diff --git a/AABB_tree/doc/AABB_tree/Concepts/AABBGeomTraits.h b/AABB_tree/doc/AABB_tree/Concepts/AABBGeomTraits.h
index c2e1e77e98b..7f0f67cca88 100644
--- a/AABB_tree/doc/AABB_tree/Concepts/AABBGeomTraits.h
+++ b/AABB_tree/doc/AABB_tree/Concepts/AABBGeomTraits.h
@@ -59,9 +59,16 @@ typedef unspecified_type Construct_sphere_3;
 
 /*!
 A functor object to compute the point on a geometric primitive which is closest from a query. Provides the operator: 
-`Point_3 operator()(const Point_3& p, const Type_2& type_2);` where `Type_2` is any type among `Segment_3` and `Triangle_3`. The operator returns the point on `type_2` which is closest to `p`. 
+`Point_3 operator()(const Type_2& type_2, const Point_3& p);` where `Type_2` is any type among `Segment_3` and `Triangle_3`. The operator returns the point on `type_2` which is closest to `p`. 
 */ 
-typedef unspecified_type Compute_closest_point_3; 
+typedef unspecified_type Construct_projected_point_3;
+
+/*!
+A functor object to compare the distance of two points wrt a third one.
+Provides the operator:
+`CGAL::Comparision_result operator()(const Point_3& p1, const Point_3& p2, const Point_3& p3)`. The operator compare the distance between `p1 and `p2`, and between `p2` and `p3`.
+*/
+typedef unspecified_type Compare_distance_3
 
 /*!
 A functor object to detect if a point lies inside a sphere or not. 
@@ -106,7 +113,12 @@ Construct_sphere_3 construct_sphere_3_object();
 /*!
 Returns the closest point constructor. 
 */ 
-Compute_closest_point_3 compute_closest_point_3_object(); 
+Construct_projected_point_3 construct_projected_point_3_object();
+
+/*!
+Returns the compare distance constructor.
+*/
+Compare_distance_3 compare_distance_3_object();
 
 /*!
 Returns the closest point constructor. 
diff --git a/AABB_tree/include/CGAL/AABB_intersections.h b/AABB_tree/include/CGAL/AABB_intersections.h
index f1df4ab0925..d7344df618a 100644
--- a/AABB_tree/include/CGAL/AABB_intersections.h
+++ b/AABB_tree/include/CGAL/AABB_intersections.h
@@ -19,6 +19,3 @@
 // Author(s)     : Camille Wormser, Pierre Alliez, Stephane Tayeb
 
 #include <CGAL/intersections.h>
-
-#include <CGAL/internal/AABB_tree/nearest_point_segment_3.h>
-#include <CGAL/internal/AABB_tree/nearest_point_triangle_3.h>
diff --git a/AABB_tree/include/CGAL/AABB_traits.h b/AABB_tree/include/CGAL/AABB_traits.h
index 39e677d3126..d17eeb10664 100644
--- a/AABB_tree/include/CGAL/AABB_traits.h
+++ b/AABB_tree/include/CGAL/AABB_traits.h
@@ -397,7 +397,12 @@ public:
 
     Point operator()(const Point& p, const Primitive& pr, const Point& bound) const
     {
-        return CGAL::nearest_point_3(p, internal::Primitive_helper<AT>::get_datum(pr,m_traits), bound);
+      GeomTraits geom_traits;
+      Point closest_point = geom_traits.construct_projected_point_3_object()(
+        internal::Primitive_helper<AT>::get_datum(pr,m_traits), p);
+      return
+        geom_traits.compare_distance_3_object()(p, closest_point, bound)==LARGER ?
+        bound : closest_point;
     }
   };