Updated

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Arrangement_on_surface_2.txt

@@ -3807,7 +3807,7 @@ disjoint line segments, defined in the input file `Europe.dat`.
 
 In the following example we use the predefined
 `Exact_predicates_exact_constructions_kernel` for instantiating our
-segment-traits class. This kernel use interval arithmetic to filter
+segment-traits class. This kernel uses interval arithmetic to filter
 the exact computations. The program reads a set of line segments with
 integer coordinates from a file and computes their arrangement. By
 default it opens the `fan_grids.dat` input-file, located in the

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/arr_polylines.h

@@ -6,13 +6,14 @@
 #include <CGAL/Arr_polyline_traits_2.h>
 #include <CGAL/Arrangement_2.h>
 
-typedef CGAL::Exact_predicates_exact_constructions_kernel  Kernel;
+typedef CGAL::Exact_predicates_exact_constructions_kernel Kernel;
-typedef Kernel::FT                                         Number_type;
+typedef Kernel::FT                                        Number_type;
 
-typedef CGAL::Arr_segment_traits_2<Kernel>                 Segment_traits;
+typedef CGAL::Arr_segment_traits_2<Kernel>                Segment_traits;
-typedef CGAL::Arr_polyline_traits_2<Segment_traits>        Traits;
+typedef CGAL::Arr_polyline_traits_2<Segment_traits>       Traits;
-typedef Traits::Point_2                                    Point;
+typedef Traits::Point_2                                   Point;
-typedef Traits::Curve_2                                    Polyline;
+typedef Traits::Segment_2                                 Segment;
-typedef CGAL::Arrangement_2<Traits>                        Arrangement;
+typedef Traits::Curve_2                                   Polyline;
+typedef CGAL::Arrangement_2<Traits>                       Arrangement;
 
 #endif

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/polycurve_bezier.cpp

@@ -48,11 +48,11 @@ int main() {
   Bezier_curve B;
 
   in_file >> n_curves;
-  unsigned int k;
+  size_t k;
   for (k = 0; k < n_curves; ++k) {
     // Read the current curve (specified by its control points).
     in_file >> B;
-    //convert it into x-monotone bezier curve.
+    // convert it into x-monotone bezier curve.
     std::vector<CGAL::Object> obj_vector;
     bezier_traits.make_x_monotone_2_object()(B, std::back_inserter(obj_vector));
     Bezier_x_monotone_curve x_seg =
@@ -61,16 +61,9 @@ int main() {
   }
 
   X_mono_polycurve polycurve = ctr_xpolycurve(x_curves.begin(), x_curves.end());
-
-  // Construct the arrangement.
   Arrangement_2 arr;
-  insert(arr, polycurve);
+  insert(arr, polycurve);               // construct the arrangement
-
+  print_arrangement_size(arr);          // print the arrangement size
-  // Print the arrangement size.
-  std::cout << "The arrangement size:" << std::endl
-            << "   V = " << arr.number_of_vertices()
-            << ",  E = " << arr.number_of_edges()
-            << ",  F = " << arr.number_of_faces() << std::endl;
 
   return 0;
 }

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/polycurves_basic.cpp

@@ -8,6 +8,8 @@
 #include <CGAL/Arr_polycurve_basic_traits_2.h>
 #include <CGAL/Arrangement_2.h>
 
+#include "arr_print.h"
+
 typedef CGAL::Exact_predicates_exact_constructions_kernel Kernel;
 typedef CGAL::Arr_directional_non_caching_segment_basic_traits_2<Kernel>
                                                             Subcurve_traits;
@@ -40,8 +42,7 @@ int main() {
   insert_non_intersecting_curve(arr, pc1);
   insert_non_intersecting_curve(arr, pc2);
 
-  std::cout << "# vertices: " << arr.number_of_vertices() << std::endl;;
+  print_arrangement_size(arr);          // print the arrangement size
-  std::cout << "# halfedges: " << arr.number_of_halfedges() << std::endl;;
+
-  std::cout << "# faces: " << arr.number_of_faces() << std::endl;;
   return 0;
 }

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/polylines.cpp

@@ -7,58 +7,42 @@
 #include "arr_polylines.h"
 #include "arr_print.h"
 
-/* Define the Arrangement traits class to be used. You can either use some user
- * defined kernel and Segment_traits_2 or the defaults.
- */
-
-// Instantiate the traits class using a user-defined kernel and Segment_traits_2.
-// Identical instantiation can be achieved using the default Kernel:
-// typedef CGAL::Arr_polyline_traits_2<>                    Geom_traits_2;
-typedef CGAL::Exact_predicates_exact_constructions_kernel Kernel;
-typedef CGAL::Arr_segment_traits_2<Kernel>                Segment_traits_2;
-typedef CGAL::Arr_polyline_traits_2<Segment_traits_2>     Geom_traits_2;
-
-typedef Geom_traits_2::Point_2                            Point_2;
-typedef Geom_traits_2::Segment_2                          Segment_2;
-typedef Geom_traits_2::Curve_2                            Polyline_2;
-typedef CGAL::Arrangement_2<Geom_traits_2>                Arrangement_2;
-
 int main() {
-  Geom_traits_2 traits;
+  Traits traits;
-  Arrangement_2 arr(&traits);
+  Arrangement arr(&traits);
 
   auto polyline_construct = traits.construct_curve_2_object();
 
-  Point_2 points1[5];
+  Point points1[5];
-  points1[0] = Point_2(0, 0);
+  points1[0] = Point(0, 0);
-  points1[1] = Point_2(2, 4);
+  points1[1] = Point(2, 4);
-  points1[2] = Point_2(3, 0);
+  points1[2] = Point(3, 0);
-  points1[3] = Point_2(4, 4);
+  points1[3] = Point(4, 4);
-  points1[4] = Point_2(6, 0);
+  points1[4] = Point(6, 0);
-  Polyline_2 pi1 = polyline_construct(&points1[0], &points1[5]);
+  Polyline pi1 = polyline_construct(&points1[0], &points1[5]);
 
-  std::list<Point_2> points2;
+  std::list<Point> points2;
-  points2.push_back(Point_2(1, 3));
+  points2.push_back(Point(1, 3));
-  points2.push_back(Point_2(0, 2));
+  points2.push_back(Point(0, 2));
-  points2.push_back(Point_2(1, 0));
+  points2.push_back(Point(1, 0));
-  points2.push_back(Point_2(2, 1));
+  points2.push_back(Point(2, 1));
-  points2.push_back(Point_2(3, 0));
+  points2.push_back(Point(3, 0));
-  points2.push_back(Point_2(4, 1));
+  points2.push_back(Point(4, 1));
-  points2.push_back(Point_2(5, 0));
+  points2.push_back(Point(5, 0));
-  points2.push_back(Point_2(6, 2));
+  points2.push_back(Point(6, 2));
-  points2.push_back(Point_2(5, 3));
+  points2.push_back(Point(5, 3));
-  points2.push_back(Point_2(4, 2));
+  points2.push_back(Point(4, 2));
-  Polyline_2 pi2 = polyline_construct(points2.begin(), points2.end());
+  Polyline pi2 = polyline_construct(points2.begin(), points2.end());
 
-  std::vector<Segment_2> segs;
+  std::vector<Segment> segs;
-  segs.push_back(Segment_2(Point_2(0, 2), Point_2(1, 2)));
+  segs.push_back(Segment(Point(0, 2), Point(1, 2)));
-  segs.push_back(Segment_2(Point_2(1, 2), Point_2(3, 6)));
+  segs.push_back(Segment(Point(1, 2), Point(3, 6)));
-  segs.push_back(Segment_2(Point_2(3, 6), Point_2(5, 2)));
+  segs.push_back(Segment(Point(3, 6), Point(5, 2)));
-  Polyline_2 pi3 = polyline_construct(segs.begin(), segs.end());
+  Polyline pi3 = polyline_construct(segs.begin(), segs.end());
 
   insert(arr, pi1);
   insert(arr, pi2);
   insert(arr, pi3);
-  print_arrangement(arr);
+  print_arrangement_size(arr);          // print the arrangement size
   return 0;
 }

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/predefined_kernel.cpp

@@ -2,19 +2,14 @@
 // Constructing an arrangement of intersecting line segments using the
 // predefined kernel with exact constructions and exact predicates.
 
-#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
-#include <CGAL/Arr_segment_traits_2.h>
-#include <CGAL/Arrangement_2.h>
-#include <CGAL/Timer.h>
 #include <list>
-#include <fstream>
 
-typedef CGAL::Exact_predicates_exact_constructions_kernel  Kernel;
+#include <CGAL/basic.h>
-typedef Kernel::FT                                         Number_type;
+#include <CGAL/Timer.h>
-typedef CGAL::Arr_segment_traits_2<Kernel>                 Traits_2;
+
-typedef Traits_2::Point_2                                  Point_2;
+#include "arr_exact_construction_segments.h"
-typedef Traits_2::X_monotone_curve_2                       Segment_2;
+#include "arr_print.h"
-typedef CGAL::Arrangement_2<Traits_2>                      Arrangement_2;
+#include "read_objects.h"
 
 int main (int argc, char* argv[]) {
   // Get the name of the input file from the command line, or use the default
@@ -23,48 +18,26 @@ int main (int argc, char* argv[]) {
 
   // Open the input file.
   std::ifstream in_file(filename);
-
   if (! in_file.is_open()) {
     std::cerr << "Failed to open " << filename << " ..." << std::endl;
     return (1);
   }
 
-  // Read the segments from the file.
+  std::list<Segment> segments;
-  // The input file format should be (all coordinate values are integers):
+  read_objects<Segment>(filename, std::back_inserter(segments));
-  // <n>                                 // number of segments.
-  // <sx_1> <sy_1>  <tx_1> <ty_1>        // source and target of segment #1.
-  // <sx_2> <sy_2>  <tx_2> <ty_2>        // source and target of segment #2.
-  //   :      :       :      :
-  // <sx_n> <sy_n>  <tx_n> <ty_n>        // source and target of segment #n.
-
-  std::list<Segment_2>  segments;
-
-  unsigned int n;
-  in_file >> n;
-  unsigned int i;
-  for (i = 0; i < n; ++i) {
-    int sx, sy, tx, ty;
-    in_file >> sx >> sy >> tx >> ty;
-    segments.push_back (Segment_2 (Point_2 (Number_type(sx), Number_type(sy)),
-                                   Point_2 (Number_type(tx), Number_type(ty))));
-  }
-  in_file.close();
 
   // Construct the arrangement by aggregately inserting all segments.
-  Arrangement_2 arr;
+  Arrangement arr;
   CGAL::Timer timer;
 
   std::cout << "Performing aggregated insertion of "
-            << n << " segments." << std::endl;
+            << segments.size() << " segments." << std::endl;
 
   timer.start();
   insert(arr, segments.begin(), segments.end());
   timer.stop();
 
-  // Print the arrangement dimensions.
+  print_arrangement_size(arr);
-  std::cout << "V = " << arr.number_of_vertices()
-            << ",  E = " << arr.number_of_edges()
-            << ",  F = " << arr.number_of_faces() << std::endl;
 
   std::cout << "Construction took " << timer.time()
             << " seconds." << std::endl;

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/spherical_degenerate_insert1.cpp

@@ -1,75 +1,52 @@
 //! \file examples/Arrangement_on_surface_2/aggregated_insertion.cpp
 // Using the global aggregated insertion functions.
 
-#include <list>
+#include "arr_geodesic_on_sphere.h"
-
+#include "arr_print.h"
-#include <CGAL/basic.h>
-#include <CGAL/Exact_rational.h>
-#include <CGAL/Cartesian.h>
-#include <CGAL/Arrangement_on_surface_2.h>
-#include <CGAL/Arr_geodesic_arc_on_sphere_traits_2.h>
-#include <CGAL/Arr_spherical_topology_traits_2.h>
-
-typedef CGAL::Exact_rational                               Number_type;
-typedef CGAL::Cartesian<Number_type>                       Kernel;
-typedef CGAL::Arr_geodesic_arc_on_sphere_traits_2<Kernel>  Geom_traits_2;
-typedef Geom_traits_2::Point_2                             Point_2;
-typedef Geom_traits_2::X_monotone_curve_2                  X_monotone_curve_2;
-typedef CGAL::Arr_spherical_topology_traits_2<Geom_traits_2> Topol_traits_2;
-typedef CGAL::Arrangement_on_surface_2<Geom_traits_2, Topol_traits_2>
-                                                           Arrangement_2;
-typedef Arrangement_2::Vertex_handle                       Vertex_handle;
 
 int main() {
-  Geom_traits_2 traits;
+  Geom_traits traits;
-  Geom_traits_2::Construct_point_2 ctr_p = traits.construct_point_2_object();
+  auto ctr_p = traits.construct_point_2_object();
-  Geom_traits_2::Construct_x_monotone_curve_2 ctr_xcv =
+  auto ctr_xcv = traits.construct_x_monotone_curve_2_object();
-    traits.construct_x_monotone_curve_2_object();
 
-  Arrangement_2 arr(&traits);
+  Arrangement arr(&traits);
-  Point_2 sp = ctr_p(0, 0, -1);
+  Point sp = ctr_p(0, 0, -1);
-  Point_2 np = ctr_p(0, 0, 1);
+  Point np = ctr_p(0, 0, 1);
   Vertex_handle spv = arr.insert_in_face_interior(sp, arr.reference_face());
   Vertex_handle npv = arr.insert_in_face_interior(np, arr.reference_face());
 
 #if 1
-  Point_2 p1 = ctr_p(-1,  0, 0);
+  Point p1 = ctr_p(-1,  0, 0);
 #else
-  Point_2 p1 = ctr_p(-1,  -1, 0);
+  Point p1 = ctr_p(-1,  -1, 0);
 #endif
-  Point_2 p2 = ctr_p( 0, -1, 0);
+  Point p2 = ctr_p( 0, -1, 0);
-  Point_2 p3 = ctr_p( 1,  0, 0);
+  Point p3 = ctr_p( 1,  0, 0);
 
   Vertex_handle v1 = arr.insert_in_face_interior(p1, arr.reference_face());
   Vertex_handle v2 = arr.insert_in_face_interior(p2, arr.reference_face());
   Vertex_handle v3 = arr.insert_in_face_interior(p3, arr.reference_face());
 
-  X_monotone_curve_2 xcv_sp1 = ctr_xcv(sp, p1);
+  X_monotone_curve xcv_sp1 = ctr_xcv(sp, p1);
-  X_monotone_curve_2 xcv_sp2 = ctr_xcv(sp, p2);
+  X_monotone_curve xcv_sp2 = ctr_xcv(sp, p2);
-  X_monotone_curve_2 xcv_sp3 = ctr_xcv(sp, p3);
+  X_monotone_curve xcv_sp3 = ctr_xcv(sp, p3);
   arr.insert_at_vertices(xcv_sp1, spv, v1);
   arr.insert_at_vertices(xcv_sp2, spv, v2);
   arr.insert_at_vertices(xcv_sp3, spv, v3);
 
-  X_monotone_curve_2 xcv_np1 = ctr_xcv(np, p1);
+  X_monotone_curve xcv_np1 = ctr_xcv(np, p1);
-  X_monotone_curve_2 xcv_np2 = ctr_xcv(np, p2);
+  X_monotone_curve xcv_np2 = ctr_xcv(np, p2);
-  X_monotone_curve_2 xcv_np3 = ctr_xcv(np, p3);
+  X_monotone_curve xcv_np3 = ctr_xcv(np, p3);
   arr.insert_at_vertices(xcv_np1, npv, v1);
   arr.insert_at_vertices(xcv_np2, npv, v2);
   arr.insert_at_vertices(xcv_np3, npv, v3);
 
-  X_monotone_curve_2 xcv_12 = ctr_xcv(p1, p2);
+  X_monotone_curve xcv_12 = ctr_xcv(p1, p2);
-  X_monotone_curve_2 xcv_23 = ctr_xcv(p2, p3);
+  X_monotone_curve xcv_23 = ctr_xcv(p2, p3);
   arr.insert_at_vertices(xcv_12, v1, v2);
   arr.insert_at_vertices(xcv_23, v2, v3);
 
-  // Print the size of the arrangement.
+  print_arrangement_size(arr);          // print the arrangement size
-  std::cout << "The arrangement size:" << std::endl
-            << "   V = " << arr.number_of_vertices()
-            << ",  E = " << arr.number_of_edges()
-            << ",  F = " << arr.number_of_faces() << std::endl;
-
-  // std::cout << "arr: " << arr << std::endl;
 
   return 0;
 }

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/spherical_degenerate_insert2.cpp

@@ -4,61 +4,39 @@
 // #define CGAL_IDENTIFICATION_XY CGAL_X_MINUS_11_Y_7
 // #define CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE 1
 
-#include <list>
+#include "arr_geodesic_on_sphere.h"
-
+#include "arr_print.h"
-#include <CGAL/basic.h>
-#include <CGAL/Exact_rational.h>
-#include <CGAL/Cartesian.h>
-#include <CGAL/Arrangement_on_surface_2.h>
-#include <CGAL/Arr_geodesic_arc_on_sphere_traits_2.h>
-#include <CGAL/Arr_spherical_topology_traits_2.h>
-
-typedef CGAL::Exact_rational                               Number_type;
-typedef CGAL::Cartesian<Number_type>                       Kernel;
-typedef CGAL::Arr_geodesic_arc_on_sphere_traits_2<Kernel>  Geom_traits_2;
-typedef Geom_traits_2::Point_2                             Point_2;
-typedef Geom_traits_2::X_monotone_curve_2                  X_monotone_curve_2;
-typedef CGAL::Arr_spherical_topology_traits_2<Geom_traits_2> Topol_traits_2;
-typedef CGAL::Arrangement_on_surface_2<Geom_traits_2, Topol_traits_2>
-                                                           Arrangement_2;
-typedef Arrangement_2::Vertex_handle                       Vertex_handle;
 
 int main() {
-  Geom_traits_2 traits;
+  Geom_traits traits;
   auto ctr_p = traits.construct_point_2_object();
   auto ctr_xcv = traits.construct_x_monotone_curve_2_object();
-  Arrangement_2 arr(&traits);
+  Arrangement arr(&traits);
-  Point_2 sp = ctr_p(0, 0, -1);
+  Point sp = ctr_p(0, 0, -1);
-  Point_2 np = ctr_p(0, 0, 1);
+  Point np = ctr_p(0, 0, 1);
 
-  Point_2 p1 = ctr_p(-1,  0, -1);
+  Point p1 = ctr_p(-1,  0, -1);
-  Point_2 p2 = ctr_p(-1,  0,  1);
+  Point p2 = ctr_p(-1,  0,  1);
-  X_monotone_curve_2 xcv_sp_p2 = ctr_xcv(sp, p2);
+  X_monotone_curve xcv_sp_p2 = ctr_xcv(sp, p2);
-  X_monotone_curve_2 xcv_np_p1 = ctr_xcv(np, p1);
+  X_monotone_curve xcv_np_p1 = ctr_xcv(np, p1);
   // std::cout << "Inserting " << xcv_sp_p2 << std::endl;
   insert(arr, xcv_sp_p2);
   // std::cout << "Inserting " << xcv_np_p1 << std::endl;
   insert(arr, xcv_np_p1);
 
-  Point_2 q1 = ctr_p(-1,  -1, -1);
+  Point q1 = ctr_p(-1,  -1, -1);
-  Point_2 q2 = ctr_p(-1,  -1,  1);
+  Point q2 = ctr_p(-1,  -1,  1);
-  X_monotone_curve_2 xcv_sp_q2 = ctr_xcv(sp, q2);
+  X_monotone_curve xcv_sp_q2 = ctr_xcv(sp, q2);
-  X_monotone_curve_2 xcv_np_q1 = ctr_xcv(np, q1);
+  X_monotone_curve xcv_np_q1 = ctr_xcv(np, q1);
   insert(arr, xcv_sp_q2);
   insert(arr, xcv_np_q1);
 
-  X_monotone_curve_2 xcv_p1_q1 = ctr_xcv(p1, q1);
+  X_monotone_curve xcv_p1_q1 = ctr_xcv(p1, q1);
-  X_monotone_curve_2 xcv_p2_q2 = ctr_xcv(p2, q2);
+  X_monotone_curve xcv_p2_q2 = ctr_xcv(p2, q2);
   insert(arr, xcv_p1_q1);
   insert(arr, xcv_p2_q2);
 
-  // Print the size of the arrangement.
+  print_arrangement_size(arr);          // print the arrangement size
-  std::cout << "The arrangement size:" << std::endl
-            << "   V = " << arr.number_of_vertices()
-            << ",  E = " << arr.number_of_edges()
-            << ",  F = " << arr.number_of_faces() << std::endl;
-
-  // std::cout << "arr: " << arr << std::endl;
 
   return 0;
 }

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/spherical_degenerate_sweep.cpp

@@ -7,15 +7,14 @@
 
 #include <vector>
 
-#include <CGAL/config.h>
+#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
-#include <CGAL/Exact_rational.h>
-#include <CGAL/Cartesian.h>
 #include <CGAL/Arrangement_on_surface_2.h>
 #include <CGAL/Arr_geodesic_arc_on_sphere_traits_2.h>
 #include <CGAL/Arr_spherical_topology_traits_2.h>
 
-typedef CGAL::Exact_rational                               Number_type;
+#include "arr_print.h"
-typedef CGAL::Cartesian<Number_type>                       Kernel;
+
+typedef CGAL::Exact_predicates_exact_constructions_kernel  Kernel;
 
 #if 0
 typedef CGAL::Arr_geodesic_arc_on_sphere_traits_2<Kernel, -8, 6>  Geom_traits_2;
@@ -125,11 +124,7 @@ int main() {
       CGAL::insert_empty(arr, xcvs_sub.begin(), xcvs_sub.end(),
                          points_sub.begin(), points_sub.end());
 
-      // Print the size of the arrangement.
+      print_arrangement_size(arr);          // print the arrangement size
-      std::cout << "The arrangement size:" << std::endl
-                << "   V = " << arr.number_of_vertices()
-                << ",  E = " << arr.number_of_edges()
-                << ",  F = " << arr.number_of_faces() << std::endl;
 
       std::cout << "======================================================="
                 << std::endl << std::endl << std::endl;

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/spherical_insert.cpp

@@ -10,19 +10,17 @@
 
 #include "arr_print.h"
 
-typedef CGAL::Exact_predicates_exact_constructions_kernel    Kernel;
+typedef CGAL::Exact_predicates_exact_constructions_kernel         Kernel;
-typedef CGAL::Arr_geodesic_arc_on_sphere_traits_2<Kernel, -11, 7>
+typedef CGAL::Arr_geodesic_arc_on_sphere_traits_2<Kernel, -11, 7> Geom_traits;
-                                                             Geom_traits_2;
+typedef Geom_traits::Point_2                                      Point;
-typedef Geom_traits_2::Point_2                               Point_2;
+typedef Geom_traits::Curve_2                                      Curve;
-typedef Geom_traits_2::Curve_2                               Curve_2;
+typedef CGAL::Arr_spherical_topology_traits_2<Geom_traits>        Topol_traits;
-typedef CGAL::Arr_spherical_topology_traits_2<Geom_traits_2> Topol_traits_2;
+typedef CGAL::Arrangement_on_surface_2<Geom_traits, Topol_traits> Arrangement;
-typedef CGAL::Arrangement_on_surface_2<Geom_traits_2, Topol_traits_2>
-                                                             Arrangement_2;
 
 int main() {
   // Construct the arrangement from 12 geodesic arcs.
-  Geom_traits_2 traits;
+  Geom_traits traits;
-  Arrangement_2 arr(&traits);
+  Arrangement arr(&traits);
 
   auto ctr_p = traits.construct_point_2_object();
   auto ctr_cv = traits.construct_curve_2_object();
@@ -31,7 +29,7 @@ int main() {
   // point (-11, 7, 0). The curve (-1,0,0),(0,1,0) intersects the identification
   // curve.
 
-  std::list<Curve_2> arcs;
+  std::list<Curve> arcs;
 
   arcs.push_back(ctr_cv(ctr_p(1, 0, 0), ctr_p(0, 0, -1)));
   arcs.push_back(ctr_cv(ctr_p(1, 0, 0), ctr_p(0, 0, 1)));
@@ -47,7 +45,7 @@ int main() {
   arcs.push_back(ctr_cv(ctr_p(-1, 0, 0), ctr_p(0, -1, 0)));
 
   CGAL::insert(arr, arcs.begin(), arcs.end());
-  print_arrangement_size(arr);
+  print_arrangement_size(arr);          // print the arrangement size
   print_arrangement(arr);
 
   return 0;

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/spherical_overlay.cpp

@@ -1,47 +1,38 @@
 //! \file examples/Arrangement_on_surface_2/spherical_overlay.cpp
 // Overlay of two arrangements embedded on the sphere.
 
-#include <CGAL/Exact_predicates_exact_constructions_kernel.h>
+#include <CGAL/basic.h>
-#include <CGAL/Arrangement_on_surface_2.h>
-#include <CGAL/Arr_geodesic_arc_on_sphere_traits_2.h>
-#include <CGAL/Arr_spherical_topology_traits_2.h>
 #include <CGAL/Arr_overlay_2.h>
 #include <CGAL/Arr_default_overlay_traits.h>
 
-typedef CGAL::Exact_predicates_exact_constructions_kernel    Kernel;
+#include "arr_geodesic_on_sphere.h"
-typedef CGAL::Arr_geodesic_arc_on_sphere_traits_2<Kernel>    Geom_traits_2;
+
-typedef Geom_traits_2::Point_2                               Point_2;
+typedef CGAL::Arr_default_overlay_traits<Arrangement>         Overlay_traits;
-typedef Geom_traits_2::X_monotone_curve_2                    X_monotone_curve_2;
-typedef CGAL::Arr_spherical_topology_traits_2<Geom_traits_2> Topol_traits_2;
-typedef CGAL::Arrangement_on_surface_2<Geom_traits_2, Topol_traits_2>
-                                                             Arrangement_2;
-typedef CGAL::Arr_default_overlay_traits<Arrangement_2>      Overlay_traits;
 
 int main() {
-  Geom_traits_2 traits;
+  Geom_traits traits;
-  Geom_traits_2::Construct_point_2 ctr_p = traits.construct_point_2_object();
+  auto ctr_p = traits.construct_point_2_object();
-  Geom_traits_2::Construct_x_monotone_curve_2 ctr_xcv =
+  auto ctr_xcv = traits.construct_x_monotone_curve_2_object();
-    traits.construct_x_monotone_curve_2_object();
 
   Kernel::Direction_3 dir1(0, 0, 1);
-  X_monotone_curve_2 g11 = ctr_xcv(ctr_p(-1, 0, 0), ctr_p(0, -1, 0), dir1);
+  X_monotone_curve g11 = ctr_xcv(ctr_p(-1, 0, 0), ctr_p(0, -1, 0), dir1);
-  X_monotone_curve_2 g12 = ctr_xcv(ctr_p(0, -1, 0), ctr_p(-1, 0, 0), dir1);
+  X_monotone_curve g12 = ctr_xcv(ctr_p(0, -1, 0), ctr_p(-1, 0, 0), dir1);
 
-  Arrangement_2 arr1(&traits);
+  Arrangement arr1(&traits);
   CGAL::insert(arr1, g11);
   CGAL::insert(arr1, g12);
   std::cout << "No. of vertices: " << arr1.number_of_vertices() << std::endl;
 
   Kernel::Direction_3 dir2(0, 0, -1);
-  X_monotone_curve_2 g21 = ctr_xcv(ctr_p(-1, 0, 0), ctr_p(0, 1, 0), dir2);
+  X_monotone_curve g21 = ctr_xcv(ctr_p(-1, 0, 0), ctr_p(0, 1, 0), dir2);
-  X_monotone_curve_2 g22 = ctr_xcv(ctr_p(0, 1, 0), ctr_p(-1, 0, 0), dir2);
+  X_monotone_curve g22 = ctr_xcv(ctr_p(0, 1, 0), ctr_p(-1, 0, 0), dir2);
 
-  Arrangement_2 arr2(&traits);
+  Arrangement arr2(&traits);
   CGAL::insert(arr2, g21);
   CGAL::insert(arr2, g22);
   std::cout << "No. of vertices: " << arr2.number_of_vertices() << std::endl;
 
-  Arrangement_2 overlay_arr;
+  Arrangement overlay_arr;
   Overlay_traits overlay_traits;
   overlay(arr1, arr2, overlay_arr, overlay_traits);
   std::cout << "No. of vertices: " << overlay_arr.number_of_vertices()