diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h b/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h
index 7cb44d5ab76..604b4d9ee92 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h
@@ -71,8 +71,6 @@ public:
   Vector_3       to_vector() const;
   Vector_3       vector() const { return to_vector(); }
 
-  Direction_3    operator-() const;
-
   const FT & delta(int i) const;
   const FT & dx() const
   {
@@ -131,14 +129,6 @@ DirectionC3<R>::to_vector() const
   return Vector_3(dx(), dy(), dz());
 }
 
-template < class R >
-inline
-typename DirectionC3<R>::Direction_3
-DirectionC3<R>::operator-() const
-{
-  return DirectionC3<R>(-dx(), -dy(), -dz());
-}
-
 template < class R >
 const typename DirectionC3<R>::FT &
 DirectionC3<R>::delta(int i) const
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h b/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h
index 206f38de812..17d76ffd4a5 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h
@@ -37,7 +37,6 @@ class SphereC3
   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 Triple<Point_3, FT, Orientation>         Rep;
   typedef typename R_::template Handle<Rep>::type  Base;
@@ -119,19 +118,6 @@ public:
       return get(base).third;
   }
 
-  Sphere_3 orthogonal_transform(const Aff_transformation_3 &t) const
-  {
-    // FIXME: precond: t.is_orthogonal() (*UNDEFINED*)
-    Vector_3 vec(FT(1), FT(0), FT(0));        // unit vector
-    vec = vec.transform(t);                   // transformed
-    FT sq_scale = vec.squared_length();       // squared scaling factor
-
-    return SphereC3(t.transform(center()),
-                    sq_scale * squared_radius(),
-                    t.is_even() ? orientation()
-                                : CGAL::opposite(orientation()));
-  }
-
   // A circle is degenerate if its (squared) radius is null or negative
   bool is_degenerate() const;
 
@@ -303,17 +289,6 @@ SphereC3<R>::bbox() const
 		maxx.sup(), maxy.sup(), maxz.sup());
 }
 
-/*
-template < class R >
-inline
-EllipseC3<SphereC3<R>::FT> SphereC3<R>::i
-transform(const Aff_transformationC3<SphereC3<R>::FT> &t) const
-{
-  return SphereC3<R>(t.transform(center()),
-                      squared_radius(),
-                      orientation());
-}
-*/
 
 #ifndef CGAL_NO_OSTREAM_INSERT_SPHEREC3
 template < class R >
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h b/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h
index 210a7ec9aa2..5d66dced742 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h
@@ -72,7 +72,6 @@ public:
   bool       has_on_unbounded_side(const Point_3 &p) const;
 
   bool       is_degenerate() const;
-  FT         volume() const;
 };
 
 template < class R >
@@ -136,14 +135,6 @@ operator[](int i) const
   return vertex(i);
 }
 
-template < class R >
-CGAL_KERNEL_MEDIUM_INLINE
-typename TetrahedronC3<R>::FT
-TetrahedronC3<R>::volume() const
-{
-    return R().compute_volume_3_object()(*this);
-}
-
 template < class R >
 Orientation
 TetrahedronC3<R>::
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/function_objects.h b/Cartesian_kernel/include/CGAL/Cartesian/function_objects.h
index 5d734149449..da5bd44845c 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/function_objects.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/function_objects.h
@@ -2872,6 +2872,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3        Point_3;
     typedef typename K::Vector_3       Vector_3;
+    typedef typename K::Tetrahedron_3  Tetrahedron_3;
   public:
     typedef typename K::Orientation    result_type;
     typedef Arity_tag< 4 >             Arity;
@@ -2893,6 +2894,12 @@ namespace CartesianKernelFunctors {
 			   v.x(), v.y(), v.z(),
 			   w.x(), w.y(), w.z());
     }
+
+    result_type
+    operator()( const Tetrahedron_3& t) const
+    { 
+      return t.rep().orientation();
+    }
   };
 
   template <typename K>
diff --git a/Homogeneous_kernel/include/CGAL/Homogeneous/DirectionH3.h b/Homogeneous_kernel/include/CGAL/Homogeneous/DirectionH3.h
index cf0fb079737..4bd85d9887b 100644
--- a/Homogeneous_kernel/include/CGAL/Homogeneous/DirectionH3.h
+++ b/Homogeneous_kernel/include/CGAL/Homogeneous/DirectionH3.h
@@ -74,9 +74,6 @@ public:
       base = Rep(-x,-y,-z,-w);
   }
 
-  DirectionH3<R>
-        operator-() const;
-
   bool  is_degenerate() const;
 
   bool  operator==( const DirectionH3<R>& d) const;
@@ -137,12 +134,6 @@ bool
 DirectionH3<R>::is_degenerate() const
 { return ((hx() == RT(0)) && (hy() == RT(0)) && (hz() == RT(0))); }
 
-template <class R >
-inline
-DirectionH3<R>
-DirectionH3<R>::operator-() const
-{ return DirectionH3<R>(- hx(),- hy(),- hz() ); }
-
 template <class R >
 inline
 typename DirectionH3<R>::Vector_3
diff --git a/Homogeneous_kernel/include/CGAL/Homogeneous/SphereH3.h b/Homogeneous_kernel/include/CGAL/Homogeneous/SphereH3.h
index 9f2d3de8c78..2cce983b638 100644
--- a/Homogeneous_kernel/include/CGAL/Homogeneous/SphereH3.h
+++ b/Homogeneous_kernel/include/CGAL/Homogeneous/SphereH3.h
@@ -36,7 +36,6 @@ class SphereH3
    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 Triple<Point_3, FT, Orientation>         Rep;
    typedef typename R_::template Handle<Rep>::type  Base;
@@ -77,8 +76,6 @@ public:
 
       Orientation orientation() const;
 
-      SphereH3<R> orthogonal_transform(const Aff_transformation_3& t) const;
-
       bool is_degenerate() const;
 
       SphereH3<R> opposite() const;
@@ -254,29 +251,6 @@ SphereH3<R>::bbox() const
 		maxx.sup(), maxy.sup(), maxz.sup());
 }
 
-template <class R>
-SphereH3<R>
-SphereH3<R>::
-orthogonal_transform(const typename SphereH3<R>::Aff_transformation_3& t) const
-{
-  typename R::Vector_3 vec( RT(1), RT(0), RT(0) );   // unit vector
-  vec = vec.transform(t);                     // transformed
-  FT  sq_scale = FT( vec*vec );               // squared scaling factor
-
-  if ( t.is_even() )
-  {
-      return SphereH3<R>(t.transform(center() ),
-                             sq_scale * squared_radius(),
-                             orientation() );
-  }
-  else
-  {
-      return SphereH3<R>(t.transform(center() ),
-                             sq_scale * squared_radius(),
-                             CGAL::opposite( orientation()) );
-  }
-}
-
 
 #ifndef CGAL_NO_OSTREAM_INSERT_SPHEREH3
 template < class R >
diff --git a/Homogeneous_kernel/include/CGAL/Homogeneous/function_objects.h b/Homogeneous_kernel/include/CGAL/Homogeneous/function_objects.h
index 3ed6b8e706a..844a8d4dd26 100644
--- a/Homogeneous_kernel/include/CGAL/Homogeneous/function_objects.h
+++ b/Homogeneous_kernel/include/CGAL/Homogeneous/function_objects.h
@@ -3560,8 +3560,9 @@ namespace HomogeneousKernelFunctors {
   template <typename K>
   class Orientation_3
   {
-    typedef typename K::Point_3      Point_3;
-    typedef typename K::Vector_3     Vector_3;
+    typedef typename K::Point_3        Point_3;
+    typedef typename K::Vector_3       Vector_3;
+    typedef typename K::Tetrahedron_3  Tetrahedron_3;
   public:
     typedef typename K::Orientation  result_type;
     typedef Arity_tag< 4 >           Arity;
@@ -3586,6 +3587,12 @@ namespace HomogeneousKernelFunctors {
 				v.hx(), v.hy(), v.hz(),
 				w.hx(), w.hy(), w.hz()));
     }
+
+    result_type
+    operator()( const Tetrahedron_3& t) const
+    { 
+      return t.rep().orientation();
+    }
   };
 
 
diff --git a/Kernel_23/include/CGAL/Direction_2.h b/Kernel_23/include/CGAL/Direction_2.h
index 810a35f295f..844c670e0de 100644
--- a/Kernel_23/include/CGAL/Direction_2.h
+++ b/Kernel_23/include/CGAL/Direction_2.h
@@ -135,7 +135,7 @@ public:
   Direction_2
   operator-() const
   {
-    return Direction_2(-dx(), -dy());
+    return R().construct_opposite_direction_2_object()(*this);
   } 
   
   Vector_2 vector() const
diff --git a/Kernel_23/include/CGAL/Direction_3.h b/Kernel_23/include/CGAL/Direction_3.h
index 8d9d8195934..c8da4b4852a 100644
--- a/Kernel_23/include/CGAL/Direction_3.h
+++ b/Kernel_23/include/CGAL/Direction_3.h
@@ -76,6 +76,13 @@ public:
   {
     return t.transform(*this);
   }
+ 
+  Direction_3
+  operator-() const
+  {
+    return R().construct_opposite_direction_3_object()(*this);
+  } 
+  
 
 };
 
diff --git a/Kernel_23/include/CGAL/Kernel/function_objects.h b/Kernel_23/include/CGAL/Kernel/function_objects.h
index fbcee0a5453..bf183411cf9 100644
--- a/Kernel_23/include/CGAL/Kernel/function_objects.h
+++ b/Kernel_23/include/CGAL/Kernel/function_objects.h
@@ -557,7 +557,7 @@ namespace CommonKernelFunctors {
   };
 
   template <typename K>
-  class Construct_center_3
+  class Construct_center_3 : Has_qrt
   {
     typedef typename K::Point_3   Point_3;
     typedef typename K::Sphere_3  Sphere_3;
@@ -565,9 +565,9 @@ namespace CommonKernelFunctors {
     typedef Point_3          result_type;
     typedef Arity_tag< 1 >   Arity;
 
-    Point_3
+    const Point_3 &
     operator()(const Sphere_3& s) const
-    { return s.center(); }
+    { return s.rep().center(); }
   };
 
   template <typename K>
@@ -770,27 +770,29 @@ namespace CommonKernelFunctors {
   template <typename K>
   class Construct_opposite_direction_2
   {
-    typedef typename K::Direction_2  Direction_2;
+    typedef typename K::Direction_2    Direction_2;
+    typedef typename Direction_2::Rep  Rep;
   public:
     typedef Direction_2      result_type;
     typedef Arity_tag< 1 >   Arity;
 
     Direction_2
     operator()( const Direction_2& d) const
-    { return -d; }
+    {  return Rep(-d.dx(), -d.dy()); }
   };
 
   template <typename K>
   class Construct_opposite_direction_3
   {
-    typedef typename K::Direction_3  Direction_3;
+    typedef typename K::Direction_3    Direction_3;
+    typedef typename Direction_3::Rep  Rep;
   public:
     typedef Direction_3      result_type;
     typedef Arity_tag< 1 >   Arity;
 
     Direction_3
     operator()( const Direction_3& d) const
-    { return -d; }
+    {  return Rep(-d.dx(), -d.dy(), -d.dz()); }
   };
 
   template <typename K>
@@ -829,7 +831,7 @@ namespace CommonKernelFunctors {
 
     Plane_3
     operator()( const Plane_3& p) const
-    { return p.opposite(); }
+    { return p.rep().opposite(); }
   };
 
   template <typename K>
@@ -1306,7 +1308,7 @@ namespace CommonKernelFunctors {
 
     Plane_3
     operator()( const Triangle_3& t) const
-    { return t.supporting_plane(); }
+    { return t.rep().supporting_plane(); }
   };
 
   template <typename K>
@@ -1372,7 +1374,7 @@ namespace CommonKernelFunctors {
 
     const Point_3 &
     operator()( const Triangle_3& t, int i) const
-    { return t.vertex(i); }
+    { return t.rep().vertex(i); }
 
     Point_3
     operator()( const Iso_cuboid_3& r, int i) const
@@ -1380,7 +1382,7 @@ namespace CommonKernelFunctors {
 
     const Point_3 &
     operator()( const Tetrahedron_3& t, int i) const
-    { return t.vertex(i); }
+    { return t.rep().vertex(i); }
   };
 
 
@@ -1765,7 +1767,7 @@ namespace CommonKernelFunctors {
 
     result_type
     operator()( const Tetrahedron_3& t, const Point_3& p) const
-    { return t.has_on_boundary(p); }
+    { return t.rep().has_on_boundary(p); }
 
     result_type
     operator()( const Iso_cuboid_3& c, const Point_3& p) const
@@ -1813,7 +1815,7 @@ namespace CommonKernelFunctors {
 
     result_type
     operator()( const Tetrahedron_3& t, const Point_3& p) const
-    { return t.has_on_bounded_side(p); }
+    { return t.rep().has_on_bounded_side(p); }
 
     result_type
     operator()( const Iso_cuboid_3& c, const Point_3& p) const
@@ -1861,7 +1863,7 @@ namespace CommonKernelFunctors {
 
     result_type
     operator()( const Tetrahedron_3& t, const Point_3& p) const
-    { return t.has_on_negative_side(p); }
+    { return t.rep().has_on_negative_side(p); }
 
     result_type
     operator()( const Plane_3& pl, const Point_3& p) const
@@ -1909,7 +1911,7 @@ namespace CommonKernelFunctors {
 
     result_type
     operator()( const Tetrahedron_3& t, const Point_3& p) const
-    { return t.has_on_positive_side(p); }
+    { return t.rep().has_on_positive_side(p); }
 
     result_type
     operator()( const Plane_3& pl, const Point_3& p) const
@@ -1959,7 +1961,7 @@ namespace CommonKernelFunctors {
 
     result_type
     operator()( const Tetrahedron_3& t, const Point_3& p) const
-    { return t.has_on_unbounded_side(p); }
+    { return t.rep().has_on_unbounded_side(p); }
 
     result_type
     operator()( const Iso_cuboid_3& c, const Point_3& p) const
@@ -2104,11 +2106,11 @@ namespace CommonKernelFunctors {
 
     result_type
     operator()( const Triangle_3& t) const
-    { return t.is_degenerate(); }
+    { return t.rep().is_degenerate(); }
 
     result_type
     operator()( const Tetrahedron_3& t) const
-    { return t.is_degenerate(); }
+    { return t.rep().is_degenerate(); }
   };
 
   template <typename K>
@@ -2233,7 +2235,7 @@ namespace CommonKernelFunctors {
 
     result_type
     operator()( const Tetrahedron_3& t, const Point_3& p) const
-    { return t.oriented_side(p); }
+    { return t.rep().oriented_side(p); }
   };
 
 } // namespace CommonKernelFunctors
diff --git a/Kernel_23/include/CGAL/Plane_3.h b/Kernel_23/include/CGAL/Plane_3.h
index b61223ce317..d34993fbfc5 100644
--- a/Kernel_23/include/CGAL/Plane_3.h
+++ b/Kernel_23/include/CGAL/Plane_3.h
@@ -84,6 +84,11 @@ public:
   {
     return t.transform(*this);
   }
+
+  Plane_3 opposite() const
+  {
+    return R().construct_opposite_plane_3_object()(*this);
+  }
 };
 
 #ifndef CGAL_NO_OSTREAM_INSERT_PLANE_3
diff --git a/Kernel_23/include/CGAL/Sphere_3.h b/Kernel_23/include/CGAL/Sphere_3.h
index ccb67346781..07d20af5878 100644
--- a/Kernel_23/include/CGAL/Sphere_3.h
+++ b/Kernel_23/include/CGAL/Sphere_3.h
@@ -31,6 +31,7 @@ class Sphere_3 : public R_::Kernel_base::Sphere_3
 {
   typedef typename R_::FT                    FT;
   typedef typename R_::Point_3               Point_3;
+  typedef typename R_::Aff_transformation_3  Aff_transformation_3;
   typedef typename R_::Kernel_base::Sphere_3  RSphere_3;
 public:
 
@@ -71,8 +72,39 @@ public:
 
       Sphere_3(const Point_3& p, const Orientation& o = COUNTERCLOCKWISE)
        : RSphere_3(p, o) {}
+
+    Sphere_3 orthogonal_transform(const Aff_transformation_3 &t) const;
+
+    // FIXME : why doesn't Qrt work here ?  We loose optimization !
+    //typename Qualified_result_of<typename R::Construct_center_3, Sphere_3>::type
+    Point_3
+    center() const
+    {
+      return R().construct_center_3_object()(*this);
+    }
+
 };
 
+template <class R_>
+Sphere_3<R_>
+Sphere_3<R_>::
+orthogonal_transform(const typename Sphere_3<R_>::Aff_transformation_3& t) const
+{
+    typedef typename  R_::RT  RT;
+    typedef typename  R_::FT  FT;
+    typedef typename  R_::Vector_3  Vector_3;
+
+    // FIXME: precond: t.is_orthogonal() (*UNDEFINED*)
+    Vector_3 vec(RT(1), RT(0), RT(0));        // unit vector
+    vec = vec.transform(t);                   // transformed
+    FT sq_scale = vec.squared_length();       // squared scaling factor
+
+    return Sphere_3(t.transform(this->center()),
+                    sq_scale * this->squared_radius(),
+                    t.is_even() ? this->orientation()
+                                : CGAL::opposite(this->orientation()));
+}
+
 CGAL_END_NAMESPACE
 
 #endif // CGAL_SPHERE_3_H
diff --git a/Kernel_23/include/CGAL/Tetrahedron_3.h b/Kernel_23/include/CGAL/Tetrahedron_3.h
index 8af2e8e40e6..07372c5d228 100644
--- a/Kernel_23/include/CGAL/Tetrahedron_3.h
+++ b/Kernel_23/include/CGAL/Tetrahedron_3.h
@@ -67,6 +67,73 @@ public:
                          t.transform(this->vertex(3)));
   }
 
+  // FIXME TODO : Why doesn't Qrt work here ???
+  //typename Qualified_result_of<typename R::Construct_vertex_3, Tetrahedron_3, int>::type
+  Point_3
+  vertex(int i) const
+  {
+    return R().construct_vertex_3_object()(*this,i);
+  }
+
+  //typename Qualified_result_of<typename R::Construct_vertex_3, Tetrahedron_3, int>::type
+  Point_3
+  operator[](int i) const
+  {
+    return vertex(i);
+  }
+
+  bool
+  is_degenerate() const
+  {
+    return R().is_degenerate_3_object()(*this);
+  }
+
+  Orientation orientation() const
+  {
+    return R().orientation_3_object()(*this);
+  }
+
+  Bounded_side bounded_side(const Point_3 &p) const
+  {
+    return R().bounded_side_3_object()(*this, p);
+  }
+
+  Oriented_side oriented_side(const Point_3 &p) const
+  {
+    return R().oriented_side_3_object()(*this, p);
+  }
+
+  bool has_on_positive_side(const Point_3 &p) const
+  {
+    return R().has_on_positive_side_3_object()(*this, p);
+  }
+
+  bool has_on_negative_side(const Point_3 &p) const
+  {
+    return R().has_on_negative_side_3_object()(*this, p);
+  }
+
+  bool has_on_boundary(const Point_3 &p) const
+  {
+    return R().has_on_boundary_3_object()(*this, p);
+  }
+
+  bool has_on_bounded_side(const Point_3 &p) const
+  {
+    return R().has_on_bounded_side_3_object()(*this, p);
+  }
+
+  bool has_on_unbounded_side(const Point_3 &p) const
+  {
+    return R().has_on_unbounded_side_3_object()(*this, p);
+  }
+
+  typename Qualified_result_of<typename R::Compute_volume_3, Tetrahedron_3>::type
+  volume() const
+  {
+    return R().compute_volume_3_object()(*this);
+  }
+
 };
 
 #ifndef CGAL_NO_OSTREAM_INSERT_TETRAHEDRON_3
diff --git a/Kernel_23/include/CGAL/Triangle_3.h b/Kernel_23/include/CGAL/Triangle_3.h
index bdb5d4a9096..46e7605a7b1 100644
--- a/Kernel_23/include/CGAL/Triangle_3.h
+++ b/Kernel_23/include/CGAL/Triangle_3.h
@@ -30,7 +30,9 @@ template <class R_>
 class Triangle_3 : public R_::Kernel_base::Triangle_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_::Aff_transformation_3  Aff_transformation_3;
   typedef typename R_::Kernel_base::Triangle_3  RTriangle_3;
 public:
@@ -63,6 +65,37 @@ public:
                       t.transform(this->vertex(1)),
                       t.transform(this->vertex(2)));
   }
+
+  Plane_3 supporting_plane() const
+  {
+    return R().construct_supporting_plane_3_object()(*this);
+  }
+
+  bool has_on(const Point_3 &p) const
+  {
+    return R().has_on_3_object()(*this, p);
+  }
+
+  Point_3 vertex(int i) const // TODO : use Qrt
+  {
+    return R().construct_vertex_3_object()(*this, i);
+  }
+
+  Point_3 operator[](int i) const // TODO : use Qrt
+  {
+    return vertex(i);
+  }
+
+  bool is_degenerate() const
+  {
+    return R().is_degenerate_3_object()(*this);
+  }
+
+  FT squared_area() const // TODO : use Qrt
+  {
+    return R().compute_squared_area_3_object()(*this);
+  }
+
 };
 
 #ifndef CGAL_NO_OSTREAM_INSERT_TRIANGLE_3