diff --git a/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/internal/Side_of_triangle_mesh/Ray_3_Triangle_3_traversal_traits.h b/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/internal/Side_of_triangle_mesh/Ray_3_Triangle_3_traversal_traits.h
index 285017791a5..130c2732ea1 100644
--- a/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/internal/Side_of_triangle_mesh/Ray_3_Triangle_3_traversal_traits.h
+++ b/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/internal/Side_of_triangle_mesh/Ray_3_Triangle_3_traversal_traits.h
@@ -216,6 +216,60 @@ public:
   }
 };
 
+//special case when ray query is from another Kernel K1 is the kernel compatible with the AABB-tree
+template<typename AABBTraits, class K1, class K2, class Helper>
+class K2_Ray_3_K1_Triangle_3_traversal_traits
+{
+  //the status indicates whether the query point is strictly inside the polyhedron, and the number of intersected triangles if yes
+  std::pair<boost::logic::tribool,std::size_t>& m_status;
+  bool m_stop;
+  const AABBTraits& m_aabb_traits;
+  typedef typename AABBTraits::Primitive Primitive;
+  typedef CGAL::AABB_node<AABBTraits> Node;
+  Helper m_helper;
+  CGAL::Cartesian_converter<K1,K2> to_K2;
+
+public:
+  K2_Ray_3_K1_Triangle_3_traversal_traits(std::pair<boost::logic::tribool,std::size_t>& status,
+                                          const AABBTraits& aabb_traits,
+                                          const Helper& h)
+    :m_status(status), m_stop(false), m_aabb_traits(aabb_traits), m_helper(h)
+  {m_status.first=true;}
+
+  bool go_further() const { return !m_stop; }
+
+  template<class Query>
+  void intersection(const Query& query, const Primitive& primitive)
+  {
+    Intersections::internal::r3t3_do_intersect_endpoint_position_visitor visitor;
+    std::pair<bool,Intersections::internal::R3T3_intersection::type> res=
+      Intersections::internal::do_intersect(to_K2(m_helper.get_primitive_datum(primitive, m_aabb_traits)),
+                                            query, K2(), visitor);
+
+    if (res.first){
+      switch (res.second){
+        case Intersections::internal::R3T3_intersection::CROSS_FACET:
+          ++m_status.second;
+        break;
+        case Intersections::internal::R3T3_intersection::ENDPOINT_IN_TRIANGLE:
+          m_status.first=false;
+          m_stop=true;
+        break;
+        default:
+          m_status.first=boost::logic::indeterminate;
+          m_stop=true;
+      }
+    }
+  }
+
+  template<class Query>
+  bool do_intersect(const Query& query, const Node& node) const
+  {
+    return CGAL::do_intersect(query, m_helper.get_node_bbox(node));
+  }
+};
+
+
 }// namespace internal
 }// namespace CGAL
 
diff --git a/Polygon_mesh_processing/include/CGAL/Side_of_triangle_mesh.h b/Polygon_mesh_processing/include/CGAL/Side_of_triangle_mesh.h
index 20ec968d50a..f0c23942f60 100644
--- a/Polygon_mesh_processing/include/CGAL/Side_of_triangle_mesh.h
+++ b/Polygon_mesh_processing/include/CGAL/Side_of_triangle_mesh.h
@@ -242,6 +242,81 @@ public:
     }
   }
 
+#ifndef DOXYGEN_RUNNING
+  template <class K2>
+  Bounded_side operator()(const typename K2::Point_3& point, const K2& k2) const
+  {
+    if(point.x() < box.xmin()
+       || point.x() > box.xmax()
+       || point.y() < box.ymin()
+       || point.y() > box.ymax()
+       || point.z() < box.zmin()
+       || point.z() > box.zmax())
+    {
+      return CGAL::ON_UNBOUNDED_SIDE;
+    }
+
+#ifdef CGAL_HAS_THREADS
+    AABB_tree_* tree_ptr =
+      const_cast<AABB_tree_*>(atomic_tree_ptr.load(std::memory_order_acquire));
+#endif
+    // Lazily build the tree only when needed
+    if (tree_ptr==nullptr)
+    {
+#ifdef CGAL_HAS_THREADS
+      CGAL_SCOPED_LOCK(tree_mutex);
+      tree_ptr = const_cast<AABB_tree_*>(atomic_tree_ptr.load(std::memory_order_relaxed));
+#endif
+      CGAL_assertion(tm_ptr != nullptr && opt_vpm!=boost::none);
+      if (tree_ptr==nullptr)
+      {
+        tree_ptr = new AABB_tree(faces(*tm_ptr).first,
+                                 faces(*tm_ptr).second,
+                                 *tm_ptr, *opt_vpm);
+        const_cast<AABB_tree_*>(tree_ptr)->build();
+#ifdef CGAL_HAS_THREADS
+      atomic_tree_ptr.store(tree_ptr, std::memory_order_release);
+#endif
+      }
+    }
+
+    typedef typename Kernel_traits<Point>::Kernel K1;
+    typedef typename AABB_tree::AABB_traits AABB_traits;
+    typedef internal::Default_tree_helper<AABB_tree> Helper;
+    Helper helper;
+    typename AABB_traits::Bounding_box bbox = helper.get_tree_bbox(*tree_ptr);
+
+    static const unsigned int seed = 1340818006;
+    CGAL::Random rg(seed); // seed some value for make it easy to debug
+    Random_points_on_sphere_3<typename K2::Point_3> random_point(1.,rg);
+
+    typename K2::Construct_ray_3 ray = k2.construct_ray_3_object();
+    typename K2::Construct_vector_3 vector = k2.construct_vector_3_object();
+
+    do { //retry with a random ray
+      typename K2::Ray_3 query = ray(point, vector(CGAL::ORIGIN,*random_point++));
+
+       std::pair<boost::logic::tribool,std::size_t>
+          status( boost::logic::tribool(boost::logic::indeterminate), 0);
+
+      internal::K2_Ray_3_K1_Triangle_3_traversal_traits<AABB_traits, K1, K2, Helper>
+        traversal_traits(status, tree_ptr->traits(), helper);
+
+      tree_ptr->traversal(query, traversal_traits);
+
+      if ( !boost::logic::indeterminate(status.first) )
+      {
+        if (status.first)
+          return (status.second&1) == 1 ? ON_BOUNDED_SIDE : ON_UNBOUNDED_SIDE;
+        //otherwise the point is on the facet
+        return ON_BOUNDARY;
+      }
+    } while (true);
+    return ON_BOUNDARY; // should never be reached
+  }
+
+#endif
+
 };
 
 } // namespace CGAL