diff --git a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_adding_handles.cpp b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_adding_handles.cpp
index 308346bd9a0..4cdb60fbea8 100644
--- a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_adding_handles.cpp
+++ b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_adding_handles.cpp
@@ -1,6 +1,7 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
-#include <CGAL/Periodic_2_triangulation_filtered_traits_2.h>
+
 #include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 #include <CGAL/Triangulation_vertex_base_2.h>
 
 template < class GT, class Vb >
@@ -32,7 +33,7 @@ public:
 
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
-typedef CGAL::Periodic_2_triangulation_filtered_traits_2<K> GT;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K> GT;
 
 typedef CGAL::Periodic_2_triangulation_vertex_base_2<GT>    VbDS;
 typedef My_vertex_base<GT, VbDS>                             Vb;
diff --git a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_colored_vertices.cpp b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_colored_vertices.cpp
index 13e9a7e0efc..63097b27e00 100644
--- a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_colored_vertices.cpp
+++ b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_colored_vertices.cpp
@@ -1,11 +1,12 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
-#include <CGAL/Periodic_2_triangulation_filtered_traits_2.h>
+
 #include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 #include <CGAL/Triangulation_vertex_base_with_info_2.h>
 #include <CGAL/IO/Color.h>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
-typedef CGAL::Periodic_2_triangulation_filtered_traits_2<K> GT;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K> GT;
 
 typedef CGAL::Periodic_2_triangulation_vertex_base_2<GT>    Vb;
 typedef CGAL::Triangulation_vertex_base_with_info_2<CGAL::Color, GT, Vb> VbInfo;
diff --git a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_covering.cpp b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_covering.cpp
index 08b7ce64e60..2ecabe71d0e 100644
--- a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_covering.cpp
+++ b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_covering.cpp
@@ -1,12 +1,13 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
-#include <CGAL/Periodic_2_triangulation_traits_2.h>
+
 #include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 
 #include <iostream>
 #include <vector>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
-typedef CGAL::Periodic_2_triangulation_traits_2<K> GT;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K> GT;
 
 typedef CGAL::Periodic_2_Delaunay_triangulation_2<GT> PDT;
 
diff --git a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_find_conflicts.cpp b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_find_conflicts.cpp
index 36d4dec33cc..2061ba836e0 100644
--- a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_find_conflicts.cpp
+++ b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_find_conflicts.cpp
@@ -1,13 +1,14 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
-#include <CGAL/Periodic_2_triangulation_filtered_traits_2.h>
+
 #include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 #include <CGAL/point_generators_2.h>
 
 #include <vector>
 #include <cassert>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
-typedef CGAL::Periodic_2_triangulation_filtered_traits_2<K> GT;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K> GT;
 
 typedef CGAL::Periodic_2_Delaunay_triangulation_2<GT> Delaunay;
 typedef Delaunay::Point                               Point;
diff --git a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_geometric_access.cpp b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_geometric_access.cpp
index 4baf8d9995a..4bde2f63e79 100644
--- a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_geometric_access.cpp
+++ b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_geometric_access.cpp
@@ -1,9 +1,10 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
-#include <CGAL/Periodic_2_triangulation_traits_2.h>
+
 #include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
-typedef CGAL::Periodic_2_triangulation_traits_2<K>          PK;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K> PK;
 typedef CGAL::Periodic_2_Delaunay_triangulation_2<PK>       P2DT2;
 
 typedef PK::Point_2        Point;
diff --git a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_pair_iterator_2.cpp b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_pair_iterator_2.cpp
index 659ad936a58..81f1e1aad05 100644
--- a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_pair_iterator_2.cpp
+++ b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_pair_iterator_2.cpp
@@ -1,11 +1,14 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
+
 #include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
-#include <CGAL/Periodic_2_triangulation_traits_2.h>
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 #include <CGAL/Triangulation_vertex_base_with_info_2.h>
+
+#include <iostream>
 #include <vector>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel         K;
-typedef CGAL::Periodic_2_triangulation_traits_2<K>                  Gt;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K>         Gt;
 typedef CGAL::Triangulation_vertex_base_with_info_2<unsigned, Gt>   Vb;
 typedef CGAL::Periodic_2_triangulation_face_base_2<Gt>              Fb;
 typedef CGAL::Triangulation_data_structure_2<Vb, Fb>                 Tds;
diff --git a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_transform_iterator_2.cpp b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_transform_iterator_2.cpp
index ddedbfbf394..e2bd3dccb72 100644
--- a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_transform_iterator_2.cpp
+++ b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_transform_iterator_2.cpp
@@ -1,11 +1,14 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
+
 #include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
-#include <CGAL/Periodic_2_triangulation_traits_2.h>
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 #include <CGAL/Triangulation_vertex_base_with_info_2.h>
+
+#include <iostream>
 #include <vector>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel         K;
-typedef CGAL::Periodic_2_triangulation_traits_2<K>                  Gt;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K>         Gt;
 typedef CGAL::Triangulation_vertex_base_with_info_2<unsigned, Gt>   Vb;
 typedef CGAL::Periodic_2_triangulation_face_base_2<Gt>              Fb;
 typedef CGAL::Triangulation_data_structure_2<Vb, Fb>                 Tds;
diff --git a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_zip_iterator_2.cpp b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_zip_iterator_2.cpp
index 2f1451c29bc..a5a844ec2c8 100644
--- a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_zip_iterator_2.cpp
+++ b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_info_insert_with_zip_iterator_2.cpp
@@ -1,11 +1,14 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
+
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 #include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
-#include <CGAL/Periodic_2_triangulation_traits_2.h>
 #include <CGAL/Triangulation_vertex_base_with_info_2.h>
+
+#include <iostream>
 #include <vector>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel         K;
-typedef CGAL::Periodic_2_triangulation_traits_2<K>                  Gt;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K>         Gt;
 typedef CGAL::Triangulation_vertex_base_with_info_2<unsigned, Gt>   Vb;
 typedef CGAL::Periodic_2_triangulation_face_base_2<Gt>              Fb;
 typedef CGAL::Triangulation_data_structure_2<Vb, Fb>                 Tds;
diff --git a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_large_point_set.cpp b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_large_point_set.cpp
index 682a4c0dedf..ad4ed28c0a6 100644
--- a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_large_point_set.cpp
+++ b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_large_point_set.cpp
@@ -1,5 +1,6 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
-#include <CGAL/Periodic_2_triangulation_traits_2.h>
+
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 #include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
 
 #include <CGAL/Random.h>
@@ -10,8 +11,7 @@
 #include <vector>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
-typedef CGAL::Periodic_2_triangulation_traits_2<K> GT;
-
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K> GT;
 typedef CGAL::Periodic_2_Delaunay_triangulation_2<GT> PDT;
 
 typedef PDT::Point          Point;
diff --git a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_simple_example.cpp b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_simple_example.cpp
index 073e6b2a97e..3117d39c64b 100644
--- a/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_simple_example.cpp
+++ b/Periodic_2_triangulation_2/examples/Periodic_2_triangulation_2/p2t2_simple_example.cpp
@@ -1,6 +1,7 @@
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
-#include <CGAL/Periodic_2_triangulation_traits_2.h>
+
 #include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 
 #include <fstream>
 #include <cassert>
@@ -8,9 +9,9 @@
 #include <vector>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
-typedef CGAL::Periodic_2_triangulation_traits_2<K> GT;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K> GT;
 
-typedef CGAL::Periodic_2_Delaunay_triangulation_2<GT> PDT;
+typedef CGAL::Periodic_2_Delaunay_triangulation_2<GT>       PDT;
 
 typedef PDT::Face_handle       Face_handle;
 typedef PDT::Vertex_handle     Vertex_handle;
diff --git a/Periodic_2_triangulation_2/test/Periodic_2_triangulation_2/test_p2t2_interface_triang_2.cpp b/Periodic_2_triangulation_2/test/Periodic_2_triangulation_2/test_p2t2_interface_triang_2.cpp
index 52beb24d672..2f685935e08 100644
--- a/Periodic_2_triangulation_2/test/Periodic_2_triangulation_2/test_p2t2_interface_triang_2.cpp
+++ b/Periodic_2_triangulation_2/test/Periodic_2_triangulation_2/test_p2t2_interface_triang_2.cpp
@@ -2,26 +2,28 @@
 
 #include "types.h"
 #include "interface_test.h"
-#include <CGAL/Periodic_2_triangulation_2.h>
 
+#include <CGAL/Periodic_2_triangulation_2.h>
+#include <CGAL/Periodic_2_Delaunay_triangulation_2.h>
+#include <CGAL/Periodic_2_Delaunay_triangulation_traits_2.h>
 
 #include <CGAL/Exact_predicates_exact_constructions_kernel.h>
 typedef CGAL::Exact_predicates_exact_constructions_kernel            K2;
-typedef CGAL::Periodic_2_triangulation_traits_2<K2>                  PTT2;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K2>         PTT2;
 // Explicit instantiation of the whole class :
 template class CGAL::Periodic_2_Delaunay_triangulation_2<PTT2>;
 
 #include <CGAL/MP_Float.h>
 #include <CGAL/Simple_homogeneous.h>
 typedef CGAL::Simple_homogeneous<CGAL::MP_Float>                     K3;
-typedef CGAL::Periodic_2_triangulation_traits_2<K3>                  PTT3;
+typedef CGAL::Periodic_2_Delaunay_triangulation_traits_2<K3>         PTT3;
 // Explicit instantiation of the whole class :
 template class CGAL::Periodic_2_Delaunay_triangulation_2<PTT3>;
 
 int main()
 {
-  typedef Periodic_2_triangulation_2<Gt>              P2T2;
-  typedef Periodic_2_Delaunay_triangulation_2<Gt>     DP2T2;
+  typedef CGAL::Periodic_2_triangulation_2<P2TT>                     P2T2;
+  typedef CGAL::Periodic_2_Delaunay_triangulation_2<P2DTT>           DP2T2;
 
   test<P2T2>(false);
   test<DP2T2>(false);
diff --git a/Periodic_2_triangulation_2/test/Periodic_2_triangulation_2/test_p2t2_test_semi_static_predicates.cpp b/Periodic_2_triangulation_2/test/Periodic_2_triangulation_2/test_p2t2_test_semi_static_predicates.cpp
index ba18d70deb2..8cba96b9154 100644
--- a/Periodic_2_triangulation_2/test/Periodic_2_triangulation_2/test_p2t2_test_semi_static_predicates.cpp
+++ b/Periodic_2_triangulation_2/test/Periodic_2_triangulation_2/test_p2t2_test_semi_static_predicates.cpp
@@ -7,11 +7,11 @@ const int N = 10;
 
 void test_orientation()
 {
-  Gt traits;
-  traits.set_domain(Gt::Iso_rectangle_2(0, 0, 1, 1));
+  P2TT traits;
+  traits.set_domain(Iso_rectangle(0, 0, 1, 1));
 
-  Gt::Offset_2 o0(0, 0);
-  Gt::Offset_2 o1(0, 1);
+  P2TT::Periodic_2_offset_2 o0(0, 0);
+  P2TT::Periodic_2_offset_2 o1(0, 1);
 
   /// Near degenerate points, which cause the predicate to fail if not filtered
   Point p0(0.5 + (0.4999 / N) * 2, 0.5 + (0.4999 / N) * -5);
@@ -54,11 +54,11 @@ void test_orientation()
 
 void test_in_circle()
 {
-  Gt traits;
-  traits.set_domain(Gt::Iso_rectangle_2(0, 0, 1, 1));
+  P2DTT traits;
+  traits.set_domain(Iso_rectangle(0, 0, 1, 1));
 
-  Gt::Offset_2 o0(0, 0);
-  Gt::Offset_2 o1(0, 1);
+  P2DTT::Periodic_2_offset_2 o0(0, 0);
+  P2DTT::Periodic_2_offset_2 o1(0, 1);
 
   /// Near degenerate points, which cause the predicate to fail if not filtered
   /// On the circle with center (0.4999, 0.4999) and radius 5