typename R_::RT                   RT;
+   typedef typename R_::FT                   FT;
+   typedef typename R_::Point_3              Point_3;
+   typedef typename R_::Line_3               Line_3;
+   typedef typename R_::Direction_3          Direction_3;
+   typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
    typedef typename R_::Ray_handle_3              Ray_handle_3_;
    typedef typename Ray_handle_3_::element_type   Ray_ref_3;
diff --git a/Packages/H3/include/CGAL/Homogeneous/SegmentH3.h b/Packages/H3/include/CGAL/Homogeneous/SegmentH3.h
index fbe7ceda7d1..6684c0c9386 100644
--- a/Packages/H3/include/CGAL/Homogeneous/SegmentH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/SegmentH3.h
@@ -30,12 +30,12 @@ template < class R_ >
 class SegmentH3
   : public R_::Segment_handle_3
 {
-  typedef typename R_::RT   RT;
-  typedef typename R_::FT   FT;
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Line_3               Line_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Line_3               Line_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Segment_handle_3            Segment_handle_3_;
   typedef typename Segment_handle_3_::element_type Segment_ref_3;
diff --git a/Packages/H3/include/CGAL/Homogeneous/SphereH3.h b/Packages/H3/include/CGAL/Homogeneous/SphereH3.h
index 2f199094972..f1489ee8d1b 100644
--- a/Packages/H3/include/CGAL/Homogeneous/SphereH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/SphereH3.h
@@ -30,10 +30,10 @@ template <class R_>
 class SphereH3
   : public R_::Sphere_handle_3
 {
-   typedef typename R_::RT    RT;
-   typedef typename R_::FT    FT;
-   typedef typename R_::Kernel_base::Point_3              Point_3;
-   typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+   typedef typename R_::RT                   RT;
+   typedef typename R_::FT                   FT;
+   typedef typename R_::Point_3              Point_3;
+   typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
    typedef typename R_::Sphere_handle_3            Sphere_handle_3_;
    typedef typename Sphere_handle_3_::element_type Sphere_ref_3;
diff --git a/Packages/H3/include/CGAL/Homogeneous/TetrahedronH3.h b/Packages/H3/include/CGAL/Homogeneous/TetrahedronH3.h
index 2d87c4b80fd..391d21e7270 100644
--- a/Packages/H3/include/CGAL/Homogeneous/TetrahedronH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/TetrahedronH3.h
@@ -33,11 +33,11 @@ template < class R_ >
 class TetrahedronH3
   : public R_::Tetrahedron_handle_3
 {
-  typedef typename R_::RT    RT;
-  typedef typename R_::FT    FT;
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Tetrahedron_handle_3            Tetrahedron_handle_3_;
   typedef typename Tetrahedron_handle_3_::element_type Tetrahedron_ref_3;
diff --git a/Packages/H3/include/CGAL/Homogeneous/TriangleH3.h b/Packages/H3/include/CGAL/Homogeneous/TriangleH3.h
index 399d411901d..0a33bef35c3 100644
--- a/Packages/H3/include/CGAL/Homogeneous/TriangleH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/TriangleH3.h
@@ -30,14 +30,14 @@ template < class R_ >
 class TriangleH3
   : public R_::Triangle_handle_3
 {
-  typedef typename R_::RT    RT;
-  typedef typename R_::FT    FT;
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Plane_3              Plane_3;
-  typedef typename R_::Kernel_base::Vector_3             Vector_3;
-  typedef typename R_::Kernel_base::Segment_3            Segment_3;
-  typedef typename R_::Kernel_base::Tetrahedron_3        Tetrahedron_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Plane_3              Plane_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Segment_3            Segment_3;
+  typedef typename R_::Tetrahedron_3        Tetrahedron_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Triangle_handle_3            Triangle_handle_3_;
   typedef typename Triangle_handle_3_::element_type Triangle_ref_3;
diff --git a/Packages/H3/include/CGAL/Homogeneous/VectorH3.h b/Packages/H3/include/CGAL/Homogeneous/VectorH3.h
index 59621a27db5..8d9a3a3d20a 100644
--- a/Packages/H3/include/CGAL/Homogeneous/VectorH3.h
+++ b/Packages/H3/include/CGAL/Homogeneous/VectorH3.h
@@ -31,11 +31,12 @@ template < class R_ >
 class VectorH3
   : public R_::Vector_handle_3
 {
-  typedef typename R_::RT     RT;
-  typedef typename R_::FT     FT;
-  typedef typename R_::Kernel_base::Point_3              Point_3;
-  typedef typename R_::Kernel_base::Direction_3          Direction_3;
-  typedef typename R_::Kernel_base::Aff_transformation_3 Aff_transformation_3;
+  typedef typename R_::RT                   RT;
+  typedef typename R_::FT                   FT;
+  typedef typename R_::Point_3              Point_3;
+  typedef typename R_::Direction_3          Direction_3;
+  typedef typename R_::Vector_3             Vector_3;
+  typedef typename R_::Aff_transformation_3 Aff_transformation_3;
 
   typedef typename R_::Vector_handle_3                   Vector_handle_3_;
   typedef typename Vector_handle_3_::element_type        Vector_ref_3;
@@ -78,13 +79,10 @@ public:
 
   int   dimension() const { return 3; };
 
-  Direction_3
-        direction() const;
-  VectorH3<R>
-        transform(const Aff_transformation_3& t ) const;
+  Direction_3 direction() const;
+  Vector_3 transform(const Aff_transformation_3& t ) const;
 
-  VectorH3<R>
-        operator-() const;
+  Vector_3 operator-() const;
 
   bool  operator==( const VectorH3<R>& v) const;
   bool  operator!=( const VectorH3<R>& v) const;
@@ -109,7 +107,7 @@ VectorH3<R>::VectorH3(const RT& x, const RT& y, const RT& z, const RT& w)
 
 template < class R >
 CGAL_KERNEL_INLINE
-typename VectorH3<R>::FT
+typename R::FT
 VectorH3<R>::cartesian(int i) const
 {
   CGAL_kernel_precondition(i == 0 || i == 1 || i == 2);
@@ -123,7 +121,7 @@ VectorH3<R>::cartesian(int i) const
 
 template < class R >
 CGAL_KERNEL_INLINE
-const typename VectorH3<R>::RT &
+const typename R::RT &
 VectorH3<R>::homogeneous(int i) const
 {
   CGAL_kernel_precondition(i == 0 || i == 1 || i == 2 || i == 3);
@@ -138,7 +136,7 @@ VectorH3<R>::homogeneous(int i) const
 
 template < class R >
 inline
-typename VectorH3<R>::Direction_3
+typename R::Direction_3
 VectorH3<R>::direction() const
 { return Direction_3(*this); }
 
@@ -160,19 +158,19 @@ VectorH3<R>::operator!=( const VectorH3<R>& v) const
 
 template < class R >
 inline
-typename VectorH3<R>::FT
+typename R::FT
 VectorH3<R>::operator[](int i) const
 { return cartesian(i); }
 
 template < class R >
 CGAL_KERNEL_INLINE
-VectorH3<R>
+typename R::Vector_3
 VectorH3<R>::operator-() const
-{ return VectorH3<R>( - hx(), - hy(), -hz(), hw() ); }
+{ return Vector_3( - hx(), - hy(), -hz(), hw() ); }
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH3<R>
+typename R::Vector_3
 operator+(const VectorH3<R>& u, const VectorH3<R>& v)
 {
   return VectorH3<R>(u.hx()*v.hw() + v.hx()*u.hw(),
@@ -183,7 +181,7 @@ operator+(const VectorH3<R>& u, const VectorH3<R>& v)
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH3<R>
+typename R::Vector_3
 operator-(const VectorH3<R>& u, const VectorH3<R>& v)
 {
   return VectorH3<R>(u.hx()*v.hw() - v.hx()*u.hw(),
@@ -194,7 +192,7 @@ operator-(const VectorH3<R>& u, const VectorH3<R>& v)
 
 template <class R>
 CGAL_KERNEL_INLINE
-typename VectorH3<R>::FT
+typename R::FT
 VectorH3<R>::operator*(const VectorH3<R>& v) const
 {
   typedef typename R::RT RT;
@@ -207,32 +205,39 @@ VectorH3<R>::operator*(const VectorH3<R>& v) const
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH3<R>
+typename R::Vector_3
 operator/(const VectorH3<R>& v, const typename R::RT& f)
 { return VectorH3<R>( v.hx(), v.hy(), v.hz(), v.hw()*f ); }
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH3<R>
+typename VectorH3<R>::Vector_3
+operator/(const VectorH3<R>& v, const typename R::FT& f)
+{ return VectorH3<R>( v.hx()*f.denominator(), v.hy()*f.denominator(),
+		      v.hz()*f.denominator(), v.hw()*f.numerator() ); }
+
+template <class R>
+CGAL_KERNEL_INLINE
+typename R::Vector_3
 operator*(const VectorH3<R>& v, const typename R::RT& f)
 { return VectorH3<R>( v.hx()*f, v.hy()*f, v.hz()*f, v.hw() ); }
 
-#ifdef __SUNPRO_CC
-template <class RT, class R>
-CGAL_KERNEL_INLINE
-VectorH3<R>
-operator*(const RT& f, const VectorH3<R>& v)
-#else
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH3<R>
+typename VectorH3<R>::Vector_3
+operator*(const VectorH3<R>& v, const typename R::FT& f)
+{ return VectorH3<R>( v.hx()*f.numerator(), v.hy()*f.numerator(),
+		      v.hz()*f.numerator(), v.hw()*f.denominator() ); }
+
+template <class R>
+CGAL_KERNEL_INLINE
+typename R::Vector_3
 operator*(const typename R::RT& f, const VectorH3<R>& v)
-#endif
 { return VectorH3<R>( v.hx()*f, v.hy()*f, v.hz()*f, v.hw() ); }
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH3<R>
+typename R::Vector_3
 cross_product(const VectorH3<R>& a, const VectorH3<R>& b)
 {
  return VectorH3<R>(a.hy()*b.hz() - a.hz()*b.hy(),
@@ -243,31 +248,31 @@ cross_product(const VectorH3<R>& a, const VectorH3<R>& b)
 
 template <class R>
 inline
-PointH3<R>
+typename R::Point_3
 operator+(const Origin& , const VectorH3<R>& v)
 { return PointH3<R>( v ); }
 
 template <class R>
 inline
-PointH3<R>
+typename R::Point_3
 operator-(const Origin& , const VectorH3<R>& v)
 { return  PointH3<R>(-v ); }
 
 template <class R>
 inline
-VectorH3<R>
+typename R::Vector_3
 operator-(const PointH3<R>& p, const Origin& )
 { return VectorH3<R>( p ); }
 
 template <class R>
 inline
-VectorH3<R>
+typename R::Vector_3
 operator-(const Origin& , const PointH3<R>& p)
 { return  - VectorH3<R>( p ); }
 
 template <class R>
 CGAL_KERNEL_INLINE
-PointH3<R>
+typename R::Point_3
 operator+(const PointH3<R>& p, const VectorH3<R>& v)
 {
   return PointH3<R>(p.hx()*v.hw() + v.hx()*p.hw(),
@@ -278,7 +283,7 @@ operator+(const PointH3<R>& p, const VectorH3<R>& v)
 
 template <class R>
 CGAL_KERNEL_INLINE
-PointH3<R>
+typename R::Point_3
 operator-(const PointH3<R>& p, const VectorH3<R>& v)
 {
   return PointH3<R>( p.hx()*v.hw() - v.hx()*p.hw(),
@@ -289,10 +294,10 @@ operator-(const PointH3<R>& p, const VectorH3<R>& v)
 
 template <class R>
 CGAL_KERNEL_INLINE
-VectorH3<R>
+typename R::Vector_3
 operator-(const PointH3<R>& p, const PointH3<R>& q)
 {
-  return PointH3<R>( p.hx()*q.hw() - q.hx()*p.hw(),
+  return VectorH3<R>(p.hx()*q.hw() - q.hx()*p.hw(),
                      p.hy()*q.hw() - q.hy()*p.hw(),
                      p.hz()*q.hw() - q.hz()*p.hw(),
                      p.hw()*q.hw() );
@@ -304,7 +309,7 @@ operator-(const PointH3<R>& p, const PointH3<R>& q)
 
 template < class R >
 inline
-VectorH3<R>
+typename R::Vector_3
 VectorH3<R>::
 transform(const typename VectorH3<R>::Aff_transformation_3&t ) const
 { return t.transform(*this); }