diff --git a/Surface_mesh_approximation/benchmark/Surface_mesh_approximation/vsa_autoinit_timing_benchmark.cpp b/Surface_mesh_approximation/benchmark/Surface_mesh_approximation/vsa_autoinit_timing_benchmark.cpp
index 17a124e0ead..4ef9f2b2dfe 100644
--- a/Surface_mesh_approximation/benchmark/Surface_mesh_approximation/vsa_autoinit_timing_benchmark.cpp
+++ b/Surface_mesh_approximation/benchmark/Surface_mesh_approximation/vsa_autoinit_timing_benchmark.cpp
@@ -11,18 +11,12 @@
 #include <CGAL/Timer.h>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
-typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
 typedef Kernel::FT FT;
-typedef Kernel::Point_3 Point_3;
-typedef Kernel::Vector_3 Vector_3;
 typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
-typedef Polyhedron_3::Facet_handle Facet_handle;
-typedef boost::associative_property_map<std::map<Facet_handle, FT> > FacetAreaMap;
-typedef boost::associative_property_map<std::map<Facet_handle, Vector_3> > FacetNormalMap;
 
 typedef CGAL::PlaneProxy<Polyhedron_3> PlaneProxy;
-typedef CGAL::L21Metric<Polyhedron_3, FacetNormalMap, FacetAreaMap> L21Metric;
-typedef CGAL::L21ProxyFitting<Polyhedron_3, FacetNormalMap, FacetAreaMap> L21ProxyFitting;
+typedef CGAL::L21Metric<Polyhedron_3> L21Metric;
+typedef CGAL::L21ProxyFitting<Polyhedron_3> L21ProxyFitting;
 typedef CGAL::VSA_approximation<Polyhedron_3, PlaneProxy, L21Metric, L21ProxyFitting> VSAL21;
 
 typedef CGAL::Timer Timer;
@@ -46,25 +40,8 @@ int main(int argc, char *argv[])
   }
   std::cerr << "#triangles " << mesh.size_of_facets() << std::endl;
 
-  std::map<Facet_handle, Vector_3> facet_normals;
-  std::map<Facet_handle, FT> facet_areas;
-  for (Polyhedron_3::Facet_iterator fitr = mesh.facets_begin();
-    fitr != mesh.facets_end(); ++fitr) {
-    Polyhedron_3::Halfedge_handle he = fitr->halfedge();
-    const Point_3 &p0 = he->opposite()->vertex()->point();
-    const Point_3 &p1 = he->vertex()->point();
-    const Point_3 &p2 = he->next()->vertex()->point();
-
-    Vector_3 normal = CGAL::unit_normal(p0, p1, p2);
-    facet_normals.insert(std::pair<Facet_handle, Vector_3>(fitr, normal));
-    FT area(std::sqrt(CGAL::to_double(CGAL::squared_area(p0, p1, p2))));
-    facet_areas.insert(std::pair<Facet_handle, FT>(fitr, area));
-  }
-  FacetNormalMap normal_pmap(facet_normals);
-  FacetAreaMap area_pmap(facet_areas);
-
-  L21Metric l21_metric(normal_pmap, area_pmap);
-  L21ProxyFitting l21_fitting(normal_pmap, area_pmap);
+  L21Metric l21_metric(mesh);
+  L21ProxyFitting l21_fitting(mesh);
 
   // algorithm instance
   VSAL21 vsa_l21(l21_metric, l21_fitting);
diff --git a/Surface_mesh_approximation/benchmark/Surface_mesh_approximation/vsa_timing_benchmark.cpp b/Surface_mesh_approximation/benchmark/Surface_mesh_approximation/vsa_timing_benchmark.cpp
index c632a3245a8..cfbeeffe013 100644
--- a/Surface_mesh_approximation/benchmark/Surface_mesh_approximation/vsa_timing_benchmark.cpp
+++ b/Surface_mesh_approximation/benchmark/Surface_mesh_approximation/vsa_timing_benchmark.cpp
@@ -12,17 +12,10 @@
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
 typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
-typedef Kernel::FT FT;
-typedef Kernel::Point_3 Point_3;
-typedef Kernel::Vector_3 Vector_3;
-typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
-typedef Polyhedron_3::Facet_handle Facet_handle;
-typedef boost::associative_property_map<std::map<Facet_handle, FT> > FacetAreaMap;
-typedef boost::associative_property_map<std::map<Facet_handle, Vector_3> > FacetNormalMap;
 
 typedef CGAL::PlaneProxy<Polyhedron_3> PlaneProxy;
-typedef CGAL::L21Metric<Polyhedron_3, FacetNormalMap, FacetAreaMap> L21Metric;
-typedef CGAL::L21ProxyFitting<Polyhedron_3, FacetNormalMap, FacetAreaMap> L21ProxyFitting;
+typedef CGAL::L21Metric<Polyhedron_3> L21Metric;
+typedef CGAL::L21ProxyFitting<Polyhedron_3> L21ProxyFitting;
 typedef CGAL::VSA_approximation<Polyhedron_3, PlaneProxy, L21Metric, L21ProxyFitting> VSAL21;
 
 typedef CGAL::Timer Timer;
@@ -48,25 +41,8 @@ int main(int argc, char *argv[])
   }
   std::cerr << "#triangles " << mesh.size_of_facets() << std::endl;
 
-  std::map<Facet_handle, Vector_3> facet_normals;
-  std::map<Facet_handle, FT> facet_areas;
-  for (Polyhedron_3::Facet_iterator fitr = mesh.facets_begin();
-    fitr != mesh.facets_end(); ++fitr) {
-    Polyhedron_3::Halfedge_handle he = fitr->halfedge();
-    const Point_3 &p0 = he->opposite()->vertex()->point();
-    const Point_3 &p1 = he->vertex()->point();
-    const Point_3 &p2 = he->next()->vertex()->point();
-
-    Vector_3 normal = CGAL::unit_normal(p0, p1, p2);
-    facet_normals.insert(std::pair<Facet_handle, Vector_3>(fitr, normal));
-    FT area(std::sqrt(CGAL::to_double(CGAL::squared_area(p0, p1, p2))));
-    facet_areas.insert(std::pair<Facet_handle, FT>(fitr, area));
-  }
-  FacetNormalMap normal_pmap(facet_normals);
-  FacetAreaMap area_pmap(facet_areas);
-
-  L21Metric l21_metric(normal_pmap, area_pmap);
-  L21ProxyFitting l21_fitting(normal_pmap, area_pmap);
+  L21Metric l21_metric(mesh);
+  L21ProxyFitting l21_fitting(mesh);
 
   // algorithm instance
   VSAL21 vsa_l21(l21_metric, l21_fitting);
diff --git a/Surface_mesh_approximation/examples/Surface_mesh_approximation/vsa_class_interface_example.cpp b/Surface_mesh_approximation/examples/Surface_mesh_approximation/vsa_class_interface_example.cpp
index fba837a0142..7edb4441ddb 100644
--- a/Surface_mesh_approximation/examples/Surface_mesh_approximation/vsa_class_interface_example.cpp
+++ b/Surface_mesh_approximation/examples/Surface_mesh_approximation/vsa_class_interface_example.cpp
@@ -11,14 +11,11 @@
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
 typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
-typedef Kernel::FT FT;
 typedef Kernel::Point_3 Point_3;
 typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
-typedef Polyhedron_3::Facet_handle Facet_handle;
-typedef boost::associative_property_map<std::map<Facet_handle, FT> > FacetAreaMap;
 
 typedef CGAL::PlaneProxy<Polyhedron_3> PlaneProxy;
-typedef CGAL::L2Metric<Polyhedron_3, FacetAreaMap> L2Metric;
+typedef CGAL::L2Metric<Polyhedron_3> L2Metric;
 typedef CGAL::L2ProxyFitting<Polyhedron_3> L2ProxyFitting;
 typedef CGAL::VSA_approximation<Polyhedron_3, PlaneProxy, L2Metric, L2ProxyFitting> VSA;
 
@@ -35,20 +32,6 @@ int main(int argc, char *argv[])
     return EXIT_FAILURE;
   }
 
-  // facet area map
-  std::map<Facet_handle, FT> facet_areas;
-  for (Polyhedron_3::Facet_iterator fitr = mesh.facets_begin();
-    fitr != mesh.facets_end(); ++fitr) {
-    Polyhedron_3::Halfedge_handle he = fitr->halfedge();
-    const Point_3 &p0 = he->opposite()->vertex()->point();
-    const Point_3 &p1 = he->vertex()->point();
-    const Point_3 &p2 = he->next()->vertex()->point();
-
-    FT farea(std::sqrt(CGAL::to_double(CGAL::squared_area(p0, p1, p2))));
-    facet_areas.insert(std::pair<Facet_handle, FT>(fitr, farea));
-  }
-  FacetAreaMap area_pmap(facet_areas);
-
   const std::size_t num_proxies = std::atoi(argv[3]);
   const std::size_t num_iterations = std::atoi(argv[4]);
   std::vector<int> tris;
@@ -57,7 +40,7 @@ int main(int argc, char *argv[])
   if (init < 0 || init > 2)
     return EXIT_FAILURE;
 
-  L2Metric metric(mesh, area_pmap);
+  L2Metric metric(mesh);
   L2ProxyFitting proxy_fitting(mesh);
 
   // create VSA L2 metric approximation algorithm instance
diff --git a/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_class_interface_test.cpp b/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_class_interface_test.cpp
index 7babeeca346..a08c0a1faf2 100644
--- a/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_class_interface_test.cpp
+++ b/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_class_interface_test.cpp
@@ -10,16 +10,15 @@
 #include <CGAL/VSA_approximation.h>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
-typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
 typedef Kernel::FT FT;
 typedef Kernel::Point_3 Point_3;
+
 typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
 typedef Polyhedron_3::Facet_handle Facet_handle;
-typedef boost::associative_property_map<std::map<Facet_handle, FT> > FacetAreaMap;
 typedef boost::associative_property_map<std::map<Facet_handle, std::size_t> > FacetProxyMap;
 
 typedef CGAL::PlaneProxy<Polyhedron_3> PlaneProxy;
-typedef CGAL::L2Metric<Polyhedron_3, FacetAreaMap> L2Metric;
+typedef CGAL::L2Metric<Polyhedron_3> L2Metric;
 typedef CGAL::L2ProxyFitting<Polyhedron_3> L2ProxyFitting;
 typedef CGAL::VSA_approximation<Polyhedron_3, PlaneProxy, L2Metric, L2ProxyFitting> VSA;
 
@@ -37,24 +36,13 @@ int main()
   }
 
   // facet area map
-  std::map<Facet_handle, FT> facet_areas;
   std::map<Facet_handle, std::size_t> facet_index;
   for (Polyhedron_3::Facet_iterator fitr = mesh.facets_begin();
-    fitr != mesh.facets_end(); ++fitr) {
-    Polyhedron_3::Halfedge_handle he = fitr->halfedge();
-    const Point_3 &p0 = he->opposite()->vertex()->point();
-    const Point_3 &p1 = he->vertex()->point();
-    const Point_3 &p2 = he->next()->vertex()->point();
-
-    FT farea(std::sqrt(CGAL::to_double(CGAL::squared_area(p0, p1, p2))));
-    facet_areas.insert(std::pair<Facet_handle, FT>(fitr, farea));
-
+    fitr != mesh.facets_end(); ++fitr)
     facet_index.insert(std::pair<Facet_handle, std::size_t>(fitr, 0));
-  }
-  FacetAreaMap area_pmap(facet_areas);
   FacetProxyMap proxy_pmap(facet_index);
 
-  L2Metric metric(mesh, area_pmap);
+  L2Metric metric(mesh);
   L2ProxyFitting proxy_fitting(mesh);
 
   // create VSA L2 metric approximation algorithm instance
diff --git a/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_correctness_test.cpp b/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_correctness_test.cpp
index 25c46e251eb..9ab05fe1d38 100644
--- a/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_correctness_test.cpp
+++ b/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_correctness_test.cpp
@@ -10,18 +10,13 @@
 #include <CGAL/VSA_approximation.h>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
-typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
 typedef Kernel::FT FT;
 typedef Kernel::Point_3 Point_3;
-typedef Kernel::Vector_3 Vector_3;
 typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
-typedef Polyhedron_3::Facet_handle Facet_handle;
-typedef boost::associative_property_map<std::map<Facet_handle, FT> > FacetAreaMap;
-typedef boost::associative_property_map<std::map<Facet_handle, Vector_3> > FacetNormalMap;
 
 typedef CGAL::PlaneProxy<Polyhedron_3> PlaneProxy;
-typedef CGAL::L21Metric<Polyhedron_3, FacetNormalMap, FacetAreaMap> L21Metric;
-typedef CGAL::L21ProxyFitting<Polyhedron_3, FacetNormalMap, FacetAreaMap> L21ProxyFitting;
+typedef CGAL::L21Metric<Polyhedron_3> L21Metric;
+typedef CGAL::L21ProxyFitting<Polyhedron_3> L21ProxyFitting;
 typedef CGAL::VSA_approximation<Polyhedron_3, PlaneProxy, L21Metric, L21ProxyFitting> VSAL21;
 
 bool test_shape(const char *file_name, const std::size_t target_num_proxies)
@@ -33,25 +28,8 @@ bool test_shape(const char *file_name, const std::size_t target_num_proxies)
     return false;
   }
 
-  std::map<Facet_handle, Vector_3> facet_normals;
-  std::map<Facet_handle, FT> facet_areas;
-  for (Polyhedron_3::Facet_iterator fitr = mesh.facets_begin();
-    fitr != mesh.facets_end(); ++fitr) {
-    Polyhedron_3::Halfedge_handle he = fitr->halfedge();
-    const Point_3 &p0 = he->opposite()->vertex()->point();
-    const Point_3 &p1 = he->vertex()->point();
-    const Point_3 &p2 = he->next()->vertex()->point();
-
-    Vector_3 normal = CGAL::unit_normal(p0, p1, p2);
-    facet_normals.insert(std::pair<Facet_handle, Vector_3>(fitr, normal));
-    FT area(std::sqrt(CGAL::to_double(CGAL::squared_area(p0, p1, p2))));
-    facet_areas.insert(std::pair<Facet_handle, FT>(fitr, area));
-  }
-  FacetNormalMap normal_pmap(facet_normals);
-  FacetAreaMap area_pmap(facet_areas);
-
-  L21Metric l21_metric(normal_pmap, area_pmap);
-  L21ProxyFitting l21_fitting(normal_pmap, area_pmap);
+  L21Metric l21_metric(mesh);
+  L21ProxyFitting l21_fitting(mesh);
 
   // algorithm instance
   VSAL21 vsa_l21(l21_metric, l21_fitting);
diff --git a/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_error_decrease_test.cpp b/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_error_decrease_test.cpp
index a7b665feaff..4441547f13b 100644
--- a/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_error_decrease_test.cpp
+++ b/Surface_mesh_approximation/test/Surface_mesh_approximation/vsa_error_decrease_test.cpp
@@ -10,18 +10,12 @@
 #include <CGAL/VSA_approximation.h>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
-typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
 typedef Kernel::FT FT;
-typedef Kernel::Point_3 Point_3;
-typedef Kernel::Vector_3 Vector_3;
 typedef CGAL::Polyhedron_3<Kernel> Polyhedron_3;
-typedef Polyhedron_3::Facet_handle Facet_handle;
-typedef boost::associative_property_map<std::map<Facet_handle, FT> > FacetAreaMap;
-typedef boost::associative_property_map<std::map<Facet_handle, Vector_3> > FacetNormalMap;
 
 typedef CGAL::PlaneProxy<Polyhedron_3> PlaneProxy;
-typedef CGAL::L21Metric<Polyhedron_3, FacetNormalMap, FacetAreaMap> L21Metric;
-typedef CGAL::L21ProxyFitting<Polyhedron_3, FacetNormalMap, FacetAreaMap> L21ProxyFitting;
+typedef CGAL::L21Metric<Polyhedron_3> L21Metric;
+typedef CGAL::L21ProxyFitting<Polyhedron_3> L21ProxyFitting;
 typedef CGAL::VSA_approximation<Polyhedron_3, PlaneProxy, L21Metric, L21ProxyFitting> VSAL21;
 
 bool check_strict_ordering(const std::vector<FT> &error)
@@ -50,25 +44,8 @@ int main()
     return EXIT_FAILURE;
   }
 
-  std::map<Facet_handle, Vector_3> facet_normals;
-  std::map<Facet_handle, FT> facet_areas;
-  for (Polyhedron_3::Facet_iterator fitr = mesh.facets_begin();
-    fitr != mesh.facets_end(); ++fitr) {
-    Polyhedron_3::Halfedge_handle he = fitr->halfedge();
-    const Point_3 &p0 = he->opposite()->vertex()->point();
-    const Point_3 &p1 = he->vertex()->point();
-    const Point_3 &p2 = he->next()->vertex()->point();
-
-    Vector_3 normal = CGAL::unit_normal(p0, p1, p2);
-    facet_normals.insert(std::pair<Facet_handle, Vector_3>(fitr, normal));
-    FT area(std::sqrt(CGAL::to_double(CGAL::squared_area(p0, p1, p2))));
-    facet_areas.insert(std::pair<Facet_handle, FT>(fitr, area));
-  }
-  FacetNormalMap normal_pmap(facet_normals);
-  FacetAreaMap area_pmap(facet_areas);
-
-  L21Metric l21_metric(normal_pmap, area_pmap);
-  L21ProxyFitting l21_fitting(normal_pmap, area_pmap);
+  L21Metric l21_metric(mesh);
+  L21ProxyFitting l21_fitting(mesh);
 
   // algorithm instance
   VSAL21 vsa_l21(l21_metric, l21_fitting);