diff --git a/Surface_mesh_deformation/include/CGAL/Surface_mesh_deformation.h b/Surface_mesh_deformation/include/CGAL/Surface_mesh_deformation.h
index a715accf733..a536e13ddae 100644
--- a/Surface_mesh_deformation/include/CGAL/Surface_mesh_deformation.h
+++ b/Surface_mesh_deformation/include/CGAL/Surface_mesh_deformation.h
@@ -86,8 +86,7 @@ struct Types_selectors;
 template<typename TriangleMesh, typename VertexPointMap>
 struct Types_selectors<TriangleMesh, VertexPointMap, CGAL::SPOKES_AND_RIMS>
 {
-  typedef SC_on_the_fly_pmap<VertexPointMap> Wrapped_VertexPointMap;
-  typedef CGAL::Weights::Single_cotangent_weight<TriangleMesh, Wrapped_VertexPointMap> Weight_calculator;
+  typedef CGAL::Weights::Single_cotangent_weight<TriangleMesh> Weight_calculator;
 
   struct ARAP_visitor
   {
@@ -107,8 +106,7 @@ struct Types_selectors<TriangleMesh, VertexPointMap, CGAL::SPOKES_AND_RIMS>
 template<class TriangleMesh, typename VertexPointMap>
 struct Types_selectors<TriangleMesh, VertexPointMap, CGAL::ORIGINAL_ARAP>
 {
-  typedef SC_on_the_fly_pmap<VertexPointMap> Wrapped_VertexPointMap;
-  typedef CGAL::Weights::Cotangent_weight<TriangleMesh, Wrapped_VertexPointMap> Weight_calculator;
+  typedef CGAL::Weights::Cotangent_weight<TriangleMesh> Weight_calculator;
 
   typedef typename Types_selectors<TriangleMesh, VertexPointMap, CGAL::SPOKES_AND_RIMS>::ARAP_visitor ARAP_visitor;
 };
@@ -116,8 +114,7 @@ struct Types_selectors<TriangleMesh, VertexPointMap, CGAL::ORIGINAL_ARAP>
 template<class TriangleMesh, typename VertexPointMap>
 struct Types_selectors<TriangleMesh, VertexPointMap, CGAL::SRE_ARAP>
 {
-  typedef SC_on_the_fly_pmap<VertexPointMap> Wrapped_VertexPointMap;
-  typedef CGAL::Weights::Cotangent_weight<TriangleMesh, Wrapped_VertexPointMap> Weight_calculator;
+  typedef CGAL::Weights::Cotangent_weight<TriangleMesh> Weight_calculator;
 
   class ARAP_visitor
   {
@@ -384,7 +381,7 @@ public:
                                vertex_index_map,
                                hedge_index_map,
                                vertex_point_map,
-                               Weight_calculator(triangle_mesh, internal::SC_on_the_fly_pmap<Vertex_point_map>(vertex_point_map)))
+                               Weight_calculator())
   { }
 
   Surface_mesh_deformation(Triangle_mesh& triangle_mesh,
@@ -437,9 +434,10 @@ public:
 private:
   void init()
   {
+    typedef internal::SC_on_the_fly_pmap<Vertex_point_map> Wrapper;
     hedge_weight.reserve(num_halfedges(m_triangle_mesh));
     for(halfedge_descriptor he : halfedges(m_triangle_mesh))
-      hedge_weight.push_back(this->weight_calculator(he));
+      hedge_weight.push_back(this->weight_calculator(he, m_triangle_mesh, Wrapper(vertex_point_map)));
 
     arap_visitor.init(m_triangle_mesh, vertex_point_map);
   }
@@ -823,6 +821,7 @@ public:
    */
   void overwrite_initial_geometry()
   {
+    typedef internal::SC_on_the_fly_pmap<Vertex_point_map> Wrapper;
     if(roi.empty()) { return; } // no ROI to overwrite
 
     region_of_solution(); // the roi should be preprocessed since we are using original_position vec
@@ -843,13 +842,13 @@ public:
         std::size_t id_e = id(he);
         if(is_weight_computed[id_e]) { continue; }
 
-        hedge_weight[id_e] = weight_calculator(he);
+        hedge_weight[id_e] = weight_calculator(he, m_triangle_mesh, Wrapper(vertex_point_map));
         is_weight_computed[id_e] = true;
 
         halfedge_descriptor e_opp = opposite(he, m_triangle_mesh);
         std::size_t id_e_opp = id(e_opp);
 
-        hedge_weight[id_e_opp] = weight_calculator(e_opp);
+        hedge_weight[id_e_opp] = weight_calculator(e_opp, m_triangle_mesh, Wrapper(vertex_point_map));
         is_weight_computed[id_e_opp] = true;
       }
     }
diff --git a/Weights/include/CGAL/Weights/cotangent_weights.h b/Weights/include/CGAL/Weights/cotangent_weights.h
index d462c3f5174..ce3471550b5 100644
--- a/Weights/include/CGAL/Weights/cotangent_weights.h
+++ b/Weights/include/CGAL/Weights/cotangent_weights.h
@@ -147,44 +147,38 @@ typename Kernel::FT cotangent_weight(const CGAL::Point_3<Kernel>& p0,
 // For border edges it returns zero.
 // This version is currently used in:
 // Surface_mesh_deformation -> Surface_mesh_deformation.h
-template<typename PolygonMesh,
-         typename VertexPointMap,
-         typename GeomTraits = typename Kernel_traits<
-                                 typename boost::property_traits<VertexPointMap>::value_type>::type>
+template<typename PolygonMesh>
 class Single_cotangent_weight
 {
   using vertex_descriptor = typename boost::graph_traits<PolygonMesh>::vertex_descriptor;
   using halfedge_descriptor = typename boost::graph_traits<PolygonMesh>::halfedge_descriptor;
 
-  using Point_ref = typename boost::property_traits<VertexPointMap>::reference;
-  using FT = typename GeomTraits::FT;
-
-private:
-  const PolygonMesh& m_pmesh;
-  const VertexPointMap m_vpm;
-  const GeomTraits m_traits;
-
 public:
-  Single_cotangent_weight(const PolygonMesh& pmesh,
-                          const VertexPointMap vpm,
-                          const GeomTraits& traits = GeomTraits())
-    : m_pmesh(pmesh), m_vpm(vpm), m_traits(traits)
-  { }
-
-  decltype(auto) operator()(const halfedge_descriptor he) const
+  // Returns the cotangent of the opposite angle of the edge
+  // 0 for border edges (which does not have an opposite angle).
+  template <typename VPM>
+  auto operator()(halfedge_descriptor he,
+                  PolygonMesh& pmesh,
+                  VPM vpm)
   {
-    if (is_border(he, m_pmesh))
+    using Point = typename boost::property_traits<VPM>::value_type;
+    using Point_ref = typename boost::property_traits<VPM>::reference;
+
+    using GeomTraits = typename Kernel_traits<Point>::type;
+    using FT = typename GeomTraits::FT;
+
+    if(is_border(he, pmesh))
       return FT{0};
 
-    const vertex_descriptor v0 = target(he, m_pmesh);
-    const vertex_descriptor v1 = source(he, m_pmesh);
-    const vertex_descriptor v2 = target(next(he, m_pmesh), m_pmesh);
+    const vertex_descriptor v0 = target(he, pmesh);
+    const vertex_descriptor v1 = source(he, pmesh);
+    const vertex_descriptor v2 = target(next(he, pmesh), pmesh);
 
-    const Point_ref p0 = get(m_vpm, v0);
-    const Point_ref p1 = get(m_vpm, v1);
-    const Point_ref p2 = get(m_vpm, v2);
+    const Point_ref p0 = get(vpm, v0);
+    const Point_ref p1 = get(vpm, v1);
+    const Point_ref p2 = get(vpm, v2);
 
-    return cotangent_3(p0, p2, p1, m_traits);
+    return cotangent(p0, p2, p1);
   }
 };
 
@@ -200,7 +194,7 @@ public:
 // Surface_mesh_parameterizer -> Orbifold_Tutte_parameterizer_3.h (default version)
 // Surface_mesh_skeletonization -> Mean_curvature_flow_skeletonization.h (clamped version)
 template<typename PolygonMesh,
-         typename VertexPointMap,
+         typename VertexPointMap = typename GetVertexPointMap<PolygonMesh>::type,
          typename GeomTraits = typename Kernel_traits<
                                  typename boost::property_traits<VertexPointMap>::value_type>::type>
 class Cotangent_weight
@@ -208,11 +202,14 @@ class Cotangent_weight
   using vertex_descriptor = typename boost::graph_traits<PolygonMesh>::vertex_descriptor;
   using halfedge_descriptor = typename boost::graph_traits<PolygonMesh>::halfedge_descriptor;
 
-  using Point_ref = typename boost::property_traits<VertexPointMap>::reference;
   using FT = typename GeomTraits::FT;
 
 private:
-  const PolygonMesh& m_pmesh;
+  // These class members are used only when the constructor initializing them
+  // is used, but Surface_mesh_deformation has its own weight API locked
+  // by the concept SurfaceMeshDeformationWeights.
+  // A bit awkward, but better than duplicating code...
+  PolygonMesh* const* m_pmesh_ptr;
   const VertexPointMap m_vpm;
   const GeomTraits m_traits;
 
@@ -220,39 +217,41 @@ private:
   bool m_bound_from_below;
 
 public:
-  Cotangent_weight(const PolygonMesh& pmesh,
-                   const VertexPointMap vpm,
-                   const GeomTraits& traits = GeomTraits(),
-                   const bool use_clamped_version = false,
-                   const bool bound_from_below = true)
-    : m_pmesh(pmesh), m_vpm(vpm), m_traits(traits),
-      m_use_clamped_version(use_clamped_version),
-      m_bound_from_below(bound_from_below)
+  Cotangent_weight()
+    : m_pmesh_ptr(nullptr), m_vpm(), m_traits(), m_use_clamped_version(false), m_bound_from_below(true)
   { }
 
-  decltype(auto) operator()(const halfedge_descriptor he) const
+  // 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
   {
-    if(is_border(he, m_pmesh))
+    using Point_ref = typename boost::property_traits<VPM>::reference;
+
+    if(is_border(he, pmesh))
       return FT{0};
 
     auto half_weight = [&] (const halfedge_descriptor he) -> FT
     {
-      if(is_border(he, m_pmesh))
+      if(is_border(he, pmesh))
         return FT{0};
 
-      const vertex_descriptor v0 = target(he, m_pmesh);
-      const vertex_descriptor v1 = source(he, m_pmesh);
-      const vertex_descriptor v2 = target(next(he, m_pmesh), m_pmesh);
+      const vertex_descriptor v0 = target(he, pmesh);
+      const vertex_descriptor v1 = source(he, pmesh);
+      const vertex_descriptor v2 = target(next(he, pmesh), pmesh);
 
-      const Point_ref p0 = get(m_vpm, v0);
-      const Point_ref p1 = get(m_vpm, v1);
-      const Point_ref p2 = get(m_vpm, v2);
+      const Point_ref p0 = get(vpm, v0);
+      const Point_ref p1 = get(vpm, v1);
+      const Point_ref p2 = get(vpm, v2);
 
       FT weight = 0;
       if (m_use_clamped_version)
-        weight = cotangent_3_clamped(p1, p2, p0, m_traits);
+        weight = cotangent_3_clamped(p1, p2, p0, traits);
       else
-        weight = cotangent_3(p1, p2, p0, m_traits);
+        weight = cotangent_3(p1, p2, p0, traits);
 
       if(m_bound_from_below)
         weight = (CGAL::max)(FT(0), weight);
@@ -260,9 +259,38 @@ public:
       return weight / FT(2);
     };
 
-    FT weight = half_weight(he) + half_weight(opposite(he, m_pmesh));
+    FT weight = half_weight(he) + half_weight(opposite(he, pmesh));
     return weight;
   }
+
+  // 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
+  {
+    using Point = typename boost::property_traits<VPM>::value_type;
+    using GT = typename Kernel_traits<Point>::type;
+    return this->operator()(he, pmesh, vpm, GT());
+  }
+
+public:
+  // This is the "normal" API: give all info to the constructor, and operator()(halfedge)
+  Cotangent_weight(const PolygonMesh& pmesh,
+                   const VertexPointMap vpm,
+                   const GeomTraits& traits = GeomTraits(),
+                   const bool use_clamped_version = false,
+                   const bool bound_from_below = true)
+    : m_pmesh_ptr(&pmesh), m_vpm(vpm), m_traits(traits),
+      m_use_clamped_version(use_clamped_version),
+      m_bound_from_below(bound_from_below)
+  { }
+
+  FT operator()(const halfedge_descriptor he) const
+  {
+    CGAL_precondition(m_pmesh_ptr != nullptr);
+    return this->operator()(he, *m_pmesh_ptr, m_vpm, m_traits);
+  }
 };
 
 // Undocumented cotangent weight class.
diff --git a/Weights/include/CGAL/Weights/internal/pmp_weights_deprecated.h b/Weights/include/CGAL/Weights/internal/pmp_weights_deprecated.h
index 7a7cd14a4c2..40984d8bc0f 100644
--- a/Weights/include/CGAL/Weights/internal/pmp_weights_deprecated.h
+++ b/Weights/include/CGAL/Weights/internal/pmp_weights_deprecated.h
@@ -51,7 +51,7 @@ struct Cotangent_value_Meyer_impl
   double operator()(vertex_descriptor v0,
                     vertex_descriptor v1,
                     vertex_descriptor v2,
-                    const VertexPointMap& ppmap)
+                    VertexPointMap ppmap)
   {
     typedef typename Kernel_traits<
       typename boost::property_traits<VertexPointMap>::value_type >::Kernel::Vector_3 Vector;
@@ -94,17 +94,17 @@ protected:
   typedef typename boost::property_traits<Point_property_map>::value_type Point;
   typedef typename Kernel_traits<Point>::Kernel::Vector_3 Vector;
 
-  PolygonMesh& pmesh_;
+  const PolygonMesh&  pmesh_;
   Point_property_map ppmap_;
 
 public:
-  Cotangent_value_Meyer(PolygonMesh& pmesh_,
+  Cotangent_value_Meyer(const PolygonMesh& pmesh_,
                         VertexPointMap vpmap_)
     : pmesh_(pmesh_), ppmap_(vpmap_)
   { }
 
-  PolygonMesh& pmesh() { return pmesh_; }
-  Point_property_map& ppmap() { return ppmap_; }
+  const PolygonMesh& pmesh() { return pmesh_; }
+  Point_property_map ppmap() { return ppmap_; }
 
   double operator()(vertex_descriptor v0,
                     vertex_descriptor v1,
@@ -128,13 +128,13 @@ class Cotangent_value_Meyer_secure
   Point_property_map ppmap_;
 
 public:
-  Cotangent_value_Meyer_secure(PolygonMesh& pmesh_,
+  Cotangent_value_Meyer_secure(const PolygonMesh& pmesh_,
                                VertexPointMap vpmap_)
     : pmesh_(pmesh_), ppmap_(vpmap_)
   { }
 
-  PolygonMesh& pmesh() { return pmesh_; }
-  Point_property_map& ppmap() { return ppmap_; }
+  const PolygonMesh& pmesh() { return pmesh_; }
+  Point_property_map ppmap() { return ppmap_; }
 
   double operator()(vertex_descriptor v0,
                     vertex_descriptor v1,
@@ -165,13 +165,13 @@ class Cotangent_value_clamped : CotangentValue
   Cotangent_value_clamped() { }
 
 public:
-  Cotangent_value_clamped(PolygonMesh& pmesh_,
+  Cotangent_value_clamped(const PolygonMesh& pmesh_,
                           VertexPointMap vpmap_)
     : CotangentValue(pmesh_, vpmap_)
   { }
 
-  PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
-  VertexPointMap& ppmap() { return CotangentValue::ppmap(); }
+  const PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
+  VertexPointMap ppmap() { return CotangentValue::ppmap(); }
 
   typedef typename boost::graph_traits<PolygonMesh>::vertex_descriptor vertex_descriptor;
 
@@ -194,13 +194,13 @@ class Cotangent_value_clamped_2 : CotangentValue
   Cotangent_value_clamped_2() { }
 
 public:
-  Cotangent_value_clamped_2(PolygonMesh& pmesh_,
+  Cotangent_value_clamped_2(const PolygonMesh& pmesh_,
                             VertexPointMap vpmap_)
     : CotangentValue(pmesh_, vpmap_)
   { }
 
-  PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
-  VertexPointMap& ppmap() { return CotangentValue::ppmap(); }
+  const PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
+  VertexPointMap ppmap() { return CotangentValue::ppmap(); }
 
   typedef typename boost::graph_traits<PolygonMesh>::vertex_descriptor vertex_descriptor;
 
@@ -238,17 +238,16 @@ template<typename PolygonMesh,
 typename CotangentValue = Cotangent_value_Meyer<PolygonMesh, VertexPointMap> >
 class Cotangent_value_minimum_zero : CotangentValue
 {
-  Cotangent_value_minimum_zero()
-  { }
-
 public:
-  Cotangent_value_minimum_zero(PolygonMesh& pmesh_,
+  Cotangent_value_minimum_zero() { }
+
+  Cotangent_value_minimum_zero(const PolygonMesh& pmesh_,
                                VertexPointMap vpmap_)
     : CotangentValue(pmesh_, vpmap_)
   { }
 
-  PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
-  VertexPointMap& ppmap() { return CotangentValue::ppmap(); }
+  const PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
+  VertexPointMap ppmap() { return CotangentValue::ppmap(); }
 
   typedef typename boost::graph_traits<PolygonMesh>::vertex_descriptor vertex_descriptor;
 
@@ -268,13 +267,13 @@ class Voronoi_area
   : CotangentValue
 {
 public:
-  Voronoi_area(PolygonMesh& pmesh_,
+  Voronoi_area(const PolygonMesh& pmesh_,
                VertexPointMap vpmap_)
     : CotangentValue(pmesh_, vpmap_)
   { }
 
-  PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
-  VertexPointMap& ppmap() { return CotangentValue::ppmap(); }
+  const PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
+  VertexPointMap ppmap() { return CotangentValue::ppmap(); }
 
   typedef typename boost::graph_traits<PolygonMesh>::vertex_descriptor vertex_descriptor;
   typedef typename boost::graph_traits<PolygonMesh>::in_edge_iterator in_edge_iterator;
@@ -348,13 +347,13 @@ class Cotangent_value_area_weighted
   Cotangent_value_area_weighted() { }
 
 public:
-  Cotangent_value_area_weighted(PolygonMesh& pmesh_,
+  Cotangent_value_area_weighted(const PolygonMesh& pmesh_,
                                 VertexPointMap vpmap_) :
     CotangentValue(pmesh_, vpmap_)
   { }
 
-  PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
-  VertexPointMap& ppmap() { return CotangentValue::ppmap(); }
+  const PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
+  VertexPointMap ppmap() { return CotangentValue::ppmap(); }
 
   typedef typename boost::graph_traits<PolygonMesh>::vertex_descriptor vertex_descriptor;
 
@@ -385,7 +384,7 @@ struct Cotangent_weight_impl
   template<class VertexPointMap>
   double operator()(halfedge_descriptor he,
                     PolygonMesh& pmesh,
-                    const VertexPointMap& ppmap)
+                    VertexPointMap ppmap)
   {
     const vertex_descriptor v0 = target(he, pmesh);
     const vertex_descriptor v1 = source(he, pmesh);
@@ -422,20 +421,20 @@ template<typename PolygonMesh,
 class Cotangent_weight
   : CotangentValue
 {
+public:
   Cotangent_weight() { }
 
-public:
-  Cotangent_weight(PolygonMesh& pmesh_,
+  Cotangent_weight(const PolygonMesh& pmesh_,
                    VertexPointMap vpmap_)
     : CotangentValue(pmesh_, vpmap_)
   { }
 
-  Cotangent_weight(PolygonMesh& pmesh_)
+  Cotangent_weight(const PolygonMesh& pmesh_)
     : CotangentValue(pmesh_, get(CGAL::vertex_point, pmesh_))
   { }
 
-  PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
-  VertexPointMap& ppmap() { return CotangentValue::ppmap(); }
+  const PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
+  VertexPointMap ppmap() { return CotangentValue::ppmap(); }
 
   typedef typename boost::graph_traits<PolygonMesh>::halfedge_descriptor halfedge_descriptor;
   typedef typename boost::graph_traits<PolygonMesh>::vertex_descriptor vertex_descriptor;
@@ -512,13 +511,13 @@ class Single_cotangent_weight
   Single_cotangent_weight() { }
 
 public:
-  Single_cotangent_weight(PolygonMesh& pmesh_,
+  Single_cotangent_weight(const PolygonMesh& pmesh_,
                           VertexPointMap vpmap_)
     : CotangentValue(pmesh_, vpmap_)
   { }
 
-  PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
-  VertexPointMap& ppmap() { return CotangentValue::ppmap(); }
+  const PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
+  VertexPointMap ppmap() { return CotangentValue::ppmap(); }
 
   typedef typename boost::graph_traits<PolygonMesh>::halfedge_descriptor halfedge_descriptor;
   typedef typename boost::graph_traits<PolygonMesh>::vertex_descriptor vertex_descriptor;
@@ -557,13 +556,13 @@ class Cotangent_weight_with_triangle_area
   Cotangent_weight_with_triangle_area() { }
 
 public:
-  Cotangent_weight_with_triangle_area(PolygonMesh& pmesh_,
+  Cotangent_weight_with_triangle_area(const PolygonMesh& pmesh_,
                                       VertexPointMap vpmap_)
     : CotangentValue(pmesh_, vpmap_)
   { }
 
-  PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
-  VertexPointMap& ppmap() { return CotangentValue::ppmap(); }
+  const PolygonMesh& pmesh() { return CotangentValue::pmesh(); }
+  VertexPointMap ppmap() { return CotangentValue::ppmap(); }
 
   double operator()(halfedge_descriptor he)
   {
@@ -614,18 +613,16 @@ template<typename PolygonMesh,
          typename VertexPointMap = typename boost::property_map<PolygonMesh, CGAL::vertex_point_t>::type>
 class Mean_value_weight
 {
-  // Mean_value_weight() {}
-
-  PolygonMesh& pmesh_;
+  const PolygonMesh& pmesh_;
   VertexPointMap vpmap_;
 
 public:
-  Mean_value_weight(PolygonMesh& pmesh_,
+  Mean_value_weight(const PolygonMesh& pmesh_,
                     VertexPointMap vpmap)
     : pmesh_(pmesh_), vpmap_(vpmap)
   { }
 
-  PolygonMesh& pmesh() { return pmesh_; }
+  const PolygonMesh& pmesh() { return pmesh_; }
 
   typedef typename boost::graph_traits<PolygonMesh>::halfedge_descriptor halfedge_descriptor;
   typedef typename boost::graph_traits<PolygonMesh>::vertex_descriptor vertex_descriptor;
@@ -722,11 +719,11 @@ class Hybrid_weight
   Hybrid_weight() { }
 
 public:
-  Hybrid_weight(PolygonMesh& pmesh_)
+  Hybrid_weight(const PolygonMesh& pmesh_)
     : primary(pmesh_), secondary(pmesh_)
   { }
 
-  PolygonMesh& pmesh() { return primary.pmesh(); }
+  const PolygonMesh& pmesh() { return primary.pmesh(); }
 
   typedef typename boost::graph_traits<PolygonMesh>::halfedge_descriptor halfedge_descriptor;
 
@@ -751,14 +748,14 @@ public:
 template<class PolygonMesh>
 class Scale_dependent_weight_fairing
 {
-  PolygonMesh& pmesh_;
+  const PolygonMesh& pmesh_;
 
 public:
-  Scale_dependent_weight_fairing(PolygonMesh& pmesh_)
+  Scale_dependent_weight_fairing(const PolygonMesh& pmesh_)
     : pmesh_(pmesh_)
   { }
 
-  PolygonMesh& pmesh() { return pmesh_; }
+  const PolygonMesh& pmesh() { return pmesh_; }
 
   typedef typename boost::graph_traits<PolygonMesh>::halfedge_descriptor halfedge_descriptor;
   typedef typename boost::graph_traits<PolygonMesh>::vertex_descriptor vertex_descriptor;
@@ -800,7 +797,7 @@ public:
 
   Cotangent_weight_with_voronoi_area_fairing(PM& pmesh_,
                                              VPMap vpmap_)
-    :  voronoi_functor(pmesh_, vpmap_),
+    : voronoi_functor(pmesh_, vpmap_),
       cotangent_functor(pmesh_, vpmap_)
   { }
 
@@ -867,7 +864,7 @@ public:
   typedef typename boost::graph_traits<PolygonMesh>::halfedge_descriptor halfedge_descriptor;
   typedef typename boost::graph_traits<PolygonMesh>::vertex_descriptor vertex_descriptor;
 
-  Uniform_weight_fairing(PolygonMesh&) { }
+  Uniform_weight_fairing(const PolygonMesh&) { }
 
   double w_ij(halfedge_descriptor /* e */) { return 1.0; }
   double w_i(vertex_descriptor /* v_i */) { return 1.0; }
diff --git a/Weights/include/CGAL/Weights/internal/utils.h b/Weights/include/CGAL/Weights/internal/utils.h
index 744af8be2d3..850fccff1e8 100644
--- a/Weights/include/CGAL/Weights/internal/utils.h
+++ b/Weights/include/CGAL/Weights/internal/utils.h
@@ -214,7 +214,6 @@ typename GeomTraits::Point_3 rotate_point_3(const double angle_rad,
                                             const GeomTraits& traits)
 {
   using FT = typename GeomTraits::FT;
-  using Point_3 = typename GeomTraits::Point_3;
 
   auto point_3 = traits.construct_point_3_object();
 
@@ -274,7 +273,6 @@ typename GeomTraits::Point_2 to_2d(const typename GeomTraits::Vector_3& b1,
                                    const GeomTraits& traits)
 {
   using FT = typename GeomTraits::FT;
-  using Point_2 = typename GeomTraits::Point_2;
   using Vector_3 = typename GeomTraits::Vector_3;
 
   auto dot_product_3 = traits.compute_scalar_product_3_object();
@@ -447,7 +445,6 @@ typename GeomTraits::FT area_3(const typename GeomTraits::Point_3& p,
                                const typename GeomTraits::Point_3& r,
                                const GeomTraits& traits)
 {
-  using FT = typename GeomTraits::FT;
   using Point_2 = typename GeomTraits::Point_2;
   using Point_3 = typename GeomTraits::Point_3;
   using Vector_3 = typename GeomTraits::Vector_3;
@@ -493,7 +490,6 @@ typename GeomTraits::FT positive_area_3(const typename GeomTraits::Point_3& p,
                                         const typename GeomTraits::Point_3& r,
                                         const GeomTraits& traits)
 {
-  using FT = typename GeomTraits::FT;
   using Get_sqrt = Get_sqrt<GeomTraits>;
   auto sqrt = Get_sqrt::sqrt_object(traits);