diff --git a/Alpha_shapes_2/include/CGAL/Alpha_shape_2.h b/Alpha_shapes_2/include/CGAL/Alpha_shape_2.h
index a2ba8615f6c..1d0b4454d36 100644
--- a/Alpha_shapes_2/include/CGAL/Alpha_shape_2.h
+++ b/Alpha_shapes_2/include/CGAL/Alpha_shape_2.h
@@ -68,13 +68,20 @@ public:
   typedef typename Dt::Geom_traits Gt;
   typedef typename Dt::Triangulation_data_structure Tds;
 
-  typedef typename internal::Alpha_nt_selector_2<Gt,ExactAlphaComparisonTag>::Type_of_alpha Type_of_alpha;
-  //check simplices are correctly instantiated
-  CGAL_static_assertion( (boost::is_same<Type_of_alpha,typename Dt::Face::FT>::value) );
-  CGAL_static_assertion( (boost::is_same<Type_of_alpha,typename Dt::Vertex::FT>::value) );
+  typedef typename internal::Alpha_nt_selector_2<
+    Gt, ExactAlphaComparisonTag, typename Dt::Weighted_tag>::Type_of_alpha  Type_of_alpha;
+  typedef typename internal::Alpha_nt_selector_2<
+    Gt, ExactAlphaComparisonTag, typename Dt::Weighted_tag>::Compute_squared_radius_2 Compute_squared_radius_2;
+  typedef typename internal::Alpha_nt_selector_2<
+    Gt, ExactAlphaComparisonTag, typename Dt::Weighted_tag>::Side_of_bounded_circle_2 Side_of_bounded_circle_2;
+
+  typedef Type_of_alpha               NT;
+  typedef Type_of_alpha               FT;
+
+  //check simplices are correctly instantiated
+  CGAL_static_assertion( (boost::is_same<NT, typename Dt::Face::NT>::value) );
+  CGAL_static_assertion( (boost::is_same<NT, typename Dt::Vertex::NT>::value) );
 
-  typedef Type_of_alpha NT;
-  typedef Type_of_alpha FT;
   typedef typename Gt::Point_2 Point;
   typedef typename Gt::Segment_2 Segment;
   typedef typename Gt::Line_2 Line;
@@ -711,33 +718,30 @@ private:
   //---------------------- PREDICATES ------------------------------------
 
 private:
-  
-  bool is_attached(const Face_handle& f, int i) const 
-    {
-      Bounded_side b =
-        this->geom_traits().side_of_bounded_circle_2_object()(f->vertex(cw(i))->point(),
-                                                              f->vertex(ccw(i))->point(),
-                                                              f->vertex(i)->point());
 
-      return (b == ON_BOUNDED_SIDE) ? true : false;
-    }
-  
+  bool is_attached(const Face_handle& f, int i) const
+  {
+    Bounded_side b = Side_of_bounded_circle_2()(*this)(f->vertex(cw(i))->point(),
+                                                       f->vertex(ccw(i))->point(),
+                                                       f->vertex(i)->point());
+
+    return (b == ON_BOUNDED_SIDE) ? true : false;
+  }
+
   //-------------------- GEOMETRIC PRIMITIVES ----------------------------
 
-  Type_of_alpha squared_radius(const Face_handle& f) const 
-    {
-      return 
-        this->geom_traits().compute_squared_radius_2_object()(f->vertex(0)->point(),
-                                                              f->vertex(1)->point(),
-                                                              f->vertex(2)->point());
-    }
+  Type_of_alpha squared_radius(const Face_handle& f) const
+  {
+    return Compute_squared_radius_2()(*this)(f->vertex(0)->point(),
+                                             f->vertex(1)->point(),
+                                             f->vertex(2)->point());
+  }
 
-  Type_of_alpha squared_radius(const Face_handle& f, int i) const 
-    {
-      return 
-        this->geom_traits().compute_squared_radius_2_object()(f->vertex(ccw(i))->point(),
-                                                              f->vertex(cw(i))->point());
-    }
+  Type_of_alpha squared_radius(const Face_handle& f, int i) const
+  {
+    return Compute_squared_radius_2()(*this)(f->vertex(ccw(i))->point(),
+                                             f->vertex(cw(i))->point());
+  }
 
   //---------------------------------------------------------------------
 
diff --git a/Alpha_shapes_2/include/CGAL/internal/Lazy_alpha_nt_2.h b/Alpha_shapes_2/include/CGAL/internal/Lazy_alpha_nt_2.h
index 1e2968d44a9..1d44d9ea9a9 100644
--- a/Alpha_shapes_2/include/CGAL/internal/Lazy_alpha_nt_2.h
+++ b/Alpha_shapes_2/include/CGAL/internal/Lazy_alpha_nt_2.h
@@ -387,8 +387,20 @@ template <class GeomTraits, class ExactAlphaComparisonTag, class Weighted_tag>
 struct Alpha_nt_selector_2
   : public Alpha_nt_selector_impl_2<
              GeomTraits,
+             // We check for two things in addition to the value of ExactAlphaComparisonTag:
+             // - if the base traits is already exact (then we don't need to do anything,
+             //   and we can simply directly use the traits class)
+             // - if the traits class' Point_2 can be converted to the underlying
+             //   kernel of the traits class, which is a necessary precondition to
+             //   be able to use the Cartesian converter and the exact kernels
              Boolean_tag<boost::is_floating_point<typename GeomTraits::FT>::value &&
-                         ExactAlphaComparisonTag::value > >
+                         boost::is_convertible<
+                           typename GeomTraits::Point_2,
+                           typename Kernel_traits<
+                             typename GeomTraits::Point_2>::Kernel::Point_2
+                         >::value &&
+                         ExactAlphaComparisonTag::value >,
+             Weighted_tag>
 { };
 
 } // namespace internal
diff --git a/Alpha_shapes_3/include/CGAL/internal/Lazy_alpha_nt_3.h b/Alpha_shapes_3/include/CGAL/internal/Lazy_alpha_nt_3.h
index bd031983d08..a159735cac0 100644
--- a/Alpha_shapes_3/include/CGAL/internal/Lazy_alpha_nt_3.h
+++ b/Alpha_shapes_3/include/CGAL/internal/Lazy_alpha_nt_3.h
@@ -356,12 +356,24 @@ struct Alpha_nt_selector_impl_3<GeomTraits,Tag_true,Tag_true>
 };
 
 template <class GeomTraits,class ExactAlphaComparisonTag,class Weighted_tag>
-struct Alpha_nt_selector_3:
-  public Alpha_nt_selector_impl_3<GeomTraits,
-              Boolean_tag< boost::is_floating_point<typename GeomTraits::FT>::value && ExactAlphaComparisonTag::value >,
-              Weighted_tag>
-{};
-
+struct Alpha_nt_selector_3
+  : public Alpha_nt_selector_impl_3<
+             GeomTraits,
+             // We check for two things in addition to the value of ExactAlphaComparisonTag:
+             // - if the base traits is already exact (then we don't need to do anything,
+             //   and we can simply directly use the traits class)
+             // - if the traits class' Point_3 can be converted to the underlying
+             //   kernel of the traits class, which is a necessary precondition to
+             //   be able to use the Cartesian converter and the exact kernels
+             Boolean_tag<boost::is_floating_point<typename GeomTraits::FT>::value &&
+                         boost::is_convertible<
+                           typename GeomTraits::Point_3,
+                           typename Kernel_traits<
+                             typename GeomTraits::Point_3>::Kernel::Point_3
+                         >::value &&
+                         ExactAlphaComparisonTag::value >,
+             Weighted_tag>
+{ };
 
 } //namespace internal