diff --git a/BGL/include/CGAL/boost/graph/properties_OpenMesh.h b/BGL/include/CGAL/boost/graph/properties_OpenMesh.h
index 5b252c00ee1..9261d01b7f9 100644
--- a/BGL/include/CGAL/boost/graph/properties_OpenMesh.h
+++ b/BGL/include/CGAL/boost/graph/properties_OpenMesh.h
@@ -13,6 +13,8 @@
 #include <CGAL/assertions.h>
 #include <CGAL/boost/graph/properties.h>
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
+#include <CGAL/Kernel_traits.h>
+
 #include <boost/mpl/if.hpp>
 
 #ifndef OPEN_MESH_CLASS
@@ -231,8 +233,8 @@ public:
 #if defined(CGAL_USE_OM_POINTS)
     const_cast<OpenMesh&>(*pm.sm_).set_point(v,p);
 #else
-    const_cast<OpenMesh&>(*pm.sm_).set_point
-      (v, typename OpenMesh::Point((float)p[0], (float)p[1], (float)p[2]));
+    typedef typename CGAL::Kernel_traits<typename OpenMesh::Point>::type FT;
+    const_cast<OpenMesh&>(*pm.sm_).set_point(v, typename OpenMesh::Point(FT(p[0]), FT(p[1]), FT(p[2])));
 #endif
   }
 
diff --git a/Linear_cell_complex/benchmark/Linear_cell_complex_2/openmesh_performance.h b/Linear_cell_complex/benchmark/Linear_cell_complex_2/openmesh_performance.h
index 71ff72ea813..afdb700c75b 100644
--- a/Linear_cell_complex/benchmark/Linear_cell_complex_2/openmesh_performance.h
+++ b/Linear_cell_complex/benchmark/Linear_cell_complex_2/openmesh_performance.h
@@ -22,19 +22,10 @@ public:
     mesh.request_face_normals();
   }
 
-
 private:
-
-  struct MyTraits : public OpenMesh::DefaultTraits
-  {
-    typedef OpenMesh::Vec3d Point;
-    typedef OpenMesh::Vec3d Normal;
-  };
-
-  typedef OpenMesh::TriMesh_ArrayKernelT<MyTraits> Mesh;
+  typedef OpenMesh::TriMesh_ArrayKernelT<OpenMesh::DefaultTraitsDouble> Mesh;
   Mesh mesh;
 
-
 private:
   void display_info()
   {
diff --git a/Surface_mesh_deformation/test/Surface_mesh_deformation/Cactus_deformation_session_OpenMesh.cpp b/Surface_mesh_deformation/test/Surface_mesh_deformation/Cactus_deformation_session_OpenMesh.cpp
index fe1aded57c0..06817a78b71 100644
--- a/Surface_mesh_deformation/test/Surface_mesh_deformation/Cactus_deformation_session_OpenMesh.cpp
+++ b/Surface_mesh_deformation/test/Surface_mesh_deformation/Cactus_deformation_session_OpenMesh.cpp
@@ -13,17 +13,10 @@
 
 #include <CGAL/Timer.h>
 
-struct DoubleTraits : public OpenMesh::DefaultTraits
-{
-  typedef OpenMesh::Vec3d Point;
-  typedef OpenMesh::Vec3d Normal;
-};
-
-
-typedef OpenMesh::PolyMesh_ArrayKernelT<DoubleTraits>               Mesh;
-typedef Mesh::Point                                                 Point;
-typedef boost::graph_traits<Mesh>::vertex_descriptor    vertex_descriptor;
-typedef boost::graph_traits<Mesh>::vertex_iterator        vertex_iterator;
+typedef OpenMesh::PolyMesh_ArrayKernelT<OpenMesh::DefaultTraitsDouble> Mesh;
+typedef Mesh::Point                                                    Point;
+typedef boost::graph_traits<Mesh>::vertex_descriptor                   vertex_descriptor;
+typedef boost::graph_traits<Mesh>::vertex_iterator                     vertex_iterator;
 
 typedef CGAL::Surface_mesh_deformation<Mesh, CGAL::Default, CGAL::Default, CGAL::ORIGINAL_ARAP>  Deform_mesh_arap;
 typedef CGAL::Surface_mesh_deformation<Mesh, CGAL::Default, CGAL::Default, CGAL::SPOKES_AND_RIMS> Deform_mesh_spoke;
diff --git a/Weights/include/CGAL/Weights/cotangent_weights.h b/Weights/include/CGAL/Weights/cotangent_weights.h
index ba128a98502..dc070c5c493 100644
--- a/Weights/include/CGAL/Weights/cotangent_weights.h
+++ b/Weights/include/CGAL/Weights/cotangent_weights.h
@@ -193,6 +193,9 @@ public:
 // Surface_mesh_deformation -> Surface_mesh_deformation.h (default version)
 // Surface_mesh_parameterizer -> Orbifold_Tutte_parameterizer_3.h (default version)
 // Surface_mesh_skeletonization -> Mean_curvature_flow_skeletonization.h (clamped version)
+//
+// The API is a bit awkward: the template parameters VertexPointMap and GeomTraits
+// are only meaningful in the API that calls the operator with a single parameter.
 template<typename PolygonMesh,
          typename VertexPointMap = typename GetVertexPointMap<PolygonMesh>::type,
          typename GeomTraits = typename Kernel_traits<
@@ -202,8 +205,6 @@ class Cotangent_weight
   using vertex_descriptor = typename boost::graph_traits<PolygonMesh>::vertex_descriptor;
   using halfedge_descriptor = typename boost::graph_traits<PolygonMesh>::halfedge_descriptor;
 
-  using FT = typename GeomTraits::FT;
-
 private:
   // These class members are used only when the constructor initializing them
   // is used, but Surface_mesh_deformation has its own weight API locked
@@ -224,12 +225,14 @@ public:
   // Common API whether mesh/vpm/traits are initialized in the constructor,
   // or passed in the operator()
   template <typename VPM, typename GT>
-  FT operator()(const halfedge_descriptor he,
-                const PolygonMesh& pmesh,
-                const VPM vpm,
-                const GT& traits) const
+  typename GT::FT
+  operator()(const halfedge_descriptor he,
+             const PolygonMesh& pmesh,
+             const VPM vpm,
+             const GT& traits) const
   {
     using Point_ref = typename boost::property_traits<VPM>::reference;
+    using FT = typename GT::FT;
 
     if(is_border(he, pmesh))
       return FT{0};
@@ -265,9 +268,10 @@ public:
 
   // That is the API called by Surface_mesh_deformation
   template <typename VPM>
-  FT operator()(const halfedge_descriptor he,
-                const PolygonMesh& pmesh,
-                const VPM vpm) const
+  auto // kernel_traits<VPM::value_type>::type::FT
+  operator()(const halfedge_descriptor he,
+             const PolygonMesh& pmesh,
+             const VPM vpm) const
   {
     using Point = typename boost::property_traits<VPM>::value_type;
     using GT = typename Kernel_traits<Point>::type;
@@ -286,7 +290,7 @@ public:
       m_bound_from_below(bound_from_below)
   { }
 
-  FT operator()(const halfedge_descriptor he) const
+  typename GeomTraits::FT operator()(const halfedge_descriptor he) const
   {
     CGAL_precondition(m_pmesh_ptr != nullptr);
     return this->operator()(he, *m_pmesh_ptr, m_vpm, m_traits);