diff --git a/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3DelaunayTriangulationTraits_3.h b/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3DelaunayTriangulationTraits_3.h
index 64016500753..881ae37a8c7 100644
--- a/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3DelaunayTriangulationTraits_3.h
+++ b/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3DelaunayTriangulationTraits_3.h
@@ -34,12 +34,7 @@ public:
 /// @{
 
 /*!
-A predicate object that must provide the function operators
-
-`Oriented_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Point_3 t)`,
-
-which determines on which side of the oriented sphere circumscribing
-`p, q, r, s` the point `t` lies and
+A predicate object that must provide the function operator
 
 `Oriented_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s, Periodic_3_offset_3 o_t)`,
 
@@ -51,12 +46,7 @@ which determines on which side of the oriented sphere circumscribing
 typedef unspecified_type Side_of_oriented_sphere_3;
 
 /*!
-A predicate object that must provide the function operators
-
-`Comparison_result operator()(Point_3 p, Point_3 q, Point_3 r)`,
-
-which compares the distance between `p` and `q` to the distance
-between `p` and `r` and
+A predicate object that must provide the function operator
 
 `Comparison_result operator()(Point_3 p, Point_3 q, Point_3 r, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r)`,
 
@@ -73,13 +63,7 @@ typedef unspecified_type Compare_distance_3;
 /// @{
 
 /*!
-A predicate object that must provide the function operators
-
-`Orientation operator()(Point_3 p, Point_3 q, Point_3 r)`,
-
-which returns `COLLINEAR`, if the points are collinear; otherwise
-it must return a consistent orientation for any three points chosen in
-a same plane and
+A predicate object that must provide the function operator
 
 `Orientation operator()(Point_3 p, Point_3 q, Point_3 r Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r)`,
 
@@ -91,12 +75,7 @@ three point-offset pairs chosen in a same plane.
 typedef unspecified_type Coplanar_orientation_3;
 
 /*!
-A predicate object that must provide the function operators
-
-`Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s)`,
-
-which determines the bounded side of the circle defined by `p, q`,
-and `r` on which the point `s` lies and
+A predicate object that must provide the function operator
 
 `Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s)`,
 
@@ -115,42 +94,19 @@ typedef unspecified_type Coplanar_side_of_bounded_circle_3;
 /// @{
 
 /*!
-A predicate object that must provide the function operators
-
-`Bounded_side operator()(Point_3 p, Point_3 q, Point_3 t)`,
-
-which returns the position of the point `t` relative to the sphere
-that has `pq` as its diameter,
+A predicate object that must provide the function operator
 
 `Bounded_side operator()(Point_3 p, Point_3 q, Point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_t)`,
 
 which returns the position of the point-offset pair `(t,o_t)`
 relative to the sphere that has `(p,o_p)(q,o_q)` as its diameter,
 
-`Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 t)`,
-
-which returns the position of the point `t` relative to the sphere
-passing through `p, q`, and `r` and whose center is in the
-plane defined by these three points,
-
 `Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_q)`,
 
 which returns the position of the point-offset pair `(t,o_t)`
 relative to the sphere passing through `(p,o_p), (q,o_q)`, and
 `(r,o_r)` and whose center is in the plane defined by these three
-point-offset pairs,
-
-`Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Point_3 t)`,
-
-which returns the relative position of point `t` to the sphere
-defined by `p, q, r`, and `s`; the order of the points `p, q, r`, and `s` does not matter, and
-
-`Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Point_3 t, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s, Periodic_3_offset_3 o_q)`,
-
-which returns the relative position of the point-offset pair
-`(t,o_t)` to the sphere defined by `(p,o_p), (q,o_q), (r,o_r)`, and `(s,o_s)`; the order of the point-offset pairs
-`(p,o_p), (q,o_q), (r,o_r)`, and `(s,o_s)` does not matter.
-\pre `p, q, r`, and `s` are not coplanar, `(p,o_p), (q,o_q), (r,o_r)`, and `(s,o_s)` are not coplanar, `p`, `q`, `r`, `s`, `t` lie inside the domain.
+point-offset pairs.
 */
 typedef unspecified_type Side_of_bounded_sphere_3;
 
@@ -162,11 +118,7 @@ typedef unspecified_type Side_of_bounded_sphere_3;
 /// @{
 
 /*!
-A constructor object that must provide the function operators
-
-`Point_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s)`,
-
-which constructs the circumcenter of four points and
+A constructor object that must provide the function operator
 
 `Point_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s)`,
 
diff --git a/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3RegularTriangulationTraits_3.h b/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3RegularTriangulationTraits_3.h
index 0c2e6e0d919..9f7ffb809eb 100644
--- a/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3RegularTriangulationTraits_3.h
+++ b/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3RegularTriangulationTraits_3.h
@@ -34,58 +34,41 @@ public:
 /// @{
 
 /*!
-A predicate object that must provide the function operators:
-
-`Oriented_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s, Weighted_point_3 t)`,
-
-which determines the position of `t` with respect to the power sphere of `p, q, r, s`.
+A predicate object that must provide the function operators
 
 `Oriented_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s, Weighted_point_3 t,
 Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s, Periodic_3_offset_3 o_t)`,
 
-which is the same for the point-offset pair `(t,o_t)` with respect to the power sphere of the point-offset pairs
-`(p,o_p), (q,o_q), (r,o_r), (s,o_s)`.
+which determines the position of the point-offset pair `(t,o_t)` with respect
+to the power sphere of the point-offset pairs `(p,o_p), (q,o_q), (r,o_r), (s,o_s)`.
 \pre `p`, `q`, `r`, `s`, `t` lie inside the domain and `p, q, r, s` are not coplanar.
 
 <HR WIDTH=50%>
 
-When vertex removal is used, the predicate must in addition provide the following function operators:
+When vertex removal is used, the predicate must in addition provide the function operators
 
-`Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 t)`,
+`Oriented_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 t,
+                          Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_t)`,
 
 which has a definition similar to the previous method, for coplanar points,
 with the power circle of `p,q,r`.
+\pre `p`, `q`, `r`, `t` lie inside the domain, `p, q, r` are not collinear,
+and `(p,o_p), (q,o_q), (r,o_r), (t,o_t)` are coplanar.
 
-`Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 t,
-Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_t)`,
+`Oriented_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 t,
+                          Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_t)`,
 
-which is the same for point-offset pairs.
+which is the same for collinear points, and the power segment of `(p,o_p)` and `(q,o_q)`,
 
-\pre `p`, `q`, `r`, `t` lie inside the domain, `p, q, r` are not collinear, and `p, q, r, t` are coplanar.
+\pre `p`, `q`, `t` lie inside the domain, `p` and `q` have different Bare_points, and
+`(p,o_p), (q,o_q), (t,o_t)` are collinear.
 
+`Oriented_side operator()(Weighted_point_3 p, Weighted_point_3 q,
+                          Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q)`,
 
-`Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 t)`,
-
-which is the same for collinear points, and the power segment of `p` and `q`,
-
-`Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 t,
-Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_t)`,
-
-which is the same for point-offset pairs.
-
-\pre `p`, `q`, `t` lie inside the domain, `p` and `q` have different Bare_points, and `p, q, t` are collinear.
-
-
-`Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q)`,
-
-which is the same for equal points, that is when `p` and `q`
+which is the same for equal points, that is when `(p,o_p)` and `(q,o_q)`
 have equal coordinates, then it returns the comparison of the weights.
 
-`Oriented_side operator()( Weighted_point_3 p, Weighted_point_3 q,
-Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q)`,
-
-which is the same for point-offset pairs.
-
 \pre `p` and `q` lie inside the domain and have equal Bare_points.
 
 */
@@ -118,15 +101,11 @@ typedef unspecified_type Compare_weighted_squared_radius_3;
 A predicate object, model of `ComparePowerDistance_3`, that must provide
 the function operator
 
-`Comparison_result operator()(Point_3 p, Weighted_point_3 q, Weighted_point_3 r)`,
-
-which compares the power distance between `p` and `q` to the power distance
-between `p` and `r` and
-
 `Comparison_result operator()(Point_3 p, Weighted_point_3 q, Weighted_point_3 r,
-Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r)`,
+                              Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r)`,
 
-which is the same for point-offset pairs.
+which compares the power distance between `(p,o_p)` and `(q,o_q)` to the power distance
+between `(p,o_p)` and `(r,o_r)`.
 
 \note This predicate is required if a call to `nearest_power_vertex()` or
 `nearest_power_vertex_in_cell()` is issued.*/
@@ -157,23 +136,51 @@ typedef unspecified_type Coplanar_orientation_3;
 
 /// @}
 
+/// \name
+/// When `is_Gabriel` functions are used, the traits class must
+/// in addition provide the following predicate object:
+/// @{
+
+/*!
+A predicate object that must provide the function operator
+
+`Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 t,
+                         Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_t)`,
+
+which returns the sign of the power test of `(t,o_t)` with respect to the smallest
+sphere orthogonal to `(p,o_p)` (which is the sphere with center `(p,o_p)` and squared
+radius `-w_p` with `w_p` the weight of `p`),
+
+`Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 t,
+                         Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_t)`,
+
+which returns the sign of the power test of `(t,o_t)` with respect to the smallest
+sphere orthogonal to `(p,o_p)` and `(q,o_q)`,
+
+`Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 t,
+                         Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_q)`,
+
+which returns the sign of the power test of `(t,o_t)` with respect to the smallest
+sphere orthogonal to `(p,o_p)`, `(q,o_q)`, and `(r,o_r)`.
+*/
+typedef unspecified_type Power_side_of_bounded_power_sphere_3;
+
+/// @}
+
 /// \name
 /// When the dual operations are used, the traits
 /// class must in addition provide the following constructor object:
 /// @{
 
 /*!
-A constructor object that must provide the function operators:
-
-`Weighted_point_3 operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s)`,
-
-which constructs the weighted circumcenter of four points and
+A constructor object that must provide the function operator
 
 `Weighted_point_3 operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 s,
 Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s)`,
 
 which constructs the weighted circumcenter of four point-offset pairs.
-\pre `p`, `q`, `r` and `s` as well as `(p,o_p)`, `(q,o_q)`, `(r,o_r)` and `(s,o_s)` must be non coplanar. `p`, `q`, `r`, `s` lie inside the domain.
+\pre `p`, `q`, `r`, `s` lie inside the domain.  `p`, `q`, `r` and `s`,
+as well as `(p,o_p)`, `(q,o_q)`, `(r,o_r)` and `(s,o_s)` must be non coplanar.
 */
 typedef unspecified_type Construct_weighted_circumcenter_3;
 
diff --git a/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3TriangulationTraits_3.h b/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3TriangulationTraits_3.h
index d18e42be86d..ff7448b2e32 100644
--- a/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3TriangulationTraits_3.h
+++ b/Periodic_3_triangulation_3/doc/Periodic_3_triangulation_3/Concepts/Periodic_3TriangulationTraits_3.h
@@ -80,11 +80,7 @@ of `Kernel::Tetrahedron_3`.
 typedef unspecified_type Tetrahedron_3;
 
 /*!
-A predicate object that must provide the function operators
-
-`Comparison_result operator()(Point_3 p, Point_3 q)`,
-
-which returns `EQUAL` if the two points are equal and
+A predicate object that must provide the function operator
 
 `Comparison_result operator()(Point_3 p, Point_3 q, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q)`,
 
@@ -96,14 +92,7 @@ in a same line.
 typedef unspecified_type Compare_xyz_3;
 
 /*!
-A predicate object that must provide the function operators
-
-`Orientation operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s)`,
-
-which returns `POSITIVE`, if `s` lies on the positive side of
-the oriented plane `h` defined by `p`, `q`, and `r`,
-returns `NEGATIVE` if `s` lies on the negative side of
-`h`, and returns `COPLANAR` if `s` lies on `h` and
+A predicate object that must provide the function operator
 
 `Orientation operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s)`,
 
@@ -134,11 +123,7 @@ which constructs a point from a point-offset pair.
 typedef unspecified_type Construct_point_3;
 
 /*!
-A constructor object that must provide the function operators
-
-`Segment_3 operator()(Point_3 p, Point_3 q)`,
-
-which constructs a segment from two points and
+A constructor object that must provide the function operator
 
 `Segment_3 operator()(Point_3 p, Point_3 q, Periodic_3_offset_3 o_p, Periodic_3_offset_3 o_q)`,
 
@@ -148,11 +133,7 @@ which constructs a segment from two point-offset pairs.
 typedef unspecified_type Construct_segment_3;
 
 /*!
-A constructor object that must provide the function operators
-
-`Triangle_3 operator()(Point_3 p, Point_3 q, Point_3 r )`,
-
-which constructs a triangle from three points and
+A constructor object that must provide the function operator
 
 `Triangle_3 operator()(Point_3 p, Point_3 q, Point_3 r, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r)`,
 
@@ -162,11 +143,7 @@ which constructs a triangle from three point-offset pairs.
 typedef unspecified_type Construct_triangle_3;
 
 /*!
-A constructor object that must provide the function operators
-
-`Tetrahedron_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s)`,
-
-which constructs a tetrahedron from four points and
+A constructor object that must provide the function operator
 
 `Tetrahedron_3 operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 s, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_q, Periodic_3_offset_3 o_r, Periodic_3_offset_3 o_s)`,
 
diff --git a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h
index 17198cb4c86..8c0edf178d0 100644
--- a/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h
+++ b/Periodic_3_triangulation_3/include/CGAL/Periodic_3_regular_triangulation_3.h
@@ -733,37 +733,96 @@ public:
 
   Vertex_handle nearest_power_vertex(const Bare_point& p, Cell_handle start) const
   {
+    CGAL_triangulation_precondition(p.x() < domain().xmax());
+    CGAL_triangulation_precondition(p.y() < domain().ymax());
+    CGAL_triangulation_precondition(p.z() < domain().zmax());
+    CGAL_triangulation_precondition(p.x() >= domain().xmin());
+    CGAL_triangulation_precondition(p.y() >= domain().ymin());
+    CGAL_triangulation_precondition(p.z() >= domain().zmin());
+
     if(number_of_vertices() == 0)
       return Vertex_handle();
 
+    typename Gt::Construct_weighted_point_3 p2wp =
+        geom_traits().construct_weighted_point_3_object();
+
     Locate_type lt;
     int li, lj;
+    Offset query_offset;
+    Cell_handle c = locate(p2wp(p), query_offset, lt, li, lj, start);
 
-    typename Gt::Construct_weighted_point_3 p2wp =
-      geom_traits().construct_weighted_point_3_object();
-    Cell_handle c = locate(p2wp(p), lt, li, lj, start);
-    if(lt == Tr_Base::VERTEX)
-      return c->vertex(li);
-    const Conflict_tester tester(p2wp(p), this);
-    Offset o = combine_offsets(Offset(), get_location_offset(tester, c));
+#ifdef CGAL_PERIODIC_DEBUG_NEAREST_POWER_VERTEX
+    std::cout << "nearest power vertex: " << p << std::endl;
+    std::cout << "vertices: " << number_of_vertices() << std::endl;
+    std::cout << "stored vertices: " << this->number_of_stored_vertices() << std::endl;
+    std::cout << "Locate: " << p << std::endl;
+    std::cout << "Cell: " << &*c << std::endl;
+    std::cout << this->point(c, 0) << std::endl;
+    std::cout << this->point(c, 1) << std::endl;
+    std::cout << this->point(c, 2) << std::endl;
+    std::cout << this->point(c, 3) << std::endl;
+    std::cout << "offset query: " << query_offset << std::endl;
+    std::cout << "bounded side : " << geom_traits().bounded_side_3_object()(
+                   Tetrahedron(this->point(c, 0).point(), this->point(c, 1).point(),
+                               this->point(c, 2).point(), this->point(c, 3).point()), p) << std::endl;
+    std::cout << "power side: " << geom_traits().power_side_of_bounded_power_sphere_3_object()(
+                   this->point(c, 0), this->point(c, 1),
+                   this->point(c, 2), this->point(c, 3), p2wp(p)) << std::endl;
+    std::cout << "power distance: " << geom_traits().compute_power_distance_to_power_sphere_3_object()(
+                   this->point(c, 0), this->point(c, 1),
+                   this->point(c, 2), this->point(c, 3), p2wp(p)) << std::endl;
+#endif
 
     // - start with the closest vertex from the located cell.
     // - repeatedly take the nearest of its incident vertices if any
     // - if not, we're done.
-    Vertex_handle nearest = nearest_vertex_in_cell(c, p, o);
+
+    // Take the opposite because periodic_locate() returns the offset such that
+    // cell + offset contains 'p' but here we need to move 'p'
+    query_offset = this->combine_offsets(Offset(), -query_offset);
+
+    Vertex_handle nearest = nearest_vertex_in_cell(c, p, query_offset);
+    Offset offset_of_nearest = get_min_dist_offset(p, query_offset, nearest);
+
+#ifdef CGAL_PERIODIC_DEBUG_NEAREST_POWER_VERTEX
+    std::cout << "nearest vertex in cell : " << &*nearest << " : " << nearest->point() << std::endl;
+    std::cout << "offset_of_nearest: " << offset_of_nearest << std::endl;
+#endif
+
     std::vector<Vertex_handle> vs;
     vs.reserve(32);
     while(true)
     {
       Vertex_handle tmp = nearest;
-      Offset tmp_off = get_min_dist_offset(p, o, tmp);
+#ifdef CGAL_PERIODIC_DEBUG_NEAREST_POWER_VERTEX
+      std::cout << "tmp set to : " << &*nearest << " : " << nearest->point()
+                << " || offset: " << nearest->offset() << std::endl;
+#endif
+
       adjacent_vertices(nearest, std::back_inserter(vs));
-      for(typename std::vector<Vertex_handle>::const_iterator vsit = vs.begin(); vsit != vs.end(); ++vsit)
-        tmp = (compare_distance(p, tmp->point(), (*vsit)->point(),
-                                o, tmp_off, get_min_dist_offset(p, o, *vsit))
-                 == SMALLER) ? tmp : *vsit;
+      for(typename std::vector<Vertex_handle>::const_iterator vsit = vs.begin();
+          vsit != vs.end(); ++vsit)
+      {
+        // Can happen in 27-sheeted triangulations composed of few points
+        if((*vsit)->point() == nearest->point())
+          continue;
+
+        const Offset min_dist_offset = get_min_dist_offset(p, query_offset, *vsit);
+        if(compare_distance(p, (*vsit)->point(), tmp->point(),
+                            query_offset, min_dist_offset, offset_of_nearest) == SMALLER)
+        {
+          tmp = *vsit;
+          offset_of_nearest = min_dist_offset;
+#ifdef CGAL_PERIODIC_DEBUG_NEAREST_POWER_VERTEX
+          std::cout << " Closer adjacent vertex: " << &*tmp << " : " << tmp->point()
+                    << " || offset " << offset_of_nearest << std::endl;
+#endif
+        }
+      }
+
       if(tmp == nearest)
         break;
+
       vs.clear();
       nearest = tmp;
     }
@@ -834,7 +893,7 @@ public:
     return true;
   }
 
-  bool is_Gabriel(const Facet& f)const
+  bool is_Gabriel(const Facet& f) const
   {
     return is_Gabriel(f.first, f.second);
   }
@@ -846,8 +905,20 @@ public:
 
   bool is_Gabriel(Vertex_handle v) const
   {
-    return nearest_power_vertex(
-             geom_traits().construct_point_3_object()(v->point()), v->cell()) == v;
+    typename Geom_traits::Power_side_of_bounded_power_sphere_3
+      side_of_bounded_orthogonal_sphere =
+      geom_traits().power_side_of_bounded_power_sphere_3_object();
+
+    const Bare_point& bp = geom_traits().construct_point_3_object()(v->point());
+
+    Vertex_handle nearest_v = nearest_power_vertex(bp, v->cell());
+
+    // Need to find the offset such that power distance v->point()
+    // to nearest_v->point() is minimum
+    Offset nearest_v_off = get_min_dist_offset_general(bp, nearest_v);
+
+    return (side_of_bounded_orthogonal_sphere(
+              v->point(), nearest_v->point(), Offset(), nearest_v_off) != CGAL::ON_BOUNDED_SIDE);
   }
 
   Offset get_min_dist_offset(const Bare_point& p, const Offset & o,
@@ -878,6 +949,36 @@ public:
     return combine_offsets(mdo,min_off);
   }
 
+  // In this version, `p` must be in the domain, and we check all possible
+  // offsets to find the minimum
+  Offset get_min_dist_offset_general(const Bare_point& p, const Vertex_handle vh) const
+  {
+    CGAL_triangulation_precondition(p.x() < domain().xmax());
+    CGAL_triangulation_precondition(p.y() < domain().ymax());
+    CGAL_triangulation_precondition(p.z() < domain().zmax());
+    CGAL_triangulation_precondition(p.x() >= domain().xmin());
+    CGAL_triangulation_precondition(p.y() >= domain().ymin());
+    CGAL_triangulation_precondition(p.z() >= domain().zmin());
+
+    Offset min_off = Offset(0,0,0);
+
+    for(int i=-1; i<=1; ++i) {
+      for(int j=-1; j<=1; ++j) {
+        for(int k=-1; k<=1; ++k)
+        {
+          if(i==0 && j==0 && k==0)
+            continue;
+
+          Offset loc_off(i, j, k);
+          if(compare_distance(p, vh->point(), vh->point(), Offset(), min_off, loc_off) == LARGER)
+            min_off = loc_off;
+        }
+      }
+    }
+
+    return min_off;
+  }
+
   Vertex_handle nearest_vertex_in_cell(const Cell_handle& c, const Bare_point& p,
                                        const Offset & o) const
   {
diff --git a/Periodic_3_triangulation_3/test/Periodic_3_triangulation_3/test_periodic_3_regular_triangulation_3.cpp b/Periodic_3_triangulation_3/test/Periodic_3_triangulation_3/test_periodic_3_regular_triangulation_3.cpp
index d7555237f02..95600fca004 100644
--- a/Periodic_3_triangulation_3/test/Periodic_3_triangulation_3/test_periodic_3_regular_triangulation_3.cpp
+++ b/Periodic_3_triangulation_3/test/Periodic_3_triangulation_3/test_periodic_3_regular_triangulation_3.cpp
@@ -42,6 +42,7 @@ public:
   typedef typename P3RT3::Facet                               Facet;
   typedef typename P3RT3::Cell                                Cell;
   typedef typename P3RT3::Vertex_iterator                     Vertex_iterator;
+  typedef typename P3RT3::Unique_vertex_iterator              Unique_vertex_iterator;
   typedef typename P3RT3::Cell_iterator                       Cell_iterator;
   typedef typename P3RT3::Segment                             Segment;
   typedef typename P3RT3::Triangle                            Triangle;
@@ -77,6 +78,100 @@ public:
     assert(p3rt3.is_valid());
   }
 
+  static void test_is_gabriel()
+  {
+    std::cout << "--- test is_gabriel" << std::endl;
+
+    P3RT3 p3rt3;
+
+    typename P3RT3::Geom_traits::Power_side_of_bounded_power_sphere_3
+      side_of_bounded_power_sphere =
+      p3rt3.geom_traits().power_side_of_bounded_power_sphere_3_object();
+
+    Weighted_point_3 t(Point_3(0.5,0.5,0.45), 0.01);
+    Weighted_point_3 s(Point_3(0.5,0.5,0.49), 0.006);
+    Weighted_point_3 q(Point_3(0.5,0.5,0.5), 0.015);
+    Weighted_point_3 p(Point_3(0.95,0.95,0.96), 0.001);
+    Weighted_point_3 r(Point_3(0.01,0.008,0.01), 0.002);
+    p3rt3.insert(p);
+    p3rt3.insert(q);
+    p3rt3.insert(r);
+    p3rt3.insert(s);
+    p3rt3.insert(t);
+
+    assert(p3rt3.is_valid());
+
+    std::cout << "p3rt3.number_of_vertices() " << p3rt3.number_of_vertices() << std::endl;
+    assert(p3rt3.number_of_vertices() == 4);
+    assert(std::distance(p3rt3.unique_vertices_begin(), p3rt3.unique_vertices_end()) == 4);
+    assert(p3rt3.number_of_stored_vertices() == 108);
+
+    for(Unique_vertex_iterator iter = p3rt3.unique_vertices_begin(), end_iter = p3rt3.unique_vertices_end(); iter != end_iter; ++iter)
+    {
+      Vertex_handle vh((Vertex_iterator(iter)));
+      std::cout << p3rt3.is_Gabriel(vh) << std::endl;
+      if(p3rt3.is_Gabriel(vh))
+      {
+        for(Unique_vertex_iterator iter2 = p3rt3.unique_vertices_begin(), end_iter2 = p3rt3.unique_vertices_end(); iter2 != end_iter2; ++iter2)
+        {
+          Vertex_handle vh2((Vertex_iterator(iter2)));
+
+          if(vh2->point() == vh->point())
+          {
+            assert(p3rt3.is_Gabriel(vh2)); // consistency check
+          }
+          else
+          {
+            // Check that w/e the offset, the power distance is positive
+            for(int i = -1; i < 2; ++i) {
+              for(int j = -1; j < 2; ++j) {
+                for(int k = -1; k < 2; ++k)
+                {
+                  const Offset off(i, j, k);
+//                  std::cout << "power distance: " << p3rt3.geom_traits().compute_power_product_3_object()(
+//                                 Weighted_point_3(vh->point().point(), vh->point().weight()),
+//                                 p3rt3.geom_traits().construct_weighted_point_3_object()(vh2->point(), off)) << std::endl;
+                  if(!(side_of_bounded_power_sphere(vh->point(), vh2->point(),
+                                                      Offset(), off) != CGAL::ON_BOUNDED_SIDE))
+                  {
+                    assert(false);
+                  }
+                }
+              }
+            }
+          }
+        }
+      }
+      else
+      {
+        bool found = false;
+        for(Vertex_iterator iter2 = p3rt3.vertices_begin(), end_iter2 = p3rt3.vertices_end(); iter2 != end_iter2; ++iter2)
+        {
+          Vertex_handle vh2 = iter2;
+          if(vh2->point() == vh->point())
+          {
+            assert(!p3rt3.is_Gabriel(vh2)); // consistency check
+          }
+          else
+          {
+            for(int i = -1; i < 2; ++i) {
+              for(int j = -1; j < 2; ++j) {
+                for(int k = -1; k < 2; ++k)
+                {
+                  const Offset off = vh->offset() + Offset(i, j, k);
+                  if(!(side_of_bounded_power_sphere(vh->point(), vh2->point(),
+                                                    vh->offset(), off) != CGAL::ON_BOUNDED_SIDE))
+                    found = true;
+                }
+              }
+            }
+          }
+        } // iter2
+        assert(found); // must have found a point that is in the smallest orthogonal power sphere
+      } // is_gabriel(vh)
+    }
+  }
+
   static void test_insert_1 ()
   {
     std::cout << "--- test_insert_1" << std::endl;
@@ -721,6 +816,7 @@ public:
 
   static void test ()
   {
+    test_is_gabriel();
     test_find_conflicts();
     test_insert_range(800, 7);
     test_construction_and_insert_range(800, 7);
@@ -736,7 +832,6 @@ public:
     test_insert_two_points_with_the_same_position();
     test_remove();
     test_27_to_1_sheeted_covering();
-    //////    Iso_cuboid unitaire ->  0 <= weight < 0.015625
     test_insert_rnd_as_delaunay(100, 0.);
     test_insert_rnd_as_delaunay(100, 0.01);
   }
diff --git a/Triangulation_3/doc/Triangulation_3/Concepts/DelaunayTriangulationTraits_3.h b/Triangulation_3/doc/Triangulation_3/Concepts/DelaunayTriangulationTraits_3.h
index a92e75955bc..f7812c32d9f 100644
--- a/Triangulation_3/doc/Triangulation_3/Concepts/DelaunayTriangulationTraits_3.h
+++ b/Triangulation_3/doc/Triangulation_3/Concepts/DelaunayTriangulationTraits_3.h
@@ -81,6 +81,29 @@ It is only needed when using the `Fast_location` policy or the
 typedef unspecified_type Compare_distance_3; 
 
 
+/// @}
+
+/// \name
+/// When `is_Gabriel` functions are used, the traits class must
+/// in addition provide the following predicate object:
+/// @{
+
+/*!
+A predicate object that must provide the function operators
+
+`Bounded_side operator()(Point_3 p, Point_3 q, Point_3 t)`,
+
+which returns the position of the point `t` relative to the sphere
+that has `pq` as its diameter,
+
+`Bounded_side operator()(Point_3 p, Point_3 q, Point_3 r, Point_3 t)`,
+
+which returns the position of the point `t` relative to the sphere
+passing through `p, q`, and `r` and whose center is in the
+plane defined by these three points.
+*/
+typedef unspecified_type Side_of_bounded_sphere_3;
+
 /// @}
 
 /*! \name
@@ -166,8 +189,6 @@ When using the `Fast_location` policy or the `CGAL::Delaunay_triangulation_3::ne
 */ 
 Compare_distance_3 compare_distance_3_object(); 
 
-
-
 /// @}
 
 /*! \name
diff --git a/Triangulation_3/doc/Triangulation_3/Concepts/RegularTriangulationTraits_3.h b/Triangulation_3/doc/Triangulation_3/Concepts/RegularTriangulationTraits_3.h
index bf16e8b37ef..dd6c787c8a5 100644
--- a/Triangulation_3/doc/Triangulation_3/Concepts/RegularTriangulationTraits_3.h
+++ b/Triangulation_3/doc/Triangulation_3/Concepts/RegularTriangulationTraits_3.h
@@ -210,6 +210,34 @@ typedef unspecified_type Construct_ray_3;
 
 /// @} 
 
+/// \name
+/// When `is_Gabriel` functions are used, the traits class must
+/// in addition provide the following predicate object:
+/// @{
+
+/*!
+A predicate object that must provide the function operators
+
+`Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 t)`,
+
+which returns the sign of the power test of `t` with respect to the smallest
+sphere orthogonal to `p` (which is the sphere with center `p` and squared
+radius `-w_p` with `w_p` the weight of `p`),
+
+`Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 t)`,
+
+which returns the sign of the power test of `t` with respect to the smallest
+sphere orthogonal to `p` and `q`,
+
+`Bounded_side operator()(Weighted_point_3 p, Weighted_point_3 q, Weighted_point_3 r, Weighted_point_3 t)`,
+
+which returns the sign of the power test of `t` with respect to the smallest
+sphere orthogonal to `p`, `q`, and `r`.
+*/
+typedef unspecified_type Power_side_of_bounded_power_sphere_3;
+
+/// @}
+
 /// \name Operations 
 /// @{
 
diff --git a/Triangulation_3/include/CGAL/Regular_triangulation_3.h b/Triangulation_3/include/CGAL/Regular_triangulation_3.h
index d2a06431086..7287944d586 100644
--- a/Triangulation_3/include/CGAL/Regular_triangulation_3.h
+++ b/Triangulation_3/include/CGAL/Regular_triangulation_3.h
@@ -2180,8 +2180,16 @@ namespace CGAL {
     Regular_triangulation_3<Gt,Tds,Lds>::
     is_Gabriel(Vertex_handle v) const
   {
-    return nearest_power_vertex(
-             geom_traits().construct_point_3_object()(v->point()), v->cell()) == v;
+    typename Geom_traits::Power_side_of_bounded_power_sphere_3
+      side_of_bounded_orthogonal_sphere =
+      geom_traits().power_side_of_bounded_power_sphere_3_object();
+
+    Vertex_handle nearest_v =
+      nearest_power_vertex(geom_traits().construct_point_3_object()(v->point()),
+                           v->cell());
+
+    return (side_of_bounded_orthogonal_sphere(v->point(), nearest_v->point())
+              != CGAL::ON_BOUNDED_SIDE);
   }
 
   // Returns