diff --git a/Arrangement_on_surface_2/test/Arrangement_on_surface_2/cgal_test.cmake b/Arrangement_on_surface_2/test/Arrangement_on_surface_2/cgal_test.cmake
index bc8f8c10c5c..4b7b99f25e7 100644
--- a/Arrangement_on_surface_2/test/Arrangement_on_surface_2/cgal_test.cmake
+++ b/Arrangement_on_surface_2/test/Arrangement_on_surface_2/cgal_test.cmake
@@ -118,6 +118,7 @@ set(IS_BETWEEN_CW 26)
 set(COMPARE_CW_AROUND_POINT 27)
 set(PUSH_BACK 28)
 set(PUSH_FRONT 29)
+set(APPROXIMATE 30)
 set(NUMBER_OF_POINTS 32)
 set(COMPARE_ENDPOINTS_XY 33)
 set(CONSTRUCT_OPPOSITE 34)
@@ -240,22 +241,30 @@ function(execute_commands_old_structure data_dir traits_type_name)
   # the old structure is default, so this function executes all commands
   # except the commands that are given as arguments
 
+  set(commands_indicator_APPROXIMATE 1)
+  set(commands_indicator_ARE_MERGEABLE 1)
+  set(commands_indicator_ASSERTIONS 1)
   set(commands_indicator_COMPARE 1)
-  set(commands_indicator_VERTEX 1)
-  set(commands_indicator_IS_VERTICAL 1)
   set(commands_indicator_COMPARE_Y_AT_X 1)
   set(commands_indicator_COMPARE_Y_AT_X_LEFT 1)
   set(commands_indicator_COMPARE_Y_AT_X_RIGHT 1)
-  set(commands_indicator_MAKE_X_MONOTONE 1)
-  set(commands_indicator_INTERSECT 1)
-  set(commands_indicator_SPLIT 1)
-  set(commands_indicator_ARE_MERGEABLE 1)
-  set(commands_indicator_MERGE 1)
-  set(commands_indicator_ASSERTIONS 1)
   set(commands_indicator_CONSTRUCTOR 1)
+  set(commands_indicator_INTERSECT 1)
+  set(commands_indicator_IS_VERTICAL 1)
+  set(commands_indicator_MAKE_X_MONOTONE 1)
+  set(commands_indicator_MERGE 1)
+  set(commands_indicator_SPLIT 1)
+  set(commands_indicator_VERTEX 1)
+
   foreach(arg ${ARGN})
     set(commands_indicator_${arg} 0)
   endforeach()
+
+  if(commands_indicator_APPROXIMATE)
+    run_trapped_test(test_traits
+      data/empty.zero data/${data_dir}/approximate.xcv
+      data/empty.zero data/${data_dir}/approximate ${traits_type_name})
+  endif()
   if(commands_indicator_COMPARE)
     run_trapped_test(test_traits
       data/compare.pt data/empty.zero
@@ -278,18 +287,21 @@ function(execute_commands_old_structure data_dir traits_type_name)
   endif()
   if(commands_indicator_COMPARE_Y_AT_X_LEFT)
     run_trapped_test(test_traits
-      data/${data_dir}/compare_y_at_x_left.pt data/${data_dir}/compare_y_at_x_left.xcv
+      data/${data_dir}/compare_y_at_x_left.pt
+      data/${data_dir}/compare_y_at_x_left.xcv
       data/empty.zero data/${data_dir}/compare_y_at_x_left ${traits_type_name})
   endif()
   if(commands_indicator_COMPARE_Y_AT_X_RIGHT)
     run_trapped_test(test_traits
-      data/${data_dir}/compare_y_at_x_right.pt data/${data_dir}/compare_y_at_x_right.xcv
+      data/${data_dir}/compare_y_at_x_right.pt
+      data/${data_dir}/compare_y_at_x_right.xcv
       data/empty.zero data/${data_dir}/compare_y_at_x_right ${traits_type_name})
   endif()
   if(commands_indicator_MAKE_X_MONOTONE)
     run_trapped_test(test_traits
       data/empty.zero data/${data_dir}/make_x_monotone.xcv
-      data/${data_dir}/make_x_monotone.cv data/${data_dir}/make_x_monotone ${traits_type_name})
+      data/${data_dir}/make_x_monotone.cv
+      data/${data_dir}/make_x_monotone ${traits_type_name})
   endif()
   if(commands_indicator_INTERSECT)
     run_trapped_test(test_traits
@@ -319,7 +331,8 @@ function(execute_commands_old_structure data_dir traits_type_name)
   if(commands_indicator_CONSTRUCTOR)
     run_trapped_test(test_traits
       data/empty.zero data/${data_dir}/constructor.xcv
-      data/${data_dir}/constructor.cv data/${data_dir}/constructor ${traits_type_name})
+      data/${data_dir}/constructor.cv data/${data_dir}/constructor
+      ${traits_type_name})
   endif()
 endfunction()
 
@@ -332,133 +345,166 @@ function(execute_commands_new_structure data_dir traits_type_name)
 # the new structure is not default, so this function executes only
 # commands that are given as arguments
 
+  set(commands_indicator_APPROXIMATE 0)
+  set(commands_indicator_ARE_MERGEABLE 0)
+  set(commands_indicator_ASSERTIONS 0)
   set(commands_indicator_COMPARE 0)
-  set(commands_indicator_VERTEX 0)
-  set(commands_indicator_IS_VERTICAL 0)
-  set(commands_indicator_COMPARE_X_ON_BOUNDARY 0)
+  set(commands_indicator_COMPARE_ENDPOINTS_XY 0)
   set(commands_indicator_COMPARE_X_NEAR_BOUNDARY 0)
-  set(commands_indicator_COMPARE_Y_NEAR_BOUNDARY 0)
-  set(commands_indicator_PARAMETER_SPACE_X 0)
-  set(commands_indicator_PARAMETER_SPACE_Y 0)
+  set(commands_indicator_COMPARE_X_ON_BOUNDARY 0)
   set(commands_indicator_COMPARE_Y_AT_X 0)
   set(commands_indicator_COMPARE_Y_AT_X_LEFT 0)
   set(commands_indicator_COMPARE_Y_AT_X_RIGHT 0)
-  set(commands_indicator_MAKE_X_MONOTONE 0)
-  set(commands_indicator_INTERSECT 0)
-  set(commands_indicator_SPLIT 0)
-  set(commands_indicator_ARE_MERGEABLE 0)
-  set(commands_indicator_MERGE 0)
-  set(commands_indicator_ASSERTIONS 0)
+  set(commands_indicator_COMPARE_Y_NEAR_BOUNDARY 0)
   set(commands_indicator_CONSTRUCTOR 0)
+  set(commands_indicator_CONSTRUCT_OPPOSITE 0)
   set(commands_indicator_EQUAL 0)
+  set(commands_indicator_INTERSECT 0)
+  set(commands_indicator_IS_VERTICAL 0)
+  set(commands_indicator_MAKE_X_MONOTONE 0)
+  set(commands_indicator_MERGE 0)
+  set(commands_indicator_NUMBER_OF_POINTS 0)
+  set(commands_indicator_PARAMETER_SPACE_X 0)
+  set(commands_indicator_PARAMETER_SPACE_Y 0)
   set(commands_indicator_PUSH_BACK 0)
   set(commands_indicator_PUSH_FRONT 0)
-  set(commands_indicator_NUMBER_OF_POINTS 0)
-  set(commands_indicator_COMPARE_ENDPOINTS_XY 0)
-  set(commands_indicator_CONSTRUCT_OPPOSITE 0)
+  set(commands_indicator_SPLIT 0)
   set(commands_indicator_TRIM 0)
+  set(commands_indicator_VERTEX 0)
+
   foreach(arg ${ARGN})
     set(commands_indicator_${arg} 1)
   endforeach()
+
+  if(commands_indicator_APPROXIMATE)
+    run_trapped_test(test_traits data/${data_dir}/points
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/approximate ${traits_type_name})
+  endif()
   if(commands_indicator_COMPARE)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/compare ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/compare ${traits_type_name})
   endif()
   if(commands_indicator_VERTEX)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/vertex ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/vertex ${traits_type_name})
   endif()
   if(commands_indicator_IS_VERTICAL)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/is_vertical ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/is_vertical ${traits_type_name})
   endif()
   if(commands_indicator_COMPARE_X_ON_BOUNDARY)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/compare_x_on_boundary ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/compare_x_on_boundary ${traits_type_name})
   endif()
   if(commands_indicator_COMPARE_X_NEAR_BOUNDARY)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/compare_x_near_boundary ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/compare_x_near_boundary ${traits_type_name})
   endif()
   if(commands_indicator_COMPARE_Y_NEAR_BOUNDARY)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/compare_y_near_boundary ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/compare_y_near_boundary ${traits_type_name})
   endif()
   if(commands_indicator_PARAMETER_SPACE_X)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/parameter_space_x ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/parameter_space_x ${traits_type_name})
   endif()
   if(commands_indicator_PARAMETER_SPACE_Y)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/parameter_space_y ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/parameter_space_y ${traits_type_name})
   endif()
   if(commands_indicator_COMPARE_Y_AT_X)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/compare_y_at_x ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/compare_y_at_x ${traits_type_name})
   endif()
   if(commands_indicator_COMPARE_Y_AT_X_LEFT)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/compare_y_at_x_left ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/compare_y_at_x_left ${traits_type_name})
   endif()
   if(commands_indicator_COMPARE_Y_AT_X_RIGHT)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/compare_y_at_x_right ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/compare_y_at_x_right ${traits_type_name})
   endif()
   if(commands_indicator_MAKE_X_MONOTONE)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/make_x_monotone ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/make_x_monotone ${traits_type_name})
   endif()
   if(commands_indicator_INTERSECT)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/intersect ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/intersect ${traits_type_name})
   endif()
   if(commands_indicator_SPLIT)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/split ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/split ${traits_type_name})
   endif()
   if(commands_indicator_ARE_MERGEABLE)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/are_mergeable ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/are_mergeable ${traits_type_name})
   endif()
   if(commands_indicator_MERGE)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/merge ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/merge ${traits_type_name})
   endif()
   if(commands_indicator_ASSERTIONS)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/assertions ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/assertions ${traits_type_name})
   endif()
   if(commands_indicator_CONSTRUCTOR)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/constructor ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/constructor ${traits_type_name})
   endif()
   if(commands_indicator_EQUAL)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/equal ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/equal ${traits_type_name})
   endif()
   if(commands_indicator_PUSH_BACK)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/push_back ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/push_back ${traits_type_name})
   endif()
   if(commands_indicator_PUSH_FRONT)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/push_front ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/push_front ${traits_type_name})
   endif()
   if(commands_indicator_NUMBER_OF_POINTS)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/number_of_points ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/number_of_points ${traits_type_name})
   endif()
   if(commands_indicator_COMPARE_ENDPOINTS_XY)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/compare_endpoints_xy ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/compare_endpoints_xy ${traits_type_name})
   endif()
   if(commands_indicator_CONSTRUCT_OPPOSITE)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/construct_opposite ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/construct_opposite ${traits_type_name})
   endif()
   if(commands_indicator_TRIM)
     run_trapped_test(test_traits data/${data_dir}/points
-      data/${data_dir}/xcurves data/${data_dir}/curves data/${data_dir}/trim ${traits_type_name})
+      data/${data_dir}/xcurves data/${data_dir}/curves
+      data/${data_dir}/trim ${traits_type_name})
   endif()
 endfunction()
 
@@ -467,26 +513,32 @@ function(execute_commands_traits_adaptor data_dir traits_type_name)
 # the new structure is not default, so this function executes only
 # commands that are given as arguments
 
-  set(commands_indicator_PARAMETER_SPACE_X 0)
-  set(commands_indicator_PARAMETER_SPACE_Y 0)
+  set(commands_indicator_ARE_MERGEABLE 0)
+  set(commands_indicator_COMPARE_CW_AROUND_POINT 0)
   set(commands_indicator_COMPARE_XY 0)
-  set(commands_indicator_COMPARE_X_ON_BOUNDARY 0)
+  set(commands_indicator_COMPARE_Y_AT_X_LEFT 0)
   set(commands_indicator_COMPARE_X_NEAR_BOUNDARY 0)
   set(commands_indicator_COMPARE_Y_NEAR_BOUNDARY 0)
-  set(commands_indicator_COMPARE_Y_AT_X_LEFT 0)
-  set(commands_indicator_ARE_MERGEABLE 0)
-  set(commands_indicator_MERGE 0)
-  set(commands_indicator_X_ON_IDENTIFICATION 0)
-  set(commands_indicator_Y_ON_IDENTIFICATION 0)
-  set(commands_indicator_IS_BOUNDED 0)
-  set(commands_indicator_IS_IN_X_RANGE 0)
+  set(commands_indicator_COMPARE_X_ON_BOUNDARY 0)
   set(commands_indicator_COMPARE_Y_POSITION 0)
   set(commands_indicator_IS_BETWEEN_CW 0)
-  set(commands_indicator_COMPARE_CW_AROUND_POINT 0)
+  set(commands_indicator_IS_BOUNDED 0)
+  set(commands_indicator_IS_IN_X_RANGE 0)
+  set(commands_indicator_MERGE 0)
+  set(commands_indicator_PARAMETER_SPACE_X 0)
+  set(commands_indicator_PARAMETER_SPACE_Y 0)
+  set(commands_indicator_X_ON_IDENTIFICATION 0)
+  set(commands_indicator_Y_ON_IDENTIFICATION 0)
+
   foreach(arg ${ARGN})
     set(commands_indicator_${arg} 1)
   endforeach()
 
+  if(commands_indicator_ARE_MERGEABLE)
+    run_trapped_test(test_traits_adaptor data/test_adaptor/${data_dir}/points
+      data/test_adaptor/${data_dir}/xcurves data/test_adaptor/${data_dir}/curves
+      data/test_adaptor/${data_dir}/are_mergeable ${traits_type_name})
+  endif()
   if(commands_indicator_PARAMETER_SPACE_X)
     run_trapped_test(test_traits_adaptor data/test_adaptor/${data_dir}/points
       data/test_adaptor/${data_dir}/xcurves data/test_adaptor/${data_dir}/curves
@@ -524,11 +576,6 @@ function(execute_commands_traits_adaptor data_dir traits_type_name)
       data/test_adaptor/${data_dir}/xcurves data/test_adaptor/${data_dir}/curves
       data/test_adaptor/${data_dir}/compare_y_at_x_left ${traits_type_name})
   endif()
-  if(commands_indicator_ARE_MERGEABLE)
-    run_trapped_test(test_traits_adaptor data/test_adaptor/${data_dir}/points
-      data/test_adaptor/${data_dir}/xcurves data/test_adaptor/${data_dir}/curves
-      data/test_adaptor/${data_dir}/are_mergeable ${traits_type_name})
-  endif()
   if(commands_indicator_MERGE)
     run_trapped_test(test_traits_adaptor data/test_adaptor/${data_dir}/points
       data/test_adaptor/${data_dir}/xcurves data/test_adaptor/${data_dir}/curves
@@ -582,7 +629,7 @@ function(test_segment_traits_adaptor)
 
   compile_test_with_flags(test_traits_adaptor segments "${flags}")
 #  if [ -n "${SUCCESS}" ] ; then
-  execute_commands_traits_adaptor( segments segments_traits_adaptor
+  execute_commands_traits_adaptor(segments segments_traits_adaptor
     COMPARE_XY COMPARE_Y_POSITION COMPARE_CW_AROUND_POINT COMPARE_Y_AT_X_LEFT
     ARE_MERGEABLE MERGE IS_IN_X_RANGE IS_BETWEEN_CW)
 endfunction()
@@ -598,7 +645,7 @@ function(test_linear_traits_adaptor)
 
   compile_test_with_flags( test_traits_adaptor linear "${flags}")
 
-  execute_commands_traits_adaptor( linear linear_traits_adaptor
+  execute_commands_traits_adaptor(linear linear_traits_adaptor
     COMPARE_XY COMPARE_Y_AT_X_LEFT ARE_MERGEABLE MERGE IS_IN_X_RANGE
     COMPARE_Y_POSITION IS_BETWEEN_CW COMPARE_CW_AROUND_POINT)
 endfunction()
@@ -615,7 +662,7 @@ function(test_spherical_arcs_traits_adaptor)
 
   compile_test_with_flags( test_traits_adaptor geodesic_arcs_on_sphere "${flags}")
 
-  execute_commands_traits_adaptor( spherical_arcs spherical_arcs_traits_adaptor
+  execute_commands_traits_adaptor(spherical_arcs spherical_arcs_traits_adaptor
     COMPARE_XY COMPARE_Y_AT_X_LEFT ARE_MERGEABLE MERGE IS_IN_X_RANGE
     COMPARE_Y_POSITION IS_BETWEEN_CW COMPARE_CW_AROUND_POINT)
 endfunction()
@@ -643,7 +690,7 @@ function(test_construction_segments)
   set(kernel ${CARTESIAN_KERNEL})
   set(geom_traits ${SEGMENT_GEOM_TRAITS})
   set(flags "-DTEST_NT=${nt} -DTEST_KERNEL=${kernel} -DTEST_GEOM_TRAITS=${geom_traits}")
-  compile_and_run_with_flags( test_construction segments "${flags}")
+  compile_and_run_with_flags(test_construction segments "${flags}")
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -655,7 +702,7 @@ function(test_construction_linear_curves)
   set(geom_traits ${LINEAR_GEOM_TRAITS})
   set(topol_traits ${PLANAR_UNBOUNDED_TOPOL_TRAITS})
   set(flags "-DTEST_NT=${nt} -DTEST_KERNEL=${kernel} -DTEST_GEOM_TRAITS=${geom_traits} -DTEST_TOPOL_TRAITS=${topol_traits}")
-  compile_and_run_with_flags( test_construction linear "${flags}")
+  compile_and_run_with_flags(test_construction linear "${flags}")
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -667,7 +714,7 @@ function(test_construction_spherical_arcs)
   set(geom_traits ${GEODESIC_ARC_ON_SPHERE_GEOM_TRAITS})
   set(topol_traits ${SPHERICAL_TOPOL_TRAITS})
   set(flags "-DTEST_NT=${nt} -DTEST_KERNEL=${kernel} -DTEST_GEOM_TRAITS=${geom_traits} -DTEST_TOPOL_TRAITS=${topol_traits}")
-  compile_and_run_with_flags( test_construction geodesic_arcs_on_sphere "${flags}")
+  compile_and_run_with_flags(test_construction geodesic_arcs_on_sphere "${flags}")
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -678,7 +725,7 @@ function(test_construction_polylines)
   set(kernel ${CARTESIAN_KERNEL})
   set(geom_traits ${POLYLINE_GEOM_TRAITS})
   set(flags "-DTEST_NT=${nt} -DTEST_KERNEL=${kernel} -DTEST_GEOM_TRAITS=${geom_traits}")
-  compile_and_run_with_flags( test_construction polylines "${flags}")
+  compile_and_run_with_flags(test_construction polylines "${flags}")
 endfunction()
 
 
@@ -837,14 +884,16 @@ function(test_segment_traits)
 
   compile_test_with_flags(test_traits segments "${flags}")
 
-  execute_commands_old_structure( segments segment_traits
-    VERTEX IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT CONSTRUCTOR
-    COMPARE_Y_AT_X_RIGHT ARE_MERGEABLE)
+  execute_commands_old_structure(segments segment_traits
+    APPROXIMATE ARE_MERGEABLE COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
+    COMPARE_Y_AT_X_RIGHT CONSTRUCTOR IS_VERTICAL VERTEX)
 
   execute_commands_new_structure( segments segment_traits
-    IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT ARE_MERGEABLE)
+    ARE_MERGEABLE
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
+    IS_VERTICAL)
 
-  run_trapped_test( test_traits
+  run_trapped_test(test_traits
     data/segments/vertex.pt data/segments/xcurves
     data/empty.zero data/segments/vertex segment_traits)
 endfunction()
@@ -861,11 +910,13 @@ function(test_non_caching_segment_traits)
   compile_test_with_flags(test_traits non_caching_segments "${flags}")
 
   execute_commands_old_structure(segments non_caching_segment_traits
-    VERTEX IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT CONSTRUCTOR
-    COMPARE_Y_AT_X_RIGHT ARE_MERGEABLE ASSERTIONS)
+    APPROXIMATE ARE_MERGEABLE ASSERTIONS
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT CONSTRUCTOR COMPARE_Y_AT_X_RIGHT
+    IS_VERTICAL VERTEX)
 
   execute_commands_new_structure(segments segment_traits
-    IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT)
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
+    IS_VERTICAL)
 
   run_trapped_test(test_traits
     data/segments/vertex.pt data/segments/xcurves
@@ -894,23 +945,20 @@ function(test_polycurve_conic_traits)
   # Execute_command_new_structure will only run the test on functors provided as the third, fourth and so on arguments.
   # To see how the input data directory should be structured for each functor, check the execute_commands_new_structure function in this file.
   execute_commands_new_structure(polycurves_conics polycurve_conic_traits
-    COMPARE_Y_AT_X
-    INTERSECT
-    EQUAL
-    IS_VERTICAL
-    SPLIT
     ARE_MERGEABLE
-    COMPARE_Y_AT_X_LEFT
-    COMPARE_Y_AT_X_RIGHT
+    COMPARE_ENDPOINTS_XY COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT
+    CONSTRUCT_OPPOSITE
+    EQUAL
+    INTERSECT
+    IS_VERTICAL
     MAKE_X_MONOTONE
+    MERGE
+    NUMBER_OF_POINTS
+    SPLIT
     PUSH_BACK
     PUSH_FRONT
-    NUMBER_OF_POINTS
-    VERTEX
-    CONSTRUCT_OPPOSITE
-    MERGE
-    COMPARE_ENDPOINTS_XY
-    TRIM)
+    TRIM
+    VERTEX)
 
 endfunction()
 
@@ -925,23 +973,21 @@ function(test_polycurve_circular_arc_traits)
 
   compile_test_with_flags(test_traits circular_arc_polycurve "${flags}")
 
-  execute_commands_new_structure(polycurves_circular_arcs polycurve_circular_arc_traits
-    COMPARE_Y_AT_X
-    EQUAL
-    IS_VERTICAL
-    SPLIT
+  execute_commands_new_structure(polycurves_circular_arcs
+    polycurve_circular_arc_traits
     ARE_MERGEABLE
-    COMPARE_Y_AT_X_LEFT
-    COMPARE_Y_AT_X_RIGHT
+    COMPARE_ENDPOINTS_XY COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT
+    CONSTRUCT_OPPOSITE
+    EQUAL
+    INTERSECT
+    IS_VERTICAL
     MAKE_X_MONOTONE
+    MERGE
+    NUMBER_OF_POINTS
     PUSH_BACK
     PUSH_FRONT
-    NUMBER_OF_POINTS
-    VERTEX
-    CONSTRUCT_OPPOSITE
-    MERGE
-    COMPARE_ENDPOINTS_XY
-    INTERSECT)
+    SPLIT
+    VERTEX)
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -960,15 +1006,15 @@ function(test_polycurve_bezier_traits)
   compile_test_with_flags(test_traits bezier_polycurve "${flags}")
 
   execute_commands_new_structure(polycurves_bezier test_polycurve_bezier_traits
-    MERGE
+    ARE_MERGEABLE
+    COMPARE_ENDPOINTS_XY
     EQUAL
     IS_VERTICAL
     NUMBER_OF_POINTS
+    MERGE
     PUSH_BACK
     PUSH_FRONT
     VERTEX
-    ARE_MERGEABLE
-    COMPARE_ENDPOINTS_XY
     # TODO (add data for these tests)
     # COMPARE_Y_AT_X
     # SPLIT
@@ -993,8 +1039,8 @@ function(test_polyline_traits)
   compile_test_with_flags(test_traits test_polylines "${flags}")
 
   execute_commands_old_structure(polylines polyline_traits
-    CONSTRUCTOR COMPARE_Y_AT_X_LEFT
-    COMPARE_Y_AT_X_RIGHT ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE
+    COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT CONSTRUCTOR)
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -1009,8 +1055,8 @@ function(test_non_caching_polyline_traits)
   compile_test_with_flags(test_traits non_caching_polylines "${flags}")
 
   execute_commands_old_structure(polylines non_caching_polyline_traits
-    CONSTRUCTOR COMPARE_Y_AT_X_LEFT
-    COMPARE_Y_AT_X_RIGHT ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE
+    COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT CONSTRUCTOR)
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -1025,32 +1071,39 @@ function(test_linear_traits)
   compile_test_with_flags(test_traits linear "${flags}")
 
   execute_commands_old_structure(linear/segments linear_traits.segments
-    VERTEX IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
-    COMPARE_Y_AT_X_RIGHT CONSTRUCTOR ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT CONSTRUCTOR
+    IS_VERTICAL VERTEX)
 
   execute_commands_new_structure(linear/segments linear_traits.segments
-    IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT)
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
+    IS_VERTICAL)
 
   run_trapped_test(test_traits
     data/linear/segments/vertex.pt data/linear/segments/xcurves
     data/empty.zero data/linear/segments/vertex linear_traits.segments)
 
   execute_commands_old_structure(linear/rays linear_traits.rays
-    VERTEX IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
-    COMPARE_Y_AT_X_RIGHT CONSTRUCTOR ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT CONSTRUCTOR
+    IS_VERTICAL VERTEX)
 
   execute_commands_new_structure(linear/rays linear_traits.rays
-    IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT)
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
+    IS_VERTICAL)
 
   run_trapped_test(test_traits
     data/linear/rays/vertex.pt data/linear/rays/xcurves
     data/empty.zero data/linear/rays/vertex linear_traits.rays)
 
   execute_commands_new_structure(linear/lines linear_traits.lines
-    IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT INTERSECT
-    SPLIT MERGE
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
+    COMPARE_X_ON_BOUNDARY COMPARE_X_NEAR_BOUNDARY COMPARE_Y_NEAR_BOUNDARY
+    IS_VERTICAL
+    INTERSECT
+    MERGE
     PARAMETER_SPACE_X PARAMETER_SPACE_Y
-    COMPARE_X_ON_BOUNDARY COMPARE_X_NEAR_BOUNDARY COMPARE_Y_NEAR_BOUNDARY)
+    SPLIT)
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -1069,11 +1122,13 @@ function(test_conic_traits)
   compile_test_with_flags(test_traits conics "${flags}")
 
   execute_commands_old_structure(conics conic_traits
-    INTERSECT SPLIT MERGE COMPARE_Y_AT_X_LEFT
-    COMPARE_Y_AT_X_RIGHT ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT
+    INTERSECT SPLIT MERGE)
 
   execute_commands_new_structure(conics conic_traits
-    INTERSECT SPLIT MERGE)
+    INTERSECT
+    MERGE
+    SPLIT)
 
   run_trapped_test(test_traits
     data/conics/compare.pt data/empty.zero
@@ -1092,11 +1147,13 @@ function(test_line_arc_traits)
   compile_test_with_flags(test_traits line_arcs "${flags}")
 
   execute_commands_old_structure(circular_lines line_arc_traits
-    VERTEX IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
-    ASSERTIONS COMPARE_Y_AT_X_RIGHT MERGE ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE ASSERTIONS
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT
+    IS_VERTICAL MERGE VERTEX)
 
   execute_commands_new_structure(circular_lines line_arc_traits
-    IS_VERTICAL COMPARE_Y_AT_X)
+    COMPARE_Y_AT_X
+    IS_VERTICAL)
 
   run_trapped_test(test_traits
     data/circular_lines/compare.pt data/empty.zero
@@ -1119,11 +1176,14 @@ function(test_circular_arc_traits)
   compile_test_with_flags(test_traits circular_arcs "${flags}")
 
   execute_commands_old_structure(circular_arcs circular_arc_traits
-    VERTEX IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
-    ASSERTIONS COMPARE_Y_AT_X_RIGHT MERGE ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE ASSERTIONS
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT
+    IS_VERTICAL MERGE VERTEX)
 
   execute_commands_new_structure(circular_arcs circular_arc_traits
-    VERTEX IS_VERTICAL COMPARE_Y_AT_X)
+    COMPARE_Y_AT_X
+    IS_VERTICAL
+    VERTEX)
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -1138,11 +1198,13 @@ function(test_circular_line_arc_traits)
   compile_test_with_flags(test_traits circular_line_arcs "${flags}")
 
   execute_commands_old_structure(circular_line_arcs circular_line_arc_traits
-    VERTEX IS_VERTICAL CONSTRUCTOR COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
-    ASSERTIONS COMPARE_Y_AT_X_RIGHT MERGE ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE ASSERTIONS
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT CONSTRUCTOR
+    IS_VERTICAL MERGE VERTEX)
 
   execute_commands_new_structure(circular_line_arcs circular_line_arc_traits
-    IS_VERTICAL COMPARE_Y_AT_X)
+    COMPARE_Y_AT_X
+    IS_VERTICAL)
 
   run_trapped_test(test_traits
     data/circular_line_arcs/vertex.pt data/circular_line_arcs/xcurves
@@ -1161,8 +1223,12 @@ function(test_circle_segments_traits)
   compile_test_with_flags(test_traits circle_segments "${flags}")
 
   execute_commands_old_structure(circle_segments circle_segments_traits
-    VERTEX IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
-    COMPARE_Y_AT_X_RIGHT CONSTRUCTOR ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT CONSTRUCTOR
+    IS_VERTICAL VERTEX)
+
+  execute_commands_new_structure(circle_segments circle_segments_traits
+    APPROXIMATE)
 
   run_trapped_test(test_traits
     data/circle_segments/points data/circle_segments/xcurves.8
@@ -1200,8 +1266,8 @@ function(test_bezier_traits)
   compile_test_with_flags(test_traits Bezier "${flags}")
 
   execute_commands_old_structure(bezier bezier_traits
-    COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT SPLIT
-    CONSTRUCTOR ASSERTIONS ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE ASSERTIONS
+    COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT CONSTRUCTOR SPLIT)
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -1217,14 +1283,14 @@ function(test_spherical_arc_traits)
   compile_test_with_flags(test_traits geodesic_arcs_on_sphere "${flags}")
 
   execute_commands_old_structure(spherical_arcs spherical_arc_traits
-    COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT INTERSECT
-    CONSTRUCTOR
-    COMPARE MAKE_X_MONOTONE SPLIT MERGE ASSERTIONS ARE_MERGEABLE)
+    APPROXIMATE ARE_MERGEABLE ASSERTIONS
+    COMPARE COMPARE_Y_AT_X_LEFT COMPARE_Y_AT_X_RIGHT INTERSECT CONSTRUCTOR
+    MAKE_X_MONOTONE MERGE SPLIT)
 
   execute_commands_new_structure(spherical_arcs spherical_arc_traits
-    INTERSECT
-    COMPARE_X_ON_BOUNDARY COMPARE_X_NEAR_BOUNDARY
-    COMPARE_Y_NEAR_BOUNDARY)
+    APPROXIMATE
+    COMPARE_X_ON_BOUNDARY COMPARE_X_NEAR_BOUNDARY COMPARE_Y_NEAR_BOUNDARY
+    INTERSECT)
 
   run_trapped_test(test_traits
     data/spherical_arcs/compare.pt data/spherical_arcs/compare.xcv
@@ -1251,8 +1317,12 @@ function(test_rational_arc_traits)
     data/empty.zero data/compare rational_arc_traits)
 
   execute_commands_new_structure(rational_arcs rational_arc_traits
-    VERTEX IS_VERTICAL COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT SPLIT MERGE
-    COMPARE_X_ON_BOUNDARY COMPARE_X_NEAR_BOUNDARY COMPARE_Y_NEAR_BOUNDARY)
+    COMPARE_Y_AT_X COMPARE_Y_AT_X_LEFT
+    COMPARE_X_ON_BOUNDARY COMPARE_X_NEAR_BOUNDARY COMPARE_Y_NEAR_BOUNDARY
+    IS_VERTICAL
+    MERGE
+    SPLIT
+    VERTEX)
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -1273,8 +1343,13 @@ function(test_algebraic_traits_gmp)
 
   execute_commands_new_structure(algebraic algebraic_traits_gmp
     COMPARE COMPARE_Y_AT_X COMPARE_Y_AT_X_RIGHT COMPARE_Y_AT_X_LEFT
-    MAKE_X_MONOTONE IS_VERTICAL VERTEX SPLIT MERGE INTERSECT
-    PARAMETER_SPACE_X PARAMETER_SPACE_Y)
+    IS_VERTICAL
+    INTERSECT
+    MAKE_X_MONOTONE
+    MERGE
+    PARAMETER_SPACE_X PARAMETER_SPACE_Y
+    SPLIT
+    VERTEX)
 endfunction()
 
 #---------------------------------------------------------------------#
@@ -1292,8 +1367,13 @@ function(test_algebraic_traits_leda)
 
   execute_commands_new_structure(algebraic algebraic_traits_leda
     COMPARE COMPARE_Y_AT_X COMPARE_Y_AT_X_RIGHT COMPARE_Y_AT_X_LEFT
-    MAKE_X_MONOTONE IS_VERTICAL VERTEX SPLIT MERGE INTERSECT
-    PARAMETER_SPACE_X PARAMETER_SPACE_Y)
+    IS_VERTICAL
+    INTERSECT
+    MAKE_X_MONOTONE
+    MERGE
+    PARAMETER_SPACE_X PARAMETER_SPACE_Y
+    SPLIT
+    VERTEX)
 endfunction()
 
 
@@ -1315,8 +1395,13 @@ function(test_algebraic_traits_core)
 
   execute_commands_new_structure(algebraic algebraic_traits_core
     COMPARE COMPARE_Y_AT_X COMPARE_Y_AT_X_RIGHT COMPARE_Y_AT_X_LEFT
-    MAKE_X_MONOTONE IS_VERTICAL VERTEX SPLIT MERGE INTERSECT
-    PARAMETER_SPACE_X PARAMETER_SPACE_Y)
+    IS_VERTICAL
+    INTERSECT
+    MAKE_X_MONOTONE
+    MERGE
+    PARAMETER_SPACE_X PARAMETER_SPACE_Y
+    SPLIT
+    VERTEX)
 endfunction()
 
 
@@ -1406,4 +1491,4 @@ if(CGAL_DISABLE_GMP)
   foreach(_test ${LIST_OF_TESTS})
     set_property(TEST ${_test} APPEND PROPERTY ENVIRONMENT CGAL_DISABLE_GMP=1)
   endforeach()
-endif()
+endif()
\ No newline at end of file