diff --git a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Iterative_authalic_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Iterative_authalic_parameterizer_3.h
index d8373eb54ba..d0e20ad5a0b 100644
--- a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Iterative_authalic_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Iterative_authalic_parameterizer_3.h
@@ -33,7 +33,9 @@
 #include <CGAL/Dynamic_property_map.h>
 #include <CGAL/Polygon_mesh_processing/measure.h>
 #include <CGAL/Polygon_mesh_processing/connected_components.h>
-#include <CGAL/Weights/internal/tools.h>
+// #include <CGAL/Weights/authalic_weights.h>
+#include <CGAL/Weights/cotangent_weights.h>
+#include <CGAL/Weights/mean_value_weights.h>
 #include <CGAL/number_type_config.h>
 
 #if defined(CGAL_EIGEN3_ENABLED)
@@ -721,8 +723,7 @@ private:
                                               VertexIndexMap& vimap) const
   {
     auto vpm = get_const_property_map(CGAL::vertex_point, tmesh);
-    const CGAL::Weights::internal::Mean_value_weight_wrapper<Triangle_mesh, decltype(vpm)>
-      compute_mvc(tmesh, vpm);
+    const CGAL::Weights::Mean_value_weight<Triangle_mesh, decltype(vpm)> compute_mvc(tmesh, vpm);
 
     const int i = get(vimap, v);
 
diff --git a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/MVC_post_processor_3.h b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/MVC_post_processor_3.h
index 87e54a7487e..a8053c2a9bd 100644
--- a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/MVC_post_processor_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/MVC_post_processor_3.h
@@ -21,7 +21,7 @@
 #include <CGAL/Surface_mesh_parameterization/Two_vertices_parameterizer_3.h>
 #include <CGAL/Surface_mesh_parameterization/parameterize.h>
 
-#include <CGAL/Weights/internal/tools.h>
+#include <CGAL/Weights/tangent_weights.h>
 #include <CGAL/Constrained_triangulation_2.h>
 #include <CGAL/Constrained_Delaunay_triangulation_2.h>
 #include <CGAL/Polygon_mesh_processing/border.h>
@@ -393,7 +393,7 @@ private:
     const Point_2& p = pk;
     const Point_2& q = pi;
     const Point_2& r = pj;
-    const CGAL::Weights::internal::Tangent_weight_wrapper<NT> tangent_weight(p, q, r);
+    const CGAL::Weights::Tangent_weight<NT> tangent_weight(p, q, r);
 
     // Set w_ij in matrix
     const NT w_ij = tangent_weight.get_w_r();
diff --git a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h
index e524fca930c..3f8337d0660 100644
--- a/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Surface_mesh_parameterization/Orbifold_Tutte_parameterizer_3.h
@@ -22,7 +22,8 @@
 #include <CGAL/Surface_mesh_parameterization/orbifold_enums.h>
 #include <CGAL/Surface_mesh_parameterization/Error_code.h>
 #include <CGAL/Surface_mesh_parameterization/orbifold_shortest_path.h>
-#include <CGAL/Weights/internal/tools.h>
+#include <CGAL/Weights/tangent_weights.h>
+#include <CGAL/Weights/cotangent_weights.h>
 #include <CGAL/assertions.h>
 
 #include <CGAL/assertions.h>
@@ -722,7 +723,7 @@ private:
     const Point_3& p = pk;
     const Point_3& q = pi;
     const Point_3& r = pj;
-    const CGAL::Weights::internal::Tangent_weight_wrapper<NT> tangent_weight(p, q, r);
+    const CGAL::Weights::Tangent_weight<NT> tangent_weight(p, q, r);
 
     // Set w_ij in matrix
     const NT w_ij = tangent_weight.get_w_r();
@@ -783,8 +784,7 @@ private:
       const int j = get(vimap, vj);
 
       if (i > j) continue;
-      const CGAL::Weights::internal::
-        Cotangent_weight_wrapper<SeamMesh> cotangent_weight;
+      const CGAL::Weights::Cotangent_weight<SeamMesh> cotangent_weight;
       const NT w_ij = NT(2) * cotangent_weight(hd, mesh, pmap);
 
       // ij