diff --git a/AABB_tree/examples/AABB_tree/AABB_segment_3_example.cpp b/AABB_tree/examples/AABB_tree/AABB_segment_3_example.cpp
index 93288489863..46a0c51118e 100644
--- a/AABB_tree/examples/AABB_tree/AABB_segment_3_example.cpp
+++ b/AABB_tree/examples/AABB_tree/AABB_segment_3_example.cpp
@@ -17,7 +17,7 @@ typedef K::Segment_3 Segment;
 typedef K::Triangle_3 Triangle;
 
 typedef std::list<Segment>::iterator Iterator;
-typedef CGAL::AABB_segment_primitive<Iterator> Primitive;
+typedef CGAL::AABB_segment_primitive<K, Iterator> Primitive;
 typedef CGAL::AABB_traits<K, Primitive> Traits;
 typedef CGAL::AABB_tree<Traits> Tree;
 
diff --git a/AABB_tree/examples/AABB_tree/AABB_triangle_3_example.cpp b/AABB_tree/examples/AABB_tree/AABB_triangle_3_example.cpp
index 3d21f70b0cb..1898ccc4edf 100644
--- a/AABB_tree/examples/AABB_tree/AABB_triangle_3_example.cpp
+++ b/AABB_tree/examples/AABB_tree/AABB_triangle_3_example.cpp
@@ -17,7 +17,7 @@ typedef K::Point_3 Point;
 typedef K::Triangle_3 Triangle;
 
 typedef std::list<Triangle>::iterator Iterator;
-typedef CGAL::AABB_triangle_primitive<Iterator> Primitive;
+typedef CGAL::AABB_triangle_primitive<K, Iterator> Primitive;
 typedef CGAL::AABB_traits<K, Primitive> AABB_triangle_traits;
 typedef CGAL::AABB_tree<AABB_triangle_traits> Tree;
 
diff --git a/AABB_tree/include/CGAL/AABB_segment_primitive.h b/AABB_tree/include/CGAL/AABB_segment_primitive.h
index ccd44f5db36..6dafc274081 100644
--- a/AABB_tree/include/CGAL/AABB_segment_primitive.h
+++ b/AABB_tree/include/CGAL/AABB_segment_primitive.h
@@ -24,43 +24,45 @@
 #define CGAL_AABB_SEGMENT_PRIMITIVE_H_
 
 #include <CGAL/AABB_primitive.h>
-#include <CGAL/Kernel_traits.h>
+#include <CGAL/result_of.h>
 #include <iterator>
 
 namespace CGAL {
 
 namespace internal {
-  template <class Iterator>
+  template <class GeomTraits, class Iterator>
   struct Source_of_segment_3_iterator_property_map{
     //classical typedefs
-    typedef typename CGAL::Kernel_traits< typename std::iterator_traits<Iterator>::value_type >::Kernel GeomTraits;
     typedef Iterator key_type;
     typedef typename GeomTraits::Point_3 value_type;
-    typedef const typename GeomTraits::Point_3& reference;
+    typedef typename cpp11::result_of<
+      typename GeomTraits::Construct_source_3(typename GeomTraits::Segment_3)
+    >::type reference;
     typedef boost::readable_property_map_tag category;
-  };
 
-  //get function for property map
-  template <class Iterator>
-  inline
-  typename Source_of_segment_3_iterator_property_map<Iterator>::reference
-  get(Source_of_segment_3_iterator_property_map<Iterator>, Iterator it)
-  {
-    return it->source();
-  }
+    inline friend
+    typename Source_of_segment_3_iterator_property_map<GeomTraits,Iterator>::reference
+    get(Source_of_segment_3_iterator_property_map<GeomTraits,Iterator>, Iterator it)
+    {
+      return typename GeomTraits::Construct_source_3()( *it );
+    }
+  };
 }//namespace internal
 
 
 /*!
  * \ingroup PkgAABB_tree
  * Primitive type that uses as identifier an iterator with a 3D segment as `value_type`.
- * The iterator from which the primitive is built should not be invalided 
+ * The iterator from which the primitive is built should not be invalided
  * while the AABB tree holding the primitive is in use.
  *
  * \cgalModels `AABBPrimitive`
  *
- * \tparam Iterator is a model of `ForwardIterator`, with `Segment_3<Kernel>`
- *    as value type
+ * \tparam GeomTraits is a traits class providing the nested type `Point_3` and `Segment_3`.
+ *         It also provides the functor `Construct_source_3` that has an operator taking a `Segment_3`
+ *         and returning its source as a type convertible to `Point_3`.
+ *         In addition `Construct_source_3` must support the result_of protocol.
+ * \tparam Iterator is a model of `ForwardIterator` with its value type convertible to `GeomTraits::Segment_3`
  * \tparam cache_datum is either `CGAL::Tag_true` or `CGAL::Tag_false`. In the former case,
  *           the datum is stored in the primitive, while in the latter it is
  *           constructed on the fly to reduce the memory footprint.
@@ -72,20 +74,21 @@ namespace internal {
  * \sa `AABB_HalfedgeGraph_segment_primitive<HalfedgeGraph,OneHalfedgeGraphPerTree,cache_datum>`
  * \sa `AABB_FaceGraph_triangle_primitive<FaceGraph,OneFaceGraphPerTree,cache_datum>`
  */
-template < class Iterator,
+template < class GeomTraits,
+           class Iterator,
            class cache_datum=Tag_false>
 class AABB_segment_primitive
 #ifndef DOXYGEN_RUNNING
   : public AABB_primitive<  Iterator,
                             Input_iterator_property_map<Iterator>,
-                            internal::Source_of_segment_3_iterator_property_map<Iterator>,
+                            internal::Source_of_segment_3_iterator_property_map<GeomTraits, Iterator>,
                             Tag_false,
                             cache_datum >
 #endif
 {
   typedef AABB_primitive< Iterator,
                           Input_iterator_property_map<Iterator>,
-                          internal::Source_of_segment_3_iterator_property_map<Iterator>,
+                          internal::Source_of_segment_3_iterator_property_map<GeomTraits, Iterator>,
                           Tag_false,
                           cache_datum > Base;
 public:
diff --git a/AABB_tree/include/CGAL/AABB_triangle_primitive.h b/AABB_tree/include/CGAL/AABB_triangle_primitive.h
index 514463e9df0..1d05fc4aed2 100644
--- a/AABB_tree/include/CGAL/AABB_triangle_primitive.h
+++ b/AABB_tree/include/CGAL/AABB_triangle_primitive.h
@@ -24,42 +24,45 @@
 #define CGAL_AABB_TRIANGLE_PRIMITIVE_H_
 
 #include <CGAL/AABB_primitive.h>
-#include <CGAL/property_map.h>
+#include <CGAL/result_of.h>
+#include <iterator>
 
 namespace CGAL {
 
 namespace internal {
-  template <class Iterator>
+  template <class GeomTraits, class Iterator>
   struct Point_from_triangle_3_iterator_property_map{
     //classical typedefs
-    typedef typename CGAL::Kernel_traits< typename std::iterator_traits<Iterator>::value_type >::Kernel GeomTraits;
     typedef Iterator key_type;
     typedef typename GeomTraits::Point_3 value_type;
-    typedef const typename GeomTraits::Point_3& reference;
+    typedef typename cpp11::result_of<
+      typename GeomTraits::Construct_vertex_3(typename GeomTraits::Triangle_3,int)
+    >::type reference;
     typedef boost::readable_property_map_tag category;
-  };
 
-  //get function for property map
-  template <class Iterator>
-  inline
-  typename Point_from_triangle_3_iterator_property_map<Iterator>::reference
-  get(Point_from_triangle_3_iterator_property_map<Iterator>,Iterator it)
-  {
-    return it->vertex(0);
-  }
+    inline friend
+    typename Point_from_triangle_3_iterator_property_map<GeomTraits,Iterator>::reference
+    get(Point_from_triangle_3_iterator_property_map<GeomTraits,Iterator>, Iterator it)
+    {
+      return typename GeomTraits::Construct_vertex_3()( *it, 0 );
+    }
+  };
 }//namespace internal
 
 
 /*!
  * \ingroup PkgAABB_tree
  * Primitive type that uses as identifier an iterator with a 3D triangle as `value_type`.
- * The iterator from which the primitive is built should not be invalided 
+ * The iterator from which the primitive is built should not be invalided
  * while the AABB tree holding the primitive is in use.
  *
  * \cgalModels `AABBPrimitive`
  *
- * \tparam Iterator is a model of `ForwardIterator`, with `Triangle_3<Kernel>`
- *           as value type
+ * \tparam GeomTraits is a traits class providing the nested type `Point_3` and `Triangle_3`.
+ *         It also provides the functor `Construct_vertex_3` that has an operator taking a `Triangle_3`
+ *         and an integer as parameters and returning a triangle point as a type convertible to `Point_3`.
+ *         In addition `Construct_vertex_3` must support the result_of protocol.
+ * \tparam Iterator is a model of `ForwardIterator` with its value type convertible to `GeomTraits::Triangle_3`
  * \tparam cache_datum is either `CGAL::Tag_true` or `CGAL::Tag_false`. In the former case,
  *           the datum is stored in the primitive, while in the latter it is
  *           constructed on the fly to reduce the memory footprint.
@@ -71,24 +74,25 @@ namespace internal {
  * \sa `AABB_HalfedgeGraph_segment_primitive<HalfedgeGraph,OneHalfedgeGraphPerTree,cache_datum>`
  * \sa `AABB_FaceGraph_triangle_primitive<FaceGraph,OneFaceGraphPerTree,cache_datum>`
  */
-template < class Iterator,
+template < class GeomTraits,
+           class Iterator,
            class cache_datum=Tag_false>
-class AABB_triangle_primitive 
+class AABB_triangle_primitive
 #ifndef DOXYGEN_RUNNING
-    : public AABB_primitive<  Iterator,
-                              Input_iterator_property_map<Iterator>,
-                              internal::Point_from_triangle_3_iterator_property_map<Iterator>,
-                              Tag_false,
-                              cache_datum >
+  : public AABB_primitive<  Iterator,
+                            Input_iterator_property_map<Iterator>,
+                            internal::Point_from_triangle_3_iterator_property_map<GeomTraits, Iterator>,
+                            Tag_false,
+                            cache_datum >
 #endif
 {
   typedef AABB_primitive< Iterator,
                           Input_iterator_property_map<Iterator>,
-                          internal::Point_from_triangle_3_iterator_property_map<Iterator>,
+                          internal::Point_from_triangle_3_iterator_property_map<GeomTraits, Iterator>,
                           Tag_false,
                           cache_datum > Base;
 public:
-  // constructors
+  ///Constructor from an iterator
   AABB_triangle_primitive(Iterator it) : Base(it){}
 };