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 73bca6c23bf..54760874786 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
@@ -66,14 +66,14 @@ namespace Point_set {
   class K_neighbor_query {
 
   public:
-
     /// \cond SKIP_IN_MANUAL
     using Traits = GeomTraits;
     using Input_range = InputRange;
     using Point_map = PointMap;
-
     using Point = typename Point_map::value_type;
+    /// \endcond
 
+  private:
     using Index_to_point_map =
     internal::Item_property_map<Input_range, Point_map>;
 
@@ -106,8 +106,8 @@ namespace Point_set {
 
     using Tree =
     typename Neighbor_search::Tree;
-    /// \endcond
 
+  public:
     /// \name Initialization
     /// @{
 
@@ -188,16 +188,19 @@ namespace Point_set {
 
     /// @}
 
+    /// \cond SKIP_IN_MANUAL
+    // A property map that can be used to access points when
+    // iterating over the indices counting items in the input range.
+    const Index_to_point_map& index_to_point_map() const {
+      return m_index_to_point_map;
+    }
+    /// \endcond
+
   private:
-
-    // Fields.
     const Input_range& m_input_range;
-
     const std::size_t m_number_of_neighbors;
-
     const Point_map m_point_map;
     const Index_to_point_map m_index_to_point_map;
-
     Distance m_distance;
     Tree m_tree;
   };
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 0dda7c92afc..2418077c5b0 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
@@ -69,7 +69,6 @@ namespace Point_set {
   class Least_squares_line_fit_region {
 
   public:
-
     /// \name Types
     /// @{
 
@@ -83,16 +82,18 @@ namespace Point_set {
     /// Number type.
     typedef typename GeomTraits::FT FT;
 
-    /// \cond SKIP_IN_MANUAL
+    /// @}
+
+  private:
     using Point_2 = typename Traits::Point_2;
     using Vector_2 = typename Traits::Vector_2;
     using Line_2 = typename Traits::Line_2;
 
-    using Local_traits = Exact_predicates_inexact_constructions_kernel;
-    using Local_FT = typename Local_traits::FT;
-    using Local_point_2 = typename Local_traits::Point_2;
-    using Local_line_2 = typename Local_traits::Line_2;
-    using To_local_converter = Cartesian_converter<Traits, Local_traits>;
+    using ITraits = Exact_predicates_inexact_constructions_kernel;
+    using IFT = typename ITraits::FT;
+    using IPoint_2 = typename ITraits::Point_2;
+    using ILine_2 = typename ITraits::Line_2;
+    using IConverter = Cartesian_converter<Traits, ITraits>;
 
     using Squared_length_2 = typename Traits::Compute_squared_length_2;
     using Squared_distance_2 = typename Traits::Compute_squared_distance_2;
@@ -100,10 +101,8 @@ namespace Point_set {
 
     using Get_sqrt = internal::Get_sqrt<Traits>;
     using Sqrt = typename Get_sqrt::Sqrt;
-    /// \endcond
-
-    /// @}
 
+  public:
     /// \name Initialization
     /// @{
 
@@ -161,7 +160,7 @@ namespace Point_set {
     m_squared_distance_2(traits.compute_squared_distance_2_object()),
     m_scalar_product_2(traits.compute_scalar_product_2_object()),
     m_sqrt(Get_sqrt::sqrt_object(traits)),
-    m_to_local_converter() {
+    m_iconverter() {
 
       CGAL_precondition(input_range.size() > 0);
 
@@ -264,19 +263,19 @@ namespace Point_set {
 
       } else { // update reference line and normal
 
-        std::vector<Local_point_2> points;
+        std::vector<IPoint_2> points;
         points.reserve(region.size());
 
         for (std::size_t i = 0; i < region.size(); ++i) {
           CGAL_precondition(region[i] < m_input_range.size());
 
           const auto& key = *(m_input_range.begin() + region[i]);
-          points.push_back(m_to_local_converter(get(m_point_map, key)));
+          points.push_back(m_iconverter(get(m_point_map, key)));
         }
         CGAL_postcondition(points.size() == region.size());
 
-        Local_line_2 fitted_line;
-        Local_point_2 fitted_centroid;
+        ILine_2 fitted_line;
+        IPoint_2 fitted_centroid;
 
         // The best fit line will be a line fitted to all region points with
         // its normal being perpendicular to the line.
@@ -284,8 +283,8 @@ namespace Point_set {
           points.begin(), points.end(),
           fitted_line, fitted_centroid,
           CGAL::Dimension_tag<0>(),
-          Local_traits(),
-          CGAL::Eigen_diagonalize_traits<Local_FT, 2>());
+          ITraits(),
+          CGAL::Eigen_diagonalize_traits<IFT, 2>());
 
         m_line_of_best_fit =
         Line_2(
@@ -305,8 +304,6 @@ namespace Point_set {
     /// @}
 
   private:
-
-    // Fields.
     const Input_range& m_input_range;
 
     const FT m_distance_threshold;
@@ -321,7 +318,7 @@ namespace Point_set {
     const Scalar_product_2 m_scalar_product_2;
     const Sqrt m_sqrt;
 
-    const To_local_converter m_to_local_converter;
+    const IConverter m_iconverter;
 
     Line_2 m_line_of_best_fit;
     Vector_2 m_normal_of_best_fit;
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 9fb5262239d..91a6b91677f 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
@@ -64,7 +64,6 @@ namespace Point_set {
   class Least_squares_line_fit_sorting {
 
   public:
-
     /// \name Types
     /// @{
 
@@ -86,6 +85,15 @@ namespace Point_set {
 
     /// @}
 
+  private:
+    using ITraits = Exact_predicates_inexact_constructions_kernel;
+    using IFT = typename ITraits::FT;
+    using IPoint_2 = typename ITraits::Point_2;
+    using ILine_2 = typename ITraits::Line_2;
+    using IConverter = Cartesian_converter<Traits, ITraits>;
+    using Compare_scores = internal::Compare_scores<IFT>;
+
+  public:
     /// \name Initialization
     /// @{
 
@@ -112,7 +120,7 @@ namespace Point_set {
     m_input_range(input_range),
     m_neighbor_query(neighbor_query),
     m_point_map(point_map),
-    m_to_local_converter() {
+    m_iconverter() {
 
       CGAL_precondition(input_range.size() > 0);
 
@@ -155,20 +163,17 @@ namespace Point_set {
     /// @}
 
   private:
+    const Input_range& m_input_range;
+    Neighbor_query& m_neighbor_query;
+    const Point_map m_point_map;
+    std::vector<std::size_t> m_order;
+    std::vector<IFT> m_scores;
+    const IConverter m_iconverter;
 
-    // Types.
-    using Local_traits = Exact_predicates_inexact_constructions_kernel;
-    using Local_FT = typename Local_traits::FT;
-    using Local_point_2 = typename Local_traits::Point_2;
-    using Local_line_2 = typename Local_traits::Line_2;
-    using To_local_converter = Cartesian_converter<Traits, Local_traits>;
-    using Compare_scores = internal::Compare_scores<Local_FT>;
-
-    // Functions.
     void compute_scores() {
 
       std::vector<std::size_t> neighbors;
-      std::vector<Local_point_2> points;
+      std::vector<IPoint_2> points;
 
       for (std::size_t i = 0; i < m_input_range.size(); ++i) {
 
@@ -181,31 +186,21 @@ namespace Point_set {
           CGAL_precondition(neighbors[j] < m_input_range.size());
 
           const auto& key = *(m_input_range.begin() + neighbors[j]);
-          points.push_back(m_to_local_converter(get(m_point_map, key)));
+          points.push_back(m_iconverter(get(m_point_map, key)));
         }
         CGAL_postcondition(points.size() == neighbors.size());
 
-        Local_line_2  fitted_line;
-        Local_point_2 fitted_centroid;
+        ILine_2  fitted_line;
+        IPoint_2 fitted_centroid;
 
         m_scores[i] = CGAL::linear_least_squares_fitting_2(
           points.begin(), points.end(),
           fitted_line, fitted_centroid,
           CGAL::Dimension_tag<0>(),
-          Local_traits(),
-          CGAL::Eigen_diagonalize_traits<Local_FT, 2>());
+          ITraits(),
+          CGAL::Eigen_diagonalize_traits<IFT, 2>());
       }
     }
-
-    // Fields.
-    const Input_range& m_input_range;
-    Neighbor_query& m_neighbor_query;
-    const Point_map m_point_map;
-
-    std::vector<std::size_t> m_order;
-    std::vector<Local_FT> m_scores;
-
-    const To_local_converter m_to_local_converter;
   };
 
 } // namespace Point_set
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 ca572c78f30..c00d147d327 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
@@ -69,7 +69,6 @@ namespace Point_set {
   class Least_squares_plane_fit_region {
 
   public:
-
     /// \name Types
     /// @{
 
@@ -83,16 +82,18 @@ namespace Point_set {
     /// Number type.
     typedef typename GeomTraits::FT FT;
 
-    /// \cond SKIP_IN_MANUAL
+    /// @}
+
+  private:
     using Point_3 = typename Traits::Point_3;
     using Vector_3 = typename Traits::Vector_3;
     using Plane_3 = typename Traits::Plane_3;
 
-    using Local_traits = Exact_predicates_inexact_constructions_kernel;
-    using Local_FT = typename Local_traits::FT;
-    using Local_point_3 = typename Local_traits::Point_3;
-    using Local_plane_3 = typename Local_traits::Plane_3;
-    using To_local_converter = Cartesian_converter<Traits, Local_traits>;
+    using ITraits = Exact_predicates_inexact_constructions_kernel;
+    using IFT = typename ITraits::FT;
+    using IPoint_3 = typename ITraits::Point_3;
+    using IPlane_3 = typename ITraits::Plane_3;
+    using IConverter = Cartesian_converter<Traits, ITraits>;
 
     using Squared_length_3 = typename Traits::Compute_squared_length_3;
     using Squared_distance_3 = typename Traits::Compute_squared_distance_3;
@@ -100,10 +101,8 @@ namespace Point_set {
 
     using Get_sqrt = internal::Get_sqrt<Traits>;
     using Sqrt = typename Get_sqrt::Sqrt;
-    /// \endcond
-
-    /// @}
 
+  public:
     /// \name Initialization
     /// @{
 
@@ -161,7 +160,7 @@ namespace Point_set {
     m_squared_distance_3(traits.compute_squared_distance_3_object()),
     m_scalar_product_3(traits.compute_scalar_product_3_object()),
     m_sqrt(Get_sqrt::sqrt_object(traits)),
-    m_to_local_converter() {
+    m_iconverter() {
 
       CGAL_precondition(input_range.size() > 0);
 
@@ -265,19 +264,19 @@ namespace Point_set {
 
       } else { // update reference plane and normal
 
-        std::vector<Local_point_3> points;
+        std::vector<IPoint_3> points;
         points.reserve(region.size());
 
         for (std::size_t i = 0; i < region.size(); ++i) {
           CGAL_precondition(region[i] < m_input_range.size());
 
           const auto& key = *(m_input_range.begin() + region[i]);
-          points.push_back(m_to_local_converter(get(m_point_map, key)));
+          points.push_back(m_iconverter(get(m_point_map, key)));
         }
         CGAL_postcondition(points.size() == region.size());
 
-        Local_plane_3 fitted_plane;
-        Local_point_3 fitted_centroid;
+        IPlane_3 fitted_plane;
+        IPoint_3 fitted_centroid;
 
         // The best fit plane will be a plane fitted to all region points with
         // its normal being perpendicular to the plane.
@@ -285,8 +284,8 @@ namespace Point_set {
           points.begin(), points.end(),
           fitted_plane, fitted_centroid,
           CGAL::Dimension_tag<0>(),
-          Local_traits(),
-          CGAL::Eigen_diagonalize_traits<Local_FT, 3>());
+          ITraits(),
+          CGAL::Eigen_diagonalize_traits<IFT, 3>());
 
         m_plane_of_best_fit =
         Plane_3(
@@ -306,8 +305,6 @@ namespace Point_set {
     /// @}
 
   private:
-
-    // Fields.
     const Input_range& m_input_range;
 
     const FT m_distance_threshold;
@@ -322,7 +319,7 @@ namespace Point_set {
     const Scalar_product_3 m_scalar_product_3;
     const Sqrt m_sqrt;
 
-    const To_local_converter m_to_local_converter;
+    const IConverter m_iconverter;
 
     Plane_3 m_plane_of_best_fit;
     Vector_3 m_normal_of_best_fit;
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 be22fa6946e..3919febc5fb 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
@@ -64,7 +64,6 @@ namespace Point_set {
   class Least_squares_plane_fit_sorting {
 
   public:
-
     /// \name Types
     /// @{
 
@@ -86,6 +85,15 @@ namespace Point_set {
 
     /// @}
 
+  private:
+    using ITraits = Exact_predicates_inexact_constructions_kernel;
+    using IFT = typename ITraits::FT;
+    using IPoint_3 = typename ITraits::Point_3;
+    using IPlane_3 = typename ITraits::Plane_3;
+    using IConverter = Cartesian_converter<Traits, ITraits>;
+    using Compare_scores = internal::Compare_scores<IFT>;
+
+  public:
     /// \name Initialization
     /// @{
 
@@ -112,7 +120,7 @@ namespace Point_set {
     m_input_range(input_range),
     m_neighbor_query(neighbor_query),
     m_point_map(point_map),
-    m_to_local_converter() {
+    m_iconverter() {
 
       CGAL_precondition(input_range.size() > 0);
 
@@ -155,20 +163,17 @@ namespace Point_set {
     /// @}
 
   private:
+    const Input_range& m_input_range;
+    Neighbor_query& m_neighbor_query;
+    const Point_map m_point_map;
+    std::vector<std::size_t> m_order;
+    std::vector<IFT> m_scores;
+    const IConverter m_iconverter;
 
-    // Types.
-    using Local_traits = Exact_predicates_inexact_constructions_kernel;
-    using Local_FT = typename Local_traits::FT;
-    using Local_point_3 = typename Local_traits::Point_3;
-    using Local_plane_3 = typename Local_traits::Plane_3;
-    using To_local_converter = Cartesian_converter<Traits, Local_traits>;
-    using Compare_scores = internal::Compare_scores<Local_FT>;
-
-    // Functions.
     void compute_scores() {
 
       std::vector<std::size_t> neighbors;
-      std::vector<Local_point_3> points;
+      std::vector<IPoint_3> points;
 
       for (std::size_t i = 0; i < m_input_range.size(); ++i) {
 
@@ -181,31 +186,21 @@ namespace Point_set {
           CGAL_precondition(neighbors[j] < m_input_range.size());
 
           const auto& key = *(m_input_range.begin() + neighbors[j]);
-          points.push_back(m_to_local_converter(get(m_point_map, key)));
+          points.push_back(m_iconverter(get(m_point_map, key)));
         }
         CGAL_postcondition(points.size() == neighbors.size());
 
-        Local_plane_3 fitted_plane;
-        Local_point_3 fitted_centroid;
+        IPlane_3 fitted_plane;
+        IPoint_3 fitted_centroid;
 
         m_scores[i] = CGAL::linear_least_squares_fitting_3(
           points.begin(), points.end(),
           fitted_plane, fitted_centroid,
           CGAL::Dimension_tag<0>(),
-          Local_traits(),
-          CGAL::Eigen_diagonalize_traits<Local_FT, 3>());
+          ITraits(),
+          CGAL::Eigen_diagonalize_traits<IFT, 3>());
       }
     }
-
-    // Fields.
-    const Input_range& m_input_range;
-    Neighbor_query& m_neighbor_query;
-    const Point_map m_point_map;
-
-    std::vector<std::size_t> m_order;
-    std::vector<Local_FT> m_scores;
-
-    const To_local_converter m_to_local_converter;
   };
 
 } // namespace Point_set
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 5dd67cef3d1..a70c6d42fd7 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
@@ -67,7 +67,6 @@ namespace Point_set {
   class Sphere_neighbor_query {
 
   public:
-
     /// \name Types
     /// @{
 
@@ -75,14 +74,15 @@ namespace Point_set {
     using Traits = GeomTraits;
     using Input_range = InputRange;
     using Point_map = PointMap;
-
     using Point = typename Point_map::value_type;
     /// \endcond
 
     /// Number type.
     typedef typename GeomTraits::FT FT;
 
-    /// \cond SKIP_IN_MANUAL
+    /// @}
+
+  private:
     using Index_to_point_map =
     internal::Item_property_map<Input_range, Point_map>;
 
@@ -102,10 +102,8 @@ namespace Point_set {
 
     using Tree
     = CGAL::Kd_tree<Search_traits, Splitter, CGAL::Tag_true, CGAL::Tag_true>;
-    /// \endcond
-
-    /// @}
 
+  public:
     /// \name Initialization
     /// @{
 
@@ -185,16 +183,19 @@ namespace Point_set {
 
     /// @}
 
+    /// \cond SKIP_IN_MANUAL
+    // A property map that can be used to access points when
+    // iterating over the indices counting items in the input range.
+    const Index_to_point_map& index_to_point_map() const {
+      return m_index_to_point_map;
+    }
+    /// \endcond
+
   private:
-
-    // Fields.
     const Input_range& m_input_range;
-
     const FT m_sphere_radius;
-
     const Point_map m_point_map;
     const Index_to_point_map m_index_to_point_map;
-
     Tree m_tree;
   };
 
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_region.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_region.h
index dc9caa0ff02..7dbd5f0823a 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_region.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_region.h
@@ -78,22 +78,31 @@ namespace Polygon_mesh {
   class Least_squares_plane_fit_region {
 
   public:
+    /// \name Types
+    /// @{
 
     /// \cond SKIP_IN_MANUAL
     using Traits = GeomTraits;
     using Face_graph = PolygonMesh;
     using Face_range = FaceRange;
     using Vertex_to_point_map = VertexToPointMap;
+    /// \endcond
 
+    /// Number type.
+    typedef typename GeomTraits::FT FT;
+
+    /// @}
+
+  private:
     using Point_3 = typename Traits::Point_3;
     using Vector_3 = typename Traits::Vector_3;
     using Plane_3 = typename Traits::Plane_3;
 
-    using Local_traits = Exact_predicates_inexact_constructions_kernel;
-    using Local_FT = typename Local_traits::FT;
-    using Local_point_3 = typename Local_traits::Point_3;
-    using Local_plane_3 = typename Local_traits::Plane_3;
-    using To_local_converter = Cartesian_converter<Traits, Local_traits>;
+    using ITraits = Exact_predicates_inexact_constructions_kernel;
+    using IFT = typename ITraits::FT;
+    using IPoint_3 = typename ITraits::Point_3;
+    using IPlane_3 = typename ITraits::Plane_3;
+    using IConverter = Cartesian_converter<Traits, ITraits>;
 
     using Squared_length_3 = typename Traits::Compute_squared_length_3;
     using Squared_distance_3 = typename Traits::Compute_squared_distance_3;
@@ -102,16 +111,8 @@ namespace Polygon_mesh {
 
     using Get_sqrt = internal::Get_sqrt<Traits>;
     using Sqrt = typename Get_sqrt::Sqrt;
-    /// \endcond
-
-    /// \name Types
-    /// @{
-
-    /// Number type.
-    typedef typename GeomTraits::FT FT;
-
-    /// @}
 
+  public:
     /// \name Initialization
     /// @{
 
@@ -164,7 +165,7 @@ namespace Polygon_mesh {
     m_scalar_product_3(traits.compute_scalar_product_3_object()),
     m_cross_product_3(traits.construct_cross_product_vector_3_object()),
     m_sqrt(Get_sqrt::sqrt_object(traits)),
-    m_to_local_converter() {
+    m_iconverter() {
 
       CGAL_precondition(m_face_range.size() > 0);
 
@@ -266,7 +267,7 @@ namespace Polygon_mesh {
 
       } else { // update reference plane and normal
 
-        std::vector<Local_point_3> points;
+        std::vector<IPoint_3> points;
         for (std::size_t i = 0; i < region.size(); ++i) {
 
           CGAL_precondition(region[i] < m_face_range.size());
@@ -278,13 +279,13 @@ namespace Polygon_mesh {
           for (const auto vertex : vertices) {
 
             const Point_3& tmp_point = get(m_vertex_to_point_map, vertex);
-            points.push_back(m_to_local_converter(tmp_point));
+            points.push_back(m_iconverter(tmp_point));
           }
         }
         CGAL_postcondition(points.size() > 0);
 
-        Local_plane_3 fitted_plane;
-        Local_point_3 fitted_centroid;
+        IPlane_3 fitted_plane;
+        IPoint_3 fitted_centroid;
 
         // The best fit plane will be a plane fitted to all vertices of all
         // region faces with its normal being perpendicular to the plane.
@@ -299,8 +300,8 @@ namespace Polygon_mesh {
           points.begin(), points.end(),
           fitted_plane, fitted_centroid,
           CGAL::Dimension_tag<0>(),
-          Local_traits(),
-          CGAL::Eigen_diagonalize_traits<Local_FT, 3>());
+          ITraits(),
+          CGAL::Eigen_diagonalize_traits<IFT, 3>());
 
         const Plane_3 unoriented_plane_of_best_fit =
         Plane_3(
@@ -351,6 +352,25 @@ namespace Polygon_mesh {
     /// @}
 
   private:
+    const Face_graph& m_face_graph;
+    const Face_range m_face_range;
+
+    const FT m_distance_threshold;
+    const FT m_normal_threshold;
+    const std::size_t m_min_region_size;
+
+    const Vertex_to_point_map m_vertex_to_point_map;
+
+    const Squared_length_3 m_squared_length_3;
+    const Squared_distance_3 m_squared_distance_3;
+    const Scalar_product_3 m_scalar_product_3;
+    const Cross_product_3 m_cross_product_3;
+    const Sqrt m_sqrt;
+
+    const IConverter m_iconverter;
+
+    Plane_3 m_plane_of_best_fit;
+    Vector_3 m_normal_of_best_fit;
 
     template<typename Face>
     void get_face_centroid(
@@ -444,27 +464,6 @@ namespace Polygon_mesh {
 
       return max_face_distance;
     }
-
-    // Fields.
-    const Face_graph& m_face_graph;
-    const Face_range m_face_range;
-
-    const FT m_distance_threshold;
-    const FT m_normal_threshold;
-    const std::size_t m_min_region_size;
-
-    const Vertex_to_point_map m_vertex_to_point_map;
-
-    const Squared_length_3 m_squared_length_3;
-    const Squared_distance_3 m_squared_distance_3;
-    const Scalar_product_3 m_scalar_product_3;
-    const Cross_product_3 m_cross_product_3;
-    const Sqrt m_sqrt;
-
-    const To_local_converter m_to_local_converter;
-
-    Plane_3 m_plane_of_best_fit;
-    Vector_3 m_normal_of_best_fit;
   };
 
 } // namespace Polygon_mesh
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_sorting.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_sorting.h
index 9c9b62ac44a..c30b69db2cf 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_sorting.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/Least_squares_plane_fit_sorting.h
@@ -77,7 +77,6 @@ namespace Polygon_mesh {
   class Least_squares_plane_fit_sorting {
 
   public:
-
     /// \name Types
     /// @{
 
@@ -100,6 +99,15 @@ namespace Polygon_mesh {
 
     /// @}
 
+  private:
+    using ITraits = Exact_predicates_inexact_constructions_kernel;
+    using IFT = typename ITraits::FT;
+    using IPoint_3 = typename ITraits::Point_3;
+    using IPlane_3 = typename ITraits::Plane_3;
+    using IConverter = Cartesian_converter<Traits, ITraits>;
+    using Compare_scores = internal::Compare_scores<IFT>;
+
+  public:
     /// \name Initialization
     /// @{
 
@@ -127,7 +135,7 @@ namespace Polygon_mesh {
     m_neighbor_query(neighbor_query),
     m_face_range(faces(m_face_graph)),
     m_vertex_to_point_map(vertex_to_point_map),
-    m_to_local_converter() {
+    m_iconverter() {
 
       CGAL_precondition(m_face_range.size() > 0);
 
@@ -170,20 +178,18 @@ namespace Polygon_mesh {
     /// @}
 
   private:
+    const Face_graph& m_face_graph;
+    Neighbor_query& m_neighbor_query;
+    const Face_range m_face_range;
+    const Vertex_to_point_map m_vertex_to_point_map;
+    std::vector<std::size_t> m_order;
+    std::vector<IFT> m_scores;
+    const IConverter m_iconverter;
 
-    // Types.
-    using Local_traits = Exact_predicates_inexact_constructions_kernel;
-    using Local_FT = typename Local_traits::FT;
-    using Local_point_3 = typename Local_traits::Point_3;
-    using Local_plane_3 = typename Local_traits::Plane_3;
-    using To_local_converter = Cartesian_converter<Traits, Local_traits>;
-    using Compare_scores = internal::Compare_scores<Local_FT>;
-
-    // Functions.
     void compute_scores() {
 
       std::vector<std::size_t> neighbors;
-      std::vector<Local_point_3> points;
+      std::vector<IPoint_3> points;
 
       for (std::size_t i = 0; i < m_face_range.size(); ++i) {
 
@@ -202,33 +208,22 @@ namespace Polygon_mesh {
           for (const auto vertex : vertices) {
 
             const auto& tmp_point = get(m_vertex_to_point_map, vertex);
-            points.push_back(m_to_local_converter(tmp_point));
+            points.push_back(m_iconverter(tmp_point));
           }
         }
         CGAL_postcondition(points.size() > 0);
 
-        Local_plane_3 fitted_plane;
-        Local_point_3 fitted_centroid;
+        IPlane_3 fitted_plane;
+        IPoint_3 fitted_centroid;
 
         m_scores[i] = CGAL::linear_least_squares_fitting_3(
           points.begin(), points.end(),
           fitted_plane, fitted_centroid,
           CGAL::Dimension_tag<0>(),
-          Local_traits(),
-          CGAL::Eigen_diagonalize_traits<Local_FT, 3>());
+          ITraits(),
+          CGAL::Eigen_diagonalize_traits<IFT, 3>());
       }
     }
-
-    // Fields.
-    const Face_graph& m_face_graph;
-    Neighbor_query& m_neighbor_query;
-    const Face_range m_face_range;
-    const Vertex_to_point_map m_vertex_to_point_map;
-
-    std::vector<std::size_t> m_order;
-    std::vector<Local_FT> m_scores;
-
-    const To_local_converter m_to_local_converter;
   };
 
 } // namespace Polygon_mesh
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/One_ring_neighbor_query.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/One_ring_neighbor_query.h
index ebbdb23da18..acc435e5174 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/One_ring_neighbor_query.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/Region_growing_on_polygon_mesh/One_ring_neighbor_query.h
@@ -58,15 +58,16 @@ namespace Polygon_mesh {
   class One_ring_neighbor_query {
 
   public:
-
     /// \cond SKIP_IN_MANUAL
     using Face_graph = PolygonMesh;
     using Face_range = FaceRange;
-
-    using Face_to_index_map
-    = internal::Item_to_index_property_map<Face_range>;
     /// \endcond
 
+  private:
+    using Face_to_index_map
+    = internal::Item_to_index_property_map<Face_range>;
+
+  public:
     /// \name Initialization
     /// @{
 
@@ -129,12 +130,16 @@ namespace Polygon_mesh {
 
     /// @}
 
-  private:
+    /// \cond SKIP_IN_MANUAL
+    // A property map that can be used to access indices of the input faces.
+    const Face_to_index_map& face_to_index_map() const {
+      return m_face_to_index_map;
+    }
+    /// \endcond
 
-    // Fields.
+  private:
     const Face_graph& m_face_graph;
     const Face_range m_face_range;
-
     const Face_to_index_map m_face_to_index_map;
   };
 
diff --git a/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/utils.h b/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/utils.h
index b0d25c7fb7e..ca17c3b4baa 100644
--- a/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/utils.h
+++ b/Shape_detection/include/CGAL/Shape_detection/Region_growing/internal/utils.h
@@ -108,20 +108,18 @@ namespace internal {
     using Traits = typename Kernel_traits<Plane_3>::Kernel;
     using FT = typename Traits::FT;
 
-    using Local_traits =
-    CGAL::Exact_predicates_inexact_constructions_kernel;
-    using To_local_converter =
-    Cartesian_converter<Traits, Local_traits>;
+    using ITraits = CGAL::Exact_predicates_inexact_constructions_kernel;
+    using IConverter = Cartesian_converter<Traits, ITraits>;
 
-                using Local_FT = typename Local_traits::FT;
-    using Local_point_3 = typename Local_traits::Point_3;
-                using Local_plane_3 = typename Local_traits::Plane_3;
+    using IFT = typename ITraits::FT;
+    using IPoint_3 = typename ITraits::Point_3;
+    using IPlane_3 = typename ITraits::Plane_3;
 
     planes.clear();
     planes.reserve(regions.size());
 
-    std::vector<Local_point_3> points;
-    const To_local_converter to_local_converter = To_local_converter();
+    std::vector<IPoint_3> points;
+    const IConverter iconverter = IConverter();
 
     for (const auto& region : regions) {
       CGAL_assertion(region.size() > 0);
@@ -132,21 +130,21 @@ namespace internal {
         CGAL_precondition(region[i] < input_range.size());
 
         const auto& key = *(input_range.begin() + region[i]);
-        points.push_back(to_local_converter(get(point_map, key)));
+        points.push_back(iconverter(get(point_map, key)));
       }
       CGAL_postcondition(points.size() == region.size());
 
-      Local_plane_3 fitted_plane;
-      Local_point_3 fitted_centroid;
+      IPlane_3 fitted_plane;
+      IPoint_3 fitted_centroid;
 
       CGAL::linear_least_squares_fitting_3(
         points.begin(), points.end(),
         fitted_plane, fitted_centroid,
         CGAL::Dimension_tag<0>(),
-        Local_traits(),
-        CGAL::Eigen_diagonalize_traits<Local_FT, 3>());
+        ITraits(),
+        CGAL::Eigen_diagonalize_traits<IFT, 3>());
 
-                  const Plane_3 plane = Plane_3(
+        const Plane_3 plane = Plane_3(
         static_cast<FT>(fitted_plane.a()),
         static_cast<FT>(fitted_plane.b()),
         static_cast<FT>(fitted_plane.c()),