diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/K_neighbor_query.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/K_neighbor_query.h
index 0b72f5115a4..54beee32af0 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/K_neighbor_query.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/K_neighbor_query.h
@@ -134,10 +134,10 @@ namespace Point_set {
     template<typename CGAL_NP_TEMPLATE_PARAMETERS>
     K_neighbor_query(
       const InputRange& input_range,
+      const Point_map point_map,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
+    m_point_map(point_map),
     m_index_to_point_map(m_input_range, m_point_map),
     m_distance(m_index_to_point_map),
     m_tree(
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_circle_fit_region.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_circle_fit_region.h
index 7500f19665f..339cca9cd1a 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_circle_fit_region.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_circle_fit_region.h
@@ -162,12 +162,12 @@ namespace Point_set {
     template<typename CGAL_NP_TEMPLATE_PARAMETERS>
     Least_squares_circle_fit_region(
       const InputRange& input_range,
+      const Point_map point_map,
+      const Normal_map normal_map,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
-    m_normal_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::normal_map), NormalMap())),
+    m_point_map(point_map),
+    m_normal_map(normal_map),
     m_traits(parameters::choose_parameter(parameters::get_parameter(
       np, internal_np::geom_traits), GeomTraits())),
     m_sqrt(Get_sqrt::sqrt_object(m_traits)),
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_circle_fit_sorting.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_circle_fit_sorting.h
index 2bc84eba645..0fc27825c05 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_circle_fit_sorting.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_circle_fit_sorting.h
@@ -112,15 +112,16 @@ namespace Point_set {
 
       \pre `input_range.size() > 0`
     */
-    template<typename CGAL_NP_TEMPLATE_PARAMETERS>
+    template<typename Dummy, typename CGAL_NP_TEMPLATE_PARAMETERS>
     Least_squares_circle_fit_sorting(
       const InputRange& input_range,
       NeighborQuery& neighbor_query,
+      const Point_map point_map,
+      const Dummy,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
     m_neighbor_query(neighbor_query),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
+    m_point_map(point_map),
     m_traits(parameters::choose_parameter(parameters::get_parameter(
       np, internal_np::geom_traits), GeomTraits())) {
 
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_cylinder_fit_region.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_cylinder_fit_region.h
index d0d9f3cbf1c..b6e51ca0fb7 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_cylinder_fit_region.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_cylinder_fit_region.h
@@ -162,12 +162,12 @@ namespace Point_set {
     template<typename CGAL_NP_TEMPLATE_PARAMETERS>
     Least_squares_cylinder_fit_region(
       const InputRange& input_range,
+      const Point_map point_map,
+      const Normal_map normal_map,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
-    m_normal_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::normal_map), NormalMap())),
+    m_point_map(point_map),
+    m_normal_map(normal_map),
     m_traits(parameters::choose_parameter(parameters::get_parameter(
       np, internal_np::geom_traits), GeomTraits())),
     m_sqrt(Get_sqrt::sqrt_object(m_traits)),
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_cylinder_fit_sorting.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_cylinder_fit_sorting.h
index 439b85accf3..381483f34e2 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_cylinder_fit_sorting.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_cylinder_fit_sorting.h
@@ -127,13 +127,13 @@ namespace Point_set {
     Least_squares_cylinder_fit_sorting(
       const InputRange& input_range,
       NeighborQuery& neighbor_query,
+      const Point_map point_map,
+      const Normal_map normal_map,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
     m_neighbor_query(neighbor_query),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
-    m_normal_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::normal_map), NormalMap())),
+    m_point_map(point_map),
+    m_normal_map(normal_map),
     m_traits(parameters::choose_parameter(parameters::get_parameter(
       np, internal_np::geom_traits), GeomTraits())) {
 
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_line_fit_region.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_line_fit_region.h
index 05cbb9d8420..44535aee23c 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_line_fit_region.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_line_fit_region.h
@@ -148,12 +148,12 @@ namespace Point_set {
     template<typename CGAL_NP_TEMPLATE_PARAMETERS>
     Least_squares_line_fit_region(
       const InputRange& input_range,
+      const Point_map point_map,
+      const Normal_map normal_map,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
-    m_normal_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::normal_map), NormalMap())),
+    m_point_map(point_map),
+    m_normal_map(normal_map),
     m_traits(parameters::choose_parameter(parameters::get_parameter(
       np, internal_np::geom_traits), GeomTraits())),
     m_squared_length_2(m_traits.compute_squared_length_2_object()),
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_line_fit_sorting.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_line_fit_sorting.h
index 0b8ea162493..d06cabd2d6e 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_line_fit_sorting.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_line_fit_sorting.h
@@ -112,15 +112,16 @@ namespace Point_set {
 
       \pre `input_range.size() > 0`
     */
-    template<typename CGAL_NP_TEMPLATE_PARAMETERS>
+    template<typename Dummy, typename CGAL_NP_TEMPLATE_PARAMETERS>
     Least_squares_line_fit_sorting(
       const InputRange& input_range,
       NeighborQuery& neighbor_query,
+      const Point_map point_map,
+      const Dummy,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
     m_neighbor_query(neighbor_query),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
+    m_point_map(point_map),
     m_traits(parameters::choose_parameter(parameters::get_parameter(
       np, internal_np::geom_traits), GeomTraits())) {
 
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_plane_fit_region.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_plane_fit_region.h
index 7d055d8c5d3..4452c99c28f 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_plane_fit_region.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_plane_fit_region.h
@@ -148,12 +148,12 @@ namespace Point_set {
     template<typename CGAL_NP_TEMPLATE_PARAMETERS>
     Least_squares_plane_fit_region(
       const InputRange& input_range,
+      const Point_map point_map,
+      const Normal_map normal_map,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
-    m_normal_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::normal_map), NormalMap())),
+    m_point_map(point_map),
+    m_normal_map(normal_map),
     m_traits(parameters::choose_parameter(parameters::get_parameter(
       np, internal_np::geom_traits), GeomTraits())),
     m_squared_length_3(m_traits.compute_squared_length_3_object()),
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_plane_fit_sorting.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_plane_fit_sorting.h
index 472e9a9433b..339aa538f7a 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_plane_fit_sorting.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_plane_fit_sorting.h
@@ -112,15 +112,17 @@ namespace Point_set {
 
       \pre `input_range.size() > 0`
     */
-    template<typename CGAL_NP_TEMPLATE_PARAMETERS>
+    template<typename Dummy,
+      typename CGAL_NP_TEMPLATE_PARAMETERS>
     Least_squares_plane_fit_sorting(
       const InputRange& input_range,
       NeighborQuery& neighbor_query,
+      const Point_map point_map,
+      const Dummy,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
     m_neighbor_query(neighbor_query),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
+    m_point_map(point_map),
     m_traits(parameters::choose_parameter(parameters::get_parameter(
       np, internal_np::geom_traits), GeomTraits())) {
 
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_sphere_fit_region.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_sphere_fit_region.h
index b2626570a7c..d2eb73f516d 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_sphere_fit_region.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_sphere_fit_region.h
@@ -161,12 +161,12 @@ namespace Point_set {
     template<typename CGAL_NP_TEMPLATE_PARAMETERS>
     Least_squares_sphere_fit_region(
       const InputRange& input_range,
+      const Point_map point_map,
+      const Normal_map normal_map,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
-    m_normal_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::normal_map), NormalMap())),
+    m_point_map(point_map),
+    m_normal_map(normal_map),
     m_traits(parameters::choose_parameter(parameters::get_parameter(
       np, internal_np::geom_traits), GeomTraits())),
     m_sqrt(Get_sqrt::sqrt_object(m_traits)),
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_sphere_fit_sorting.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_sphere_fit_sorting.h
index 54a4eb1996a..f9cb2cfac5c 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_sphere_fit_sorting.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Least_squares_sphere_fit_sorting.h
@@ -112,15 +112,17 @@ namespace Point_set {
 
       \pre `input_range.size() > 0`
     */
-    template<typename CGAL_NP_TEMPLATE_PARAMETERS>
+    template<typename Dummy,
+      typename CGAL_NP_TEMPLATE_PARAMETERS>
     Least_squares_sphere_fit_sorting(
       const InputRange& input_range,
       NeighborQuery& neighbor_query,
+      const Point_map point_map,
+      const Dummy,
       const CGAL_NP_CLASS& np= parameters::default_values()) :
     m_input_range(input_range),
     m_neighbor_query(neighbor_query),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
+    m_point_map(point_map),
     m_traits(parameters::choose_parameter(parameters::get_parameter(
       np, internal_np::geom_traits), GeomTraits())) {
 
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Sphere_neighbor_query.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Sphere_neighbor_query.h
index 542875bc0bb..a74280708e5 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Sphere_neighbor_query.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_point_set/Sphere_neighbor_query.h
@@ -131,10 +131,10 @@ namespace Point_set {
     template<typename CGAL_NP_TEMPLATE_PARAMETERS>
     Sphere_neighbor_query(
       const InputRange& input_range,
+      const Point_map point_map,
       const CGAL_NP_CLASS& np = parameters::default_values()) :
     m_input_range(input_range),
-    m_point_map(parameters::choose_parameter(parameters::get_parameter(
-      np, internal_np::point_map), PointMap())),
+    m_point_map(point_map),
     m_index_to_point_map(m_input_range, m_point_map),
     m_tree(
       boost::counting_iterator<std::size_t>(0),
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/free_functions.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/free_functions.h
index 797d418620f..ce2a440fc2d 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/free_functions.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/free_functions.h
@@ -16,11 +16,6 @@
 
 #include <CGAL/license/Shape_detection.h>
 
-#include <CGAL/Point_set_3.h>
-#include <CGAL/Surface_mesh.h>
-#include <CGAL/Polyhedron_3.h>
-#include <CGAL/HalfedgeDS_vector.h>
-
 #include <CGAL/Shape_detection/Region_growing/Region_growing.h>
 #include <CGAL/Shape_detection/Region_growing/Region_growing_on_point_set.h>
 #include <CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh.h>
@@ -28,7 +23,7 @@
 #include <CGAL/Shape_detection/Region_growing/Region_growing_on_segment_set.h>
 
 // TODO: this is not using generic programming for FaceGraph and PointSet
-//       as a consequence, artifical depencencies on Surface_mesh, Polyhedron,
+//       as a consequence, artificial dependencies on Surface_mesh, Polyhedron,
 //       HDS and Point_set_3 are done.
 
 namespace CGAL {
@@ -37,161 +32,79 @@ namespace internal {
 
 template<
 typename InputRange,
+typename PointMap,
+typename NormalMap,
 typename OutputIterator,
 typename CGAL_NP_TEMPLATE_PARAMETERS>
 OutputIterator region_growing_lines(
-  const InputRange& points_with_normals, OutputIterator regions, const CGAL_NP_CLASS& np = parameters::default_values()) {
+  const InputRange& points, PointMap point_map, NormalMap normal_map, OutputIterator regions, const CGAL_NP_CLASS& np = parameters::default_values()) {
 
-  using Input_range = InputRange;
-  using Point_with_normal = typename Input_range::value_type;
-  using Point_type = typename Point_with_normal::first_type;
-  using Traits = typename Kernel_traits<Point_type>::Kernel;
-  using Point_map = CGAL::First_of_pair_property_map<Point_with_normal>;
-  using Normal_map = CGAL::Second_of_pair_property_map<Point_with_normal>;
+  using Kernel = typename Kernel_traits<boost::property_traits<PointMap>::value_type>::Kernel;
+  using Neighbor_query = Point_set::K_neighbor_query<Kernel, InputRange, PointMap>;
+  using Region_type = Point_set::Least_squares_line_fit_region<Kernel, InputRange, PointMap, NormalMap>;
+  using Sorting = Point_set::Least_squares_line_fit_sorting<Kernel, InputRange, Neighbor_query, PointMap>;
+  using Region_growing = Region_growing<InputRange, Neighbor_query, Region_type, typename Sorting::Seed_map>;
 
-  using Neighbor_query = Point_set::K_neighbor_query<Traits, Input_range, Point_map>;
-  using Region_type = Point_set::Least_squares_line_fit_region<Traits, Input_range, Point_map, Normal_map>;
-  using Sorting = Point_set::Least_squares_line_fit_sorting<Traits, Input_range, Neighbor_query, Point_map>;
-  using Region_growing = Region_growing<Input_range, Neighbor_query, Region_type, typename Sorting::Seed_map>;
-
-  Neighbor_query neighbor_query(points_with_normals, np);
-  Region_type region_type(points_with_normals, np);
-  Sorting sorting(points_with_normals, neighbor_query, np);
+  Neighbor_query neighbor_query(points, point_map, np);
+  Region_type region_type(points, point_map, normal_map, np);
+  Sorting sorting(points, neighbor_query, point_map, np);
   sorting.sort();
 
   Region_growing region_growing(
-    points_with_normals, neighbor_query, region_type, sorting.seed_map());
+    points, neighbor_query, region_type, sorting.seed_map());
   region_growing.detect(regions);
   return regions;
 }
 
 template<
 typename InputRange,
+typename PointMap,
+typename NormalMap,
 typename OutputIterator,
 typename CGAL_NP_TEMPLATE_PARAMETERS>
 OutputIterator region_growing_planes(
-  const InputRange& points_with_normals, OutputIterator regions, const CGAL_NP_CLASS& np = parameters::default_values()) {
+  InputRange& points, PointMap point_map, NormalMap normal_map, OutputIterator regions, const CGAL_NP_CLASS& np = parameters::default_values()) {
 
-  using Input_range = InputRange;
-  using Point_with_normal = typename Input_range::value_type;
-  using Point_type = typename Point_with_normal::first_type;
-  using Traits = typename Kernel_traits<Point_type>::Kernel;
-  using Point_map = CGAL::First_of_pair_property_map<Point_with_normal>;
-  using Normal_map = CGAL::Second_of_pair_property_map<Point_with_normal>;
+  using Kernel = typename Kernel_traits<boost::property_traits<PointMap>::value_type>::Kernel;
+  using Neighbor_query = Point_set::K_neighbor_query<Kernel, InputRange, PointMap>;
+  using Region_type = Point_set::Least_squares_plane_fit_region<Kernel, InputRange, PointMap, NormalMap>;
+  using Sorting = Point_set::Least_squares_plane_fit_sorting<Kernel, InputRange, Neighbor_query, PointMap>;
+  using Region_growing = Region_growing<InputRange, Neighbor_query, Region_type, typename Sorting::Seed_map>;
 
-  using Neighbor_query = Point_set::K_neighbor_query<Traits, Input_range, Point_map>;
-  using Region_type = Point_set::Least_squares_plane_fit_region<Traits, Input_range, Point_map, Normal_map>;
-  using Sorting = Point_set::Least_squares_plane_fit_sorting<Traits, Input_range, Neighbor_query, Point_map>;
-  using Region_growing = Region_growing<Input_range, Neighbor_query, Region_type, typename Sorting::Seed_map>;
-
-  Neighbor_query neighbor_query(points_with_normals, np);
-  Region_type region_type(points_with_normals, np);
-  Sorting sorting(points_with_normals, neighbor_query, np);
+  Neighbor_query neighbor_query(points, point_map, np);
+  Region_type region_type(points, point_map, normal_map, np);
+  Sorting sorting(points, neighbor_query, point_map, np);
   sorting.sort();
 
   Region_growing region_growing(
-    points_with_normals, neighbor_query, region_type, sorting.seed_map());
+    points, neighbor_query, region_type, sorting.seed_map());
   region_growing.detect(regions);
   return regions;
 }
 
 template<
-typename PointType,
-typename VectorType,
-typename OutputIterator,
-typename CGAL_NP_TEMPLATE_PARAMETERS_NO_DEFAULT>
-OutputIterator region_growing_planes(
-  const CGAL::Point_set_3<PointType, VectorType>& point_set, OutputIterator regions, const CGAL_NP_CLASS& np) {
-
-  using Point_type = PointType;
-  using Vector_type = VectorType;
-  using Traits = typename Kernel_traits<Point_type>::Kernel;
-  using Point_set_3 = CGAL::Point_set_3<Point_type, Vector_type>;
-  using Point_map = typename Point_set_3::Point_map;
-  using Normal_map = typename Point_set_3::Vector_map;
-
-  using Neighbor_query = Point_set::K_neighbor_query<Traits, Point_set_3, Point_map>;
-  using Region_type = Point_set::Least_squares_plane_fit_region<Traits, Point_set_3, Point_map, Normal_map>;
-  using Sorting = Point_set::Least_squares_plane_fit_sorting<Traits, Point_set_3, Neighbor_query, Point_map>;
-  using Region_growing = Region_growing<Point_set_3, Neighbor_query, Region_type, typename Sorting::Seed_map>;
-
-  CGAL_precondition(point_set.has_normal_map());
-  Neighbor_query neighbor_query(point_set, np);
-  Region_type region_type(point_set, np);
-  Sorting sorting(point_set, neighbor_query, np);
-  sorting.sort();
-
-  Region_growing region_growing(
-    point_set, neighbor_query, region_type, sorting.seed_map());
-  region_growing.detect(regions);
-  return regions;
-}
-
-template<
-typename PointType,
-typename VectorType,
-typename OutputIterator>
-OutputIterator region_growing_planes(
-  const CGAL::Point_set_3<PointType, VectorType>& point_set, OutputIterator regions) {
-
-  return region_growing_planes(
-    point_set, regions, CGAL::parameters::
-    point_map(point_set.point_map()).
-    normal_map(point_set.normal_map()));
-}
-
-template<
-typename GeomTraits,
+typename PolygonMesh,
 typename OutputIterator,
 typename CGAL_NP_TEMPLATE_PARAMETERS>
-OutputIterator region_growing_planes(
-  const CGAL::Polyhedron_3<GeomTraits, CGAL::Polyhedron_items_3, CGAL::HalfedgeDS_vector>& polyhedron,
-  OutputIterator regions, const CGAL_NP_CLASS& np = parameters::default_values()) {
+OutputIterator region_growing_planes_polygon_mesh(
+  const PolygonMesh& polygon_mesh, OutputIterator regions, const CGAL_NP_CLASS& np = parameters::default_values()) {
 
-  using Traits = GeomTraits;
-  using Polyhedron = CGAL::Polyhedron_3<Traits, CGAL::Polyhedron_items_3, CGAL::HalfedgeDS_vector>;
-  using Face_range = typename CGAL::Iterator_range<typename boost::graph_traits<Polyhedron>::face_iterator>;
-  using Neighbor_query = Polygon_mesh::One_ring_neighbor_query<Polyhedron>;
-  using Region_type = Polygon_mesh::Least_squares_plane_fit_region<Traits, Polyhedron, Face_range>;
-  using Sorting = Polygon_mesh::Least_squares_plane_fit_sorting<Traits, Polyhedron, Neighbor_query, Face_range>;
+  using Kernel = typename Kernel_traits<typename PolygonMesh::Point>::Kernel;
+  using Face_iterator = typename boost::graph_traits<PolygonMesh>::face_iterator;
+  using Face_range = Iterator_range<typename Face_iterator>;
+
+  using Neighbor_query = Polygon_mesh::One_ring_neighbor_query<PolygonMesh>;
+  using Region_type = Polygon_mesh::Least_squares_plane_fit_region<Kernel, PolygonMesh, Face_range>;
+  using Sorting = Polygon_mesh::Least_squares_plane_fit_sorting<Kernel, PolygonMesh, Neighbor_query, Face_range>;
   using Region_growing = Region_growing<Face_range, Neighbor_query, Region_type, typename Sorting::Seed_map>;
 
-  const Face_range face_range = faces(polyhedron);
-  Neighbor_query neighbor_query(polyhedron);
-  Region_type region_type(polyhedron, np);
-  Sorting sorting(polyhedron, neighbor_query, np);
+  Neighbor_query neighbor_query(polygon_mesh);
+  Region_type region_type(polygon_mesh, np);
+  Sorting sorting(polygon_mesh, neighbor_query, np);
   sorting.sort();
 
   Region_growing region_growing(
-    face_range, neighbor_query, region_type, sorting.seed_map());
-  region_growing.detect(regions);
-  return regions;
-}
-
-template<
-typename PointType,
-typename OutputIterator,
-typename CGAL_NP_TEMPLATE_PARAMETERS>
-OutputIterator region_growing_planes(
-  const CGAL::Surface_mesh<PointType>& surface_mesh, OutputIterator regions, const CGAL_NP_CLASS& np = parameters::default_values()) {
-
-  using Point_type = PointType;
-  using Traits = typename Kernel_traits<Point_type>::Kernel;
-  using Surface_mesh = CGAL::Surface_mesh<Point_type>;
-  using Face_range = typename Surface_mesh::Face_range;
-  using Neighbor_query = Polygon_mesh::One_ring_neighbor_query<Surface_mesh>;
-  using Region_type = Polygon_mesh::Least_squares_plane_fit_region<Traits, Surface_mesh>;
-  using Sorting = Polygon_mesh::Least_squares_plane_fit_sorting<Traits, Surface_mesh, Neighbor_query>;
-  using Region_growing = Region_growing<Face_range, Neighbor_query, Region_type, typename Sorting::Seed_map>;
-
-  const Face_range face_range = faces(surface_mesh);
-  Neighbor_query neighbor_query(surface_mesh);
-  Region_type region_type(surface_mesh, np);
-  Sorting sorting(surface_mesh, neighbor_query, np);
-  sorting.sort();
-
-  Region_growing region_growing(
-    face_range, neighbor_query, region_type, sorting.seed_map());
+    faces(polygon_mesh), neighbor_query, region_type, sorting.seed_map());
   region_growing.detect(regions);
   return regions;
 }
@@ -203,15 +116,14 @@ typename CGAL_NP_TEMPLATE_PARAMETERS>
 OutputIterator region_growing_polylines(
   const InputRange& polyline, OutputIterator regions, const CGAL_NP_CLASS& np = parameters::default_values()) {
 
-  using Input_range = InputRange;
-  using Point_type = typename Input_range::value_type;
-  using Traits = typename Kernel_traits<Point_type>::Kernel;
+  using Point_type = typename InputRange::value_type;
+  using Kernel = typename Kernel_traits<Point_type>::Kernel;
   using Point_map = CGAL::Identity_property_map<Point_type>;
 
-  using Neighbor_query = Polyline::One_ring_neighbor_query<Traits, Input_range>;
-  using Region_type = Polyline::Least_squares_line_fit_region<Traits, Input_range, Point_map>;
-  using Sorting = Polyline::Least_squares_line_fit_sorting<Traits, Input_range, Neighbor_query, Point_map>;
-  using Region_growing = Region_growing<Input_range, Neighbor_query, Region_type, typename Sorting::Seed_map>;
+  using Neighbor_query = Polyline::One_ring_neighbor_query<Kernel, InputRange>;
+  using Region_type = Polyline::Least_squares_line_fit_region<Kernel, InputRange, Point_map>;
+  using Sorting = Polyline::Least_squares_line_fit_sorting<Kernel, InputRange, Neighbor_query, Point_map>;
+  using Region_growing = Region_growing<InputRange, Neighbor_query, Region_type, typename Sorting::Seed_map>;
 
   Neighbor_query neighbor_query(polyline);
   Region_type region_type(polyline, np);
diff --git a/Shape_detection/test/Shape_detection/test_region_growing_basic.cpp b/Shape_detection/test/Shape_detection/test_region_growing_basic.cpp
index de47140ae76..226d065485d 100644
--- a/Shape_detection/test/Shape_detection/test_region_growing_basic.cpp
+++ b/Shape_detection/test/Shape_detection/test_region_growing_basic.cpp
@@ -45,8 +45,8 @@ int main(int argc, char *argv[]) {
   assert(input_range.size() == 3634);
 
   // Create parameter classes.
-  Neighbor_query neighbor_query(input_range);
-  Region_type region_type(input_range);
+  Neighbor_query neighbor_query(input_range, Point_map());
+  Region_type region_type(input_range, Point_map(), Normal_map());
 
   // Run region growing.
   Region_growing region_growing(
diff --git a/Shape_detection/test/Shape_detection/test_region_growing_on_cube.cpp b/Shape_detection/test/Shape_detection/test_region_growing_on_cube.cpp
index 42fabc4a152..527eb38be26 100644
--- a/Shape_detection/test/Shape_detection/test_region_growing_on_cube.cpp
+++ b/Shape_detection/test/Shape_detection/test_region_growing_on_cube.cpp
@@ -94,7 +94,7 @@ bool test_region_growing_on_cube(int argc, char *argv[]) {
   // Test free functions and stability.
   for (std::size_t k = 0; k < 3; ++k) {
     regions.clear();
-    SD::internal::region_growing_planes(
+    SD::internal::region_growing_planes_polygon_mesh(
       polyhedron, std::back_inserter(regions),
       CGAL::parameters::
       maximum_distance(distance_threshold).
diff --git a/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_2.cpp b/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_2.cpp
index cae861cbdd1..d9384c3348e 100644
--- a/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_2.cpp
+++ b/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_2.cpp
@@ -58,9 +58,12 @@ bool test_region_growing_on_point_set_2(int argc, char *argv[]) {
 
   // Create parameter classes.
   Neighbor_query neighbor_query(
-    input_range, CGAL::parameters::sphere_radius(sphere_radius));
+    input_range, Point_map(),
+    CGAL::parameters::sphere_radius(sphere_radius));
   Region_type region_type(
     input_range,
+    Point_map(),
+    Normal_map(),
     CGAL::parameters::
     maximum_distance(distance_threshold).
     maximum_angle(angle_threshold).
diff --git a/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_2_with_sorting.cpp b/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_2_with_sorting.cpp
index b65e6fed294..ec50ed1fd9e 100644
--- a/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_2_with_sorting.cpp
+++ b/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_2_with_sorting.cpp
@@ -61,9 +61,9 @@ bool test(int argc, char** argv, const std::string name, const std::size_t minr,
 
   // Create parameter classes.
   Neighbor_query neighbor_query(
-    input_range, CGAL::parameters::k_neighbors(k));
+    input_range, Point_map(), CGAL::parameters::k_neighbors(k));
   Region_type region_type(
-    input_range,
+    input_range, Point_map(), Normal_map(),
     CGAL::parameters::
     maximum_distance(max_distance).
     maximum_angle(max_angle).
@@ -71,7 +71,7 @@ bool test(int argc, char** argv, const std::string name, const std::size_t minr,
 
   // Sort indices.
   Sorting sorting(
-    input_range, neighbor_query);
+    input_range, neighbor_query, Point_map(), Normal_map());
   sorting.sort();
 
   // Run region growing.
@@ -89,7 +89,8 @@ bool test(int argc, char** argv, const std::string name, const std::size_t minr,
   for (std::size_t k = 0; k < 3; ++k) {
     regions.clear();
     SD::internal::region_growing_lines(
-      input_range, std::back_inserter(regions),
+      input_range, Point_map(), Normal_map(),
+      std::back_inserter(regions),
       CGAL::parameters::
       maximum_distance(max_distance).
       maximum_angle(max_angle).
diff --git a/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_3.cpp b/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_3.cpp
index 83f2f17796e..a3f6290f7ea 100644
--- a/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_3.cpp
+++ b/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_3.cpp
@@ -55,18 +55,18 @@ bool test_region_growing_on_point_set_3(int argc, char *argv[]) {
 
   // Create parameter classes.
   Neighbor_query neighbor_query(
-    input_range, CGAL::parameters::
-    sphere_radius(sphere_radius).
-    point_map(input_range.point_map()));
+    input_range,
+    input_range.point_map(),
+    CGAL::parameters::
+    sphere_radius(sphere_radius));
 
   Region_type region_type(
-    input_range,
+    input_range, input_range.point_map(),
+    input_range.normal_map(),
     CGAL::parameters::
     maximum_distance(distance_threshold).
     maximum_angle(angle_threshold).
-    minimum_region_size(min_region_size).
-    point_map(input_range.point_map()).
-    normal_map(input_range.normal_map()));
+    minimum_region_size(min_region_size));
 
   // Run region growing.
   Region_growing region_growing(
diff --git a/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_3_with_sorting.cpp b/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_3_with_sorting.cpp
index 6d5d8890d51..93617fedc34 100644
--- a/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_3_with_sorting.cpp
+++ b/Shape_detection/test/Shape_detection/test_region_growing_on_point_set_3_with_sorting.cpp
@@ -65,16 +65,14 @@ bool test(
 
   // Create parameter classes.
   Neighbor_query neighbor_query(
-    input_range, CGAL::parameters::
-    k_neighbors(k).
-    point_map(input_range.point_map()));
+    input_range, input_range.point_map(), CGAL::parameters::
+    k_neighbors(k));
 
   // Sort indices.
   Sorting sorting(
     input_range, neighbor_query,
-    CGAL::parameters::
-    point_map(input_range.point_map()).
-    normal_map(input_range.normal_map()));
+    input_range.point_map(),
+    input_range.normal_map());
   sorting.sort();
 
   // Run region growing.
@@ -97,13 +95,14 @@ bool test(
   for (std::size_t k = 0; k < 3; ++k) {
     regions.clear();
     SD::internal::region_growing_planes(
-      input_range, std::back_inserter(regions),
+      input_range,
+      input_range.point_map(),
+      input_range.normal_map(),
+      std::back_inserter(regions),
       CGAL::parameters::
       maximum_distance(max_distance).
       maximum_angle(max_angle).
-      minimum_region_size(min_region_size).
-      point_map(input_range.point_map()).
-      normal_map(input_range.normal_map()));
+      minimum_region_size(min_region_size));
     assert(regions.size() == 7);
   }
 
@@ -122,12 +121,12 @@ int main(int argc, char *argv[]) {
        const FT          max_distance    = FT(2);
        const FT          max_angle       = FT(20);
        const std::size_t min_region_size = 50;
-       return Plane_region(input_range, CGAL::parameters::
-          maximum_distance(max_distance).
+       return Plane_region(input_range,
+          input_range.point_map(),
+          input_range.normal_map(),
+          CGAL::parameters::maximum_distance(max_distance).
           maximum_angle(max_angle).
-          minimum_region_size(min_region_size).
-          point_map(input_range.point_map()).
-          normal_map(input_range.normal_map()));
+          minimum_region_size(min_region_size));
     },
     [](const auto& region) -> bool {
       std::cout << "- num regions planes: " << region.size() << std::endl;
@@ -146,12 +145,12 @@ int main(int argc, char *argv[]) {
        const FT          max_distance    = FT(1) / FT(100);
        const FT          max_angle       = FT(10);
        const std::size_t min_region_size = 50;
-       return Sphere_region(input_range, CGAL::parameters::
-          maximum_distance(max_distance).
+       return Sphere_region(input_range,
+          input_range.point_map(),
+          input_range.normal_map(),
+          CGAL::parameters::maximum_distance(max_distance).
           maximum_angle(max_angle).
-          minimum_region_size(min_region_size).
-          point_map(input_range.point_map()).
-          normal_map(input_range.normal_map()));
+          minimum_region_size(min_region_size));
     },
     [](const auto& region) -> bool {
       std::cout << "- num regions spheres: " << region.size() << std::endl;
@@ -170,12 +169,12 @@ int main(int argc, char *argv[]) {
        const FT          max_distance    = FT(1) / FT(20);
        const FT          max_angle       = FT(5);
        const std::size_t min_region_size = 200;
-       return Cylinder_region(input_range, CGAL::parameters::
-          maximum_distance(max_distance).
+       return Cylinder_region(input_range,
+          input_range.point_map(),
+          input_range.normal_map(),
+          CGAL::parameters::maximum_distance(max_distance).
           maximum_angle(max_angle).
-          minimum_region_size(min_region_size).
-          point_map(input_range.point_map()).
-          normal_map(input_range.normal_map()));
+          minimum_region_size(min_region_size));
     },
     [](const auto& region) -> bool {
       std::cout << "- num regions cylinders: " << region.size() << std::endl;
diff --git a/Shape_detection/test/Shape_detection/test_region_growing_on_polygon_mesh_with_sorting.cpp b/Shape_detection/test/Shape_detection/test_region_growing_on_polygon_mesh_with_sorting.cpp
index 9b6bed4e294..ad5397b58d5 100644
--- a/Shape_detection/test/Shape_detection/test_region_growing_on_polygon_mesh_with_sorting.cpp
+++ b/Shape_detection/test/Shape_detection/test_region_growing_on_polygon_mesh_with_sorting.cpp
@@ -81,7 +81,7 @@ int main(int argc, char *argv[]) {
   // Test free functions and stability.
   for (std::size_t k = 0; k < 3; ++k) {
     regions.clear();
-    SD::internal::region_growing_planes(
+    SD::internal::region_growing_planes_polygon_mesh(
       polygon_mesh, std::back_inserter(regions),
       CGAL::parameters::
       maximum_distance(distance_threshold).
diff --git a/Shape_detection/test/Shape_detection/test_region_growing_strict.cpp b/Shape_detection/test/Shape_detection/test_region_growing_strict.cpp
index befb4ab030d..401d7b2a5a7 100644
--- a/Shape_detection/test/Shape_detection/test_region_growing_strict.cpp
+++ b/Shape_detection/test/Shape_detection/test_region_growing_strict.cpp
@@ -11,6 +11,11 @@
 #include <CGAL/assertions.h>
 #include <CGAL/property_map.h>
 
+#include <CGAL/Point_set_3.h>
+#include <CGAL/Surface_mesh.h>
+#include <CGAL/Polyhedron_3.h>
+#include <CGAL/HalfedgeDS_vector.h>
+
 #include <CGAL/Simple_cartesian.h>
 #include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
 #include <CGAL/Exact_predicates_exact_constructions_kernel.h>
@@ -45,7 +50,10 @@ bool test_lines_points_with_normals() {
   assert(points_with_normals.size() == 9);
   std::vector< std::vector<std::size_t> > regions;
   CGAL::Shape_detection::internal::region_growing_lines(
-    points_with_normals, std::back_inserter(regions));
+    points_with_normals,
+    CGAL::First_of_pair_property_map<std::pair<Point_2, Vector_2> >(),
+    CGAL::Second_of_pair_property_map<std::pair<Point_2, Vector_2> >(),
+    std::back_inserter(regions));
   assert(regions.size() == 3);
   assert(regions[0].size() == 3);
   assert(regions[1].size() == 3);
@@ -201,7 +209,7 @@ bool test_lines_segment_set_3() {
 
   assert(surface_mesh.number_of_faces() == 7320);
   std::vector< std::vector<std::size_t> > regions;
-  CGAL::Shape_detection::internal::region_growing_planes(
+  CGAL::Shape_detection::internal::region_growing_planes_polygon_mesh(
     surface_mesh, std::back_inserter(regions));
   assert(regions.size() == 9);
 
@@ -259,7 +267,10 @@ bool test_planes_points_with_normals() {
   assert(points_with_normals.size() == 9);
   std::vector< std::vector<std::size_t> > regions;
   CGAL::Shape_detection::internal::region_growing_planes(
-    points_with_normals, std::back_inserter(regions));
+    points_with_normals,
+    CGAL::First_of_pair_property_map<std::pair<Point_3, Vector_3> >(),
+    CGAL::Second_of_pair_property_map<std::pair<Point_3, Vector_3> >(),
+    std::back_inserter(regions));
   assert(regions.size() == 1);
   assert(regions[0].size() == 9);
   return true;
@@ -295,7 +306,10 @@ bool test_planes_point_set() {
 
   std::vector< std::vector<std::size_t> > regions;
   CGAL::Shape_detection::internal::region_growing_planes(
-    point_set, std::back_inserter(regions));
+    point_set,
+    point_set.point_map(),
+    point_set.normal_map(),
+    std::back_inserter(regions));
   assert(regions.size() == 1);
   assert(regions[0].size() == 9);
   return true;
@@ -318,7 +332,7 @@ bool test_planes_polyhedron() {
 
   assert(polyhedron.size_of_facets() == 4);
   std::vector< std::vector<std::size_t> > regions;
-  CGAL::Shape_detection::internal::region_growing_planes(
+  CGAL::Shape_detection::internal::region_growing_planes_polygon_mesh(
     polyhedron, std::back_inserter(regions));
   assert(regions.size() == polyhedron.size_of_facets());
   return true;
@@ -340,7 +354,7 @@ bool test_planes_surface_mesh() {
 
   assert(surface_mesh.number_of_faces() == 7320);
   std::vector< std::vector<std::size_t> > regions;
-  CGAL::Shape_detection::internal::region_growing_planes(
+  CGAL::Shape_detection::internal::region_growing_planes_polygon_mesh(
     surface_mesh, std::back_inserter(regions));
   assert(regions.size() == 9);
   return true;
diff --git a/Shape_detection/test/Shape_detection/test_validity_sampled_data.cpp b/Shape_detection/test/Shape_detection/test_validity_sampled_data.cpp
index 3a3062638c0..b5508aeff89 100644
--- a/Shape_detection/test/Shape_detection/test_validity_sampled_data.cpp
+++ b/Shape_detection/test/Shape_detection/test_validity_sampled_data.cpp
@@ -112,9 +112,10 @@ void test_copied_point_cloud (const Point_set& original_points, std::size_t nb)
   CGAL::Real_timer t;
   t.start();
   RG_query rg_query (
-    points, CGAL::parameters::sphere_radius(parameters.cluster_epsilon));
+    points, Point_map(),
+    CGAL::parameters::sphere_radius(parameters.cluster_epsilon));
   RG_region rg_region (
-    points,
+    points, Point_map(), Normal_map(),
     CGAL::parameters::
     maximum_distance(parameters.epsilon).
     cosine_value(parameters.normal_threshold).