diff --git a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/DistanceProxy.h b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/DistanceProxy.h
deleted file mode 100644
index ad31fca56d2..00000000000
--- a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/DistanceProxy.h
+++ /dev/null
@@ -1,45 +0,0 @@
-
-/*!
-\ingroup PkgTSMAConcepts
-\cgalConcept
-
-The Euclidean distance proxy.
-
-\cgalRefines `Proxy`
-\cgalHasModel `TBD`
-*/
-class DistanceProxy {
-public:
-  /*! 3D point type
-   * It must be default constructible, and can be constructed from 3 objects of type `FT`.
-   * `bool operator<(Point_3, Point_3)` to lexicographically compare two points must be available.
-   * Access to Cartesian coordinates must be possible using `Point_3::x()`, `Point_3::y(), Point_3::z()` and
-   * `FT operator[](int i)` with  `0 <= i < 3`.
-   *
-   * There must be a specialization of `CGAL::Kernel_traits` such that
-   * `CGAL::Kernel_traits<Point_3>::%Kernel` is a model implementing this concept.
-   */
-  typedef unspecified_type Point_3;
-
-  /// 3D vector type
-  typedef unspecified_type Vector_3;
-  /// 3D plane type
-  typedef unspecified_type Plane_3;
-
-  /// Triangle mesh facet descriptor.
-  typedef unspecified_type facet_descriptor;
-
-  /// @name Data members
-  /// @{
-
-  /// data member to describe the plane proxy center position.
-  Point_3 center;
-  /// data member to describe the plane proxy normal.
-  Vector_3 normal;
-  /// data member to describe the proxy seed.
-  facet_descriptor seed;
-  /// data member to describe the fitted plane.
-  Plane_3 fit_plane;
-
-  /// }
-};
\ No newline at end of file
diff --git a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/ErrorMetric.h b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/ErrorMetric.h
index 06914fd0e22..80034295b98 100644
--- a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/ErrorMetric.h
+++ b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/ErrorMetric.h
@@ -5,8 +5,8 @@
 
 This is a functor to compute the fitting error of a facet to a proxy.
 
-\cgalHasModel `L21ErrorMetric`
-\cgalHasModel `DistanceErrorMetric`
+\cgalHasModel `L21Metric`
+\cgalHasModel `L2Metric`
 
 */
 class ErrorMetric {
diff --git a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/L21ErrorMetric.h b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/L21ErrorMetric.h
deleted file mode 100644
index 1b3fbd36d1a..00000000000
--- a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/L21ErrorMetric.h
+++ /dev/null
@@ -1,39 +0,0 @@
-
-/*!
-\ingroup PkgTSMAConcepts
-\cgalConcept
-
-This is a functor to compute the L21 fitting error of a facet to a proxy.
-Is this concept needed?
-
-\cgalRefines `ErrorMetric`
-\cgalHasModel `L21Metric`
-
-*/
-class L21ErrorMetric {
-public:
-  /*!
-   * A number type model of `Field` and `RealEmbeddable`
-   */
-  typedef unspecified_type FT;
-  /// Triangle mesh facet descriptor.
-  typedef unspecified_type facet_descriptor;
-  /// Parametrized shape proxy.
-  typedef unspecified_type Proxy;
-  /// 3D vector type
-  typedef unspecified_type Vector_3;
-
-  /// @name Functors
-  /// @{
-  /// Functor for computing scaled vector. Is this required?
-  typedef unspecified_type Construct_scaled_vector_3;
-
-  /// }
-
-  /// @name Functions
-  /// @{
-  /// data member to describe the proxy seed.
-  FT operator()(const facet_descriptor &f, const Proxy &px);
-  /// }
-};
-
diff --git a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/PlaneFitting.h b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/PlaneFitting.h
new file mode 100644
index 00000000000..a5e702d6e54
--- /dev/null
+++ b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/PlaneFitting.h
@@ -0,0 +1,17 @@
+/*!
+\ingroup PkgTSMAConcepts
+\cgalConcept
+
+The function object class to fit a plane from a range of facets.
+
+\cgalHasModel `PlaneFitting`
+\cgalHasModel `PCAPlaneFitting`
+
+*/
+class PlaneFitting {
+  typedef unspecified_type Proxy;
+
+  template<FacetsIterator>
+  Proxy operator()(const FacetsIterator &beg, const FacetsIterator &end) const;
+};
+
diff --git a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/PlaneProxy.h b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/PlaneProxy.h
deleted file mode 100644
index dfdc11e2b96..00000000000
--- a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/PlaneProxy.h
+++ /dev/null
@@ -1,47 +0,0 @@
-
-/*!
-\ingroup PkgTSMAConcepts
-\cgalConcept
-
-The parameterized plane shape that is fitted.
-
-\cgalRefines `Proxy`
-\cgalHasModel `PlaneProxy`
-
-*/
-class PlaneProxy {
-public:
-  /*! 3D point type
-   * It must be default constructible, and can be constructed from 3 objects of type `FT`.
-   * `bool operator<(Point_3, Point_3)` to lexicographically compare two points must be available.
-   * Access to Cartesian coordinates must be possible using `Point_3::x()`, `Point_3::y(), Point_3::z()` and
-   * `FT operator[](int i)` with  `0 <= i < 3`.
-   *
-   * There must be a specialization of `CGAL::Kernel_traits` such that
-   * `CGAL::Kernel_traits<Point_3>::%Kernel` is a model implementing this concept.
-   */
-  typedef unspecified_type Point_3;
-
-  /// 3D vector type
-  typedef unspecified_type Vector_3;
-  /// 3D plane type
-  typedef unspecified_type Plane_3;
-
-  /// Triangle mesh facet descriptor.
-  typedef unspecified_type facet_descriptor;
-
-  /// @name Data members
-  /// @{
-
-  /// data member to describe the plane proxy center position.
-  Point_3 center;
-  /// data member to describe the plane proxy normal.
-  Vector_3 normal;
-  /// data member to describe the proxy seed.
-  facet_descriptor seed;
-  /// data member to describe the fitted plane.
-  Plane_3 fit_plane;
-
-  /// }
-};
-
diff --git a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/Proxy.h b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/Proxy.h
index dfcd15e38ba..3adee0b322d 100644
--- a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/Proxy.h
+++ b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/Proxy.h
@@ -6,7 +6,6 @@
 The parameterized shape that is fitted.
 
 \cgalHasModel `PlaneProxy`
-\cgalHasModel `DistanceProxy`
 
 */
 class Proxy {
diff --git a/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/ProxyFitting.h b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/ProxyFitting.h
new file mode 100644
index 00000000000..79cc3e86d94
--- /dev/null
+++ b/Surface_mesh_approximation/doc/Surface_mesh_approximation/Concepts/ProxyFitting.h
@@ -0,0 +1,16 @@
+/*!
+\ingroup PkgTSMAConcepts
+\cgalConcept
+
+The function object class to fit a proxy from a range of facets.
+
+\cgalHasModel `L21ProxyFitting`
+\cgalHasModel `L2ProxyFitting`
+
+*/
+class PlaneFitting {
+  typedef unspecified_type Proxy;
+
+  template<FacetsIterator>
+  Proxy operator()(const FacetsIterator &beg, const FacetsIterator &end) const;
+};
diff --git a/Surface_mesh_approximation/doc/Surface_mesh_approximation/PackageDescription.txt b/Surface_mesh_approximation/doc/Surface_mesh_approximation/PackageDescription.txt
index 0c415fea97c..3a6baf6e4a0 100644
--- a/Surface_mesh_approximation/doc/Surface_mesh_approximation/PackageDescription.txt
+++ b/Surface_mesh_approximation/doc/Surface_mesh_approximation/PackageDescription.txt
@@ -24,13 +24,13 @@
 
 ## Concepts ##
 - `Proxy`
-- `DistanceProxy`
-- `PlaneProxy`
 - `ErrorMetric`
-- `L21ErrorMetric`
+- `ProxyFitting`
+- `PlaneFitting`
 
 ## Classes and Functions##
-- `CGAL::vsa_mesh_approximation()`
-
+- `CGAL::vsa_approximate()`
+- `CGAL::vsa_extract()`
+- `CGAL::vsa_approximate_and_extract()`
 
 */
diff --git a/Surface_mesh_approximation/doc/Surface_mesh_approximation/surface_mesh_approximation.txt b/Surface_mesh_approximation/doc/Surface_mesh_approximation/surface_mesh_approximation.txt
index 587612b5604..cc4446dee46 100644
--- a/Surface_mesh_approximation/doc/Surface_mesh_approximation/surface_mesh_approximation.txt
+++ b/Surface_mesh_approximation/doc/Surface_mesh_approximation/surface_mesh_approximation.txt
@@ -18,19 +18,21 @@ This package can thus be used either with `Polyhedron_3`, `Surface_mesh`, or
 any class model of the concept `FaceGraph` described in \ref chapterBGL.
 
 Approximate a triangle mesh and extract the approximated triangle surface, using:
-- `CGAL::vsa_mesh_approximation()` : given a triangle mesh, approximate the geometry and extract the approximated triangle surface.
+- `CGAL::vsa_approximate()` : given a triangle mesh, approximate the geometry and output the facet proxy index map.
+- `CGAL::vsa_extract()` : given a triangle mesh, approximate the geometry and output the approximated triangle surface.
+- `CGAL::vsa_approximate_and_extract()` : given a triangle mesh, approximate the geometry, output the facet proxy index map and extract the approximated triangle surface.
 
 \section sma_examples Examples
 
 \subsection ApproximationExample_1 Approximate a Mesh
 
-The following example calls the functions `CGAL::vsa_mesh_approximation()` on the input triangle mesh.
+The following example calls the functions `CGAL::vsa_extract()` on the input triangle mesh.
 
 \cgalExample{Surface_mesh_approximation/vsa_extraction_example.cpp}
 
 \subsection ApproximationExample_2 User-defined Proxy
 
-We could use self-defined proxy and error for the approximation algorithm. In the following example, we define a simple point proxy to achieve a compact patch approximation of the shape:
+We could use self-defined proxy and error for the approximation algorithm. In the following example, we define a simple point-wise proxy to achieve a compact patch approximation of the shape:
 
 \cgalExample{Surface_mesh_approximation/vsa_metric_example.cpp}
 
diff --git a/Surface_mesh_approximation/include/CGAL/vsa_mesh_approximation.h b/Surface_mesh_approximation/include/CGAL/vsa_mesh_approximation.h
index 7645bddeefa..29cf8f6b323 100644
--- a/Surface_mesh_approximation/include/CGAL/vsa_mesh_approximation.h
+++ b/Surface_mesh_approximation/include/CGAL/vsa_mesh_approximation.h
@@ -40,6 +40,7 @@ namespace CGAL
  * @param pos anchor position container
  * @param vtx anchor vertex container
  * @param bdrs proxy patch boundary container
+ * @param plane_fitting plane fitting functor
  * @param fit_error error metric functor
  * @param proxy_fitting proxy fitting functor
  */
@@ -64,7 +65,7 @@ void vsa_mesh_approximation(
     AnchorPositionContainer &pos,
     AnchorVertexContainer &vtx,
     BoundaryContainer &bdrs,
-    const PlaneFitting &_plane_fitting,
+    const PlaneFitting &plane_fitting,
     const ErrorMetric &fit_error,
     const ProxyFitting &proxy_fitting) {
   // CGAL_precondition(is_pure_triangle(tm));
@@ -83,7 +84,7 @@ void vsa_mesh_approximation(
     PlaneFitting,
     VertexPointMap> VSA_mesh_extraction;
 
-  VSA_mesh_extraction extractor(tm, f_proxy_pmap, _plane_fitting, v_point_pmap);
+  VSA_mesh_extraction extractor(tm, f_proxy_pmap, plane_fitting, v_point_pmap);
 
   extractor.extract_mesh(tris);
   BOOST_FOREACH(const typename VSA_mesh_extraction::Anchor &a, extractor.collect_anchors()) {
@@ -167,6 +168,7 @@ void vsa_approximate(
  * @param tm a triangle mesh
  * @param[out] tris approximation indexed triangle soup
  * @param[out] pos anchor position container
+ * @param plane_fitting plane fitting functor
  * @param fit_error error metric functor
  * @param proxy_fitting proxy fitting functor
  * @param init select seed initialization
@@ -183,7 +185,7 @@ void vsa_extract(
     const TriangleMesh &tm,
     AnchorIndexContainer &tris,
     AnchorPositionContainer &pos,
-    const PlaneFitting &_plane_fitting,
+    const PlaneFitting &plane_fitting,
     const ErrorMetric &fit_error,
     const ProxyFitting &proxy_fitting,
     const int init,
@@ -211,7 +213,7 @@ void vsa_extract(
     FacetProxyMap,
     PlaneFitting> VSA_mesh_extraction;
 
-  VSA_mesh_extraction extractor(tm, f_proxy_pmap, _plane_fitting);
+  VSA_mesh_extraction extractor(tm, f_proxy_pmap, plane_fitting);
 
   extractor.extract_mesh(tris);
   BOOST_FOREACH(const typename VSA_mesh_extraction::Anchor &a, extractor.collect_anchors())
@@ -238,6 +240,7 @@ void vsa_extract(
  * @param[out] f_proxy_pmap facet proxy patch id property map
  * @param[out] tris approximation indexed triangle soup
  * @param[out] pos anchor position container
+ * @param plane_fitting plane fitting functor
  * @param fit_error error metric functor
  * @param proxy_fitting proxy fitting functor
  * @param init select seed initialization
@@ -256,7 +259,7 @@ void vsa_approximate_and_extract(
     FacetProxyMap f_proxy_pmap,
     AnchorIndexContainer &tris,
     AnchorPositionContainer &pos,
-    const PlaneFitting &_plane_fitting,
+    const PlaneFitting &plane_fitting,
     const ErrorMetric &fit_error,
     const ProxyFitting &proxy_fitting,
     const int init,
@@ -277,7 +280,7 @@ void vsa_approximate_and_extract(
     FacetProxyMap,
     PlaneFitting> VSA_mesh_extraction;
 
-  VSA_mesh_extraction extractor(tm, f_proxy_pmap, _plane_fitting);
+  VSA_mesh_extraction extractor(tm, f_proxy_pmap, plane_fitting);
 
   extractor.extract_mesh(tris);
   BOOST_FOREACH(const typename VSA_mesh_extraction::Anchor &a, extractor.collect_anchors())