Merge remote-tracking branch 'cgal/master' into CGAL_IO-maxGimeno

.github/workflows/build_doc.yml

@@ -0,0 +1,93 @@
+name: Documentation
+
+on:
+  issue_comment:
+      types: [created]
+
+jobs:
+  build:
+
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/github-script@v3
+        id: get_round
+        with:
+         result-encoding: string
+         script: |
+           const asso = context.payload.comment.author_association
+           if(asso == 'OWNER' || asso == 'MEMBER') {
+             const body = context.payload.comment.body
+             if(body.includes("build:")) {
+               return body.replace('build:','')
+             }
+           }
+           return 'stop'
+      - uses: actions/github-script@v3
+        if: steps.get_round.outputs.result != 'stop'
+        id: get_pr_number
+        with:
+         result-encoding: string
+         script: |
+           //get pullrequest url
+           const pr_number = context.payload.issue.number
+           return pr_number
+      - uses: actions/checkout@v2
+        name: "checkout branch"
+        if: steps.get_round.outputs.result != 'stop'
+        with:
+          repository: ${{ github.repository }}
+          ref: refs/pull/${{ steps.get_pr_number.outputs.result }}/merge
+          token: ${{ secrets.PUSH_TO_CGAL_GITHUB_IO_TOKEN }}
+
+
+      - name: install dependencies
+        if: steps.get_round.outputs.result != 'stop'
+        run: |
+          set -x
+          sudo apt-get install -y graphviz ssh
+          sudo pip install lxml pyquery
+          wget --no-verbose -O doxygen_exe https://cgal.geometryfactory.com/~mgimeno/doxygen/build_1_8_13/bin/doxygen
+          sudo mv doxygen_exe /usr/bin/doxygen
+          sudo chmod +x /usr/bin/doxygen
+          git config --global user.email "maxime.gimeno@geometryfactory.com"
+          git config --global user.name "Maxime Gimeno"
+
+      - name: configure all
+        if: steps.get_round.outputs.result != 'stop'
+        run: |
+          set -ex
+          git clone https://CGAL:${{ secrets.PUSH_TO_CGAL_GITHUB_IO_TOKEN }}@github.com/CGAL/cgal.github.io.git --depth=5
+          mkdir -p build_doc && cd build_doc && cmake ../Documentation/doc
+
+      - name: Upload Doc
+        if: steps.get_round.outputs.result != 'stop'
+        run: |
+          set -ex
+          PR_NUMBER=${{ steps.get_pr_number.outputs.result }}
+          ROUND=${{ steps.get_round.outputs.result }}
+          wget --no-verbose cgal.github.io -O tmp.html
+          if ! egrep -q "\/$PR_NUMBER\/$ROUND" tmp.html; then
+            mkdir -p cgal.github.io/${PR_NUMBER}/$ROUND
+            cd build_doc && make -j2 doc && make -j2 doc_with_postprocessing
+            cp -r ./doc_output/* ../cgal.github.io/${PR_NUMBER}/$ROUND
+            cd ../cgal.github.io
+            egrep -v " ${PR_NUMBER}\." index.html > tmp.html
+            echo "<li><a href=https://cgal.github.io/${PR_NUMBER}/$ROUND/Manual/index.html>Manual for PR ${PR_NUMBER} ($ROUND).</a></li>" >> ./tmp.html
+            mv tmp.html index.html
+            git add ${PR_NUMBER}/$ROUND && git commit -q -a -m "Add ${PR_NUMBER} $ROUND" && git push -q -u origin master
+          else
+            exit 1
+          fi
+
+      - name: Post address
+        uses: actions/github-script@v3
+        if: steps.get_round.outputs.result != 'stop'
+        with:
+          script: |
+            const address = "The documentation is built. You can find it here : https://cgal.github.io/${{ steps.get_pr_number.outputs.result }}/${{ steps.get_round.outputs.result }}/Manual/index.html"
+            github.issues.createComment({
+              owner: "CGAL",
+              repo: "cgal",
+              issue_number: ${{ github.event.issue.number }},
+              body: address
+            });

.github/workflows/delete_doc.yml

@@ -0,0 +1,27 @@
+name: Documentation Removal
+
+on:
+  pull_request_target:
+    types: [closed, removed]
+jobs:
+  build:
+
+    runs-on: ubuntu-latest
+
+    steps:
+    - uses: actions/checkout@v2.0.0
+    - name: delete directory
+      run: |
+         set -x
+         git config --global user.email "maxime.gimeno@geometryfactory.com"
+         git config --global user.name "Maxime Gimeno"
+         git clone https://maxGimeno:${{ secrets.PUSH_TO_CGAL_GITHUB_IO_TOKEN }}@github.com/CGAL/cgal.github.io.git --depth=5
+         PR_NUMBER=$(python -c "import json; import os; y = json.load(open(os.environ['GITHUB_EVENT_PATH'])); print(y[\"number\"])")
+         cd cgal.github.io/
+         egrep -v " ${PR_NUMBER}\." index.html > tmp.html
+         if [ -n "$(diff -q ./index.html ./tmp.html)" ]; then
+           mv tmp.html index.html
+         #git rm -r ${PR_NUMBER} && git commit -a -m "Remove ${PR_NUMBER}" && git push -u origin master
+           git commit -a -m "Remove ${PR_NUMBER}" && git push -u origin master
+         fi
+

.travis/build_package.sh

@@ -41,6 +41,7 @@ do
     continue
   fi
 cd $ROOT
+
 #install openmesh only if necessary
   if [ "$ARG" = "CHECK" ] || [ "$ARG" = BGL ] || [ "$ARG" = Convex_hull_3 ] ||\
      [ "$ARG" = Polygon_mesh_processing ] || [ "$ARG" = Property_map ] ||\
@@ -92,28 +93,27 @@ cd $ROOT
           exit 1
     fi
     echo "Matrix is up to date."
-    #check if non standard cgal installation works
-    cd $ROOT
-    mkdir build_test
-    cd build_test
-    mytime cmake -DCMAKE_INSTALL_PREFIX=install/ -DCGAL_BUILD_THREE_DOC=TRUE ..
-    mytime make install
-    # test install with minimal downstream example
-    mkdir installtest
-    cd installtest
-    touch main.cpp
-    mkdir build
-    echo 'project(Example)' >> CMakeLists.txt
-    echo 'set(PROJECT_SRCS ${PROJECT_SOURCE_DIR}/main.cpp)' >> CMakeLists.txt
-    echo 'find_package(CGAL REQUIRED)' >> CMakeLists.txt
-    echo 'add_executable(${PROJECT_NAME} ${PROJECT_SRCS})' >> CMakeLists.txt
-    echo 'target_link_libraries(${PROJECT_NAME} CGAL::CGAL)' >> CMakeLists.txt
-    echo '#include "CGAL/remove_outliers.h"' >> main.cpp
-    cd build
-    mytime cmake -DCMAKE_INSTALL_PREFIX=../../install -DCGAL_BUILD_THREE_DOC=TRUE ..
-    cd ..
     exit 0
   fi
+
+  if [ "$ARG" = "Installation" ]
+  then
+  mkdir build_dir
+  cd build_dir
+  cmake -DWITH_tests=ON -DBUILD_TESTING=ON ..
+  ctest -j2 -L CGAL_cmake_testsuite --output-on-failure
+  cd ..
+  rm -rf ./build_dir
+  #==-- configure all CGAL with -DWITH_examples=ON -DWITH_demos=ON -DWITH_tests=ON, and then launch CTest on a few labels. --==
+  mkdir config_dir
+  cd config_dir
+  cmake -DWITH_examples=ON -DWITH_demos=ON -DWITH_tests=ON -DBUILD_TESTING=ON ..
+  ctest -j2 -L AABB_tree --output-on-failure
+  cd ..
+  rm -rf ./config_dir
+    exit 0
+  fi
+
   IFS=$old_IFS
 
   if [ -n "$TRAVIS_PULL_REQUEST_BRANCH" ] && [ "$ARG" != Polyhedron_demo ]; then
@@ -185,6 +185,7 @@ cd $ROOT
     cd "$ROOT/$DEMO"
     build_demo
   fi
+
 done
 IFS=$old_IFS
 # Local Variables:

AABB_tree/include/CGAL/AABB_tree.h

@@ -578,7 +578,7 @@ public:
     #ifdef CGAL_HAS_THREADS
     mutable CGAL_MUTEX build_mutex; // mutex used to protect const calls inducing build() and build_kd_tree()
     #endif
-
+  public:
     const Node* root_node() const {
       CGAL_assertion(size() > 1);
 
@@ -596,7 +596,7 @@ public:
       }
       return m_p_root_node;
     }
-
+  private:
     const Primitive& singleton_data() const {
       CGAL_assertion(size() == 1);
       return *m_primitives.begin();

Arrangement_on_surface_2/archive/include/CGAL/Arr_extended_rational_arc_traits_d_1.h

@@ -660,23 +660,33 @@ public:
     return Equal_2(_traits);
   }
 
-  /*! A functor that divides a curve into continues (x-monotone) curves. */
+  //! A functor for subdividing curves into x-monotone curves.
   class Make_x_monotone_2
   {
   private:
     Traits& _traits;
+
   public:
     Make_x_monotone_2(Traits& traits) : _traits(traits) {}
-  template<class OutputIterator>
-    OutputIterator operator() (const Curve_2& cv, OutputIterator oi) const
+
+    /*! Subdivide a given curve into x-monotone subcurves and insert them into
+     * a given output iterator.
+     * \param cv the curve.
+     * \param oi an output iterator for the result. Its value type is a variant
+     *           that wraps Point_2 or an X_monotone_curve_2 objects.
+     * \return The past-the-end iterator.
+     */
+    template <class OutputIterator>
+    OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const
     {
       Object_vector res (boost::apply_visitor(Make_x_monotone_2_visitor(_traits),cv.variant()));
       re_cast_object_vector(res,oi);
       return (oi);
     }
+
   private:
-    class Make_x_monotone_2_visitor
-      : public boost::static_visitor < Object_vector >
+    class Make_x_monotone_2_visitor :
+      public boost::static_visitor < Object_vector >
     {
     private:
       typedef boost::static_visitor <Object_vector> Base;
@@ -1787,4 +1797,3 @@ public:
 #endif // CGAL_DONT_SUBMIT
 
 #endif  //CGAL_ARR_RATIONAL_ARC_TRAITS_D_1_H
-

Arrangement_on_surface_2/archive/include/CGAL/Arr_rational_arc_traits_2.h

@@ -80,7 +80,7 @@ public:
   Arr_rational_arc_traits_2 ()
   {}
 
-  /// \name Functor definitions.
+  /// \name Basic functor definitions.
   //@{
 
   /*! A functor that compares the x-coordinates of two points */
@@ -381,19 +381,22 @@ public:
   {
     return Equal_2();
   }
+  //@}
+
+  //! \name Intersections, subdivisions, and mergings
+  //@{
 
   /*! A functor that divides a curve into continues (x-monotone) curves. */
   class Make_x_monotone_2
   {
   public:
 
-    /*!
-     * Cut the given conic curve (or conic arc) into x-monotone subcurves
-     * and insert them to the given output iterator.
-     * \param cv The curve.
-     * \param oi The output iterator, whose value-type is Object. The returned
-     *           objects is a wrapper for an X_monotone_curve_2 object.
-     * \return The past-the-end iterator.
+    /*! Subdivide a given rational arc into x-monotone subcurves and insert them
+     * into a given output iterator.
+     * \param cv the arc.
+     * \param oi an output iterator for the result. Its value type is a variant
+     *           that wraps Point_2 or an X_monotone_curve_2 objects.
+     * \return the past-the-end iterator.
      */
     template<class OutputIterator>
     OutputIterator operator() (const Curve_2& cv, OutputIterator oi)

Arrangement_on_surface_2/demo/Arrangement_on_surface_2/tests/TestAlgebraic.cpp

@@ -25,26 +25,26 @@ typedef CGAL::Curved_kernel_via_analysis_2< Algebraic_kernel_d_2 > CKvA_2;
 
 Polynomial_2 makeParabola( )
 {
-    Polynomial_2 x = CGAL::shift( Polynomial_2( 1 ), 1, 0 );
-    Polynomial_2 y = CGAL::shift( Polynomial_2( 1 ), 1, 1 );
-    Polynomial_2 parabola = y - x*x;
+  Polynomial_2 x = CGAL::shift( Polynomial_2( 1 ), 1, 0 );
+  Polynomial_2 y = CGAL::shift( Polynomial_2( 1 ), 1, 1 );
+  Polynomial_2 parabola = y - x*x;
 
-    return parabola;
+  return parabola;
 }
 
 X_monotone_curve_2 makeVerticalLine( Bound x )
 {
-    Traits traits;
-    Traits::Construct_point_2 constructPoint =
-        traits.construct_point_2_object( );
-    Traits::Construct_x_monotone_segment_2 constructSegment =
-        traits.construct_x_monotone_segment_2_object( );
+  Traits traits;
+  Traits::Construct_point_2 constructPoint =
+    traits.construct_point_2_object( );
+  Traits::Construct_x_monotone_segment_2 constructSegment =
+    traits.construct_x_monotone_segment_2_object( );
 
-    std::vector< X_monotone_curve_2 > curves;
-    Point_2 p1 = constructPoint( Algebraic_real_1(x), Algebraic_real_1(Bound( -10000 )) );
-    Point_2 p2 = constructPoint( x, Bound( +10000 ) );
-    constructSegment( p1, p2, std::back_inserter( curves ) );
-    return curves[ 0 ];
+  std::vector< X_monotone_curve_2 > curves;
+  Point_2 p1 = constructPoint( Algebraic_real_1(x), Algebraic_real_1(Bound( -10000 )) );
+  Point_2 p2 = constructPoint( x, Bound( +10000 ) );
+  constructSegment( p1, p2, std::back_inserter( curves ) );
+  return curves[ 0 ];
 }
 
 typedef CGAL::Cartesian< Coefficient > Kernel;
@@ -52,40 +52,38 @@ typedef Kernel::Point_2 Kernel_point_2;
 
 int main( )
 {
-    Algebraic_real_1 real( 1 );
-    //CGAL::Qt::Converter< Algebraic_kernel_d_2 > testConverter;
-    //CGAL::Qt::Converter< CKvA_2 > testConverter;
+  Algebraic_real_1 real( 1 );
+  //CGAL::Qt::Converter< Algebraic_kernel_d_2 > testConverter;
+  //CGAL::Qt::Converter< CKvA_2 > testConverter;
 
-    //CGAL::Qt::Converter< Cartesian > testConverter;
-    Kernel_point_2 testPt( 1, 2 );
-    Point_2 testPt2( testPt.x( ), testPt.y( ) );
-    Traits traits;
-    Construct_curve_2 constructCurve = traits.construct_curve_2_object( );
-    Curve_2 curve = constructCurve( makeParabola( ) );
-    Make_x_monotone_2 mm = traits.make_x_monotone_2_object( );
-    std::vector< CGAL::Object > curves;
-    mm( curve, std::back_inserter( curves ) );
-    std::cout << curves.size( ) << std::endl;
-    X_monotone_curve_2 c1;
-    CGAL::assign( c1, curves[ 0 ] );
-    double lb = -3;
-    double ub = 3;
-    double step = 6.0 / 1000;
+  //CGAL::Qt::Converter< Cartesian > testConverter;
+  Kernel_point_2 testPt( 1, 2 );
+  Point_2 testPt2( testPt.x( ), testPt.y( ) );
+  Traits traits;
+  Construct_curve_2 constructCurve = traits.construct_curve_2_object( );
+  Curve_2 curve = constructCurve( makeParabola( ) );
+  Make_x_monotone_2 mm = traits.make_x_monotone_2_object( );
+  std::vector< CGAL::Object > curves;
+  mm( curve, std::back_inserter( curves ) );
+  std::cout << curves.size( ) << std::endl;
+  X_monotone_curve_2 c1;
+  CGAL::assign( c1, curves[ 0 ] );
+  double lb = -3;
+  double ub = 3;
+  double step = 6.0 / 1000;
 
-    for ( int i = 0; i < 1000; ++i )
-    {
-        X_monotone_curve_2 c2 = makeVerticalLine( lb + step * i );
+  for ( int i = 0; i < 1000; ++i ) {
+    X_monotone_curve_2 c2 = makeVerticalLine( lb + step * i );
 
-        CGAL::Object o;
-        CGAL::Oneset_iterator< CGAL::Object > oi( o );
-        Intersect_2 intersect = traits.intersect_2_object( );
-        intersect( c1, c2, oi );
-        std::pair< Point_2, Multiplicity > res;
-        CGAL::assign( res, o );
-        std::pair< double, double > approx = res.first.to_double( );
-        std::cout << approx.first << " " << approx.second << std::endl;
-    }
+    CGAL::Object o;
+    CGAL::Oneset_iterator< CGAL::Object > oi( o );
+    Intersect_2 intersect = traits.intersect_2_object( );
+    intersect( c1, c2, oi );
+    std::pair< Point_2, Multiplicity > res;
+    CGAL::assign( res, o );
+    std::pair< double, double > approx = res.first.to_double( );
+    std::cout << approx.first << " " << approx.second << std::endl;
+  }
 
-
-    return 0;
+  return 0;
 }

Arrangement_on_surface_2/demo/Arrangement_on_surface_2/tests/TestRayPL.cpp

@@ -6,28 +6,29 @@
 #include <CGAL/Arr_walk_along_line_point_location.h>
 #include <CGAL/Object.h>
 
-typedef CGAL::Gmpq NT;
-typedef CGAL::Cartesian< NT > Kernel;
-typedef CGAL::Arr_linear_traits_2< Kernel > Traits;
-typedef CGAL::Arr_default_dcel< Traits > Dcel;
-typedef CGAL::Arrangement_with_history_2< Traits, Dcel > Arrangement;
-typedef CGAL::Arr_walk_along_line_point_location< Arrangement > WalkAlongLinePointLocationStrategy;
-typedef Kernel::Point_2 Point_2;
-typedef Kernel::Ray_2 Ray_2;
-typedef Arrangement::Curve_2 Curve_2;
+typedef CGAL::Gmpq                                      NT;
+typedef CGAL::Cartesian<NT>                             Kernel;
+typedef CGAL::Arr_linear_traits_2<Kernel>               Traits;
+typedef CGAL::Arr_default_dcel<Traits>                  Dcel;
+typedef CGAL::Arrangement_with_history_2< Traits, Dcel> Arrangement;
+typedef CGAL::Arr_walk_along_line_point_location<Arrangement>
+  Walk_along_line_pl;
+typedef Kernel::Point_2                                 Point_2;
+typedef Kernel::Ray_2                                   Ray_2;
+typedef Arrangement::Curve_2                            Curve_2;
 
 int main( )
 {
-    Point_2 p1( 0, 0 );
-    Point_2 p2( 1, 0 );
-    Ray_2 ray( p1, p2 );
-    Curve_2 curve( ray );
+    Point_2 p1(0, 0);
+    Point_2 p2(1, 0);
+    Ray_2 ray(p1, p2);
+    Curve_2 curve(ray);
 
     Arrangement arr;
-    CGAL::insert( arr, curve );
+    CGAL::insert(arr, curve);
 
-    WalkAlongLinePointLocationStrategy pl( arr );
-    CGAL::Object o = pl.locate( Point_2( 1, -1 ) );
+    Walk_along_line_pl pl(arr);
+    auto o = pl.locate(Point_2(1, -1));
 
     return 0;
 }

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/CGAL/Arr_polycurve_traits_2.h

@@ -239,19 +239,17 @@ namespace CGAL {
                                     const Point_2& tgt) const;
     };
 
-    /*! Subdivide the given subcurve into x-monotone subcurves and insert them
-     * into the given output iterator. Since the subcurves that
-     * constitute a general polycurve are not necessarily
-     * \f$x\f$-monotone, this functor may break them.
+    /*! Subdivide a given subcurve into x-monotone subcurves and insert them
+     * into a given output iterator.
      */
     class Make_x_monotone_2 {
     public:
       /*!
       * \pre if `cv` is not empty then it must be continuous and well-oriented.
-      * \param cv The subcurve.
-      * \param oi The output iterator, whose value-type is Object. The output
-      *           object is a wrapper of a X_monotone_curve_2 objects.
-      * \return The past-the-end iterator.
+      * \param cv the subcurve.
+      * \param oi an output iterator for the result. Its value type is a variant
+      *           that wraps Point_2 or an X_monotone_curve_2 objects.
+      * \return the past-the-end iterator.
       */
       template <typename OutputIterator>
       OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const;

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/CGAL/Arr_vertical_decomposition_2.h

@@ -1,57 +1,52 @@
 namespace CGAL {
 
-/*!
-\ingroup PkgArrangementOnSurface2Funcs
-
-Produces the symbolic vertical decomposition of a
-given arrangement, performing a batched vertical ray-shooting query from
-all arrangement vertices, such that every vertex is associated with a pair
-of objects, one corresponds to the arrangement feature that lies below it,
-and the other corresponds to the feature that lies above it.
-The output of this function can be readily used for inserting vertical walls
-and physically decomposing the arrangement into pseudo-trapezoids. To do
-this, it is convenient to process the vertices in an ascending
-\f$ xy\f$-lexicographic order. The visible objects are therefore returned through
-an output iterator, which pairs each finite arrangement vertex with the two
-features it "sees", such that the vertices are given in ascending
-\f$ xy\f$-lexicographic order.
-
-Produces the symbolic vertical decomposition of the `arr` arrangement.
-More precisely, it performs a batched vertical ray-shooting query from all
-arrangement vertices, such that every vertex is associated with a pair of
-objects, one corresponding to the arrangement feature that lies below it,
-while the other corresponds to the feature that lies above it.
-The query results are returned through the output iterator, which pairs
-each finite arrangement vertex with a pair of `Object`s, the first
-represents the feature below the vertex, and the second represents the
-feature that lies above it. Each `Object` may be one of the following:
-<UL>
-<LI>`Halfedge_const_handle`, if the vertex is located above (or
-below) an edge. The given halfedge is always directed from right to left.
-In case there is no concrete edge below (or above) the vertex, and
-the arrangement is unbounded, then the object returned is a
-<I>fictitious</I> halfedge.
-<LI>`Face_const_handle`, in case there is no edge below (or above)
-the vertex, and the arrangement is bounded.
-<LI>`Vertex_const_handle`, in case the vertex is located vertically
-above (or below) another arrangement vertex.
-<LI>An empty object, in case the vertex is the top end-vertex of
-a vertical edge, we define there is no feature below it. Similarly, if
-it is the bottom end-vertex of a vertical edge, we define that there
-is no feature above it.
-</UL>
-The function returns a past-the-end iterator for its output sequence.
-
-\cgalHeading{Requirements}
-
-`OutputIterator::value_type` must be
-`pair<Arrangement_2::Vertex_const_handle, pair<Object, Object> >`.
-
-*/
+/*! \ingroup PkgArrangementOnSurface2Funcs
+ *
+ * Produces the symbolic vertical decomposition of a given arrangement,
+ * performing a batched vertical ray-shooting query from all arrangement
+ * vertices, such that every vertex is associated with a pair of objects, one
+ * corresponds to the arrangement feature that lies below it, and the other
+ * corresponds to the feature that lies above it.  The output of this function
+ * can be readily used for inserting vertical walls and physically decomposing
+ * the arrangement into pseudo-trapezoids. To do this, it is convenient to
+ * process the vertices in an ascending \f$ xy\f$-lexicographic order. The
+ * visible objects are therefore returned through an output iterator, which
+ * pairs each finite arrangement vertex with the two features it "sees", such
+ * that the vertices are given in ascending \f$ xy\f$-lexicographic order.
+ *
+ * Produces the symbolic vertical decomposition of the `arr` arrangement.  More
+ * precisely, it performs a batched vertical ray-shooting query from all
+ * arrangement vertices, such that every vertex is associated with a pair of
+ * objects, one corresponding to the arrangement feature that lies below it,
+ * while the other corresponds to the feature that lies above it.  The query
+ * results are returned through the output iterator, which pairs each finite
+ * arrangement vertex with a pair of objects, the first represents the feature
+ * below the vertex, and the second represents the feature that lies above
+ * it. Each object is an optional variant that wraps a handle to an arrangement
+ * feature. If the vertex is the top end-vertex of a vertical edge, we say that
+ * there is no feature below it; similarly, if it is the bottom end-vertex of a
+ * vertical edge, we say that there is no feature above it. In these cases the
+ * optional object is set to be empty; otherwise it is set as follows:
+ * <UL>
+ * <LI>`Halfedge_const_handle`, if the vertex is located above (or below) an
+ * edge. The given halfedge is always directed from right to left.  In case
+ * there is no concrete edge below (or above) the vertex, and the arrangement
+ * is unbounded, then the object returned is a <I>fictitious</I> halfedge.
+ * <LI>`Face_const_handle`, in case there is no edge below (or above)
+ * the vertex, and the arrangement is bounded.
+ * <LI>`Vertex_const_handle`, in case the vertex is located vertically above
+ * (or below) another arrangement vertex.
+ * </UL> The function returns a past-the-end iterator for its output sequence.
+ *
+ * \cgalHeading{Requirements}
+ *
+ * `OutputIterator::value_type` must be
+ * `pair<Arrangement_2::Vertex_const_handle, pair<Object, Object> >`.
+ *
+ */
 template<typename Traits, typename Dcel,
 typename OutputIterator>
 OutputIterator decompose (const Arrangement_2<Traits,Dcel>& arr,
 OutputIterator oi);
 
 } /* namespace CGAL */
-

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrTraits--MakeXMonotone_2.h

@@ -16,18 +16,16 @@ public:
 /// A model of this concept must provide:
 /// @{
 
-/*!
-subdivides the input curve `c` into \f$ x\f$-monotone subcurves and
-isolated points, and inserts the results into a container through the
-given output iterator. The value type of `OutputIterator` is
-`CGAL::Object`, where each `Object` wraps either an
-`ArrTraits::X_monotone_curve_2` object or a `ArrTraits::Point_2`
-object. The operator returns a past-the-end iterator for the output
-sequence.
-*/
+/*! subdivides the input curve `c` into \f$ x\f$-monotone subcurves and
+ * isolated points, and inserts the results into a range begining at the
+ * given output iterator `oi`. The type `OutputIterator` dereferences a
+ * `boost::variant` that wraps either an `ArrTraits::Point_2` object or an
+ * `ArrTraits::X_monotone_curve_2` object. The operator returns a past-the-end
+ * iterator for the output sequence.
+ */
 template <typename OutputIterator>
-OutputIterator operator()( ArrTraits::Curve_2 c,
-OutputIterator oi);
+OutputIterator operator()(ArrTraits::Curve_2 c,
+                          OutputIterator oi);
 
 /// @}
 

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/algebraic_segments.cpp

@@ -25,106 +25,102 @@ typedef CORE::BigInt Integer;
 typedef LEDA::integer Integer;
 #endif
 
-typedef CGAL::Arr_algebraic_segment_traits_2<Integer> Arr_traits_2;
-typedef CGAL::Arrangement_2<Arr_traits_2> Arrangement_2;
-typedef Arr_traits_2::Curve_2 Curve_2;
-typedef Arr_traits_2::Polynomial_2 Polynomial_2;
-typedef Arr_traits_2::Algebraic_real_1 Algebraic_real_1;
-typedef Arr_traits_2::X_monotone_curve_2 X_monotone_curve_2;
-typedef Arr_traits_2::Point_2 Point_2;
+typedef CGAL::Arr_algebraic_segment_traits_2<Integer>   Arr_traits_2;
+typedef CGAL::Arrangement_2<Arr_traits_2>               Arrangement_2;
+typedef Arr_traits_2::Curve_2                           Curve_2;
+typedef Arr_traits_2::Polynomial_2                      Polynomial_2;
+typedef Arr_traits_2::Algebraic_real_1                  Algebraic_real_1;
+typedef Arr_traits_2::X_monotone_curve_2                X_monotone_curve_2;
+typedef Arr_traits_2::Point_2                           Point_2;
 
+typedef boost::variant<Point_2, X_monotone_curve_2>     Make_x_monotone_result;
 
-int main() {
+int main()
+{
+  Arr_traits_2 arr_traits;
+  auto construct_curve = arr_traits.construct_curve_2_object();
+  auto construct_x_monotone_segment =
+    arr_traits.construct_x_monotone_segment_2_object();
+  auto construct_point = arr_traits.construct_point_2_object();
+  auto make_x_monotone = arr_traits.make_x_monotone_2_object();
 
-    Arr_traits_2 arr_traits;
+  Arrangement_2 arr(&arr_traits);
 
-    Arr_traits_2::Construct_curve_2 construct_curve
-        = arr_traits.construct_curve_2_object();
-    Arr_traits_2::Construct_x_monotone_segment_2 construct_x_monotone_segment
-        = arr_traits.construct_x_monotone_segment_2_object();
-    Arr_traits_2::Construct_point_2 construct_point
-        = arr_traits.construct_point_2_object();
-    Arr_traits_2::Make_x_monotone_2 make_x_monotone
-        = arr_traits.make_x_monotone_2_object();
+  std::vector<X_monotone_curve_2> segs;
 
-    Arrangement_2 arr(&arr_traits);
+  Polynomial_2 x = CGAL::shift(Polynomial_2(1),1,0);
+  Polynomial_2 y = CGAL::shift(Polynomial_2(1),1,1);
 
-    std::vector<X_monotone_curve_2> segs;
+  // Construct x^4+y^3-1
+  Curve_2 cv0 = construct_curve(CGAL::ipower(x,4)+CGAL::ipower(y,3)-1);
+  // Construct all x-monotone segments using the Make_x_mononotone functor
+  std::vector<Make_x_monotone_result> pre_segs;
+  make_x_monotone(cv0, std::back_inserter(pre_segs));
+  // Cast all CGAL::Objects into X_monotone_segment_2
+  // (the vector might also contain Point_2 objects for isolated points,
+  // but not for this instance
+  for(size_t i = 0; i < pre_segs.size(); ++i) {
+    auto* curr_p = boost::get<X_monotone_curve_2>(&pre_segs[i]);;
+    CGAL_assertion(curr_p);
+    segs.push_back(*curr_p);
+  }
+  // Construct an ellipse with equation 2*x^2+5*y^2-7=0
+  Curve_2 cv1 = construct_curve(2*CGAL::ipower(x,2)+5*CGAL::ipower(y,2)-7);
 
-    Polynomial_2 x = CGAL::shift(Polynomial_2(1),1,0);
-    Polynomial_2 y = CGAL::shift(Polynomial_2(1),1,1);
+  // Construct point on the upper arc (counting of arc numbers starts with 0!
+  Point_2 p11 = construct_point(Algebraic_real_1(0),cv1,1);
 
-    // Construct x^4+y^3-1
-    Curve_2 cv0 = construct_curve(CGAL::ipower(x,4)+CGAL::ipower(y,3)-1);
-    // Construct all x-monotone segments using the Make_x_mononotone functor
-    std::vector<CGAL::Object> pre_segs;
-    make_x_monotone(cv0,std::back_inserter(pre_segs));
-    // Cast all CGAL::Objects into X_monotone_segment_2
-    // (the vector might also contain Point_2 objects for isolated points,
-    // but not for this instance
-    for(size_t i = 0; i < pre_segs.size(); i++ ) {
-        X_monotone_curve_2 curr;
-        bool check = CGAL::assign(curr,pre_segs[i]);
-        assert(check); CGAL_USE(check);
-        segs.push_back(curr);
-    }
-    // Construct an ellipse with equation 2*x^2+5*y^2-7=0
-    Curve_2 cv1 = construct_curve(2*CGAL::ipower(x,2)+5*CGAL::ipower(y,2)-7);
+  construct_x_monotone_segment(cv1,p11,Arr_traits_2::POINT_IN_INTERIOR,
+                               std::back_inserter(segs));
 
-    // Construct point on the upper arc (counting of arc numbers starts with 0!
-    Point_2 p11 = construct_point(Algebraic_real_1(0),cv1,1);
+  // Construct a vertical cusp x^2-y^3=0
+  Curve_2 cv2 = construct_curve(CGAL::ipower(x,2)-CGAL::ipower(y,3));
 
-    construct_x_monotone_segment(cv1,p11,Arr_traits_2::POINT_IN_INTERIOR,
+  // Construct a segment containing the cusp point.
+  // This adds to X_monotone_curve_2 objects to the vector,
+  // because the cusp is a critical point
+  Point_2 p21 = construct_point(Algebraic_real_1(-2),cv2,0);
+  Point_2 p22 = construct_point(Algebraic_real_1(2),cv2,0);
+  construct_x_monotone_segment(cv2,p21,p22,std::back_inserter(segs));
+
+  // Construct an unbounded curve, starting at x=3
+  Point_2 p23 = construct_point(Algebraic_real_1(3),cv2,0);
+  construct_x_monotone_segment(cv2,p23,Arr_traits_2::MIN_ENDPOINT,
                                  std::back_inserter(segs));
 
-    // Construct a vertical cusp x^2-y^3=0
-    Curve_2 cv2 = construct_curve(CGAL::ipower(x,2)-CGAL::ipower(y,3));
+  // Construct another conic: y^2-x^2+1
+  Curve_2 cv3 = construct_curve(CGAL::ipower(y,2)-CGAL::ipower(x,2)+1);
 
-    // Construct a segment containing the cusp point.
-    // This adds to X_monotone_curve_2 objects to the vector,
-    // because the cusp is a critical point
-    Point_2 p21 = construct_point(Algebraic_real_1(-2),cv2,0);
-    Point_2 p22 = construct_point(Algebraic_real_1(2),cv2,0);
-    construct_x_monotone_segment(cv2,p21,p22,std::back_inserter(segs));
+  Point_2 p31 = construct_point(Algebraic_real_1(2),cv3,1);
+  construct_x_monotone_segment(cv3,p31,Arr_traits_2::MAX_ENDPOINT,
+                               std::back_inserter(segs));
 
-    // Construct an unbounded curve, starting at x=3
-    Point_2 p23 = construct_point(Algebraic_real_1(3),cv2,0);
-    construct_x_monotone_segment(cv2,p23,Arr_traits_2::MIN_ENDPOINT,
-                                 std::back_inserter(segs));
+  // Construct a vertical segment
+  Point_2 v1 = construct_point(0,0);
+  Point_2 v2 = construct_point(Algebraic_real_1(0),cv1,1);
+  construct_x_monotone_segment(v1,v2,std::back_inserter(segs));
 
-    // Construct another conic: y^2-x^2+1
-    Curve_2 cv3 = construct_curve(CGAL::ipower(y,2)-CGAL::ipower(x,2)+1);
+  CGAL::insert(arr,segs.begin(),segs.end());
 
-    Point_2 p31 = construct_point(Algebraic_real_1(2),cv3,1);
-    construct_x_monotone_segment(cv3,p31,Arr_traits_2::MAX_ENDPOINT,
-                                 std::back_inserter(segs));
+  // Add some isolated points (must be wrapped into CGAL::Object)
+  std::vector<CGAL::Object> isolated_points;
+  isolated_points.push_back
+    (CGAL::make_object(construct_point(Algebraic_real_1(2),cv3,0)));
+  isolated_points.push_back
+    (CGAL::make_object(construct_point(Integer(1),Integer(5))));
+  isolated_points.push_back
+    (CGAL::make_object(construct_point(Algebraic_real_1(-1),
+                                       Algebraic_real_1(5))));
 
-    // Construct a vertical segment
-    Point_2 v1 = construct_point(0,0);
-    Point_2 v2 = construct_point(Algebraic_real_1(0),cv1,1);
-    construct_x_monotone_segment(v1,v2,std::back_inserter(segs));
+  CGAL::insert(arr,isolated_points.begin(), isolated_points.end());
 
-    CGAL::insert(arr,segs.begin(),segs.end());
+  // 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;
 
-    // Add some isolated points (must be wrapped into CGAL::Object)
-    std::vector<CGAL::Object> isolated_points;
-    isolated_points.push_back
-        (CGAL::make_object(construct_point(Algebraic_real_1(2),cv3,0)));
-    isolated_points.push_back
-      (CGAL::make_object(construct_point(Integer(1),Integer(5))));
-    isolated_points.push_back
-      (CGAL::make_object(construct_point(Algebraic_real_1(-1),
-                                         Algebraic_real_1(5))));
-
-    CGAL::insert(arr,isolated_points.begin(), isolated_points.end());
-
-    // 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;
+  return 0;
 }
 
 #endif

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/bounded_planar_vertical_decomposition.cpp

@@ -8,16 +8,20 @@
 #include <list>
 
 typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
-typedef CGAL::Arr_segment_traits_2<Kernel>              Traits_2;
-typedef Traits_2::Point_2                               Point_2;
-typedef Traits_2::X_monotone_curve_2                    Segment_2;
-typedef CGAL::Arrangement_2<Traits_2>                   Arrangement_2;
-typedef Arrangement_2::Vertex_const_handle              Vertex_const_handle;
-typedef Arrangement_2::Halfedge_const_handle            Halfedge_const_handle;
-typedef Arrangement_2::Face_const_handle                Face_const_handle;
-typedef std::pair<CGAL::Object, CGAL::Object>           Object_pair;
-typedef std::pair<Vertex_const_handle, Object_pair>     Vert_decomp_entry;
-typedef std::list<Vert_decomp_entry>                    Vert_decomp_list;
+typedef CGAL::Arr_segment_traits_2<Kernel>               Traits_2;
+typedef Traits_2::Point_2                                Point_2;
+typedef Traits_2::X_monotone_curve_2                     Segment_2;
+typedef CGAL::Arrangement_2<Traits_2>                    Arrangement_2;
+typedef Arrangement_2::Vertex_const_handle               Vertex_const_handle;
+typedef Arrangement_2::Halfedge_const_handle             Halfedge_const_handle;
+typedef Arrangement_2::Face_const_handle                 Face_const_handle;
+
+typedef boost::variant<Vertex_const_handle, Halfedge_const_handle,
+                         Face_const_handle>              Cell_type;
+typedef boost::optional<Cell_type>                       Vert_decomp_type;
+typedef std::pair<Vert_decomp_type, Vert_decomp_type>    Vert_decomp_pair;
+typedef std::pair<Vertex_const_handle, Vert_decomp_pair> Vert_decomp_entry;
+typedef std::list<Vert_decomp_entry>                     Vert_decomp_list;
 
 int main()
 {
@@ -39,34 +43,41 @@ int main()
   CGAL::decompose(arr, std::back_inserter(vd_list));
 
   // Print the results.
-  Vert_decomp_list::const_iterator vd_iter;
-  for (vd_iter = vd_list.begin(); vd_iter != vd_list.end(); ++vd_iter) {
-    const Object_pair& curr = vd_iter->second;
+  for (auto vd_iter = vd_list.begin(); vd_iter != vd_list.end(); ++vd_iter) {
+    const Vert_decomp_pair& curr = vd_iter->second;
     std::cout << "Vertex (" << vd_iter->first->point() << ") : ";
 
-    Vertex_const_handle vh;
-    Halfedge_const_handle hh;
-    Face_const_handle fh;
-
     std::cout << " feature below: ";
-    if (CGAL::assign(hh, curr.first))
-      std::cout << '[' << hh->curve() << ']';
-    else if (CGAL::assign(vh, curr.first))
-      std::cout << '(' << vh->point() << ')';
-    else if (CGAL::assign(fh, curr.first))
-      std::cout << "NONE";
-    else
-      std::cout << "EMPTY";
+    if (! curr.first) std::cout << "EMPTY";
+    else {
+      auto* vh = boost::get<Vertex_const_handle>(&*(curr.first));
+      if (vh) std::cout << '(' << (*vh)->point() << ')';
+      else {
+        auto* hh = boost::get<Halfedge_const_handle>(&*(curr.first));
+        if (hh) std::cout << '[' << (*hh)->curve() << ']';
+        else {
+          auto* fh = boost::get<Face_const_handle>(&*(curr.first));
+          CGAL_assertion(fh);
+          std::cout << "NONE (" << (*fh)->is_unbounded() << ")";
+        }
+      }
+    }
 
     std::cout << "   feature above: ";
-    if (CGAL::assign(hh, curr.second))
-      std::cout << '[' << hh->curve() << ']' << std::endl;
-    else if (CGAL::assign(vh, curr.second))
-      std::cout << '(' << vh->point() << ')' << std::endl;
-    else if (CGAL::assign(fh, curr.second))
-      std::cout << "NONE" << std::endl;
-    else
-      std::cout << "EMPTY" << std::endl;
+    if (! curr.second) std::cout << "EMPTY" << std::endl;
+    else {
+      auto* vh = boost::get<Vertex_const_handle>(&*(curr.second));
+      if (vh) std::cout << '(' << (*vh)->point() << ')' << std::endl;
+      else {
+        auto* hh = boost::get<Halfedge_const_handle>(&*(curr.second));
+        if (hh) std::cout << '[' << (*hh)->curve() << ']' << std::endl;
+        else {
+          auto* fh = boost::get<Face_const_handle>(&*(curr.second));
+          CGAL_assertion(fh);
+          std::cout << "NONE (" << (*fh)->is_unbounded() << ")" << std::endl;
+        }
+      }
+    }
   }
 
   return 0;

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/polycurve_bezier.cpp

@@ -33,16 +33,18 @@ typedef CGAL::Arr_Bezier_curve_traits_2<Rat_kernel, Alg_kernel, Nt_traits>
 typedef Bezier_traits::Curve_2                         Bezier_curve_2;
 typedef Bezier_traits::X_monotone_curve_2              Bezier_x_curve_2;
 typedef CGAL::Arr_polycurve_traits_2<Bezier_traits>    Polycurve_bezier_traits_2;
+typedef Polycurve_bezier_traits_2::Point_2             Point_2;
 typedef Polycurve_bezier_traits_2::X_monotone_curve_2  X_mono_polycurve;
 typedef CGAL::Arrangement_2<Polycurve_bezier_traits_2> Arrangement_2;
 
+typedef boost::variant<Point_2, Bezier_x_curve_2>      Make_x_monotone_result;
+
 int main()
 {
   Polycurve_bezier_traits_2 pc_traits;
   Bezier_traits bezier_traits;
 
-  Polycurve_bezier_traits_2::Construct_x_monotone_curve_2
-    construct_x_mono_polycurve =
+  auto construct_x_mono_polycurve =
     pc_traits.construct_x_monotone_curve_2_object();
 
   std::vector<Bezier_x_curve_2> x_bezier_curves;
@@ -69,11 +71,11 @@ int main()
     // Read the current curve (specified by its control points).
     in_file >> B;
     //convert it into x-monotone bezier curve.
-    std::vector<CGAL::Object> obj_vector;
+    std::vector<Make_x_monotone_result> obj_vector;
     bezier_traits.make_x_monotone_2_object()(B, std::back_inserter(obj_vector));
-    Bezier_x_curve_2 x_seg =
-      CGAL::object_cast<Bezier_x_curve_2>((obj_vector[0]));
-    x_curves.push_back(x_seg);
+    auto* x_seg_p = boost::get<Bezier_x_curve_2>(&obj_vector[0]);
+    CGAL_assertion(x_seg_p);
+    x_curves.push_back(*x_seg_p);
   }
 
   X_mono_polycurve polycurve =

Arrangement_on_surface_2/examples/Arrangement_on_surface_2/unb_planar_vertical_decomposition.cpp

@@ -8,19 +8,22 @@
 #include <CGAL/Arr_vertical_decomposition_2.h>
 #include <list>
 
-typedef CGAL::MP_Float                                  Number_type;
-typedef CGAL::Cartesian<Number_type>                    Kernel;
-typedef CGAL::Arr_linear_traits_2<Kernel>              Traits_2;
-typedef Traits_2::Point_2                               Point_2;
-typedef Traits_2::X_monotone_curve_2                    Segment_2;
-typedef CGAL::Arrangement_2<Traits_2>                   Arrangement_2;
-typedef Arrangement_2::Vertex_const_handle              Vertex_const_handle;
-typedef Arrangement_2::Halfedge_const_handle            Halfedge_const_handle;
-typedef Arrangement_2::Face_const_handle                Face_const_handle;
+typedef CGAL::MP_Float                                   Number_type;
+typedef CGAL::Cartesian<Number_type>                     Kernel;
+typedef CGAL::Arr_linear_traits_2<Kernel>                Traits_2;
+typedef Traits_2::Point_2                                Point_2;
+typedef Traits_2::X_monotone_curve_2                     Segment_2;
+typedef CGAL::Arrangement_2<Traits_2>                    Arrangement_2;
+typedef Arrangement_2::Vertex_const_handle               Vertex_const_handle;
+typedef Arrangement_2::Halfedge_const_handle             Halfedge_const_handle;
+typedef Arrangement_2::Face_const_handle                 Face_const_handle;
 
-typedef std::pair<Vertex_const_handle, std::pair<CGAL::Object, CGAL::Object> >
-                                                        Vert_decomp_entry;
-typedef std::list<Vert_decomp_entry>                    Vert_decomp_list;
+typedef boost::variant<Vertex_const_handle, Halfedge_const_handle,
+                         Face_const_handle>              Cell_type;
+typedef boost::optional<Cell_type>                       Vert_decomp_type;
+typedef std::pair<Vert_decomp_type, Vert_decomp_type>    Vert_decomp_pair;
+typedef std::pair<Vertex_const_handle, Vert_decomp_pair> Vert_decomp_entry;
+typedef std::list<Vert_decomp_entry>                     Vert_decomp_list;
 
 int main ()
 {
@@ -43,37 +46,37 @@ int main ()
   CGAL::decompose (arr, std::back_inserter(vd_list));
 
   // Print the results.
-  Vert_decomp_list::const_iterator       vd_iter;
-  std::pair<CGAL::Object, CGAL::Object>  curr;
-  Vertex_const_handle                    vh;
-  Halfedge_const_handle                  hh;
-  Face_const_handle                      fh;
-
-  for (vd_iter = vd_list.begin(); vd_iter != vd_list.end(); ++vd_iter) {
-    curr = vd_iter->second;
+  for (auto vd_iter = vd_list.begin(); vd_iter != vd_list.end(); ++vd_iter) {
+    const Vert_decomp_pair& curr = vd_iter->second;
     std::cout << "Vertex (" << vd_iter->first->point() << ") : ";
 
     std::cout << " feature below: ";
-    if (CGAL::assign (vh, curr.first))
-          std::cout << '(' << vh->point() << ')';
-    else if (CGAL::assign (hh, curr.first))
-      if (!hh->is_fictitious())
-        std::cout << '[' << hh->curve() << ']';
-      else
-        std::cout << "NONE";
-    else
-      std::cout << "EMPTY";
+    if (! curr.first) std::cout << "EMPTY";
+    else {
+      auto* vh = boost::get<Vertex_const_handle>(&*(curr.first));;
+      if (vh) std::cout << '(' << (*vh)->point() << ')';
+      else {
+        auto* hh = boost::get<Halfedge_const_handle>(&*(curr.first));
+        CGAL_assertion(hh);
+        if (! (*hh)->is_fictitious())
+          std::cout << '[' << (*hh)->curve() << ']';
+        else std::cout << "NONE";
+      }
+    }
 
     std::cout << "   feature above: ";
-    if (CGAL::assign (vh, curr.second))
-          std::cout << '(' << vh->point() << ')' << std::endl;
-    else if (CGAL::assign (hh, curr.second))
-      if (!hh->is_fictitious())
-        std::cout << '[' << hh->curve() << ']' << std::endl;
-      else
-        std::cout << "NONE" << std::endl;
-    else
-      std::cout << "EMPTY" << std::endl;
+    if (! curr.second) std::cout << "EMPTY" << std::endl;
+    else {
+      auto* vh = boost::get<Vertex_const_handle>(&*(curr.second));;
+      if (vh) std::cout << '(' << (*vh)->point() << ')' << std::endl;
+      else {
+        auto* hh = boost::get<Halfedge_const_handle>(&*(curr.second));
+        CGAL_assertion(hh);
+        if (! (*hh)->is_fictitious())
+          std::cout << '[' << (*hh)->curve() << ']' << std::endl;
+        else std::cout << "NONE" << std::endl;
+      }
+    }
   }
 
   return 0;

Arrangement_on_surface_2/include/CGAL/Arr_Bezier_curve_traits_2.h

@@ -7,9 +7,9 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-// Author(s)     : Ron Wein     <wein@post.tau.ac.il>
-//                 Iddo Hanniel <iddoh@cs.technion.ac.il>
-//                 Waqar Khan   <wkhan@mpi-inf.mpg.de>
+// Author(s): Ron Wein     <wein@post.tau.ac.il>
+//            Iddo Hanniel <iddoh@cs.technion.ac.il>
+//            Waqar Khan   <wkhan@mpi-inf.mpg.de>
 
 #ifndef CGAL_ARR_BEZIER_CURVE_TRAITS_2_H
 #define CGAL_ARR_BEZIER_CURVE_TRAITS_2_H
@@ -163,7 +163,7 @@ public:
   }
   //@}
 
-  /// \name Functor definitions for the arrangement traits.
+  /// \name Basic functor definitions for the arrangement traits
   //@{
 
   /*! \class Compare_x_2
@@ -473,33 +473,34 @@ public:
   {
     return (Equal_2 (p_cache));
   }
+  //@}
+
+  //! \name Intersections, subdivisions, and mergings
+  //@{
 
   /*! \class Make_x_monotone_2
-   * The Make_x_monotone_2 functor.
+   * A functor for subdividing a curve into x-monotone curves.
    */
-  class Make_x_monotone_2
-  {
+  class Make_x_monotone_2 {
   private:
-    Bezier_cache         *p_cache;
+    Bezier_cache* p_cache;
 
   public:
-
     /*! Constructor. */
-    Make_x_monotone_2 (Bezier_cache *cache) :
-      p_cache (cache)
-    {}
+    Make_x_monotone_2(Bezier_cache* cache) : p_cache(cache) {}
 
-    /*!
-     * Cut the given Bezier curve into x-monotone subcurves and insert them
-     * into the given output iterator.
-     * \param cv The curve.
-     * \param oi The output iterator, whose value-type is Object. The returned
-     *           objects is a wrapper for an X_monotone_curve_2 object.
-     * \return The past-the-end iterator.
+    /*! Subdivide a given Bezier curve into x-monotone subcurves and insert them
+     * into a given output iterator.
+     * \param cv the curve.
+     * \param oi an output iterator for the result. Its value type is a variant
+     *           that wraps Point_2 or an X_monotone_curve_2 objects.
+     * \return the past-the-end iterator.
      */
-    template<class OutputIterator>
+    template <typename OutputIterator>
     OutputIterator operator() (const Curve_2& B, OutputIterator oi) const
     {
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+                                                Make_x_monotone_result;
       typedef typename Bounding_traits::Vertical_tangency_point
                                                 Vertical_tangency_point;
 
@@ -512,20 +513,17 @@ public:
                  std::back_inserter(cpts));
 
       bound_tr.compute_vertical_tangency_points
-          (cpts, std::back_inserter (vpt_bounds));
+        (cpts, std::back_inserter (vpt_bounds));
 
       // Construct Point_2 from bounded tangency points.
       std::list<Point_2>                            vpts;
       bool                                          app_ok = true;
-      typename std::list<Vertical_tangency_point>::const_iterator iter;
 
-      for (iter = vpt_bounds.begin(); iter != vpt_bounds.end(); ++iter)
-      {
+      for (auto iter = vpt_bounds.begin(); iter != vpt_bounds.end(); ++iter) {
         const typename Bounding_traits::Bez_point_bound& bound = iter->bound;
         const typename Bounding_traits::Bez_point_bbox&  bbox = iter->bbox;
 
-        if (! bound.can_refine)
-        {
+        if (! bound.can_refine) {
           // If we cannot refine the vertical-tangency bound anymore, then
           // we failed to bound the vertical tangency point.
           // \todo In the future, we might want to use the info.
@@ -534,28 +532,25 @@ public:
         }
 
         // Construct an approximate vertical tangency point.
-        Point_2   pt;
+        Point_2 pt;
 
-        if (bound.type == Bounding_traits::Bez_point_bound::RATIONAL_PT)
-        {
+        if (bound.type == Bounding_traits::Bez_point_bound::RATIONAL_PT) {
           CGAL_assertion (CGAL::compare (bound.t_min, bound.t_max) == EQUAL);
-          Rational  t0 = bound.t_min;
+          Rational t0 = bound.t_min;
 
           pt = Point_2 (B, t0);
         }
-        else
-        {
-          pt.add_originator (typename Point_2::Originator(B, bound));
+        else {
+          pt.add_originator(typename Point_2::Originator(B, bound));
         }
-        pt.set_bbox (bbox);
+        pt.set_bbox(bbox);
 
         vpts.push_back(pt);
       }
 
       // If bounding the approximated vertical-tangency points went fine,
       // use these points as endpoint for the x-monotone subcurves.
-      if (app_ok)
-      {
+      if (app_ok) {
         // Create the x-monotone subcurves with approximate endpoints.
         typename std::list<Point_2>::const_iterator pit;
         unsigned int  xid = 1;            // Serial number of the subcurve.
@@ -563,19 +558,16 @@ public:
         // Note: xid is needed in ctr of p0 (and of p1 below),
         // for handling end case of start point == end point.
 
-        for (pit = vpts.begin(); pit != vpts.end(); ++pit)
-        {
-          *oi++ = CGAL::make_object (X_monotone_curve_2 (B, xid,
-                                                         p0, *pit,
-                                                         *p_cache));
+        for (pit = vpts.begin(); pit != vpts.end(); ++pit) {
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(B, xid, p0, *pit,
+                                                            *p_cache));
           xid++;
           p0 = *pit;
         }
 
         Point_2    p1(B, xid, Rational(1)); // A rational end point.
-        *oi++ = CGAL::make_object (X_monotone_curve_2 (B, xid,
-                                                       p0, p1,
-                                                       *p_cache));
+        *oi++ = Make_x_monotone_result(X_monotone_curve_2(B, xid, p0, p1,
+                                                          *p_cache));
         return (oi);
       }
 
@@ -583,41 +575,34 @@ public:
       // points in an exact manner. We do this by considering all t-values
       // on B(t) = (X(t), Y(t)), such that X'(t) = 0.
       const typename Bezier_cache::Vertical_tangency_list&
-        vt_list = p_cache->get_vertical_tangencies (B.id(),
-                                                    B.x_polynomial(),
-                                                    B.x_norm());
+        vt_list = p_cache->get_vertical_tangencies(B.id(), B.x_polynomial(),
+                                                   B.x_norm());
 
       // Create the x-monotone subcurves.
-      Point_2                                        p1;
-      typename Bezier_cache::Vertical_tangency_iter  it;
-      unsigned int  xid = 1;            // Serial number of the subcurve.
-      Point_2                                        p0 (B, xid, Rational(0));
+      Point_2 p1;
+      unsigned int xid = 1;            // Serial number of the subcurve.
+      Point_2 p0(B, xid, Rational(0));
 
 
-      for (it = vt_list.begin(); it != vt_list.end(); ++it)
-      {
-        p1 = Point_2 (B, *it);
-        *oi++ = CGAL::make_object (X_monotone_curve_2 (B, xid,
-                                                       p0, p1,
-                                                       *p_cache));
+      for (auto it = vt_list.begin(); it != vt_list.end(); ++it) {
+        p1 = Point_2(B, *it);
+        *oi++ = Make_x_monotone_result(X_monotone_curve_2(B, xid, p0, p1,
+                                                          *p_cache));
         xid++;
         p0 = p1;
       }
 
       // Create the final subcurve.
-      p1 = Point_2 (B, xid, Rational(1));
-      *oi++ = CGAL::make_object (X_monotone_curve_2 (B, xid,
-                                                     p0, p1,
-                                                     *p_cache));
-      return (oi);
+      p1 = Point_2(B, xid, Rational(1));
+      *oi++ = Make_x_monotone_result(X_monotone_curve_2(B, xid, p0, p1,
+                                                        *p_cache));
+      return oi;
     }
   };
 
   /*! Get a Make_x_monotone_2 functor object. */
-  Make_x_monotone_2 make_x_monotone_2_object () const
-  {
-    return (Make_x_monotone_2 (p_cache));
-  }
+  Make_x_monotone_2 make_x_monotone_2_object() const
+  { return (Make_x_monotone_2 (p_cache)); }
 
   /*! \class Split_2
    * The Split_2 functor.
@@ -756,7 +741,6 @@ public:
   {
     return Merge_2(this);
   }
-
   //@}
 
   /// \name Functor definitions for the Boolean set-operation traits.

Arrangement_on_surface_2/include/CGAL/Arr_accessor.h

@@ -22,6 +22,7 @@
  */
 
 #include <CGAL/Arrangement_2/Arr_traits_adaptor_2.h>
+#include <CGAL/Arr_point_location_result.h>
 
 namespace CGAL {
 
@@ -60,6 +61,9 @@ private:
   typedef typename Arrangement_2::DInner_ccb            DInner_ccb;
   typedef typename Arrangement_2::DIso_vertex           DIso_vertex;
 
+  typedef Arr_point_location_result<Arrangement_2>      Pl_result;
+  typedef typename Pl_result::Type                      Pl_result_type;
+
 private:
   Arrangement_2* p_arr;           // The associated arrangement.
 
@@ -99,33 +103,29 @@ public:
    *         This object may wrap a Face_const_handle (the general case),
    *         or a Halfedge_const_handle (in case of an overlap).
    */
-  CGAL::Object locate_curve_end(const X_monotone_curve_2& cv,
-                                Arr_curve_end ind,
-                                Arr_parameter_space ps_x,
-                                Arr_parameter_space ps_y) const
+  Pl_result_type locate_curve_end(const X_monotone_curve_2& cv,
+                                  Arr_curve_end ind,
+                                  Arr_parameter_space ps_x,
+                                  Arr_parameter_space ps_y) const
   {
     CGAL_precondition((ps_x != ARR_INTERIOR) || (ps_y != ARR_INTERIOR));
 
     // Use the topology traits to locate the unbounded curve end.
-    CGAL::Object obj =
-      p_arr->topology_traits()->locate_curve_end(cv, ind, ps_x, ps_y);
+    auto obj = p_arr->topology_traits()->locate_curve_end(cv, ind, ps_x, ps_y);
 
     // Return a handle to the DCEL feature.
-    DFace* f;
-    if (CGAL::assign(f, obj))
-      return (CGAL::make_object(p_arr->_const_handle_for(f)));
+    DFace** f_p = boost::get<DFace*>(&obj);
+    if (f_p) return (Pl_result::make_result(p_arr->_const_handle_for(*f_p)));
 
-    DHalfedge* he;
-    if (CGAL::assign(he, obj))
-      return (CGAL::make_object(p_arr->_const_handle_for(he)));
+    DHalfedge** he_p = boost::get<DHalfedge*>(&obj);
+    if (he_p) return (Pl_result::make_result(p_arr->_const_handle_for(*he_p)));
 
-    DVertex* v;
-    if (CGAL::assign(v, obj))
-      return (CGAL::make_object(p_arr->_const_handle_for(v)));
+    DVertex** v_p = boost::get<DVertex*>(&obj);
+    if (v_p) return (Pl_result::make_result(p_arr->_const_handle_for(*v_p)));
 
     // We should never reach here:
     CGAL_error();
-    return Object();
+    return Pl_result::make_result(Vertex_const_handle());
   }
 
   /*!

Arrangement_on_surface_2/include/CGAL/Arr_algebraic_segment_traits_2.h

@@ -8,7 +8,7 @@
 // SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial
 //
 //
-// Author(s)     : Michael Kerber    <mkerber@mpi-inf.mpg.de>
+// Author(s): Michael Kerber    <mkerber@mpi-inf.mpg.de>
 //
 // ============================================================================
 
@@ -264,138 +264,141 @@ public:
 
         }
 
-        template<typename OutputIterator> OutputIterator
-        x_monotone_segment (Curve_2 cv,
-                            Point_2 p,
-                            boost::optional<Point_2> start,
-                            boost::optional<Point_2> end,
-                            OutputIterator out) const {
+      template <typename OutputIterator> OutputIterator
+      x_monotone_segment(Curve_2 cv,
+                         Point_2 p,
+                         boost::optional<Point_2> start,
+                         boost::optional<Point_2> end,
+                         OutputIterator out) const
+      {
+        typedef boost::variant<Point_2, X_monotone_curve_2>
+                                                        Make_x_monotone_result;
+        //CGAL_assertion(is_one_one(cv,p));
 
-            //CGAL_assertion(is_one_one(cv,p));
+        std::list<X_monotone_curve_2> segs;
 
-            std::list<X_monotone_curve_2> segs;
-
-            Is_on_2 on_arc
-                = this->_ckva()->is_on_2_object();
-            Construct_min_vertex_2 left
-                = this->_ckva()->construct_min_vertex_2_object();
-            Construct_max_vertex_2 right
-                = this->_ckva()->construct_max_vertex_2_object();
-            Equal_2 equal = this->_ckva()->equal_2_object();
-
-            std::vector<CGAL::Object> arcs;
-            this->_ckva()->make_x_monotone_2_object()
-                (cv,std::back_inserter(arcs));
-            typename std::vector<CGAL::Object>::const_iterator
-                it = arcs.begin(),helper;
-            X_monotone_curve_2 it_seg;
-            CGAL::assign(it_seg,*it);
-            while(it!=arcs.end()) {
-                if( on_arc(p,it_seg) ) {
-                    break;
-                }
-                CGAL_assertion(it!=arcs.end());
-                it++;
-                CGAL::assign(it_seg,*it);
-            }
-
-            bool left_on_arc = start && on_arc(start.get(),it_seg);
-            bool right_on_arc = end && on_arc(end.get(),it_seg);
-
-            if( left_on_arc && right_on_arc ) {
-                segs.push_back(it_seg.trim(start.get(),end.get()));
-            }
-            if(left_on_arc && (!right_on_arc)) {
-                if(!it_seg.is_finite(CGAL::ARR_MAX_END) ||
-                     !equal(start.get(),right(it_seg))) {
-                  if(it_seg.is_finite(CGAL::ARR_MIN_END) && equal(start.get(),left(it_seg))) {
-                        segs.push_back(it_seg);
-                    } else {
-                        X_monotone_curve_2 split1,split2;
-                        it_seg.split(start.get(),split1,split2);
-                        segs.push_back(split2);
-                    }
-                }
-            }
-            if((!left_on_arc) && right_on_arc) {
-                if(!it_seg.is_finite(CGAL::ARR_MIN_END) ||
-                   ! equal(left(it_seg),end.get())) {
-                    if(it_seg.is_finite(CGAL::ARR_MAX_END) && equal(end.get(),right(it_seg))) {
-                        segs.push_back(it_seg);
-                    } else {
-                        X_monotone_curve_2 split1,split2;
-                        it_seg.split(end.get(),split1,split2);
-                        segs.push_back(split1);
-                    }
-                }
-            }
-            if( (!left_on_arc) && (!right_on_arc)) {
-                segs.push_back(it_seg);
-            }
-            helper=it; // for later usage
-
-            if(! left_on_arc) {
-
-                Point_2 point_it;
-                while(true) {
-                    if(it_seg.is_finite(CGAL::ARR_MIN_END) &&
-                       is_one_one(cv,left(it_seg) ) ) {
-                        point_it=left(it_seg);
-                    } else {
-                        CGAL_assertion(! start);
-                        break;
-                    }
-                    CGAL_assertion(it!=arcs.begin());
-                    it--;
-                    CGAL::assign(it_seg,*it);
-                    while(! on_arc(point_it,it_seg)) {
-                        CGAL_assertion(it!=arcs.begin());
-                        it--;
-                        CGAL::assign(it_seg,*it);
-                    }
-                    if(start && on_arc(start.get(),it_seg)) {
-                        segs.push_front(it_seg.trim(start.get(),
-                                                    right(it_seg)));
-                        break;
-                    } else {
-                        segs.push_front(it_seg);
-                    }
-                }
-            }
-            if(! right_on_arc) {
-                it=helper; // reset
-                CGAL::assign(it_seg,*it);
-                Point_2 point_it;
-                while(true) {
-                    if(it_seg.is_finite(CGAL::ARR_MAX_END) &&
-                       is_one_one(cv,right(it_seg) ) ) {
-                        point_it=right(it_seg);
-                    } else {
-                        CGAL_assertion(! end);
-                        break;
-                    }
-                    it++;
-                    CGAL_assertion(it!=arcs.end());
-                    CGAL::assign(it_seg,*it);
-                    while(! on_arc(point_it,it_seg)) {
-                        it++;
-                        CGAL_assertion(it!=arcs.end());
-                        CGAL::assign(it_seg,*it);
-                    }
-                    if(end && on_arc(end.get(),it_seg)) {
-                        segs.push_back(it_seg.trim(left(it_seg),end.get()));
-                        break;
-                    } else {
-                        segs.push_back(it_seg);
-                    }
-                }
-            }
-
-            std::copy(segs.begin(),segs.end(),out);
-            return out;
+        Is_on_2 on_arc = this->_ckva()->is_on_2_object();
+        auto left = this->_ckva()->construct_min_vertex_2_object();
+        auto right = this->_ckva()->construct_max_vertex_2_object();
+        Equal_2 equal = this->_ckva()->equal_2_object();
 
+        std::vector<Make_x_monotone_result> arcs;
+        this->_ckva()->make_x_monotone_2_object()(cv, std::back_inserter(arcs));
+        auto it = arcs.begin();
+        auto helper = it;
+        const auto* it_seg_p = boost::get<X_monotone_curve_2>(&(*it));
+        while (it != arcs.end()) {
+          if ( on_arc(p, *it_seg_p) ) break;
+          it++;
+          it_seg_p = boost::get<X_monotone_curve_2>(&(*it));
         }
 
+        bool left_on_arc = start && on_arc(start.get(), *it_seg_p);
+        bool right_on_arc = end && on_arc(end.get(), *it_seg_p);
+
+        if ( left_on_arc && right_on_arc ) {
+          segs.push_back(it_seg_p->trim(start.get(),end.get()));
+        }
+        if (left_on_arc && (!right_on_arc)) {
+          if (!it_seg_p->is_finite(CGAL::ARR_MAX_END) ||
+             !equal(start.get(),right(*it_seg_p))) {
+            if (it_seg_p->is_finite(CGAL::ARR_MIN_END) &&
+                equal(start.get(),left(*it_seg_p)))
+            {
+              segs.push_back(*it_seg_p);
+            }
+            else {
+              X_monotone_curve_2 split1,split2;
+              it_seg_p->split(start.get(),split1,split2);
+              segs.push_back(split2);
+            }
+          }
+        }
+        if ((!left_on_arc) && right_on_arc) {
+          if (!it_seg_p->is_finite(CGAL::ARR_MIN_END) ||
+              ! equal(left(*it_seg_p), end.get()))
+          {
+            if (it_seg_p->is_finite(CGAL::ARR_MAX_END) &&
+                equal(end.get(), right(*it_seg_p)))
+            {
+              segs.push_back(*it_seg_p);
+            }
+            else {
+              X_monotone_curve_2 split1,split2;
+              it_seg_p->split(end.get(), split1, split2);
+              segs.push_back(split1);
+            }
+          }
+        }
+        if ( (!left_on_arc) && (!right_on_arc)) {
+          segs.push_back(*it_seg_p);
+        }
+        helper = it; // for later usage
+
+        if (! left_on_arc) {
+
+          Point_2 point_it;
+          while (true) {
+            if (it_seg_p->is_finite(CGAL::ARR_MIN_END) &&
+               is_one_one(cv, left(*it_seg_p) ) ) {
+              point_it = left(*it_seg_p);
+            } else {
+              CGAL_assertion(! start);
+              break;
+            }
+            CGAL_assertion(it != arcs.begin());
+            it--;
+            it_seg_p = boost::get<X_monotone_curve_2>(&(*it));
+            while (! on_arc(point_it, *it_seg_p)) {
+              CGAL_assertion(it != arcs.begin());
+              it--;
+              it_seg_p = boost::get<X_monotone_curve_2>(&(*it));
+            }
+            if (start && on_arc(start.get(),*it_seg_p)) {
+              segs.push_front(it_seg_p->trim(start.get(), right(*it_seg_p)));
+              break;
+            }
+            else {
+              segs.push_front(*it_seg_p);
+            }
+          }
+        }
+        if (! right_on_arc) {
+          it = helper; // reset
+          it_seg_p = boost::get<X_monotone_curve_2>(&(*it));
+          Point_2 point_it;
+          while (true) {
+            if (it_seg_p->is_finite(CGAL::ARR_MAX_END) &&
+                is_one_one(cv,right(*it_seg_p) ) )
+            {
+              point_it=right(*it_seg_p);
+            } else {
+              CGAL_assertion(! end);
+              break;
+            }
+            it++;
+            CGAL_assertion(it != arcs.end());
+            it_seg_p = boost::get<X_monotone_curve_2>(&(*it));
+            while(! on_arc(point_it, *it_seg_p)) {
+              it++;
+              CGAL_assertion(it != arcs.end());
+              it_seg_p = boost::get<X_monotone_curve_2>(&(*it));
+            }
+            if(end && on_arc(end.get(),*it_seg_p)) {
+              segs.push_back(it_seg_p->trim(left(*it_seg_p),end.get()));
+              break;
+            }
+            else {
+              segs.push_back(*it_seg_p);
+            }
+          }
+        }
+
+        std::copy(segs.begin(), segs.end(), out);
+        return out;
+
+      }
+
     public:
 
         template<typename OutputIterator> OutputIterator

Arrangement_on_surface_2/include/CGAL/Arr_batched_point_location.h

@@ -41,11 +41,9 @@ namespace Ss2 = Surface_sweep_2;
  * \param oi Output: An output iterator for the query results.
  * \pre The value-type of PointsIterator is Arrangement::Point_2,
  *      and the value-type of OutputIterator is is pair<Point_2, Result>,
- *      where Result is either
- *       (i) Object or
- *      (ii) boost::optional<boost::variant<Vertex_const_handle,
- *                                          Halfedge_const_handle,
- *                                          Face_const_handle> >.
+ *      where Result is boost::optional<boost::variant<Vertex_const_handle,
+ *                                      Halfedge_const_handle,
+ *                                      Face_const_handle> >.
  *      It represents the arrangement feature containing the point.
  */
 template <typename GeometryTraits_2, typename TopologyTraits,

Arrangement_on_surface_2/include/CGAL/Arr_bounded_planar_topology_traits_2.h

@@ -7,14 +7,15 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-//
-// Author(s) : Ron Wein        <wein@post.tau.ac.il>
-//             Efi Fogel       <efif@post.tau.ac.il>
-//             Eric Berberich  <ericb@post.tau.ac.il>
+// Author(s): Ron Wein        <wein@post.tau.ac.il>
+//            Efi Fogel       <efif@post.tau.ac.il>
+//            Eric Berberich  <ericb@post.tau.ac.il>
 
 #ifndef CGAL_ARR_BOUNDED_PLANAR_TOPOLOGY_TRAITS_2_H
 #define CGAL_ARR_BOUNDED_PLANAR_TOPOLOGY_TRAITS_2_H
 
+#include <boost/variant.hpp>
+
 #include <CGAL/license/Arrangement_on_surface_2.h>
 
 #include <CGAL/disable_warnings.h>
@@ -306,17 +307,18 @@ public:
    * \param ps_x The boundary condition of the curve end in x.
    * \param ps_y The boundary condition of the curve end in y.
    * \pre The curve has a boundary condition in either x or y.
-   * \return An object that contains the curve end.
+   * \return An object that wraps the curve end.
    */
-  CGAL::Object place_boundary_vertex(Face*,
-                                     const X_monotone_curve_2&,
-                                     Arr_curve_end,
-                                     Arr_parameter_space /* ps_x */,
-                                     Arr_parameter_space /* ps_y */)
+  boost::optional<boost::variant<Vertex*, Halfedge*> >
+  place_boundary_vertex(Face*,
+                        const X_monotone_curve_2&,
+                        Arr_curve_end,
+                        Arr_parameter_space /* ps_x */,
+                        Arr_parameter_space /* ps_y */)
   {
     // This function should never be called:
     CGAL_error();
-    return Object();
+    return boost::none;
   }
 
   /*! Locate the predecessor halfedge for the given curve around a given
@@ -349,14 +351,17 @@ public:
    * \pre The curve end is incident to the boundary.
    * \return An object that contains the curve end.
    */
-  CGAL::Object locate_curve_end(const X_monotone_curve_2&,
-                                Arr_curve_end,
-                                Arr_parameter_space /* ps_x */,
-                                Arr_parameter_space /* ps_y */)
+  boost::variant<Vertex*, Halfedge*, Face*>
+  locate_curve_end(const X_monotone_curve_2&,
+                   Arr_curve_end,
+                   Arr_parameter_space /* ps_x */,
+                   Arr_parameter_space /* ps_y */)
   {
+    typedef boost::variant<Vertex*, Halfedge*, Face*>   Result;
     // This function should never be called:
     CGAL_error();
-    return Object();
+    Vertex* v(nullptr);
+    return Result(v);
   }
 
   /*! Split a fictitious edge using the given vertex.

Arrangement_on_surface_2/include/CGAL/Arr_circle_segment_traits_2.h

@@ -378,46 +378,47 @@ public:
   }
   //@}
 
-  /// \name Functor definitions for supporting intersections.
+  /// \name Intersections, subdivisions, and mergings
   //@{
 
-  class Make_x_monotone_2
-  {
+  /*! \class
+   * A functor for subdividing a curve into x-monotone curves.
+   */
+  class Make_x_monotone_2 {
   private:
     typedef Arr_circle_segment_traits_2<Kernel_, Filter> Self;
 
     bool m_use_cache;
 
   public:
+    Make_x_monotone_2(bool use_cache = false) : m_use_cache(use_cache) {}
 
-    Make_x_monotone_2(bool use_cache = false) : m_use_cache(use_cache)
-    {}
-
-    /*!
-     * Cut the given conic curve (ocv.is_in_x_range (p)r conic arc) into x-monotone subcurves
-     * and insert them to the given output iterator.
-     * \param cv The curve.
-     * \param oi The output iterator, whose value-type is Object. The returned
-     *           objects are all wrcv.is_in_x_range (p)appers X_monotone_curve_2 objects.
-     * \return The past-the-end iterator.
+    /*! Subdivide a given circular arc or line segment into x-monotone subcurves
+     * and insert them to a given output iterator.
+     * \param cv the curve.
+     * \param oi the output iterator for the result. Its dereference type is a
+     *           variant that wraps a \c Point_2 or an \c X_monotone_curve_2
+     *           objects.
+     * \return the past-the-end iterator.
      */
-    template<class OutputIterator>
-    OutputIterator operator() (const Curve_2& cv, OutputIterator oi) const
+    template <typename OutputIterator>
+    OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const
     {
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
+
       // Increment the serial number of the curve cv, which will serve as its
       // unique identifier.
-      unsigned int  index = 0;
-      if(m_use_cache)
-        index = Self::get_index();
+      unsigned int index = 0;
+      if (m_use_cache) index = Self::get_index();
 
-      if (cv.orientation() == COLLINEAR)
-      {
+      if (cv.orientation() == COLLINEAR) {
         // The curve is a line segment.
-        *oi = make_object (X_monotone_curve_2 (cv.supporting_line(),
-                                               cv.source(), cv.target(),
-                                               index));
-        ++oi;
-        return (oi);
+        *oi++ = Make_x_monotone_result(X_monotone_curve_2(cv.supporting_line(),
+                                                          cv.source(),
+                                                          cv.target(),
+                                                          index));
+        return oi;
       }
 
       // Check the case of a degenrate circle (a point).
@@ -425,12 +426,11 @@ public:
       CGAL::Sign   sign_rad = CGAL::sign (circ.squared_radius());
       CGAL_precondition (sign_rad != NEGATIVE);
 
-      if (sign_rad == ZERO)
-      {
+      if (sign_rad == ZERO) {
         // Create an isolated point.
-        *oi = make_object (Point_2 (circ.center().x(), circ.center().y()));
-        ++oi;
-        return (oi);
+        *oi++ = Make_x_monotone_result(Point_2(circ.center().x(),
+                                               circ.center().y()));
+        return oi;
       }
 
       // The curve is circular: compute the to vertical tangency points
@@ -438,84 +438,73 @@ public:
       Point_2         vpts[2];
       unsigned int    n_vpts = cv.vertical_tangency_points (vpts);
 
-      if (cv.is_full())
-      {
+      if (cv.is_full()) {
         CGAL_assertion (n_vpts == 2);
 
         // Subdivide the circle into two arcs (an upper and a lower half).
-        *oi = make_object (X_monotone_curve_2 (circ,
-                                               vpts[0], vpts[1],
-                                               cv.orientation(),
-                                               index));
-        ++oi;
+        *oi++ = Make_x_monotone_result(X_monotone_curve_2(circ,
+                                                          vpts[0], vpts[1],
+                                                          cv.orientation(),
+                                                          index));
 
-        *oi = make_object (X_monotone_curve_2 (circ,
-                                               vpts[1], vpts[0],
-                                               cv.orientation(),
-                                               index));
-        ++oi;
+        *oi++ = Make_x_monotone_result(X_monotone_curve_2(circ,
+                                                          vpts[1], vpts[0],
+                                                          cv.orientation(),
+                                                          index));
       }
-      else
-      {
+      else {
         // Act according to the number of vertical tangency points.
-        if (n_vpts == 2)
-        {
+        if (n_vpts == 2) {
           // Subdivide the circular arc into three x-monotone arcs.
-          *oi = make_object (X_monotone_curve_2 (circ,
-                                                 cv.source(), vpts[0],
-                                                 cv.orientation(),
-                                                 index));
-          ++oi;
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(circ,
+                                                            cv.source(), vpts[0],
+                                                            cv.orientation(),
+                                                            index));
 
-          *oi = make_object (X_monotone_curve_2 (circ,
-                                                 vpts[0], vpts[1],
-                                                 cv.orientation(),
-                                                 index));
-          ++oi;
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(circ,
+                                                            vpts[0], vpts[1],
+                                                            cv.orientation(),
+                                                            index));
 
-          *oi = make_object (X_monotone_curve_2 (circ,
-                                                 vpts[1], cv.target(),
-                                                 cv.orientation(),
-                                                 index));
-          ++oi;
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(circ,
+                                                            vpts[1],
+                                                            cv.target(),
+                                                            cv.orientation(),
+                                                            index));
         }
-        else if (n_vpts == 1)
-        {
+        else if (n_vpts == 1) {
           // Subdivide the circular arc into two x-monotone arcs.
-          *oi = make_object (X_monotone_curve_2 (circ,
-                                                 cv.source(), vpts[0],
-                                                 cv.orientation(),
-                                                 index));
-          ++oi;
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(circ,
+                                                            cv.source(),
+                                                            vpts[0],
+                                                            cv.orientation(),
+                                                            index));
 
-          *oi = make_object (X_monotone_curve_2 (circ,
-                                                 vpts[0], cv.target(),
-                                                 cv.orientation(),
-                                                 index));
-          ++oi;
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(circ,
+                                                            vpts[0],
+                                                            cv.target(),
+                                                            cv.orientation(),
+                                                            index));
         }
-        else
-        {
-          CGAL_assertion (n_vpts == 0);
+        else {
+          CGAL_assertion(n_vpts == 0);
 
           // The arc is already x-monotone:
-          *oi = make_object (X_monotone_curve_2 (circ,
-                                                 cv.source(), cv.target(),
-                                                 cv.orientation(),
-                                                 index));
-          ++oi;
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(circ,
+                                                            cv.source(),
+                                                            cv.target(),
+                                                            cv.orientation(),
+                                                            index));
         }
       }
 
-      return (oi);
+      return oi;
     }
   };
 
   /*! Get a Make_x_monotone_2 functor object. */
-  Make_x_monotone_2 make_x_monotone_2_object () const
-  {
-    return Make_x_monotone_2(m_use_cache);
-  }
+  Make_x_monotone_2 make_x_monotone_2_object() const
+  { return Make_x_monotone_2(m_use_cache); }
 
   class Split_2
   {

Arrangement_on_surface_2/include/CGAL/Arr_circular_arc_traits_2.h

@@ -27,6 +27,10 @@
  * Arrangement_2 package, which it is now part of. It contains a traits
  * class for the arrangement package that handles circular curves.
  * It is based on the circular kernel.
+ *
+ * \todo Fix the circular-kernel make-x-monotone functor to use modern variant
+ *       instead of the legacy CGAL::Object. Then, eliminate the special
+ *       implementation here and directly use the kernel functor instead.
  */
 
 #include <CGAL/basic.h>
@@ -109,7 +113,7 @@ public:
   typedef unsigned int                           Multiplicity;
 
   typedef CGAL::Tag_false                        Has_left_category;
-  typedef CGAL::Tag_false                          Has_merge_category;
+  typedef CGAL::Tag_false                        Has_merge_category;
   typedef CGAL::Tag_false                        Has_do_intersect_category;
 
   typedef Arr_oblivious_side_tag                 Left_side_category;
@@ -129,7 +133,7 @@ public:
   typedef typename CircularKernel::Construct_circular_min_vertex_2
                                                         Construct_min_vertex_2;
   typedef typename CircularKernel::Equal_2              Equal_2;
-  typedef typename CircularKernel::Make_x_monotone_2    Make_x_monotone_2;
+  // typedef typename CircularKernel::Make_x_monotone_2    Make_x_monotone_2;
   typedef typename CircularKernel::Split_2              Split_2;
   typedef typename CircularKernel::Intersect_2          Intersect_2;
   typedef typename CircularKernel::Is_vertical_2        Is_vertical_2;
@@ -149,8 +153,8 @@ public:
   Equal_2 equal_2_object() const
   { return ck.equal_2_object(); }
 
-  Make_x_monotone_2 make_x_monotone_2_object() const
-  { return ck.make_x_monotone_2_object(); }
+  // Make_x_monotone_2 make_x_monotone_2_object() const
+  // { return ck.make_x_monotone_2_object(); }
 
   Split_2 split_2_object() const
   { return ck.split_2_object(); }
@@ -168,6 +172,34 @@ public:
     { return ck.is_vertical_2_object();  }
 
 
+  //! A functor for subdividing curves into x-monotone curves.
+  class Make_x_monotone_2 {
+  public:
+    template <typename OutputIterator>
+    OutputIterator operator()(const Curve_2& arc, OutputIterator oi) const
+    {
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
+
+      std::vector<CGAL::Object> objs;
+      CircularKernel().make_x_monotone_2_object()(arc, std::back_inserter(objs));
+      for (const auto& obj : objs) {
+        if (const auto* p = CGAL::object_cast<Point_2>(&obj)) {
+          *oi++ = Make_x_monotone_result(*p);
+          continue;
+        }
+        if (const auto* xcv = CGAL::object_cast<X_monotone_curve_2>(&obj)) {
+          *oi++ = Make_x_monotone_result(*xcv);
+          continue;
+        }
+        CGAL_error();
+      }
+      return oi;
+    }
+  };
+
+  Make_x_monotone_2 make_x_monotone_2_object() const
+  { return Make_x_monotone_2(); }
 };
 
 } // namespace CGAL

Arrangement_on_surface_2/include/CGAL/Arr_circular_line_arc_traits_2.h

@@ -27,6 +27,10 @@
  * Arrangement_2 package, which it is now part of. It contains a traits
  * class for the arrangement package that handles circular and linear curves.
  * It is based on the circular kernel.
+ *
+ * \todo Fix the circular-kernel make-x-monotone functor to use modern variant
+ *       instead of the legacy CGAL::Object. Then, eliminate the special
+ *       implementation here and directly use the kernel functor instead.
  */
 
 #include <vector>
@@ -39,53 +43,35 @@
 namespace CGAL {
   namespace VariantFunctors{
 
-    // Takes an iterator range of Object(Line/Circular_arc),
-    // returns an Object(Variant(Line/Circular_arc)).
-    // Do nothing for Object(Endpoint).
+    // Takes an iterator range of Object(Line/Circular_arc/Point),
+    // returns a variant of Line, Circular_arc, and Point_2.
     template <class CK, class Arc1, class Arc2, class OutputIterator>
     OutputIterator
     object_to_object_variant(const std::vector<CGAL::Object>& res1,
                              OutputIterator res2)
     {
-      for (std::vector<CGAL::Object>::const_iterator it = res1.begin();
-          it != res1.end(); ++it ) {
-        if (const Arc1 *arc = CGAL::object_cast< Arc1 >(&*it)){
-          boost::variant< Arc1, Arc2 > v =  *arc;
-          *res2++ = make_object(v);
+      typedef typename CK::Circular_arc_point_2         Point_2;
+      typedef boost::variant<Arc1, Arc2>                X_monotone_curve_2;
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
+
+      for (auto it = res1.begin(); it != res1.end(); ++it) {
+        if (const Arc1* arc = CGAL::object_cast<Arc1>(&*it)) {
+          boost::variant<Arc1, Arc2> v =  *arc;
+          *res2++ = Make_x_monotone_result(v);
         }
-        else if (const Arc2 *line = CGAL::object_cast< Arc2 >(&*it)){
-          boost::variant< Arc1, Arc2 > v =  *line;
-          *res2++ = make_object(v);
+        else if (const Arc2* line = CGAL::object_cast<Arc2>(&*it)) {
+          boost::variant<Arc1, Arc2> v =  *line;
+          *res2++ = Make_x_monotone_result(v);
         }
-        else{
-          *res2++ = *it;
+        else if (const Point_2* p = CGAL::object_cast<Point_2>(&*it)) {
+          *res2++ = Make_x_monotone_result(*p);
         }
+        else CGAL_error();
       }
       return res2;
     }
 
-    template <class CK, class Arc,
-              class IntersectionPoint, class XMonotoneCurve,
-              class InternaType, class OutputIterator>
-    OutputIterator
-    object_to_object_variant1(const std::vector<InternaType>& res,
-                              OutputIterator oi)
-    {
-      typedef IntersectionPoint                         Intersection_point;
-      typedef XMonotoneCurve                            X_monotone_curve_2;
-      typedef boost::variant<Intersection_point, X_monotone_curve_2>
-                                                        Intersection_result;
-
-      for (auto it = res.begin(); it != res.end(); ++it) {
-        if (const Arc* arc = boost::get<Arc>(&*it)) {
-          X_monotone_curve_2 cv = *arc;
-          *oi++ = Intersection_result(cv);
-        }
-        else *oi++ = Intersection_result(*it);
-      }
-      return oi;
-    }
-
     template <class CircularKernel, class Arc1, class Arc2>
     class Compare_y_to_right_2
     {
@@ -147,17 +133,12 @@ namespace CGAL {
 
     template <class CircularKernel, class Arc1, class Arc2>
     class Equal_2
-#ifndef CGAL_CFG_MATCHING_BUG_6
-      : public
-      CircularKernel::Equal_2
-#endif
+      : public CircularKernel::Equal_2
     {
     public:
       typedef boost::variant< Arc1, Arc2 >  Curve_2;
       typedef bool result_type;
-#ifndef CGAL_CFG_MATCHING_BUG_6
       using CircularKernel::Equal_2::operator();
-#else
       typedef typename CircularKernel::Circular_arc_point_2
                                                   Circular_arc_point_2;
       typedef typename CircularKernel::Line_arc_2     Line_arc_2;
@@ -185,8 +166,6 @@ namespace CGAL {
     operator() ( const Circular_arc_2 &a0, const Line_arc_2 &a1) const
     { return false; }
 
-#endif
-
       result_type
       operator()(const Curve_2 &a0, const Curve_2 &a1) const
       {
@@ -259,6 +238,7 @@ namespace CGAL {
     };
 
 
+    //! A functor for subdividing curves into x-monotone curves.
     template <class CircularKernel, class Arc1, class Arc2>
     class Make_x_monotone_2
     {
@@ -268,20 +248,21 @@ namespace CGAL {
 
       template < class OutputIterator,class Not_X_Monotone >
       OutputIterator
-      operator()(const boost::variant<Arc1, Arc2, Not_X_Monotone> &A, OutputIterator res) const
+      operator()(const boost::variant<Arc1, Arc2, Not_X_Monotone> &A,
+                 OutputIterator res) const
       {
-        if ( const Arc1* arc1 = boost::get<Arc1>( &A ) ){
+        if ( const Arc1* arc1 = boost::get<Arc1>( &A ) ) {
           std::vector<CGAL::Object> container;
-          CircularKernel()
-            .make_x_monotone_2_object()(*arc1,std::back_inserter(container));
+          CircularKernel().
+            make_x_monotone_2_object()(*arc1,std::back_inserter(container));
           return object_to_object_variant<CircularKernel, Arc1, Arc2>
                                           (container, res);
         }
         else {
           const Arc2* arc2 = boost::get<Arc2>( &A );
           std::vector<CGAL::Object> container;
-          CircularKernel()
-            .make_x_monotone_2_object()(*arc2,std::back_inserter(container));
+          CircularKernel().
+            make_x_monotone_2_object()(*arc2,std::back_inserter(container));
           return object_to_object_variant<CircularKernel, Arc1, Arc2>
                                           (container, res);
         }

Arrangement_on_surface_2/include/CGAL/Arr_conic_traits_2.h

@@ -448,135 +448,127 @@ public:
   }
   //@}
 
-  /// \name Functor definitions for supporting intersections.
+  /// \name Intersections, subdivisions, and mergings
   //@{
 
-  class Make_x_monotone_2
-  {
+  /*! \class Make_x_monotone_2
+   * A functor for subdividing curves into x-monotone curves.
+   */
+  class Make_x_monotone_2 {
     typedef Arr_conic_traits_2 <Rat_kernel_, Alg_kernel_, Nt_traits_>    Self;
-  public:
 
-    /*!
-     * Cut the given conic curve (or conic arc) into x-monotone subcurves
-     * and insert them to the given output iterator.
-     * \param cv The curve.
-     * \param oi The output iterator, whose value-type is Object. The returned
-     *           objects are all wrappers X_monotone_curve_2 objects.
-     * \return The past-the-end iterator.
+  public:
+    /*! Subdivide a given conic curve (or conic arc) into x-monotone subcurves
+     * and insert them to a given output iterator.
+     * \param cv the curve.
+     * \param oi the output iterator for the result. Its dereference type is a
+     *           variant that wraps a \c Point_2 or an \c X_monotone_curve_2
+     *           objects.
+     * \return the past-the-end iterator.
      */
-    template<class OutputIterator>
-    OutputIterator operator() (const Curve_2& cv, OutputIterator oi) const
+    template <typename OutputIterator>
+    OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const
     {
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
+
       // Increment the serial number of the curve cv, which will serve as its
       // unique identifier.
-      unsigned int  index = Self::get_index();
-      Conic_id      conic_id (index);
+      auto index = Self::get_index();
+      Conic_id conic_id(index);
 
       // Find the points of vertical tangency to cv and act accordingly.
-      typename Curve_2::Point_2  vtan_ps[2];
-      int                        n_vtan_ps;
+      typename Curve_2::Point_2 vtan_ps[2];
+      int n_vtan_ps;
 
-      n_vtan_ps = cv.vertical_tangency_points (vtan_ps);
+      n_vtan_ps = cv.vertical_tangency_points(vtan_ps);
 
-      if (n_vtan_ps == 0)
-      {
+      if (n_vtan_ps == 0) {
         // In case the given curve is already x-monotone:
-        *oi = make_object (X_monotone_curve_2 (cv, conic_id));
-        ++oi;
-        return (oi);
+        *oi++ = Make_x_monotone_result(X_monotone_curve_2(cv, conic_id));
+        return oi;
       }
 
       // Split the conic arc into x-monotone sub-curves.
-      if (cv.is_full_conic())
-      {
+      if (cv.is_full_conic()) {
         // Make sure we have two vertical tangency points.
         CGAL_assertion(n_vtan_ps == 2);
 
         // In case the curve is a full conic, split it into two x-monotone
         // arcs, one going from ps[0] to ps[1], and the other from ps[1] to
         // ps[0].
-        *oi = make_object (X_monotone_curve_2 (cv, vtan_ps[0], vtan_ps[1],
-                                               conic_id));
-        ++oi;
-        *oi = make_object (X_monotone_curve_2 (cv, vtan_ps[1], vtan_ps[0],
-                                               conic_id));
-        ++oi;
+        *oi++ = Make_x_monotone_result(X_monotone_curve_2(cv, vtan_ps[0],
+                                                          vtan_ps[1],
+                                                          conic_id));
+        *oi++ = Make_x_monotone_result(X_monotone_curve_2(cv, vtan_ps[1],
+                                                          vtan_ps[0],
+                                                          conic_id));
       }
-      else
-      {
-        if (n_vtan_ps == 1)
-        {
+      else {
+        if (n_vtan_ps == 1) {
           // Split the arc into two x-monotone sub-curves: one going from the
           // arc source to ps[0], and the other from ps[0] to the target.
-          *oi = make_object (X_monotone_curve_2 (cv, cv.source(), vtan_ps[0],
-                                                 conic_id));
-          ++oi;
-          *oi = make_object (X_monotone_curve_2 (cv, vtan_ps[0], cv.target(),
-                                                 conic_id));
-          ++oi;
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(cv, cv.source(),
+                                                            vtan_ps[0],
+                                                            conic_id));
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(cv, vtan_ps[0],
+                                                            cv.target(),
+                                                            conic_id));
         }
-        else
-        {
-          CGAL_assertion (n_vtan_ps == 2);
+        else {
+          CGAL_assertion(n_vtan_ps == 2);
 
           // Identify the first point we encounter when going from cv's source
           // to its target, and the second point we encounter. Note that the
           // two endpoints must both be below the line connecting the two
           // tangnecy points (or both lies above it).
-          int                   ind_first = 0;
-          int                   ind_second = 1;
-          Alg_kernel_           ker;
-          typename Alg_kernel_::Line_2  line =
-            ker.construct_line_2_object() (vtan_ps[0], vtan_ps[1]);
-          const Comparison_result       start_pos =
+          int ind_first = 0;
+          int ind_second = 1;
+          Alg_kernel_ ker;
+          typename Alg_kernel_::Line_2 line =
+            ker.construct_line_2_object()(vtan_ps[0], vtan_ps[1]);
+          const Comparison_result start_pos =
             ker.compare_y_at_x_2_object() (cv.source(), line);
-          const Comparison_result       order_vpts =
-            ker.compare_x_2_object() (vtan_ps[0], vtan_ps[1]);
+          const Comparison_result order_vpts =
+            ker.compare_x_2_object()(vtan_ps[0], vtan_ps[1]);
 
-          CGAL_assertion (start_pos != EQUAL &&
-                          ker.compare_y_at_x_2_object() (cv.target(),
-                                                         line) == start_pos);
-          CGAL_assertion (order_vpts != EQUAL);
+          CGAL_assertion(start_pos != EQUAL &&
+                         ker.compare_y_at_x_2_object()(cv.target(),
+                                                       line) == start_pos);
+          CGAL_assertion(order_vpts != EQUAL);
 
-          if ((cv.orientation() == COUNTERCLOCKWISE &&
-               start_pos == order_vpts) ||
-              (cv.orientation() == CLOCKWISE &&
-               start_pos != order_vpts))
+          if (((cv.orientation() == COUNTERCLOCKWISE) &&
+               (start_pos == order_vpts)) ||
+              ((cv.orientation() == CLOCKWISE) && (start_pos != order_vpts)))
           {
             ind_first = 1;
             ind_second = 0;
           }
 
           // Split the arc into three x-monotone sub-curves.
-          *oi = make_object (X_monotone_curve_2 (cv,
-                                                 cv.source(),
-                                                 vtan_ps[ind_first],
-                                                 conic_id));
-          ++oi;
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(cv, cv.source(),
+                                                            vtan_ps[ind_first],
+                                                            conic_id));
 
-          *oi = make_object (X_monotone_curve_2 (cv,
-                                                 vtan_ps[ind_first],
-                                                 vtan_ps[ind_second],
-                                                 conic_id));
-          ++oi;
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(cv,
+                                                            vtan_ps[ind_first],
+                                                            vtan_ps[ind_second],
+                                                            conic_id));
 
-          *oi = make_object (X_monotone_curve_2 (cv,
-                                                 vtan_ps[ind_second],
-                                                 cv.target(),
-                                                 conic_id));
-          ++oi;
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(cv,
+                                                            vtan_ps[ind_second],
+                                                            cv.target(),
+                                                            conic_id));
         }
       }
 
-      return (oi);
+      return oi;
     }
   };
 
   /*! Get a Make_x_monotone_2 functor object. */
-  Make_x_monotone_2 make_x_monotone_2_object () const
-  {
-    return Make_x_monotone_2();
-  }
+  Make_x_monotone_2 make_x_monotone_2_object() const
+  { return Make_x_monotone_2(); }
 
   class Split_2
   {

Arrangement_on_surface_2/include/CGAL/Arr_counting_traits_2.h

@@ -101,120 +101,120 @@ public:
   }
 
   /*! Obtain the counter of the given operation */
-  unsigned int count(Operation_id id) const
+  size_t count(Operation_id id) const
   { return m_counters[id]; }
 
-  unsigned int count_compare_x() const
+  size_t count_compare_x() const
   { return m_counters[COMPARE_X_OP]; }
 
-  unsigned int count_compare_xy() const
+  size_t count_compare_xy() const
   { return m_counters[COMPARE_XY_OP]; }
 
-  unsigned int count_construct_min_vertex() const
+  size_t count_construct_min_vertex() const
   { return m_counters[CONSTRUCT_MIN_VERTEX_OP]; }
 
-  unsigned int count_construct_max_vertex() const
+  size_t count_construct_max_vertex() const
   { return m_counters[CONSTRUCT_MAX_VERTEX_OP]; }
 
-  unsigned int count_is_vertical() const
+  size_t count_is_vertical() const
   { return m_counters[IS_VERTICAL_OP]; }
 
-  unsigned int count_compare_y_at_x() const
+  size_t count_compare_y_at_x() const
   { return m_counters[COMPARE_Y_AT_X_OP]; }
 
-  unsigned int count_equal_points() const
+  size_t count_equal_points() const
   { return m_counters[EQUAL_POINTS_OP]; }
 
-  unsigned int count_equal_curves() const
+  size_t count_equal_curves() const
   { return m_counters[EQUAL_CURVES_OP]; }
 
-  unsigned int count_compare_y_at_x_left() const
+  size_t count_compare_y_at_x_left() const
   { return m_counters[COMPARE_Y_AT_X_LEFT_OP]; }
 
-  unsigned int count_compare_y_at_x_right() const
+  size_t count_compare_y_at_x_right() const
   { return m_counters[COMPARE_Y_AT_X_RIGHT_OP]; }
 
-  unsigned int count_make_x_monotone() const
+  size_t count_make_x_monotone() const
   { return m_counters[MAKE_X_MONOTONE_OP]; }
 
-  unsigned int count_split() const
+  size_t count_split() const
   { return m_counters[SPLIT_OP]; }
 
-  unsigned int count_intersect() const
+  size_t count_intersect() const
   { return m_counters[INTERSECT_OP]; }
 
-  unsigned int count_are_mergeable() const
+  size_t count_are_mergeable() const
   { return m_counters[ARE_MERGEABLE_OP]; }
 
-  unsigned int count_merge() const
+  size_t count_merge() const
   { return m_counters[MERGE_OP]; }
 
-  unsigned int count_construct_opposite() const
+  size_t count_construct_opposite() const
   { return m_counters[CONSTRUCT_OPPOSITE_OP]; }
 
-  unsigned int count_compare_endpoints_xy() const
+  size_t count_compare_endpoints_xy() const
   { return m_counters[COMPARE_ENDPOINTS_XY_OP]; }
 
   // left-right
 
-  unsigned int count_parameter_space_in_x_curve_end() const
+  size_t count_parameter_space_in_x_curve_end() const
   { return m_counters[PARAMETER_SPACE_IN_X_CURVE_END_OP]; }
 
-  unsigned int count_parameter_space_in_x_curve() const
+  size_t count_parameter_space_in_x_curve() const
   { return m_counters[PARAMETER_SPACE_IN_X_CURVE_OP]; }
 
-  unsigned int count_parameter_space_in_x_point() const
+  size_t count_parameter_space_in_x_point() const
   { return m_counters[PARAMETER_SPACE_IN_X_POINT_OP]; }
 
-  unsigned int count_is_on_x_identification_point() const
+  size_t count_is_on_x_identification_point() const
   { return m_counters[IS_ON_X_IDENTIFICATION_POINT_OP]; }
 
-  unsigned int count_is_on_x_identification_curve() const
+  size_t count_is_on_x_identification_curve() const
   { return m_counters[IS_ON_X_IDENTIFICATION_CURVE_OP]; }
 
-  unsigned int count_compare_y_on_boundary() const
+  size_t count_compare_y_on_boundary() const
   { return m_counters[COMPARE_Y_ON_BOUNDARY_OP]; }
 
-  unsigned int count_compare_y_near_boundary() const
+  size_t count_compare_y_near_boundary() const
   { return m_counters[COMPARE_Y_NEAR_BOUNDARY_OP]; }
 
 
   // bottom-top
 
-  unsigned int count_parameter_space_in_y_curve_end() const
+  size_t count_parameter_space_in_y_curve_end() const
   { return m_counters[PARAMETER_SPACE_IN_Y_CURVE_END_OP]; }
 
-  unsigned int count_parameter_space_in_y_curve() const
+  size_t count_parameter_space_in_y_curve() const
   { return m_counters[PARAMETER_SPACE_IN_Y_CURVE_OP]; }
 
-  unsigned int count_parameter_space_in_y_point() const
+  size_t count_parameter_space_in_y_point() const
   { return m_counters[PARAMETER_SPACE_IN_Y_POINT_OP]; }
 
-  unsigned int count_is_on_y_identification_point() const
+  size_t count_is_on_y_identification_point() const
   { return m_counters[IS_ON_Y_IDENTIFICATION_POINT_OP]; }
 
-  unsigned int count_is_on_y_identification_curve() const
+  size_t count_is_on_y_identification_curve() const
   { return m_counters[IS_ON_Y_IDENTIFICATION_CURVE_OP]; }
 
-  unsigned int count_compare_x_at_limit_point_curve_end() const
+  size_t count_compare_x_at_limit_point_curve_end() const
   { return m_counters[COMPARE_X_AT_LIMIT_POINT_CURVE_END_OP]; }
 
-  unsigned int count_compare_x_at_limit_curve_ends() const
+  size_t count_compare_x_at_limit_curve_ends() const
   { return m_counters[COMPARE_X_AT_LIMIT_CURVE_ENDS_OP]; }
 
-  unsigned int count_compare_x_near_limit() const
+  size_t count_compare_x_near_limit() const
   { return m_counters[COMPARE_X_NEAR_LIMIT_OP]; }
 
-  unsigned int count_compare_x_on_boundary_points() const
+  size_t count_compare_x_on_boundary_points() const
   { return m_counters[COMPARE_X_ON_BOUNDARY_POINTS_OP]; }
 
-  unsigned int count_compare_x_on_boundary_point_curve_end() const
+  size_t count_compare_x_on_boundary_point_curve_end() const
   { return m_counters[COMPARE_X_ON_BOUNDARY_POINT_CURVE_END_OP]; }
 
-  unsigned int count_compare_x_on_boundary_curve_ends() const
+  size_t count_compare_x_on_boundary_curve_ends() const
   { return m_counters[COMPARE_X_ON_BOUNDARY_CURVE_ENDS_OP]; }
 
-  unsigned int count_compare_x_near_boundary() const
+  size_t count_compare_x_near_boundary() const
   { return m_counters[COMPARE_X_NEAR_BOUNDARY_OP]; }
 
   /// \name Types and functors inherited from the base
@@ -242,11 +242,11 @@ public:
   class Compare_x_2 {
   private:
     typename Base::Compare_x_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Compare_x_2(const Base* base, unsigned int& counter) :
+    Compare_x_2(const Base* base, size_t& counter) :
       m_object(base->compare_x_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -258,11 +258,11 @@ public:
   class Compare_xy_2 {
   private:
     typename Base::Compare_xy_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Compare_xy_2(const Base* base, unsigned int& counter) :
+    Compare_xy_2(const Base* base, size_t& counter) :
       m_object(base->compare_xy_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -274,11 +274,11 @@ public:
   class Construct_min_vertex_2 {
   private:
     typename Base::Construct_min_vertex_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Construct_min_vertex_2(const Base* base, unsigned int& counter) :
+    Construct_min_vertex_2(const Base* base, size_t& counter) :
       m_object(base->construct_min_vertex_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -290,11 +290,11 @@ public:
   class Construct_max_vertex_2 {
   private:
     typename Base::Construct_max_vertex_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Construct_max_vertex_2(const Base* base, unsigned int& counter) :
+    Construct_max_vertex_2(const Base* base, size_t& counter) :
       m_object(base->construct_max_vertex_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -306,11 +306,11 @@ public:
   class Is_vertical_2 {
   private:
     typename Base::Is_vertical_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Is_vertical_2(const Base* base, unsigned int& counter) :
+    Is_vertical_2(const Base* base, size_t& counter) :
       m_object(base->is_vertical_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -324,11 +324,11 @@ public:
   class Compare_y_at_x_2 {
   private:
     typename Base::Compare_y_at_x_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Compare_y_at_x_2(const Base* base, unsigned int& counter) :
+    Compare_y_at_x_2(const Base* base, size_t& counter) :
       m_object(base->compare_y_at_x_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -343,12 +343,12 @@ public:
   class Equal_2 {
   private:
     typename Base::Equal_2 m_object;
-    unsigned int& m_counter1;
-    unsigned int& m_counter2;
+    size_t& m_counter1;
+    size_t& m_counter2;
 
   public:
     /*! Construct */
-    Equal_2(const Base* base, unsigned int& counter1, unsigned int& counter2) :
+    Equal_2(const Base* base, size_t& counter1, size_t& counter2) :
       m_object(base->equal_2_object()),
       m_counter1(counter1), m_counter2(counter2)
     {}
@@ -369,11 +369,11 @@ public:
   class Compare_y_at_x_left_2 {
   private:
     typename Base::Compare_y_at_x_left_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Compare_y_at_x_left_2(const Base* base, unsigned int& counter) :
+    Compare_y_at_x_left_2(const Base* base, size_t& counter) :
       m_object(base->compare_y_at_x_left_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -389,11 +389,11 @@ public:
   class Compare_y_at_x_right_2 {
   private:
     typename Base::Compare_y_at_x_right_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Compare_y_at_x_right_2(const Base* base, unsigned int& counter) :
+    Compare_y_at_x_right_2(const Base* base, size_t& counter) :
       m_object(base->compare_y_at_x_right_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -403,19 +403,27 @@ public:
     { ++m_counter; return m_object(xc1, xc2, p); }
   };
 
-  /*! A functor that divides a curve into x-monotone curves. */
+  /*! \class Make_x_monotone_2
+   * A functor that subdivides a curve into x-monotone curves.
+   */
   class Make_x_monotone_2 {
   private:
     typename Base::Make_x_monotone_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Make_x_monotone_2(const Base* base, unsigned int& counter) :
+    Make_x_monotone_2(const Base* base, size_t& counter) :
       m_object(base->make_x_monotone_2_object()), m_counter(counter) {}
 
-    /*! Operate */
-    template<class OutputIterator>
+    /*! Subdivide a given curve into x-monotone subcurves and insert them into
+     * a given output iterator.
+     * \param cv the curve.
+     * \param oi the output iterator for the result. Its value type is a variant
+     *           that wraps Point_2 or an X_monotone_curve_2 objects.
+     * \return The past-the-end iterator.
+     */
+    template <typename OutputIterator>
     OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const
     { ++m_counter; return m_object(cv, oi); }
   };
@@ -424,11 +432,11 @@ public:
   class Split_2 {
   private:
     typename Base::Split_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Split_2(const Base* base, unsigned int& counter) :
+    Split_2(const Base* base, size_t& counter) :
       m_object(base->split_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -441,11 +449,11 @@ public:
   class Intersect_2 {
   private:
     typename Base::Intersect_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Intersect_2(const Base* base, unsigned int& counter) :
+    Intersect_2(const Base* base, size_t& counter) :
       m_object(base->intersect_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -460,11 +468,11 @@ public:
   class Are_mergeable_2 {
   private:
     typename Base::Are_mergeable_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Are_mergeable_2(const Base* base, unsigned int& counter) :
+    Are_mergeable_2(const Base* base, size_t& counter) :
       m_object(base->are_mergeable_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -477,11 +485,11 @@ public:
   class Merge_2 {
   private:
     typename Base::Merge_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Merge_2(const Base* base, unsigned int& counter) :
+    Merge_2(const Base* base, size_t& counter) :
       m_object(base->merge_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -495,11 +503,11 @@ public:
   class Construct_opposite_2 {
   private:
     typename Base::Construct_opposite_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Construct_opposite_2(const Base* base, unsigned int& counter) :
+    Construct_opposite_2(const Base* base, size_t& counter) :
       m_object(base->construct_opposite_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -513,11 +521,11 @@ public:
   class Compare_endpoints_xy_2 {
   private:
     typename Base::Compare_endpoints_xy_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Compare_endpoints_xy_2(const Base* base, unsigned int& counter) :
+    Compare_endpoints_xy_2(const Base* base, size_t& counter) :
       m_object(base->compare_endpoints_xy_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -533,14 +541,14 @@ public:
   class Parameter_space_in_x_2 {
   private:
     typename Base::Parameter_space_in_x_2 m_object;
-    unsigned int& m_counter1;
-    unsigned int& m_counter2;
-    unsigned int& m_counter3;
+    size_t& m_counter1;
+    size_t& m_counter2;
+    size_t& m_counter3;
 
   public:
     /*! Construct */
-    Parameter_space_in_x_2(const Base* base, unsigned int& counter1,
-                           unsigned int& counter2, unsigned int& counter3) :
+    Parameter_space_in_x_2(const Base* base, size_t& counter1,
+                           size_t& counter2, size_t& counter3) :
       m_object(base->parameter_space_in_x_2_object()),
       m_counter1(counter1),
       m_counter2(counter2),
@@ -569,13 +577,13 @@ public:
   class Is_on_x_identification_2 {
   private:
     typename Base::Is_on_x_identificiation_2 m_object;
-    unsigned int& m_counter1;
-    unsigned int& m_counter2;
+    size_t& m_counter1;
+    size_t& m_counter2;
 
   public:
     /*! Construct */
     Is_on_x_identification_2(const Base* base,
-                             unsigned int& counter1, unsigned int& counter2) :
+                             size_t& counter1, size_t& counter2) :
       m_object(base->is_on_x_identificiation_2_object()),
       m_counter1(counter1),
       m_counter2(counter2) {}
@@ -597,11 +605,11 @@ public:
   class Compare_y_on_boundary_2 {
   private:
     typename Base::Compare_y_on_boundary_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Compare_y_on_boundary_2(const Base* base, unsigned int& counter) :
+    Compare_y_on_boundary_2(const Base* base, size_t& counter) :
       m_object(base->compare_y_on_boundary_2_object()),
       m_counter(counter)
     {}
@@ -617,11 +625,11 @@ public:
   class Compare_y_near_boundary_2 {
   private:
     typename Base::Compare_y_near_boundary_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Compare_y_near_boundary_2(const Base* base, unsigned int& counter) :
+    Compare_y_near_boundary_2(const Base* base, size_t& counter) :
       m_object(base->compare_y_near_boundary_2_object()), m_counter(counter) {}
 
     /*! Operate */
@@ -639,14 +647,14 @@ public:
   class Parameter_space_in_y_2 {
   private:
     typename Base::Parameter_space_in_y_2 m_object;
-    unsigned int& m_counter1;
-    unsigned int& m_counter2;
-    unsigned int& m_counter3;
+    size_t& m_counter1;
+    size_t& m_counter2;
+    size_t& m_counter3;
 
   public:
     /*! Construct */
-    Parameter_space_in_y_2(const Base* base, unsigned int& counter1,
-                           unsigned int& counter2, unsigned int& counter3) :
+    Parameter_space_in_y_2(const Base* base, size_t& counter1,
+                           size_t& counter2, size_t& counter3) :
       m_object(base->parameter_space_in_y_2_object()),
       m_counter1(counter1),
       m_counter2(counter2),
@@ -673,13 +681,13 @@ public:
   class Is_on_y_identification_2 {
   private:
     typename Base::Is_on_y_identificiation_2 m_object;
-    unsigned int& m_counter1;
-    unsigned int& m_counter2;
+    size_t& m_counter1;
+    size_t& m_counter2;
 
   public:
     /*! Construct */
     Is_on_y_identification_2(const Base* base,
-                             unsigned int& counter1, unsigned int& counter2) :
+                             size_t& counter1, size_t& counter2) :
       m_object(base->is_on_y_identificiation_2_object()),
       m_counter1(counter1),
       m_counter2(counter2) {}
@@ -701,13 +709,13 @@ public:
   class Compare_x_at_limit_2 {
   private:
     typename Base::Compare_x_at_limit_2 m_object;
-    unsigned int& m_counter1;
-    unsigned int& m_counter2;
+    size_t& m_counter1;
+    size_t& m_counter2;
 
   public:
     /*! Construct */
     Compare_x_at_limit_2(const Base* base,
-                         unsigned int& counter1, unsigned int& counter2) :
+                         size_t& counter1, size_t& counter2) :
       m_object(base->compare_x_at_limit_2_object()),
       m_counter1(counter1),
       m_counter2(counter2) {}
@@ -733,11 +741,11 @@ public:
   class Compare_x_near_limit_2 {
   private:
     typename Base::Compare_x_near_limit_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
-    Compare_x_near_limit_2(const Base* base, unsigned int& counter) :
+    Compare_x_near_limit_2(const Base* base, size_t& counter) :
       m_object(base->compare_x_near_limit_2_object()),
       m_counter(counter) {}
 
@@ -755,14 +763,14 @@ public:
   class Compare_x_on_boundary_2 {
   private:
     typename Base::Compare_x_on_boundary_2 m_object;
-    unsigned int& m_counter1;
-    unsigned int& m_counter2;
-    unsigned int& m_counter3;
+    size_t& m_counter1;
+    size_t& m_counter2;
+    size_t& m_counter3;
 
   public:
     /*! Construct */
-  Compare_x_on_boundary_2(const Base* base,  unsigned int& counter1,
-                          unsigned int& counter2, unsigned int& counter3 ) :
+  Compare_x_on_boundary_2(const Base* base,  size_t& counter1,
+                          size_t& counter2, size_t& counter3 ) :
       m_object(base->compare_x_on_boundary_2_object()),
       m_counter1(counter1),
       m_counter2(counter2),
@@ -794,12 +802,12 @@ public:
   class Compare_x_near_boundary_2 {
   private:
     typename Base::Compare_x_near_boundary_2 m_object;
-    unsigned int& m_counter;
+    size_t& m_counter;
 
   public:
     /*! Construct */
     Compare_x_near_boundary_2(const Base* base,
-                              unsigned int& counter) :
+                              size_t& counter) :
       m_object(base->compare_x_near_boundary_2_object()),
       m_counter(counter) {}
 
@@ -945,12 +953,12 @@ public:
    * \param doit indicates whethet to actually inceremnt the counter or not
    * \return the counter at the end of the operation
    */
-  static unsigned int increment(bool doit = true)
+  static size_t increment(bool doit = true)
   {
 #ifdef CGAL_NO_ATOMIC
-    static unsigned int counter;
+    static size_t counter;
 #else
-    static CGAL::cpp11::atomic<unsigned int> counter;
+    static CGAL::cpp11::atomic<size_t> counter;
 #endif
     if (doit) ++counter;
     return counter;
@@ -962,7 +970,7 @@ public:
 
 private:
   /*! The operation counters */
-  mutable unsigned int m_counters[NUMBER_OF_OPERATIONS];
+  mutable size_t m_counters[NUMBER_OF_OPERATIONS];
 };
 
 template <class Out_stream, class Base_traits>
@@ -971,8 +979,8 @@ Out_stream& operator<<(Out_stream& os,
                        const Arr_counting_traits_2<Base_traits>& traits)
 {
   typedef Arr_counting_traits_2<Base_traits>            Traits;
-  unsigned int sum = 0;
-  unsigned int i;
+  size_t sum = 0;
+  size_t i;
   for (i = 0; i < Traits::NUMBER_OF_OPERATIONS; ++i)
     sum += traits.count(static_cast<typename Traits::Operation_id>(i));
   os << "# of COMPARE_X operation = "

Arrangement_on_surface_2/include/CGAL/Arr_curve_data_traits_2.h

@@ -27,8 +27,8 @@
 #include <boost/utility/enable_if.hpp>
 #include <boost/mpl/has_xxx.hpp>
 
-#include <CGAL/Object.h>
 #include <CGAL/tags.h>
+#include <CGAL/assertions.h>
 #include <CGAL/Arr_tags.h>
 #include <CGAL/Arr_geometry_traits/Curve_data_aux.h>
 
@@ -101,6 +101,12 @@ public:
   /// \name Overriden functors.
   //@{
 
+  //! \name Intersections & subdivisions
+  //@{
+
+  /*! \class Make_x_monotone_2
+   * A functor for subdividing a curve into x-monotone curves.
+   */
   class Make_x_monotone_2 {
   private:
     const Base_traits_2& m_base;
@@ -109,36 +115,36 @@ public:
     /*! Constructor. */
     Make_x_monotone_2(const Base_traits_2& base) : m_base(base) {}
 
-    /*! Cut the given curve into x-monotone subcurves and insert them to the
-     * given output iterator. As segments are always x_monotone, only one
-     * x-monotone curve will be contained in the iterator.
-     * \param cv The curve.
-     * \param oi The output iterator, whose value-type is X_monotone_curve_2.
-     * \return The past-the-end iterator.
+    /*! Subdivide a given curve into x-monotone subcurves and insert them into
+     * a given output iterator.
+     * \param cv the curve.
+     * \param oi the output iterator for the result. Its value type is a variant
+     *           that wraps Point_2 or an X_monotone_curve_2 objects.
+     * \return the past-the-end iterator.
      */
-    template<typename OutputIterator>
+    template <typename OutputIterator>
     OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const
     {
+      typedef boost::variant<Point_2, Base_x_monotone_curve_2>
+        Base_make_x_monotone_result;
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
+
       // Make the original curve x-monotone.
-      std::list<CGAL::Object> base_objects;
+      std::list<Base_make_x_monotone_result> base_objects;
       m_base.make_x_monotone_2_object()(cv, std::back_inserter(base_objects));
 
       // Attach the data to each of the resulting x-monotone curves.
-      const Base_x_monotone_curve_2* base_x_curve;
       X_monotone_curve_data xdata = Convert()(cv.data());
-      for (typename std::list<CGAL::Object>::const_iterator it =
-             base_objects.begin(); it != base_objects.end(); ++it)
-      {
-        base_x_curve = object_cast<Base_x_monotone_curve_2>(&(*it));
-        if (base_x_curve != nullptr) {
-          // Current object is an x-monotone curve: Attach data to it.
-          *oi++ = make_object(X_monotone_curve_2(*base_x_curve, xdata));
-        }
-        else {
-          // Current object is an isolated point: Leave it as is.
-          CGAL_assertion(object_cast<Point_2>(&(*it)) != nullptr);
-          *oi++ = *it;
+      for (const auto& base_obj : base_objects) {
+        if (const auto* bxcv = boost::get<Base_x_monotone_curve_2>(&base_obj)) {
+          *oi++ = Make_x_monotone_result(X_monotone_curve_2(*bxcv, xdata));
+          continue;
         }
+        // Current object is an isolated point: Leave it as is.
+        const auto* bp = boost::get<Point_2>(&base_obj);
+        CGAL_assertion(bp);
+        *oi++ = Make_x_monotone_result(*bp);
       }
 
       return oi;
@@ -359,6 +365,8 @@ public:
   /*! Obtain a Merge_2 functor object. */
   Merge_2 merge_2_object() const { return Merge_2(*this); }
 
+  //@}
+
   class Construct_x_monotone_curve_2 {
   private:
     const Base_traits_2& m_base;

Arrangement_on_surface_2/include/CGAL/Arr_geodesic_arc_on_sphere_traits_2.h

@@ -1358,8 +1358,9 @@ public:
   /// \name Functor definitions for supporting intersections.
   //@{
 
-  /*! A functor that divides an arc into x-monotone arcs. That are, arcs that
-   * do not cross the identification arc.
+  /*! \class Make_x_monotone_2
+   * A functor for subdividing arcs into x-monotone arcs that do not cross the
+   * identification arc.
    */
   class Make_x_monotone_2 {
   protected:
@@ -1376,21 +1377,24 @@ public:
     friend class Arr_geodesic_arc_on_sphere_traits_2<Kernel>;
 
   public:
-    /*! Cut the given curve into x-monotone subcurves and insert them into the
-     * given output iterator. As spherical_arcs are always x_monotone, only one
-     * object will be contained in the iterator.
+    /*! Subdivide a given curve into x-monotone subcurves and insert them into
+     * a given output iterator. As spherical_arcs are always x_monotone, only
+     * one object will be contained in the iterator.
      * \param xc the curve.
-     * \param oi the output iterator, whose value-type is Object. The output
-     *           object is a wrapper of either an X_monotone_curve_2, or - in
-     *           case the input spherical_arc is degenerate - a Point_2 object.
+     * \param oi the output iterator for the result. Its dereference type is a
+     *           variant that wraps a \c Point_2 or an \c X_monotone_curve_2
+     *           objects.
      * \return the past-the-end iterator.
      */
-    template<typename OutputIterator>
+    template <typename OutputIterator>
     OutputIterator operator()(const Curve_2& c, OutputIterator oi) const
     {
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
+
       if (c.is_degenerate()) {
         // The spherical_arc is a degenerate point - wrap it with an object:
-        *oi++ = make_object(c.right());
+        *oi++ = Make_x_monotone_result(c.right());
         return oi;
       }
 
@@ -1398,7 +1402,7 @@ public:
         // The spherical arc is monotone - wrap it with an object:
         // *oi++ = make_object(X_monotone_curve_2(c));
         const X_monotone_curve_2* xc = &c;
-        *oi++ = make_object(*xc);
+        *oi++ = Make_x_monotone_result(*xc);
         return oi;
       }
 
@@ -1410,14 +1414,14 @@ public:
           const Direction_3& pp = m_traits->pos_pole();
           X_monotone_curve_2 xc1(np, pp, c.normal(), true, true);
           X_monotone_curve_2 xc2(pp, np, c.normal(), true, false);
-          *oi++ = make_object(xc1);
-          *oi++ = make_object(xc2);
+          *oi++ = Make_x_monotone_result(xc1);
+          *oi++ = Make_x_monotone_result(xc2);
           return oi;
         }
 #if defined(CGAL_FULL_X_MONOTONE_GEODESIC_ARC_ON_SPHERE_IS_SUPPORTED)
         // The arc is not vertical => break it at the discontinuity arc:
         const X_monotone_curve_2 xc(c.normal());
-        *oi++ = make_object(xc);
+        *oi++ = Make_x_monotone_result(xc);
 #else
         // Full x-monotone arcs are not supported!
         // Split the arc at the intersection point with the complement of the
@@ -1428,8 +1432,8 @@ public:
         Direction_3 d2(normal.dz(), 0, -(normal.dx()));
         X_monotone_curve_2 xc1(d1, d2, normal, false, directed_right);
         X_monotone_curve_2 xc2(d2, d1, normal, false, directed_right);
-        *oi++ = make_object(xc1);
-        *oi++ = make_object(xc2);
+        *oi++ = Make_x_monotone_result(xc1);
+        *oi++ = Make_x_monotone_result(xc2);
 #endif
         return oi;
       }
@@ -1447,16 +1451,16 @@ public:
         if (source.is_min_boundary() || target.is_min_boundary()) {
           X_monotone_curve_2 xc1(source, pp, normal, true, true);
           X_monotone_curve_2 xc2(pp, target, normal, true, false);
-          *oi++ = make_object(xc1);
-          *oi++ = make_object(xc2);
+          *oi++ = Make_x_monotone_result(xc1);
+          *oi++ = Make_x_monotone_result(xc2);
           return oi;
         }
 
         if (source.is_max_boundary() || target.is_max_boundary()) {
           X_monotone_curve_2 xc1(source, np, normal, true, false);
           X_monotone_curve_2 xc2(np, target, normal, true, true);
-          *oi++ = make_object(xc1);
-          *oi++ = make_object(xc2);
+          *oi++ = Make_x_monotone_result(xc1);
+          *oi++ = Make_x_monotone_result(xc2);
           return oi;
         }
 
@@ -1476,19 +1480,19 @@ public:
                     (!plane_is_positive && !s_is_positive));
         const Point_2& pole1 = (ccw) ? pp : np;
         X_monotone_curve_2 xc1(source, pole1, normal, true, ccw);
-        *oi++ = make_object(xc1);
+        *oi++ = Make_x_monotone_result(xc1);
         if (s_is_positive != t_is_positive) {
           // Construct 1 more arc:
           X_monotone_curve_2 xc2(pole1, target, normal, true, !ccw);
-          *oi++ = make_object(xc2);
+          *oi++ = Make_x_monotone_result(xc2);
           return oi;
         }
         // Construct 2 more arcs:
         const Point_2& pole2 = (ccw) ? np : pp;
         X_monotone_curve_2 xc2(pole1, pole2, normal, true, !ccw);
-        *oi++ = make_object(xc2);
+        *oi++ = Make_x_monotone_result(xc2);
         X_monotone_curve_2 xc3(pole2, target, normal, true, ccw);
-        *oi++ = make_object(xc3);
+        *oi++ = Make_x_monotone_result(xc3);
         return oi;
       }
 
@@ -1514,8 +1518,8 @@ public:
 
       X_monotone_curve_2 xc1(source, p, normal, false, directed_right);
       X_monotone_curve_2 xc2(p, target, normal, false, directed_right);
-      *oi++ = make_object(xc1);
-      *oi++ = make_object(xc2);
+      *oi++ = Make_x_monotone_result(xc1);
+      *oi++ = Make_x_monotone_result(xc2);
       return oi;
     }
   };

Arrangement_on_surface_2/include/CGAL/Arr_geometry_traits/Arr_plane_3.h

@@ -197,49 +197,51 @@ public:
 /*! Intersect 2 planes
  * \param plane1 the first plane
  * \param plane2 the second plane
- * \return a geometric object that represents the intersection. Could be
- * the line of intersection, or a plane in case plane1 and plane2 coincide.
+ * \return a variant that represents the intersection. It wraps a line of
+ * intersection or a plane in case plane1 and plane2 coincide.
  */
-template <class Kernel>
-CGAL::Object intersect(const Arr_plane_3<Kernel> & plane1,
+template <typname Kernel>
+boost::variant<typename Kernel::Line_3, Arr_plane_3<Kernel> >
+intersect(const Arr_plane_3<Kernel> & plane1,
                        const Arr_plane_3<Kernel> & plane2)
 {
   typedef typename Kernel::Point_3      Point_3;
   typedef typename Kernel::Direction_3  Direction_3;
   typedef typename Kernel::Line_3       Line_3;
   typedef typename Kernel::FT           FT;
+  typedef boost::variant<Line_3, Arr_plane_3<Kernel> >  Intersection_result;
 
   // We know that the plane goes throgh the origin
-  const FT & a1 = plane1.a();
-  const FT & b1 = plane1.b();
-  const FT & c1 = plane1.c();
+  const FT& a1 = plane1.a();
+  const FT& b1 = plane1.b();
+  const FT& c1 = plane1.c();
 
-  const FT & a2 = plane2.a();
-  const FT & b2 = plane2.b();
-  const FT & c2 = plane2.c();
+  const FT& a2 = plane2.a();
+  const FT& b2 = plane2.b();
+  const FT& c2 = plane2.c();
 
   FT det = a1*b2 - a2*b1;
   if (det != 0) {
     Point_3 is_pt = Point_3(0, 0, 0, det);
     Direction_3 is_dir = Direction_3(b1*c2 - c1*b2, a2*c1 - a1*c2, det);
-    return make_object(Line_3(is_pt, is_dir));
+    return Intersection_result(Line_3(is_pt, is_dir));
   }
 
   det = a1*c2 - a2*c1;
   if (det != 0) {
     Point_3 is_pt = Point_3(0, 0, 0, det);
     Direction_3 is_dir = Direction_3(c1*b2 - b1*c2, det, a2*b1 - a1*b2);
-    return make_object(Line_3(is_pt, is_dir));
+    return Intersection_result(Line_3(is_pt, is_dir));
   }
   det = b1*c2 - c1*b2;
   if (det != 0) {
     Point_3 is_pt = Point_3(0, 0, 0, det);
     Direction_3 is_dir = Direction_3(det, c1*a2 - a1*c2, a1*b2 - b1*a2);
-    return make_object(Line_3(is_pt, is_dir));
+    return Intersection_result(Line_3(is_pt, is_dir));
   }
 
   // degenerate case
-  return make_object(plane1);
+  return Intersection_result(plane1);
 }
 
 /*! Compute the image point of the projection of p under an affine

Arrangement_on_surface_2/include/CGAL/Arr_geometry_traits/Bezier_bounding_rational_traits.h

@@ -1262,36 +1262,26 @@ private:
         // the intersection of the two skewed bounding boxes.
         Bez_point_bbox  ipt_bbox;
         Control_points  aux_vec;
-        CGAL::Object    res;
-        Point_2         p;
 
-        res = f_intersect (skew1a, skew2a);
-        if (! assign (p, res))
-        {
-          CGAL_error();
-        }
-        aux_vec.push_back(p);
+        auto res1 = f_intersect(skew1a, skew2a);
+        const Point_2* p1 = boost::get<Point_2>(&*res1);
+        if (! p1) CGAL_error();
+        aux_vec.push_back(*p1);
 
-        res = f_intersect (skew1a, skew2b);
-        if (! assign(p, res))
-        {
-          CGAL_error();
-        }
-        aux_vec.push_back(p);
+        auto res2 = f_intersect(skew1a, skew2b);
+        const Point_2* p2 = boost::get<Point_2>(&*res2);
+        if (! p2) CGAL_error();
+        aux_vec.push_back(*p2);
 
-        res = f_intersect (skew1b, skew2a);
-        if (! assign(p, res))
-        {
-          CGAL_error();
-        }
-        aux_vec.push_back(p);
+        auto res3 = f_intersect(skew1b, skew2a);
+        const Point_2* p3 = boost::get<Point_2>(&*res3);
+        if (! p3) CGAL_error();
+        aux_vec.push_back(*p3);
 
-        res = f_intersect (skew1b, skew2b);
-        if (!assign(p, res))
-        {
-          CGAL_error();
-        }
-        aux_vec.push_back(p);
+        auto res4 = f_intersect (skew1b, skew2b);
+        const Point_2* p4 = boost::get<Point_2>(&*res4);
+        if (! p4) CGAL_error();
+        aux_vec.push_back(*p4);
 
         construct_bbox (aux_vec, ipt_bbox);
 

Arrangement_on_surface_2/include/CGAL/Arr_line_arc_traits_2.h

@@ -27,6 +27,10 @@
  * Arrangement_2 package, which it is now part of. It contains a traits
  * class for the arrangement package that handles linear curves.
  * It is based on the circular kernel.
+ *
+ * \todo Fix the circular-kernel make-x-monotone functor to use modern variant
+ *       instead of the legacy CGAL::Object. Then, eliminate the special
+ *       implementation here and directly use the kernel functor instead.
  */
 
 #include <CGAL/basic.h>
@@ -73,7 +77,7 @@ public:
   typedef typename CircularKernel::Compare_y_at_x_2     Compare_y_at_x_2;
   typedef typename CircularKernel::Compare_y_to_right_2 Compare_y_at_x_right_2;
   typedef typename CircularKernel::Equal_2              Equal_2;
-  typedef typename CircularKernel::Make_x_monotone_2    Make_x_monotone_2;
+  // typedef typename CircularKernel::Make_x_monotone_2    Make_x_monotone_2;
   typedef typename CircularKernel::Split_2              Split_2;
   typedef typename CircularKernel::Construct_circular_min_vertex_2
                                                         Construct_min_vertex_2;
@@ -97,8 +101,8 @@ public:
   Equal_2 equal_2_object() const
   { return ck.equal_2_object(); }
 
-  Make_x_monotone_2 make_x_monotone_2_object() const
-  { return ck.make_x_monotone_2_object(); }
+  // Make_x_monotone_2 make_x_monotone_2_object() const
+  // { return ck.make_x_monotone_2_object(); }
 
   Split_2 split_2_object() const
   { return ck.split_2_object(); }
@@ -115,7 +119,20 @@ public:
   Is_vertical_2 is_vertical_2_object() const
   { return ck.is_vertical_2_object();}
 
+  //! A functor for subdividing curves into x-monotone curves.
+  class Make_x_monotone_2 {
+  public:
+    template <typename OutputIterator>
+    OutputIterator operator()(const Curve_2& line, OutputIterator oi) const
+    {
+      typedef boost::variant<Point_2, X_monotone_curve_2> Make_x_monotone_result;
+      *oi++ = Make_x_monotone_result(line);
+      return oi;
+    }
+  };
 
+  Make_x_monotone_2 make_x_monotone_2_object() const
+  { return Make_x_monotone_2(); }
 };
 
 } // namespace CGAL

Arrangement_on_surface_2/include/CGAL/Arr_linear_traits_2.h

@@ -1239,16 +1239,18 @@ public:
      * given output iterator. As segments are always x_monotone, only one
      * object will be contained in the iterator.
      * \param cv The curve.
-     * \param oi The output iterator, whose value-type is Object. The output
-     *           object is a wrapper of an X_monotone_curve_2 which is
-     *           essentially the same as the input curve.
+     * \param oi an output iterator for the result. Its dereference type is a
+     *           variant that wraps a \c Point_2 or an \c X_monotone_curve_2
+     *           objects.
      * \return The past-the-end iterator.
      */
     template <typename OutputIterator>
     OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const
     {
-      // Wrap the curve with an object.
-      *oi++ = make_object(cv);
+      // Wrap the segment with a variant.
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
+      *oi++ = Make_x_monotone_result(cv);
       return oi;
     }
   };

Arrangement_on_surface_2/include/CGAL/Arr_non_caching_segment_basic_traits_2.h

@@ -7,14 +7,14 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-// Author(s)     : Efi Fogel <efif@post.tau.ac.il>
-//                 Ron Wein  <wein@post.tau.ac.il>
-//                 (based on old version by: Iddo Hanniel,
-//                                           Eyal Flato,
-//                                           Oren Nechushtan,
-//                                           Ester Ezra,
-//                                           Shai Hirsch,
-//                                           and Eugene Lipovetsky)
+// Author(s): Efi Fogel <efif@post.tau.ac.il>
+//            Ron Wein  <wein@post.tau.ac.il>
+//            (based on old version by: Iddo Hanniel,
+//                                      Eyal Flato,
+//                                      Oren Nechushtan,
+//                                      Ester Ezra,
+//                                      Shai Hirsch,
+//                                      and Eugene Lipovetsky)
 
 #ifndef CGAL_ARR_NON_CACHING_SEGMENT_BASIC_TRAITS_2_H
 #define CGAL_ARR_NON_CACHING_SEGMENT_BASIC_TRAITS_2_H

Arrangement_on_surface_2/include/CGAL/Arr_non_caching_segment_traits_2.h

@@ -113,24 +113,27 @@ public:
   // Traits types:
   typedef X_monotone_curve_2                    Curve_2;
 
-  /*! \class
-   * A functor for splitting curves into x-monotone pieces.
+  /*! \class Make_x_monotone_2
+   * A functor for subdividing a curve into x-monotone curves.
    */
   class Make_x_monotone_2 {
   public:
-
-    /*! Cut the given segment into x-monotone subcurves and insert them into
-     * the given output iterator. As segments are always x_monotone, only one
+    /*! Subdivide a given curve into x-monotone subcurves and insert them into
+     * a given output iterator. As segments are always x_monotone, only one
      * x-monotone curve is inserted into the output iterator.
-     * \param cv The segment.
-     * \param oi The output iterator, whose value-type is Object. The output
-     *           object is a wrapper of an X_monotone_curve_2 object.
-     * \return The past-the-end iterator.
+     * \param cv the segment.
+     * \param oi the output iterator for the result. Its dereference type is a
+     *           variant that wraps a \c Point_2 or an \c X_monotone_curve_2
+     *           objects.
+     * \return the past-the-end iterator.
      */
     template <typename OutputIterator>
     OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const
     {
-      *oi++ = make_object(cv);
+      // Wrap the segment with a variant.
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
+      *oi++ = Make_x_monotone_result(cv);
       return oi;
     }
   };

Arrangement_on_surface_2/include/CGAL/Arr_point_location/Arr_batched_point_location_traits_2.h

@@ -42,6 +42,7 @@ public:
 
   typedef typename Base_traits_2::X_monotone_curve_2   Base_x_monotone_curve_2;
   typedef typename Base_traits_2::Point_2              Base_point_2;
+  typedef typename Base_traits_2::Multiplicity         Multiplicity;
 
   typedef typename Base_traits_2::Construct_min_vertex_2
                                                     Base_construct_min_vertex_2;

Arrangement_on_surface_2/include/CGAL/Arr_point_location/Arr_landmarks_pl_impl.h

@@ -56,15 +56,15 @@ Arr_landmarks_point_location<Arr, Gen>::locate(const Point_2& p) const
   const Vertex_const_handle*   vh;
   const Halfedge_const_handle* hh;
   const Face_const_handle*     fh;
-  if ( ( vh = Result().template assign<Vertex_const_handle>(lm_location_obj) ) )
+  if ( ( vh = Result().template assign<Vertex_const_handle>(&lm_location_obj) ) )
     out_obj = _walk_from_vertex(*vh, p, crossed_edges);
-  else if ( ( hh = Result().template assign<Halfedge_const_handle>(lm_location_obj) ) )
+  else if ( ( hh = Result().template assign<Halfedge_const_handle>(&lm_location_obj) ) )
     out_obj = _walk_from_edge(*hh, landmark_point, p, crossed_edges);
-  else if ( ( fh =  Result().template assign<Face_const_handle>(lm_location_obj) ) )
+  else if ( ( fh =  Result().template assign<Face_const_handle>(&lm_location_obj) ) )
     out_obj = _walk_from_face(*fh, landmark_point, p, crossed_edges);
   else CGAL_error_msg("lm_location_obj of an unknown type.");
 
-  if ( ( fh = Result().template assign<Face_const_handle>(out_obj) ) ) {
+  if ( ( fh = Result().template assign<Face_const_handle>(&out_obj) ) ) {
     // If we reached here, we did not locate the query point in any of the
     // holes inside the current face, so we conclude it is contained in this
     // face.
@@ -159,7 +159,7 @@ _walk_from_vertex(Vertex_const_handle nearest_vertex,
     if (new_vertex) {
       // We found a vertex closer to p; Continue using this vertex.
       const Vertex_const_handle* p_vh =
-        Result().template assign<Vertex_const_handle>(obj);
+        Result().template assign<Vertex_const_handle>(&obj);
       CGAL_assertion(p_vh != nullptr);
       vh = *p_vh;
       continue;
@@ -167,11 +167,12 @@ _walk_from_vertex(Vertex_const_handle nearest_vertex,
 
     // If p is located on an edge or on a vertex, return the object
     // that wraps this arrangement feature.
-    if (Result().template assign<Halfedge_const_handle>(obj) ||
-        Result().template assign<Vertex_const_handle>(obj))
+    if (Result().template assign<Halfedge_const_handle>(&obj) ||
+        Result().template assign<Vertex_const_handle>(&obj))
       return obj;
 
-    const Face_const_handle* p_fh = Result().template assign<Face_const_handle>(obj);
+    const Face_const_handle* p_fh =
+      Result().template assign<Face_const_handle>(&obj);
     if (p_fh)
       // Walk to p from the face we have located:
       return _walk_from_face(*p_fh, vh->point(), p, crossed_edges);

Arrangement_on_surface_2/include/CGAL/Arr_point_location/Arr_simple_point_location_impl.h

@@ -44,10 +44,8 @@ Arr_simple_point_location<Arrangement>::locate(const Point_2& p) const
 
   // Go over arrangement halfedges and check whether one of them contains
   // the query point in its interior.
-  typename Traits_adaptor_2::Is_in_x_range_2  is_in_x_range =
-    m_geom_traits->is_in_x_range_2_object();
-  typename Traits_adaptor_2::Compare_y_at_x_2 cmp_y_at_x =
-    m_geom_traits->compare_y_at_x_2_object();
+  auto is_in_x_range = m_geom_traits->is_in_x_range_2_object();
+  auto cmp_y_at_x = m_geom_traits->compare_y_at_x_2_object();
 
   typename Arrangement::Edge_const_iterator   eit;
   for (eit = m_arr->edges_begin(); eit != m_arr->edges_end(); ++eit) {
@@ -71,14 +69,14 @@ Arr_simple_point_location<Arrangement>::locate(const Point_2& p) const
   // In case the ray-shooting returned a vertex, we have to locate the first
   // halfedge whose source vertex is v, rotating clockwise around the vertex
   // from "6 o'clock", and to return its incident face.
-  const Vertex_const_handle* vh = Result().template assign<Vertex_const_handle>(obj);
+  const auto* vh = Result::template assign<Vertex_const_handle>(&obj);
   if (vh) {
     Halfedge_const_handle hh = _first_around_vertex(*vh);
     Face_const_handle fh = hh->face();
     return make_result(fh);
   }
 
-  const Halfedge_const_handle* hh = Result().template assign<Halfedge_const_handle>(obj);
+  const auto* hh = Result::template assign<Halfedge_const_handle>(&obj);
   if (hh) {
     // Make sure that the edge is directed from right to left, so that p
     // (which lies below it) is contained in its incident face. If necessary,
@@ -288,14 +286,13 @@ Arr_simple_point_location<Arrangement>::_vertical_ray_shoot(const Point_2& p,
 
   if (! optional_empty(optional_obj)) {
     const Result_type& obj = optional_assign(optional_obj);
-    const Vertex_const_handle* p_vh = Result().template assign<Vertex_const_handle>(obj);
+    const auto* p_vh = Result::template assign<Vertex_const_handle>(&obj);
     if (p_vh) {
       found_vertex = true;
       closest_v = *p_vh;
     }
     else {
-      const Halfedge_const_handle* p_hh =
-        Result().template assign<Halfedge_const_handle>(obj);
+      const auto* p_hh = Result::template assign<Halfedge_const_handle>(&obj);
       CGAL_assertion(p_hh != nullptr);
       found_halfedge = true;
       closest_he = *p_hh;

Arrangement_on_surface_2/include/CGAL/Arr_point_location/Arr_walk_along_line_pl_impl.h

@@ -618,7 +618,8 @@ _is_in_connected_component (const Point_2& p,
       // The current curve is not vertical. Check the query point is in the
       // semi-open x-range (source, target] of this curve and lies below it.
       if (source_res != EQUAL) {
-        if (closest_he == invalid_he || (closest_he->twin() == curr)) {
+        if ((closest_he == invalid_he) ||
+            (closest_he->twin() == Halfedge_const_handle(curr))) {
           // 1. If we have no closests halfedge, we have just found one.
           // 2. If the closest halfedge is the twin of our current halfedge,
           // we can take our halfedge to be the closest one. This covers the
@@ -701,7 +702,7 @@ _is_in_connected_component (const Point_2& p,
           do {
             next_non_vert = next_non_vert->next();
 
-            CGAL_assertion (next_non_vert != curr);
+            CGAL_assertion(next_non_vert != Halfedge_const_handle(curr));
           }
           while ((! next_non_vert->is_fictitious() &&
                   is_vertical(next_non_vert->curve())) ||

Arrangement_on_surface_2/include/CGAL/Arr_point_location_result.h

@@ -78,32 +78,34 @@ struct Arr_point_location_result {
   // lead to conversion overhead, and so we rather go for the real type.
   // Overloads for empty returns are also provided.
 #if CGAL_ARR_POINT_LOCATION_VERSION < 2
-  template<typename T>
+  template <typename T>
   static
-  inline CGAL::Object make_result(T t) { return CGAL::make_object(t); }
+  inline Type make_result(T t) { return CGAL::make_object(t); }
 
   static
   inline CGAL::Object empty_optional_result() { return CGAL::Object(); }
 
-  template<typename T>
-  const T* assign(CGAL::Object obj) const { return CGAL::object_cast<T>(&obj); }
+  template <typename T>
+  static
+  inline const T* assign(const Type* obj) { return CGAL::object_cast<T>(obj); }
 #else
-  template<typename T>
+  template <typename T>
   static
   inline Type make_result(T t) { return Type(t); }
 
-  inline
   static
-  boost::optional<Type> empty_optional_result() { return boost::optional<Type>(); }
+  inline boost::optional<Type> empty_optional_result()
+  { return boost::optional<Type>(); }
 
-  template<typename T>
-  const T* assign(const Type& obj) const { return boost::get<T>(&obj); }
+  template <typename T>
+  static
+  inline const T* assign(const Type* obj) { return boost::get<T>(obj); }
 #endif // CGAL_ARR_POINT_LOCATION_VERSION < 2
 
   //this one is only to remove warnings in functions
   static
   inline Type default_result(){
-    CGAL_error_msg("This functions should never have been called!");
+    CGAL_error_msg("This functions should have never been called!");
     return Type();
   }
 };

Arrangement_on_surface_2/include/CGAL/Arr_polycurve_basic_traits_2.h

@@ -63,6 +63,7 @@ public:
 
   typedef typename Subcurve_traits_2::Point_2              Point_2;
   typedef typename Subcurve_traits_2::X_monotone_curve_2   X_monotone_subcurve_2;
+  typedef typename Subcurve_traits_2::Multiplicity         Multiplicity;
 
   //@}
 

Arrangement_on_surface_2/include/CGAL/Arr_polycurve_traits_2.h

@@ -7,10 +7,10 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-// Author(s)     : Efi Fogel <efif@post.tau.ac.il>
-//                 Ron Wein  <wein@post.tau.ac.il>
-//                 Dror Atariah <dror.atariah@fu-berlin.de>
-//                 Waqar Khan <wkhan@mpi-inf.mpg.de>
+// Author(s): Efi Fogel <efif@post.tau.ac.il>
+//            Ron Wein  <wein@post.tau.ac.il>
+//            Dror Atariah <dror.atariah@fu-berlin.de>
+//            Waqar Khan <wkhan@mpi-inf.mpg.de>
 
 #ifndef CGAL_ARR_POLYCURVE_TRAITS_2_H
 #define CGAL_ARR_POLYCURVE_TRAITS_2_H
@@ -160,10 +160,8 @@ public:
 #ifndef DOXYGEN_RUNNING
   class Push_back_2;
 #endif
-  /*! \class
-   * A functor that divides an arc into x-monotone arcs. That are, arcs that
-   * do not cross the identification arc.
-   */
+
+  //! A functor for subdividing curves into x-monotone curves.
   class Make_x_monotone_2 {
   protected:
     typedef Arr_polycurve_traits_2<Subcurve_traits_2>     Polycurve_traits_2;
@@ -176,21 +174,24 @@ public:
       m_poly_traits(traits)
     {}
 
-    /*! Cut the given curve into x-monotone sub-curves and insert them into the
-     * given output iterator.
+    /*! Subdivide a given curve into x-monotone sub-curves and insert them into
+     * a given output iterator.
      *
      * \pre if `cv` is not empty then it must be continuous and well-oriented.
-     * \param cv The curve.
-     * \param oi The output iterator, whose value-type is Object. The output
-     *           object is a wrapper of a X_monotone_curve_2.
-     * \return The past-the-end iterator.
+     * \param cv the curve.
+     * \param oi an output iterator for the result. Its value type is a variant
+     *           that wraps Point_2 or an X_monotone_curve_2 objects.
+     * \return the past-the-end iterator.
      */
   private:
     template <typename OutputIterator>
     OutputIterator operator_impl(const Curve_2& cv, OutputIterator oi,
                                  Arr_all_sides_oblivious_tag) const
     {
-       typedef typename Curve_2::Subcurve_const_iterator const_seg_iterator;
+      typedef boost::variant<Point_2, X_monotone_subcurve_2>
+        Make_x_monotone_subresult;
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
 
       // If the polycurve is empty, return.
       if (cv.number_of_subcurves() == 0) return oi;
@@ -204,33 +205,29 @@ public:
         m_poly_traits.subcurve_traits_2()->compare_endpoints_xy_2_object();
 
 #ifdef CGAL_ALWAYS_LEFT_TO_RIGHT
-      typename Subcurve_traits_2::Construct_opposite_2 ctr_seg_opposite =
+      auto ctr_seg_opposite =
         m_poly_traits.subcurve_traits_2()->construct_opposite_2_object();
 #endif
 
-      // Convert the input polycurve to a sequence of CGAL objects, such
-      // that each Object wraps an x-monotone subcurve.
-      std::vector<Object> x_seg_objects;
-      const_seg_iterator it_segs;
-      for (it_segs = cv.subcurves_begin(); it_segs != cv.subcurves_end();
-           ++it_segs)
-        make_seg_x_monotone(*it_segs, std::back_inserter(x_seg_objects));
-      typename std::vector<Object>::iterator it = x_seg_objects.begin();
-      X_monotone_subcurve_2 x_seg;
-#if defined (CGAL_NO_ASSERTIONS)
-      CGAL::assign(x_seg, *it);
-#else
-      bool assign_res = CGAL::assign(x_seg, *it);
-      CGAL_assertion(assign_res);
+      // Convert the input polycurve to a sequence of variant objects, such
+      // that each object wraps an x-monotone subcurve.
+      std::vector<Make_x_monotone_subresult> x_seg_objects;
+      for (auto its = cv.subcurves_begin(); its != cv.subcurves_end(); ++its)
+        make_seg_x_monotone(*its, std::back_inserter(x_seg_objects));
+      auto it = x_seg_objects.begin();
+      const auto* x_seg_p = boost::get<X_monotone_subcurve_2>(&(*it));
+#if ! defined (CGAL_NO_ASSERTIONS)
+      CGAL_assertion(x_seg_p != nullptr);
 #endif
 
       // If the polycurve consists of a single x-monotone subcurve, return.
       if (x_seg_objects.size() == 1) {
 #ifdef CGAL_ALWAYS_LEFT_TO_RIGHT
         if (cmp_seg_endpts(x_seg) == LARGER)
-          x_seg = ctr_seg_opposite(x_seg);
+          *oi++ = Make_x_monotone_result(ctr_x_curve(ctr_seg_opposite(*x_seg_p)));        else *oi++ = Make_x_monotone_result(ctr_x_curve(*x_seg_p));
+#else
+        *oi++ = Make_x_monotone_result(ctr_x_curve(*x_seg_p));
 #endif
-        *oi++ = make_object(ctr_x_curve(x_seg));
         x_seg_objects.clear();
         return oi;
       }
@@ -244,39 +241,38 @@ public:
          auto max_seg_v =
            m_poly_traits.subcurve_traits_2()->construct_max_vertex_2_object();
          auto equal = m_poly_traits.subcurve_traits_2()->equal_2_object();
-         Point_2 last_target = (cmp_seg_endpts(x_seg) == SMALLER) ?
-           max_seg_v(x_seg) : min_seg_v(x_seg);
+         Point_2 last_target = (cmp_seg_endpts(*x_seg_p) == SMALLER) ?
+           max_seg_v(*x_seg_p) : min_seg_v(*x_seg_p);
          Point_2 next_src;
          );
 
       // The polycurve consists of at least 2 x-monotone subcurves:
       Push_back_2 push_back = m_poly_traits.push_back_2_object();
-      typename Subcurve_traits_2::Is_vertical_2 is_seg_vertical =
+      auto is_seg_vertical =
         m_poly_traits.subcurve_traits_2()->is_vertical_2_object();
 
-      bool is_start_vertical = is_seg_vertical(x_seg);
-      Comparison_result start_dir = cmp_seg_endpts(x_seg);
+      bool is_start_vertical = is_seg_vertical(*x_seg_p);
+      Comparison_result start_dir = cmp_seg_endpts(*x_seg_p);
 
 #ifdef CGAL_ALWAYS_LEFT_TO_RIGHT
       Push_front_2 push_front = m_poly_traits.push_front_2_object();
-      if (cmp_seg_endpts(x_seg) == LARGER) x_seg = ctr_seg_opposite(x_seg);
-#endif
-      X_monotone_curve_2 x_polycurve = ctr_x_curve(x_seg);
-
-      for (++it; it != x_seg_objects.end(); ++it){
-        X_monotone_subcurve_2 x_seg;
-#if defined (CGAL_NO_ASSERTIONS)
-        CGAL::assign(x_seg, *it);
+      X_monotone_curve_2 x_polycurve = (cmp_seg_endpts(x_seg) == LARGER) ?
+        ctr_x_curve(ctr_seg_opposite(*x_seg_p)) : ctr_x_curve(*x_seg_p);
 #else
-        bool assign_res = CGAL::assign(x_seg, *it);
-        CGAL_assertion(assign_res);
+      X_monotone_curve_2 x_polycurve = ctr_x_curve(*x_seg_p);
+#endif
+
+      for (++it; it != x_seg_objects.end(); ++it) {
+        const auto* x_seg_p = boost::get<X_monotone_subcurve_2>(&(*it));
+#if ! defined (CGAL_NO_ASSERTIONS)
+        CGAL_assertion(x_seg_p != nullptr);
 #endif
 
         // Test that cv is continuous and well-oriented.
         CGAL_precondition_code
           (
-           next_src = (cmp_seg_endpts(x_seg) == SMALLER) ?
-             min_seg_v(x_seg) : max_seg_v(x_seg);
+           next_src = (cmp_seg_endpts(*x_seg_p) == SMALLER) ?
+             min_seg_v(*x_seg_p) : max_seg_v(*x_seg_p);
            );
         CGAL_precondition_msg
           (
@@ -285,46 +281,50 @@ public:
            );
         CGAL_precondition_code
           (
-           last_target = (cmp_seg_endpts(x_seg) == SMALLER) ?
-             max_seg_v(x_seg) : min_seg_v(x_seg);
+           last_target = (cmp_seg_endpts(*x_seg_p) == SMALLER) ?
+             max_seg_v(*x_seg_p) : min_seg_v(*x_seg_p);
            );
 
-        if ((cmp_seg_endpts(x_seg) != start_dir) ||
-            (is_seg_vertical(x_seg) != is_start_vertical))
+        if ((cmp_seg_endpts(*x_seg_p) != start_dir) ||
+            (is_seg_vertical(*x_seg_p) != is_start_vertical))
         {
             // Construct an x-monotone curve from the sub-range which was found
-          *oi++ = make_object(x_polycurve);
-          is_start_vertical = is_seg_vertical(x_seg);
-          start_dir = cmp_seg_endpts(x_seg);
+          *oi++ = Make_x_monotone_result(x_polycurve);
+          is_start_vertical = is_seg_vertical(*x_seg_p);
+          start_dir = cmp_seg_endpts(*x_seg_p);
 #ifdef CGAL_ALWAYS_LEFT_TO_RIGHT
-          if (cmp_seg_endpts(x_seg) == LARGER) x_seg = ctr_seg_opposite(x_seg);
+          x_polycurve = (cmp_seg_endpts(*x_seg_p) == LARGER) ?
+            ctr_x_curve(ctr_seg_opposite(*x_seg_p)) : ctr_x_curve(*x_seg_p);
+#else
+          x_polycurve = ctr_x_curve(*x_seg_p);
 #endif
-          x_polycurve = ctr_x_curve(x_seg);
         }
         else {
 #ifdef CGAL_ALWAYS_LEFT_TO_RIGHT
-          if (cmp_seg_endpts(x_seg) == LARGER) {
-            x_seg = ctr_seg_opposite(x_seg);
-              push_front(x_polycurve, x_seg);
-          }
+          if (cmp_seg_endpts(*x_seg_p) == LARGER)
+            push_front(x_polycurve, ctr_seg_opposite(*x_seg_p));
           else
-            push_back(x_polycurve, x_seg);
+            push_back(x_polycurve, *x_seg_p);
 #else
-          push_back(x_polycurve, x_seg);
+          push_back(x_polycurve, *x_seg_p);
 #endif
         }
 
       } // for loop
       if (x_polycurve.number_of_subcurves() != 0)
-        *oi++ = make_object(x_polycurve);
+        *oi++ = Make_x_monotone_result(x_polycurve);
       x_seg_objects.clear();
       return oi;
     }
+
     template <typename OutputIterator>
     OutputIterator operator_impl(const Curve_2& cv, OutputIterator oi,
                                  Arr_not_all_sides_oblivious_tag) const
     {
-      typedef typename Curve_2::Subcurve_const_iterator const_seg_iterator;
+      typedef boost::variant<Point_2, X_monotone_subcurve_2>
+        Make_x_monotone_subresult;
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
 
       // If the polycurve is empty, return.
       if (cv.number_of_subcurves() == 0) return oi;
@@ -347,29 +347,25 @@ public:
         m_poly_traits.subcurve_traits_2()->construct_opposite_2_object();
 #endif
 
-      // Convert the input polycurve to a sequence of CGAL objects, such
-      // that each Object wraps an x-monotone subcurve.
-      std::vector<Object> x_seg_objects;
-      const_seg_iterator it_segs;
-      for (it_segs = cv.subcurves_begin(); it_segs != cv.subcurves_end();
-           ++it_segs)
-        make_seg_x_monotone(*it_segs, std::back_inserter(x_seg_objects));
-      typename std::vector<Object>::iterator it = x_seg_objects.begin();
-      X_monotone_subcurve_2 x_seg;
-#if defined (CGAL_NO_ASSERTIONS)
-      CGAL::assign(x_seg, *it);
-#else
-      bool assign_res = CGAL::assign(x_seg, *it);
-      CGAL_assertion(assign_res);
+      // Convert the input polycurve to a sequence of objects, such that
+      // each object wraps an x-monotone subcurve.
+      std::vector<Make_x_monotone_subresult> x_seg_objects;
+      for (auto its = cv.subcurves_begin(); its != cv.subcurves_end(); ++its)
+        make_seg_x_monotone(*its, std::back_inserter(x_seg_objects));
+      auto it = x_seg_objects.begin();
+      const auto* x_seg_p = boost::get<X_monotone_subcurve_2>(&(*it));
+#if ! defined (CGAL_NO_ASSERTIONS)
+      CGAL_assertion(x_seg_p != nullptr);
 #endif
 
       // If the polycurve consists of a single x-monotone subcurve, return.
       if (x_seg_objects.size() == 1) {
 #ifdef CGAL_ALWAYS_LEFT_TO_RIGHT
         if (cmp_seg_endpts(x_seg) == LARGER)
-          x_seg = ctr_seg_opposite(x_seg);
+          *oi++ = Make_x_monotone_result(ctr_x_curve(ctr_seg_opposite(*x_seg_p)));        else *oi++ = Make_x_monotone_result(ctr_x_curve(*x_seg_p));
+#else
+        *oi++ = Make_x_monotone_result(ctr_x_curve(*x_seg_p));
 #endif
-        *oi++ = make_object(ctr_x_curve(x_seg));
         x_seg_objects.clear();
         return oi;
       }
@@ -383,39 +379,38 @@ public:
          auto max_seg_v =
            m_poly_traits.subcurve_traits_2()->construct_max_vertex_2_object();
          auto equal = m_poly_traits.subcurve_traits_2()->equal_2_object();
-         Point_2 last_target = (cmp_seg_endpts(x_seg) == SMALLER) ?
-           max_seg_v(x_seg) : min_seg_v(x_seg);
+         Point_2 last_target = (cmp_seg_endpts(*x_seg_p) == SMALLER) ?
+           max_seg_v(*x_seg_p) : min_seg_v(*x_seg_p);
          Point_2 next_src;
          );
 
       // The polycurve consists of at least 2 x-monotone subcurves:
       Push_back_2 push_back = m_poly_traits.push_back_2_object();
-      typename Subcurve_traits_2::Is_vertical_2 is_seg_vertical =
+      auto is_seg_vertical =
         m_poly_traits.subcurve_traits_2()->is_vertical_2_object();
 
-      bool is_start_vertical = is_seg_vertical(x_seg);
-      Comparison_result start_dir = cmp_seg_endpts(x_seg);
+      bool is_start_vertical = is_seg_vertical(*x_seg_p);
+      Comparison_result start_dir = cmp_seg_endpts(*x_seg_p);
 
 #ifdef CGAL_ALWAYS_LEFT_TO_RIGHT
       Push_front_2 push_front = m_poly_traits.push_front_2_object();
-      if (cmp_seg_endpts(x_seg) == LARGER) x_seg = ctr_seg_opposite(x_seg);
-#endif
-      X_monotone_curve_2 x_polycurve = ctr_x_curve(x_seg);
-
-      for (++it; it != x_seg_objects.end(); ++it){
-        X_monotone_subcurve_2 x_seg;
-#if defined (CGAL_NO_ASSERTIONS)
-        CGAL::assign(x_seg, *it);
+      X_monotone_curve_2 x_polycurve = (cmp_seg_endpts(x_seg) == LARGER) ?
+        ctr_x_curve(ctr_seg_opposite(*x_seg_p)) : ctr_x_curve(*x_seg_p);
 #else
-        bool assign_res = CGAL::assign(x_seg, *it);
-        CGAL_assertion(assign_res);
+      X_monotone_curve_2 x_polycurve = ctr_x_curve(*x_seg_p);
+#endif
+
+      for (++it; it != x_seg_objects.end(); ++it) {
+        const auto* x_seg_p = boost::get<X_monotone_subcurve_2>(&(*it));
+#if ! defined (CGAL_NO_ASSERTIONS)
+        CGAL_assertion(x_seg_p != nullptr);
 #endif
 
         // Test that cv is continuous and well-oriented.
         CGAL_precondition_code
           (
-           next_src = (cmp_seg_endpts(x_seg) == SMALLER) ?
-             min_seg_v(x_seg) : max_seg_v(x_seg);
+           next_src = (cmp_seg_endpts(*x_seg_p) == SMALLER) ?
+             min_seg_v(*x_seg_p) : max_seg_v(*x_seg_p);
            );
         CGAL_precondition_msg
           (
@@ -424,59 +419,61 @@ public:
            );
         CGAL_precondition_code
           (
-           last_target = (cmp_seg_endpts(x_seg) == SMALLER) ?
-             max_seg_v(x_seg) : min_seg_v(x_seg);
+           last_target = (cmp_seg_endpts(*x_seg_p) == SMALLER) ?
+             max_seg_v(*x_seg_p) : min_seg_v(*x_seg_p);
            );
 
-          Arr_curve_end polycurve_target =
-            (cmp_seg_endpts(x_polycurve[0]) == SMALLER) ?
-            ARR_MAX_END : ARR_MIN_END;
-          Arr_curve_end seg_source = (cmp_seg_endpts(x_seg) == SMALLER) ?
-            ARR_MIN_END : ARR_MAX_END;
-          unsigned int num_segs = x_polycurve.number_of_subcurves();
+        Arr_curve_end polycurve_target =
+          (cmp_seg_endpts(x_polycurve[0]) == SMALLER) ?
+          ARR_MAX_END : ARR_MIN_END;
+        Arr_curve_end seg_source = (cmp_seg_endpts(*x_seg_p) == SMALLER) ?
+          ARR_MIN_END : ARR_MAX_END;
+        auto num_segs = x_polycurve.number_of_subcurves();
 
-        if ((cmp_seg_endpts(x_seg) != start_dir) ||
-            (is_seg_vertical(x_seg) != is_start_vertical))
+        if ((cmp_seg_endpts(*x_seg_p) != start_dir) ||
+            (is_seg_vertical(*x_seg_p) != is_start_vertical))
         {
             // Construct an x-monotone curve from the sub-range which was found
-          *oi++ = make_object(x_polycurve);
-          is_start_vertical = is_seg_vertical(x_seg);
-          start_dir = cmp_seg_endpts(x_seg);
+          *oi++ = Make_x_monotone_result(x_polycurve);
+          is_start_vertical = is_seg_vertical(*x_seg_p);
+          start_dir = cmp_seg_endpts(*x_seg_p);
 #ifdef CGAL_ALWAYS_LEFT_TO_RIGHT
-          if (cmp_seg_endpts(x_seg) == LARGER) x_seg = ctr_seg_opposite(x_seg);
+          x_polycurve (cmp_seg_endpts(*x_seg_p) == LARGER) ?
+            ctr_x_curve(ctr_seg_opposite(*x_seg_p)) : ctr_x_curve(*x_seg_p);
+#else
+          x_polycurve = ctr_x_curve(*x_seg_p);
 #endif
-          x_polycurve = ctr_x_curve(x_seg);
         }
         else if (ps_x(x_polycurve[num_segs-1], polycurve_target) !=
                  ARR_INTERIOR ||
-                 ps_x(x_seg, seg_source) != ARR_INTERIOR)
+                 (ps_x(*x_seg_p, seg_source) != ARR_INTERIOR))
         {
-          *oi++ = make_object(x_polycurve);
+          *oi++ = Make_x_monotone_result(x_polycurve);
 #ifdef CGAL_ALWAYS_LEFT_TO_RIGHT
-          if (cmp_seg_endpts(x_seg) == LARGER) x_seg = ctr_seg_opposite(x_seg);
+          x_polycurve = (cmp_seg_endpts(*x_seg_p) == LARGER) ?
+            ctr_seg_opposite(*x_seg_p) : ctr_x_curve(*x_seg_p);
 #endif
-            x_polycurve = ctr_x_curve(x_seg);
+          x_polycurve = ctr_x_curve(*x_seg_p);
         }
 
         else {
 #ifdef CGAL_ALWAYS_LEFT_TO_RIGHT
-          if (cmp_seg_endpts(x_seg) == LARGER) {
-            x_seg = ctr_seg_opposite(x_seg);
-            push_front(x_polycurve, x_seg);
-          }
+          if (cmp_seg_endpts(*x_seg_p) == LARGER)
+            push_front(x_polycurve, ctr_seg_opposite(*x_seg_p));
           else
-            push_back(x_polycurve, x_seg);
+            push_back(x_polycurve, *x_seg_p);
 #else
-          push_back(x_polycurve, x_seg);
+          push_back(x_polycurve, *x_seg_p);
 #endif
         }
       } // for loop
       if (x_polycurve.number_of_subcurves() != 0)
-        *oi++ = make_object(x_polycurve);
+        *oi++ = Make_x_monotone_result(x_polycurve);
       x_seg_objects.clear();
       return oi;
     }
-public:
+
+  public:
     template <typename OutputIterator>
     OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const
     { return operator_impl(cv, oi, Are_all_sides_oblivious_tag()); }

Arrangement_on_surface_2/include/CGAL/Arr_rational_function_traits_2.h

@@ -808,45 +808,41 @@ public:
     return Equal_2(this);
   }
 
-  /*! A functor that divides a curve into continues (x-monotone) curves. */
-  class Make_x_monotone_2
-  {
-  public:
+  //! \name Intersections & subdivisions
+  //@{
 
-    /*!
-     * Cut the given conic curve (or conic arc) into x-monotone subcurves
-     * and insert them to the given output iterator.
-     * \param cv The curve.
-     * \param oi The output iterator, whose value-type is Object. The returned
-     *           objects is a wrapper for an X_monotone_curve_2 object.
-     * \return The past-the-end iterator.
+  /*! \class Make_x_monotone_2
+   * A functor for subdividing a curve into continues x-monotone curves.
+   */
+  class Make_x_monotone_2 {
+  public:
+    /*! Subdivide a given rational-function curve into x-monotone subcurves
+     * and insert them to a given output iterator.
+     * \param cv the curve.
+     * \param oi the output iterator for the result. Its dereference type is a
+     *           variant that wraps a \c Point_2 or an \c X_monotone_curve_2
+     *           objects.
+     * \return the past-the-end iterator.
      */
-    template<typename OutputIterator>
+    template <typename OutputIterator>
     OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const
     {
-      // Make the rational arc continuous.
-      std::list<X_monotone_curve_2>                           arcs;
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
 
+      // Make the rational arc continuous.
+      std::list<X_monotone_curve_2> arcs;
       cv.make_continuous(std::back_inserter(arcs));
 
       // Create objects.
-      typename std::list<X_monotone_curve_2>::const_iterator  iter;
-
-      for (iter = arcs.begin(); iter != arcs.end(); ++iter)
-      {
-        *oi = make_object (*iter);
-        ++oi;
-      }
-
-      return (oi);
+      for (const auto& arc : arcs) *oi++ = Make_x_monotone_result(arc);
+      return oi;
     }
   };
 
   /*! Obtain a Make_x_monotone_2 functor object. */
   Make_x_monotone_2 make_x_monotone_2_object() const
-  {
-    return Make_x_monotone_2();
-  }
+  { return Make_x_monotone_2(); }
 
   /*! A functor that splits a curve at a point. */
   class Split_2
@@ -975,6 +971,8 @@ public:
     return Merge_2(this);
   }
 
+  //@}
+
   /// \name Functor definitions to handle boundaries
   //@{
 

Arrangement_on_surface_2/include/CGAL/Arr_segment_traits_2.h

@@ -543,23 +543,30 @@ public:
 
   //@}
 
-  /// \name Functor definitions for supporting intersections.
+  //! \name Intersections, subdivisions, and mergings
   //@{
 
+  /*! \class Make_x_monotone_2
+   * A functor for subdividing a curve into x-monotone curves.
+   */
   class Make_x_monotone_2 {
   public:
-    /*! Cut the given curve into x-monotone subcurves and insert them into the
-     * given output iterator. As segments are always x_monotone, only one
-     * object will be contained in the iterator.
+    /*! Subdivide a given curve into x-monotone subcurves and insert them into
+     * a given output iterator. As segments are always x_monotone a single
+     * object is inserted.
      * \param cv the curve.
-     * \param oi the output iterator, whose value-type is variant<....
-     * \return the past-the-end iterator.
+     * \param oi the output iterator for the result. Its dereference type is a
+     *           variant that wraps a \c Point_2 or an \c X_monotone_curve_2
+     *           objects.
+     * \return the past-the-end output iterator.
      */
     template <typename OutputIterator>
     OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const
     {
-      // Wrap the segment with an object.
-      *oi++ = make_object(cv);
+      // Wrap the segment with a variant.
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
+      *oi++ = Make_x_monotone_result(cv);
       return oi;
     }
   };
@@ -1161,6 +1168,7 @@ public:
 //! \brief constructs default.
 template <typename Kernel>
 Arr_segment_traits_2<Kernel>::_Segment_cached_2::_Segment_cached_2() :
+  m_is_directed_right(false),
   m_is_vert(false),
   m_is_degen(true)
 {}

Arrangement_on_surface_2/include/CGAL/Arr_spherical_gaussian_map_3/Arr_spherical_gaussian_map_3.h

@@ -151,23 +151,25 @@ public:
   OutputIterator insert(const Vector_3 & normal1, const Vector_3 & normal2,
                         OutputIterator oi)
   {
-    std::list<CGAL::Object> x_objects;
+    typedef boost::variant<Point_2, X_monotone_curve_2>
+      Make_x_monotone_result;
+
+    std::list<Make_x_monotone_result> x_objects;
     make_x_monotone(normal1, normal2, std::back_inserter(x_objects));
 
-    typename std::list<CGAL::Object>::iterator it = x_objects.begin();
-    const X_monotone_curve_2 * xc = object_cast<X_monotone_curve_2>(&(*it));
+    auto it = x_objects.begin();
+    const auto* xc = boost::get<X_monotone_curve_2>(&(*it));
 #if CGAL_ARR_SPHERICAL_GAUSSIAN_MAP_3_DEBUG==1
     std::cout << "1.a. insert_in_face_interior(" << *xc << ")" << std::endl;
 #endif
-    Halfedge_handle he =
-      m_sgm.insert_in_face_interior(*xc, m_sgm.faces_begin());
-    if (!xc->is_directed_right()) he = he->twin();
+    Halfedge_handle he = m_sgm.insert_in_face_interior(*xc, m_sgm.faces_begin());
+    if (! xc->is_directed_right()) he = he->twin();
     *oi++ = he;
 
     ++it;
     if (it == x_objects.end()) return oi;
 
-    xc = object_cast<X_monotone_curve_2>(&(*it));
+    xc = boost::get<X_monotone_curve_2>(&(*it));
 #if CGAL_ARR_SPHERICAL_GAUSSIAN_MAP_3_DEBUG==1
     std::cout << "1.b. insert_from_vertex(" << *xc << ")" << std::endl;
 #endif
@@ -190,11 +192,14 @@ public:
                         const Vector_3 & normal2,
                         OutputIterator oi)
   {
-    std::list<CGAL::Object> x_objects;
+    typedef boost::variant<Point_2, X_monotone_curve_2>
+      Make_x_monotone_result;
+
+    std::list<Make_x_monotone_result> x_objects;
     make_x_monotone(normal1, normal2, std::back_inserter(x_objects));
 
-    typename std::list<CGAL::Object>::iterator it = x_objects.begin();
-    const X_monotone_curve_2 * xc = object_cast<X_monotone_curve_2>(&(*it));
+    auto it = x_objects.begin();
+    const auto* xc = boost::get<X_monotone_curve_2>(&(*it));
 #if CGAL_ARR_SPHERICAL_GAUSSIAN_MAP_3_DEBUG==1
     std::cout << "2.a. insert_from_vertex(" << *xc << ", "
               << vertex1->point() << ")" << std::endl;
@@ -208,7 +213,7 @@ public:
     ++it;
     if (it == x_objects.end()) return oi;
 
-    xc = object_cast<X_monotone_curve_2>(&(*it));
+    xc = boost::get<X_monotone_curve_2>(&(*it));
 #if CGAL_ARR_SPHERICAL_GAUSSIAN_MAP_3_DEBUG==1
     std::cout << "2.b. insert_from_vertex(" << *xc << ")" << std::endl;
 #endif
@@ -231,12 +236,15 @@ public:
                         const Vector_3 & normal2, Vertex_handle vertex2,
                         OutputIterator oi)
   {
-    std::list<CGAL::Object> x_objects;
+    typedef boost::variant<Point_2, X_monotone_curve_2>
+      Make_x_monotone_result;
+
+    std::list<Make_x_monotone_result> x_objects;
     make_x_monotone(normal1, normal2, std::back_inserter(x_objects));
 
-    typename std::list<CGAL::Object>::iterator it = x_objects.begin();
+    auto it = x_objects.begin();
     if (x_objects.size() == 1) {
-      const X_monotone_curve_2 * xc = object_cast<X_monotone_curve_2>(&(*it));
+      const auto* xc = boost::get<X_monotone_curve_2>(&(*it));
 #if CGAL_ARR_SPHERICAL_GAUSSIAN_MAP_3_DEBUG==1
       std::cout << "3. insert_from_vertex(" << *xc << ")" << std::endl;
 #endif
@@ -247,8 +255,8 @@ public:
       return oi;
     }
 
-    const X_monotone_curve_2 * xc1 = object_cast<X_monotone_curve_2>(&(*it++));
-    const X_monotone_curve_2 * xc2 = object_cast<X_monotone_curve_2>(&(*it));
+    const X_monotone_curve_2* xc1 = boost::get<X_monotone_curve_2>(&(*it++));
+    const X_monotone_curve_2* xc2 = boost::get<X_monotone_curve_2>(&(*it));
 
 #if CGAL_ARR_SPHERICAL_GAUSSIAN_MAP_3_DEBUG==1
     std::cout << "3.a. insert_from_vertex(" << *xc2 << ")" << std::endl;
@@ -283,11 +291,13 @@ public:
                         const Vector_3 & normal2, Vertex_handle vertex2,
                         OutputIterator oi)
   {
-    std::list<CGAL::Object> x_objects;
+    typedef boost::variant<Point_2, X_monotone_curve_2>
+      Make_x_monotone_result;
+    std::list<Make_x_monotone_result> x_objects;
     make_x_monotone(normal1, normal2, std::back_inserter(x_objects));
-    typename std::list<CGAL::Object>::iterator it = x_objects.begin();
+    auto it = x_objects.begin();
     if (x_objects.size() == 1) {
-      const X_monotone_curve_2 * xc = object_cast<X_monotone_curve_2>(&(*it));
+      const auto* xc = boost::get<X_monotone_curve_2>(&(*it));
 #if CGAL_ARR_SPHERICAL_GAUSSIAN_MAP_3_DEBUG==1
       std::cout << "4. insert_at_vertices(" << *xc << ")" << std::endl;
 #endif
@@ -295,8 +305,8 @@ public:
       return oi;
     }
 
-    const X_monotone_curve_2 * xc1 = object_cast<X_monotone_curve_2>(&(*it++));
-    const X_monotone_curve_2 * xc2 = object_cast<X_monotone_curve_2>(&(*it));
+    const X_monotone_curve_2 * xc1 = boost::get<X_monotone_curve_2>(&(*it++));
+    const X_monotone_curve_2 * xc2 = boost::get<X_monotone_curve_2>(&(*it));
 
 #if CGAL_ARR_SPHERICAL_GAUSSIAN_MAP_3_DEBUG==1
     std::cout << "4.a. insert_from_vertex(" << *xc1

Arrangement_on_surface_2/include/CGAL/Arr_spherical_gaussian_map_3/Arr_transform_on_sphere.h

@@ -35,6 +35,7 @@ void Arr_transform_on_sphere(Arrangement & arr,
   typedef typename Arrangement::Geometry_traits_2         Geometry_traits_2;
   typedef typename Arrangement::Topology_traits           Topology_traits;
 
+  typedef typename Geometry_traits_2::Point_2             Point_2;
   typedef typename Geometry_traits_2::Curve_2             Curve_2;
   typedef typename Geometry_traits_2::X_monotone_curve_2  X_monotone_curve_2;
 
@@ -45,6 +46,8 @@ void Arr_transform_on_sphere(Arrangement & arr,
   typedef typename Arrangement::Halfedge_around_vertex_circulator
     Halfedge_around_vertex_circulator;
 
+  typedef boost::variant<Point_2, X_monotone_curve_2>     Make_x_monotone_result;
+
   const Geometry_traits_2 * geom_traits = arr.geometry_traits();
   Topology_traits * topol_traits = arr.topology_traits();
 
@@ -53,8 +56,7 @@ void Arr_transform_on_sphere(Arrangement & arr,
   // Preprocessing loop - merge all the edges that were splited
   // (meaning have a common endpoint that lies on the boundary and their degree
   // is 2) on the identification curve.
-  for (Vertex_handle vi1 = arr.vertices_begin() ; vi1 != arr.vertices_end() ;)
-  {
+  for (auto vi1 = arr.vertices_begin(); vi1 != arr.vertices_end(); ) {
     Vertex_handle v_temp = vi1;
     ++vi1;
 
@@ -63,9 +65,9 @@ void Arr_transform_on_sphere(Arrangement & arr,
     Arr_parameter_space by =
       geom_traits->parameter_space_in_y_2_object()(v_temp->point());
 
-    // use vertex->parameter_space_in_x() != interior || vertex->parameter_space_in_y() != interior)
-    if ((bx != ARR_INTERIOR || by != ARR_INTERIOR) && (v_temp->degree() == 2))
-    {
+    // use vertex->parameter_space_in_x() != interior ||
+    //     vertex->parameter_space_in_y() != interior)
+    if ((bx != ARR_INTERIOR || by != ARR_INTERIOR) && (v_temp->degree() == 2)) {
       Curve_2 merged_cv;
       Halfedge_around_vertex_circulator havc = v_temp->incident_halfedges();
       Halfedge_around_vertex_circulator havc_next = havc;
@@ -80,16 +82,13 @@ void Arr_transform_on_sphere(Arrangement & arr,
       // Compare the direction of the edges.
       bool eq_direction = havc->direction() == havc_next->twin()->direction();
 
-      if (point_eq1 && normal_eq1 && eq_direction)
-      {
-        if (havc->source()->point() == havc->curve().source())
-        {
+      if (point_eq1 && normal_eq1 && eq_direction) {
+        if (havc->source()->point() == havc->curve().source()) {
           merged_cv = Curve_2(havc->source()->point(),
                               havc_next->twin()->target()->point(),
                               havc->curve().normal());
         }
-        else if (havc->source()->point() == havc->curve().target())
-        {
+        else if (havc->source()->point() == havc->curve().target()) {
           merged_cv = Curve_2(havc_next->twin()->target()->point(),
                               havc->source()->point(),
                               havc->curve().normal());
@@ -111,20 +110,17 @@ void Arr_transform_on_sphere(Arrangement & arr,
   }
 
   //Rotate all the vertices.
-  for (Vertex_handle vi1 = arr.vertices_begin(); vi1 != arr.vertices_end() ;
-       ++vi1)
-  {
+  for (auto vi1 = arr.vertices_begin(); vi1 != arr.vertices_end(); ++vi1) {
     m_arr_access.modify_vertex_ex(vi1, aff.transform(vi1->point()));
   }
 
-  unsigned int num_of_edges = arr.number_of_edges();
+  size_t num_of_edges = arr.number_of_edges();
   Edge_iterator ei1 = arr.edges_begin();
 
   // Rotate all the halfedges.
   // The loop is over the initial edges , since new edges are created and
   // added to the edges list.
-  for (unsigned int i=0 ; i < num_of_edges ; ++i)
-  {
+  for (size_t i = 0; i < num_of_edges; ++i) {
     Curve_2 new_cv;
 
     Halfedge_handle hei1 = ei1;
@@ -133,8 +129,7 @@ void Arr_transform_on_sphere(Arrangement & arr,
     // Take only the halfedge that its source is equal to the source of
     // the curve.
     bool eq1 = hei1->source()->point() == aff.transform(hei1->curve().source());
-    if (!eq1)
-    {
+    if (!eq1) {
       hei1 = hei1->twin();
       eq1 = hei1->source()->point() == aff.transform(hei1->curve().source());
     }
@@ -150,31 +145,28 @@ void Arr_transform_on_sphere(Arrangement & arr,
     // Modify the edge with the new curve.
     m_arr_access.modify_edge_ex(hei1, new_cv);
 
-    std::list<CGAL::Object> objects;
+    std::list<Make_x_monotone_result> objects;
     // Try to split the curve into x_monotone pieces.
     geom_traits->make_x_monotone_2_object()(new_cv , std::back_inserter(objects));
 
     // If the curve is not x-monotone - split it into 2 x_monotone parts.
     // Since the curves were x_monotone before , can assume that it will be
     // splited into 2 parts max.
-    if (objects.size() == 2)
-    {
-      typename std::list<CGAL::Object>::iterator it = objects.begin();
+    if (objects.size() == 2) {
+      auto it = objects.begin();
 
       // The curve that its left vertex lies on the identification curve
-      const X_monotone_curve_2 * sub_cv1 =
-        object_cast<X_monotone_curve_2>(&(*it));
+      const auto* sub_cv1 = boost::get<X_monotone_curve_2>(&(*it));
       ++it;
       //The curve that its rigth vertex lies on the identification curve
-      const X_monotone_curve_2 * sub_cv2 =
-        object_cast<X_monotone_curve_2>(&(*it));
+      const auto* sub_cv2 = boost::get<X_monotone_curve_2>(&(*it));
 
       bool eq1 = (*sub_cv1).source() == hei1->source()->point();
       bool eq2 = (*sub_cv2).target() == hei1->target()->point();
 
       if (eq1 && eq2)
-        m_arr_access.split_edge_ex (hei1, (*sub_cv1).target(), *sub_cv1,
-                                    *sub_cv2);
+        m_arr_access.split_edge_ex(hei1, (*sub_cv1).target(), *sub_cv1,
+                                   *sub_cv2);
       else
         CGAL_error_msg
           ("The new curve endpoints should be equal to the original ones");
@@ -183,25 +175,20 @@ void Arr_transform_on_sphere(Arrangement & arr,
 
   // Update all the vertices that located on the boundary after the rotation
   // with boundary conditions
-  for (Vertex_handle vi = arr.vertices_begin() ; vi != arr.vertices_end() ;
-       ++vi)
-  {
+  for (auto vi = arr.vertices_begin(); vi != arr.vertices_end(); ++vi) {
     Arr_parameter_space bx =
       geom_traits->parameter_space_in_x_2_object()(vi->point());
     Arr_parameter_space by =
       geom_traits->parameter_space_in_y_2_object()(vi->point());
 
-    if (bx != ARR_INTERIOR || by != ARR_INTERIOR)
-    {
+    if (bx != ARR_INTERIOR || by != ARR_INTERIOR) {
       // The target of the Halfedge_around_vertex_circulator is the relevant
       // point.
       Halfedge_around_vertex_circulator havc = vi->incident_halfedges();
 
-      Arr_curve_end ind;
-          if (geom_traits->construct_min_vertex_2_object()(havc->curve()) == vi->point())
-                ind = ARR_MIN_END;
-          else
-                ind = ARR_MAX_END;
+      Arr_curve_end ind =
+        (geom_traits->construct_min_vertex_2_object()(havc->curve()) ==
+             vi->point()) ? ARR_MIN_END : ARR_MAX_END;
 
       // Check if it was already added.
       if (topol_traits->discontinuity_vertex(havc->curve(), ind)== nullptr &&
@@ -219,7 +206,7 @@ void Arr_transform_on_sphere(Arrangement & arr,
   }
 
   // Transform the faces with the suitable transformation traits.
-  for(Face_handle f1 = arr.faces_begin() ; f1 != arr.faces_end() ; ++f1)
+  for (auto f1 = arr.faces_begin(); f1 != arr.faces_end(); ++f1)
     tran_tr.rotate_face(f1, aff);
 }
 

Arrangement_on_surface_2/include/CGAL/Arr_spherical_topology_traits_2.h

@@ -7,8 +7,8 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-// Author(s)     : Efi Fogel         <efif@post.tau.ac.il>
-//                 Eric Berberich    <ericb@post.tau.ac.il>
+// Author(s): Efi Fogel         <efif@post.tau.ac.il>
+//            Eric Berberich    <ericb@post.tau.ac.il>
 
 #ifndef CGAL_ARR_SPHERICAL_TOPOLOGY_TRAITS_2_H
 #define CGAL_ARR_SPHERICAL_TOPOLOGY_TRAITS_2_H
@@ -491,11 +491,12 @@ public:
    * \pre The curve has a boundary condition in either x or y.
    * \return An object that contains the curve end.
    */
-  CGAL::Object place_boundary_vertex(Face* f,
-                                     const X_monotone_curve_2& xc,
-                                     Arr_curve_end ind,
-                                     Arr_parameter_space ps_x,
-                                     Arr_parameter_space ps_y);
+  boost::optional<boost::variant<Vertex*, Halfedge*> >
+  place_boundary_vertex(Face* f,
+                        const X_monotone_curve_2& xc,
+                        Arr_curve_end ind,
+                        Arr_parameter_space ps_x,
+                        Arr_parameter_space ps_y);
 
   /*! Locate the predecessor halfedge for the given curve around a given
    * vertex with boundary conditions.
@@ -522,9 +523,10 @@ public:
    * \pre The curve end is incident to the boundary.
    * \return An object that contains the curve end.
    */
-  CGAL::Object locate_curve_end(const X_monotone_curve_2& xc, Arr_curve_end ce,
-                                Arr_parameter_space ps_x,
-                                Arr_parameter_space ps_y);
+  boost::variant<Vertex*, Halfedge*, Face*>
+  locate_curve_end(const X_monotone_curve_2& xc, Arr_curve_end ce,
+                   Arr_parameter_space ps_x,
+                   Arr_parameter_space ps_y);
 
   /*! Split a fictitious edge using the given vertex.
    * \param e The edge to split (one of the pair of halfedges).

Arrangement_on_surface_2/include/CGAL/Arr_topology_traits/Arr_bounded_planar_vert_decomp_helper.h

@@ -7,9 +7,8 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-//
-// Author(s)     : Ron Wein <wein@post.tau.ac.il>
-//                 Efi Fogel <efif@post.tau.ac.il>
+// Author(s): Ron Wein <wein@post.tau.ac.il>
+//            Efi Fogel <efif@post.tau.ac.il>
 
 #ifndef CGAL_ARR_BOUNDED_PLANAR_VERT_DEOCMP_HELPER_H
 #define CGAL_ARR_BOUNDED_PLANAR_VERT_DEOCMP_HELPER_H
@@ -43,9 +42,15 @@ private:
   typedef Geometry_traits_2                             Gt2;
 
 public:
+  typedef typename Arrangement_2::Vertex_const_handle   Vertex_const_handle;
+  typedef typename Arrangement_2::Halfedge_const_handle Halfedge_const_handle;
   typedef typename Arrangement_2::Face_const_handle     Face_const_handle;
   typedef typename Arrangement_2::Topology_traits       Topology_traits;
 
+  typedef boost::variant<Vertex_const_handle, Halfedge_const_handle,
+                         Face_const_handle>             Cell_type;
+  typedef boost::optional<Cell_type>                    Vert_type;
+
 protected:
   // Data members:
   const Topology_traits* m_top_traits;  // The topology-traits class.
@@ -76,10 +81,10 @@ public:
   //@}
 
   /*! Get the current top object. */
-  CGAL::Object top_object() const { return CGAL::make_object(m_unb_face); }
+  Vert_type top_object() const { return Vert_type(m_unb_face); }
 
   /*! Get the current bottom object. */
-  CGAL::Object bottom_object() const { return CGAL::make_object(m_unb_face); }
+  Vert_type bottom_object() const { return Vert_type(m_unb_face); }
 };
 
 } // namespace CGAL

Arrangement_on_surface_2/include/CGAL/Arr_topology_traits/Arr_spherical_topology_traits_2_impl.h

@@ -7,8 +7,8 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-// Author(s)     : Efi Fogel         <efif@post.tau.ac.il>
-//                 Ron Wein          <wein@post.tau.ac.il>
+// Author(s): Efi Fogel         <efif@post.tau.ac.il>
+//            Ron Wein          <wein@post.tau.ac.il>
 
 #ifndef CGAL_ARR_SPHERICAL_TOPOLOGY_TRAITS_2_IMPL_H
 #define CGAL_ARR_SPHERICAL_TOPOLOGY_TRAITS_2_IMPL_H
@@ -514,7 +514,10 @@ let_me_decide_the_outer_ccb(std::pair< CGAL::Sign, CGAL::Sign> signs1,
  * the interior of the curve, find a place for a boundary vertex that will
  * represent the curve end along the face boundary */
 template <typename GeomTraits, typename Dcel>
-CGAL::Object
+boost::optional
+  <boost::variant
+    <typename Arr_spherical_topology_traits_2<GeomTraits, Dcel>::Vertex*,
+     typename Arr_spherical_topology_traits_2<GeomTraits, Dcel>::Halfedge*> >
 Arr_spherical_topology_traits_2<GeomTraits, Dcel>::
 place_boundary_vertex(Face* /* f */,
                       const X_monotone_curve_2& xc, Arr_curve_end ind,
@@ -525,15 +528,18 @@ place_boundary_vertex(Face* /* f */,
                       ,
                       Arr_parameter_space ps_y)
 {
+  typedef boost::variant<Vertex*, Halfedge*>    Non_optional_result;
+  typedef boost::optional<Non_optional_result>  Result;
+
   // std::cout << "place_boundary_vertex()" << std::endl;
   if (ps_y == ARR_BOTTOM_BOUNDARY) {
-    if (m_south_pole == nullptr) return Object();
-    return CGAL::make_object(m_south_pole);
+    if (m_south_pole == nullptr) return boost::none;
+    return Result(Non_optional_result(m_south_pole));
   }
 
   if (ps_y == ARR_TOP_BOUNDARY) {
-    if (m_north_pole == nullptr) return Object();
-    return CGAL::make_object(m_north_pole);
+    if (m_north_pole == nullptr) return boost::none;
+    return Result(Non_optional_result(m_north_pole));
   }
 
   CGAL_assertion((ps_x == ARR_LEFT_BOUNDARY) || (ps_x == ARR_RIGHT_BOUNDARY));
@@ -541,15 +547,14 @@ place_boundary_vertex(Face* /* f */,
   const Point_2& key = (ind == ARR_MIN_END) ?
     m_geom_traits->construct_min_vertex_2_object()(xc) :
     m_geom_traits->construct_max_vertex_2_object()(xc);
-  typename Vertex_map::iterator it = m_boundary_vertices.find(key);
-
+  auto it = m_boundary_vertices.find(key);
   if (it != m_boundary_vertices.end()) {
     Vertex* v = it->second;
-    return CGAL::make_object(v);
+    return Result(Non_optional_result(v));
   }
 
   // The vertex hasn't been created yet, return a null object:
-  return Object();
+  return boost::none;
 }
 
 /*! \brief locate the predecessor halfedge for the given curve around a given
@@ -585,7 +590,11 @@ locate_around_boundary_vertex(Vertex* v,
 
 /*! \brief locates a DCEL feature that contains a given curve end. */
 template <typename GeomTraits, typename Dcel>
-CGAL::Object Arr_spherical_topology_traits_2<GeomTraits, Dcel>::
+boost::variant
+<typename Arr_spherical_topology_traits_2<GeomTraits, Dcel>::Vertex*,
+ typename Arr_spherical_topology_traits_2<GeomTraits, Dcel>::Halfedge*,
+ typename Arr_spherical_topology_traits_2<GeomTraits, Dcel>::Face*>
+Arr_spherical_topology_traits_2<GeomTraits, Dcel>::
 locate_curve_end(const X_monotone_curve_2& xc, Arr_curve_end ind,
                  Arr_parameter_space
 #if !defined(CGAL_NO_ASSERTIONS)
@@ -594,13 +603,14 @@ locate_curve_end(const X_monotone_curve_2& xc, Arr_curve_end ind,
                  ,
                  Arr_parameter_space ps_y)
 {
+  typedef boost::variant<Vertex*, Halfedge*, Face*>     Result;
   // Act according to the boundary conditions.
   if (ps_y == ARR_TOP_BOUNDARY) {
     // In case the curve end coincides with the north pole, return the vertex
     // representing the north pole, if one exists. Otherwise, return the face
     // containing this pole (the spherical face).
-    if (m_north_pole != nullptr) return CGAL::make_object(m_north_pole);
-    return CGAL::make_object(m_spherical_face);
+    if (m_north_pole != nullptr) return Result(m_north_pole);
+    return Result(m_spherical_face);
   }
 
   typename Vertex_map::iterator it;
@@ -610,7 +620,7 @@ locate_curve_end(const X_monotone_curve_2& xc, Arr_curve_end ind,
     // In case the curve end coincides with the south pole, return the vertex
     // representing the south pole, if one exists. Otherwise, search for the
     // face containing this pole.
-    if (m_south_pole != nullptr) return CGAL::make_object(m_south_pole);
+    if (m_south_pole != nullptr) return Result(m_south_pole);
     it = m_boundary_vertices.begin();
   }
   else {
@@ -625,7 +635,7 @@ locate_curve_end(const X_monotone_curve_2& xc, Arr_curve_end ind,
     it = m_boundary_vertices.find(key);
     if (it != m_boundary_vertices.end()) {
       v = it->second;
-      return CGAL::make_object(v);
+      return Result(v);
     }
 
     it = m_boundary_vertices.lower_bound(key);
@@ -635,11 +645,10 @@ locate_curve_end(const X_monotone_curve_2& xc, Arr_curve_end ind,
   // discontinuity that is strictly above the curve end. If there is none,
   // we know the curve end is contained in the spherical face. Otherwise,
   // we return the face that lies below the vertex v.
-  if (it == m_boundary_vertices.end())
-    return CGAL::make_object(m_spherical_face);
+  if (it == m_boundary_vertices.end()) return Result(m_spherical_face);
 
   v = it->second;
-  return CGAL::make_object(_face_below_vertex_on_discontinuity(v));
+  return Result(_face_below_vertex_on_discontinuity(v));
 }
 
 /*! \brief determines whether a given boundary vertex is redundant */

Arrangement_on_surface_2/include/CGAL/Arr_topology_traits/Arr_spherical_vert_decomp_helper.h

@@ -7,8 +7,8 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-// Author(s)     : Ron Wein <wein@post.tau.ac.il>
-//                 Efi Fogel <efif@post.tau.ac.il>
+// Author(s): Ron Wein <wein@post.tau.ac.il>
+//            Efi Fogel <efif@post.tau.ac.il>
 
 #ifndef CGAL_ARR_SPHERICAL_VERT_DECOMP_HELPER_H
 #define CGAL_ARR_SPHERICAL_VERT_DECOMP_HELPER_H
@@ -44,8 +44,13 @@ private:
 public:
   typedef typename Gt2::X_monotone_curve_2              X_monotone_curve_2;
 
-  typedef typename Arrangement_2::Face_const_handle     Face_const_handle;
   typedef typename Arrangement_2::Vertex_const_handle   Vertex_const_handle;
+  typedef typename Arrangement_2::Halfedge_const_handle Halfedge_const_handle;
+  typedef typename Arrangement_2::Face_const_handle     Face_const_handle;
+
+  typedef boost::variant<Vertex_const_handle, Halfedge_const_handle,
+                         Face_const_handle>             Cell_type;
+  typedef boost::optional<Cell_type>                    Vert_type;
 
 protected:
   typedef typename Arrangement_2::Topology_traits       Topology_traits;
@@ -80,17 +85,17 @@ public:
   //@}
 
   /*! Get the current top object. */
-  CGAL::Object top_object () const
+  Vert_type top_object () const
   {
     return (m_valid_north_pole) ?
-      CGAL::make_object (m_north_pole) : CGAL::make_object (m_north_face);
+      Vert_type(m_north_pole) : Vert_type(m_north_face);
   }
 
   /*! Get the current bottom object. */
-  CGAL::Object bottom_object () const
+  Vert_type bottom_object () const
   {
     return (m_valid_south_pole) ?
-      CGAL::make_object(m_south_pole) : CGAL::make_object(m_south_face);
+      Vert_type(m_south_pole) : Vert_type(m_south_face);
   }
 };
 

Arrangement_on_surface_2/include/CGAL/Arr_topology_traits/Arr_unb_planar_topology_traits_2_impl.h

@@ -8,8 +8,8 @@
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
 //
-// Author(s)     : Ron Wein  <wein@post.tau.ac.il>
-//                 Efi Fogel <efif@post.tau.ac.il>
+// Author(s): Ron Wein  <wein@post.tau.ac.il>
+//            Efi Fogel <efif@post.tau.ac.il>
 
 #ifndef CGAL_ARR_UNB_PLANAR_TOPOLOGY_TRAITS_2_IMPL_H
 #define CGAL_ARR_UNB_PLANAR_TOPOLOGY_TRAITS_2_IMPL_H
@@ -296,27 +296,31 @@ are_equal(const Vertex *v,
 // represent the curve end along the face boundary.
 //
 template <typename GeomTraits, typename Dcel_>
-CGAL::Object
+boost::optional
+  <boost::variant
+    <typename Arr_unb_planar_topology_traits_2<GeomTraits, Dcel_>::Vertex*,
+     typename Arr_unb_planar_topology_traits_2<GeomTraits, Dcel_>::Halfedge*> >
 Arr_unb_planar_topology_traits_2<GeomTraits, Dcel_>::
-place_boundary_vertex(Face *f,
+place_boundary_vertex(Face* f,
                       const X_monotone_curve_2& cv, Arr_curve_end ind,
                       Arr_parameter_space ps_x, Arr_parameter_space ps_y)
 {
-  // Get a halfedge on the outer CCB of f and start traversing the CCB.
-  Halfedge           *first = *(f->outer_ccbs_begin());
-  Halfedge           *curr = first;
-  bool                eq_source, eq_target;
+  typedef boost::variant<Vertex*, Halfedge*>    Non_optional_result;
+  typedef boost::optional<Non_optional_result>  Result;
 
-  do
-  {
+  // Get a halfedge on the outer CCB of f and start traversing the CCB.
+  Halfedge* first = *(f->outer_ccbs_begin());
+  Halfedge* curr = first;
+  bool eq_source, eq_target;
+
+  do {
     // Note we consider only fictitious halfedges and check whether they
     // contain the relevant curve end.
     if (curr->has_null_curve() &&
-        _is_on_fictitious_edge (cv, ind, ps_x, ps_y, curr,
-                                eq_source, eq_target))
+        _is_on_fictitious_edge(cv, ind, ps_x, ps_y, curr, eq_source, eq_target))
     {
-      CGAL_assertion (! eq_source && ! eq_target);
-      return (CGAL::make_object (curr));
+      CGAL_assertion(! eq_source && ! eq_target);
+      return Result(curr);
     }
 
     // Move to the next halfegde along the CCB.
@@ -327,59 +331,58 @@ place_boundary_vertex(Face *f,
   // If we reached here, we did not find a suitable halfegde, which should
   // never happen.
   CGAL_error();
-  return CGAL::Object();
+  return boost::none;
 }
 
 //-----------------------------------------------------------------------------
 // Locate a DCEL feature that contains the given unbounded curve end.
 //
 template <typename GeomTraits, typename Dcel_>
-CGAL::Object Arr_unb_planar_topology_traits_2<GeomTraits, Dcel_>::
+boost::variant
+<typename Arr_unb_planar_topology_traits_2<GeomTraits, Dcel_>::Vertex*,
+ typename Arr_unb_planar_topology_traits_2<GeomTraits, Dcel_>::Halfedge*,
+ typename Arr_unb_planar_topology_traits_2<GeomTraits, Dcel_>::Face*>
+Arr_unb_planar_topology_traits_2<GeomTraits, Dcel_>::
 locate_curve_end (const X_monotone_curve_2& cv, Arr_curve_end ind,
                   Arr_parameter_space ps_x, Arr_parameter_space ps_y)
 {
+  typedef boost::variant<Vertex*, Halfedge*, Face*>     Result;
+
   // Start traversing the inner CCB of the fictitious face and try to locate
   // a feature that contains the curve end.
-  Halfedge   *first = *(fict_face->inner_ccbs_begin());
-  Halfedge   *curr = first;
-  bool        eq_source, eq_target;
+  Halfedge* first = *(fict_face->inner_ccbs_begin());
+  Halfedge* curr = first;
+  bool eq_source, eq_target;
 
-  do
-  {
-    if (_is_on_fictitious_edge (cv, ind, ps_x, ps_y, curr,
-                                eq_source, eq_target))
+  do {
+    if (_is_on_fictitious_edge(cv, ind, ps_x, ps_y, curr, eq_source, eq_target))
     {
-      if (eq_source)
-      {
+      if (eq_source) {
         // cv's end coincides with the source vertex of the current
         // fictitious halfedge. This means that cv overlaps the curve that
         // is associated with the only non-fictitious halfedge incident to
         // this vertex. We therefore return a pointer to this halfedge.
-        Halfedge     *he = curr->opposite()->next();
-
-        CGAL_assertion (! he->has_null_curve());
-        return (CGAL::make_object (he));
+        Halfedge* he = curr->opposite()->next();
+        CGAL_assertion(! he->has_null_curve());
+        return Result(he);
       }
-      else if (eq_target)
-      {
+      else if (eq_target) {
         // cv's end coincides with the target vertex of the current
         // fictitious halfedge. This means that cv overlaps the curve that
         // is associated with the only non-fictitious halfedge incident to
         // this vertex. We therefore return a pointer to this halfedge.
-        Halfedge     *he = curr->opposite()->prev();
-
-        CGAL_assertion (! he->has_null_curve());
-        return (CGAL::make_object (he));
+        Halfedge* he = curr->opposite()->prev();
+        CGAL_assertion(! he->has_null_curve());
+        return Result(he);
       }
 
       // The current ficitious edge contains cv's end in its interior.
       // Note we use curr's twin, whose incident face is a valid
       // unbounded face (whereas the incident face of curr is the fictitious
       // face).
-      Face      *uf = curr->opposite()->outer_ccb()->face();
-
+      Face* uf = curr->opposite()->outer_ccb()->face();
       CGAL_assertion (uf->is_unbounded() && ! uf->is_fictitious());
-      return (CGAL::make_object (uf));
+      return Result(uf);
     }
 
     curr = curr->next();
@@ -388,7 +391,8 @@ locate_curve_end (const X_monotone_curve_2& cv, Arr_curve_end ind,
 
   // We should never reach here.
   CGAL_error();
-  return Object();
+  Vertex* v(nullptr);
+  return Result(v);
 }
 
 //-----------------------------------------------------------------------------

Arrangement_on_surface_2/include/CGAL/Arr_topology_traits/Arr_unb_planar_vert_decomp_helper.h

@@ -7,9 +7,8 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-//
-// Author(s)     : Ron Wein   <wein@post.tau.ac.il>
-//                 Efi Fogel  <efif@post.tau.ac.il>
+// Author(s): Ron Wein   <wein@post.tau.ac.il>
+//            Efi Fogel  <efif@post.tau.ac.il>
 
 #ifndef CGAL_ARR_UNB_PLANAR_VERT_DECOMP_HELPER_H
 #define CGAL_ARR_UNB_PLANAR_VERT_DECOMP_HELPER_H
@@ -43,12 +42,16 @@ private:
   typedef Geometry_traits_2                             Gt2;
 
 public:
+  typedef typename Arrangement_2::Vertex_const_handle   Vertex_const_handle;
+  typedef typename Arrangement_2::Halfedge_const_handle Halfedge_const_handle;
   typedef typename Arrangement_2::Face_const_handle     Face_const_handle;
 
+  typedef boost::variant<Vertex_const_handle, Halfedge_const_handle,
+                         Face_const_handle>             Cell_type;
+  typedef boost::optional<Cell_type>                    Vert_type;
+
 protected:
   typedef typename Arrangement_2::Topology_traits       Topology_traits;
-  typedef typename Arrangement_2::Halfedge_const_handle Halfedge_const_handle;
-  typedef typename Arrangement_2::Vertex_const_handle   Vertex_const_handle;
 
   // Data members:
   const Topology_traits* m_top_traits;  // The topology-traits class.
@@ -77,10 +80,10 @@ public:
   //@}
 
   /*! Get the current top object. */
-  CGAL::Object top_object() const { return CGAL::make_object(m_top_fict); }
+  Vert_type top_object() const { return Vert_type(m_top_fict); }
 
   /*! Get the current bottom object. */
-  CGAL::Object bottom_object() const { return CGAL::make_object(m_bottom_fict); }
+  Vert_type bottom_object() const { return Vert_type(m_bottom_fict); }
 };
 
 //-----------------------------------------------------------------------------

Arrangement_on_surface_2/include/CGAL/Arr_tracing_traits_2.h

@@ -504,7 +504,12 @@ public:
     }
   };
 
-  /*! A functor that divides a curve into x-monotone curves. */
+  //! \name Intersections & subdivisions
+  //@{
+
+  /*! \class Make_x_monotone_2
+   * A functor for subdividing curves into x-monotone curves.
+   */
   class Make_x_monotone_2 {
   private:
     typename Base::Make_x_monotone_2 m_object;
@@ -515,39 +520,43 @@ public:
     Make_x_monotone_2(const Base * base, bool enabled = true) :
       m_object(base->make_x_monotone_2_object()), m_enabled(enabled) {}
 
-    /*! Operate
-     * \param cv the curve
-     * \param oi an output iterator that contains the result. It's value
-     * type is CGAL::Object, which wraps either an x-monotone curve or a point
-     * \return the output iterator
+    /*! Subdivide a given curve into x-monotone subcurves and insert them into
+     * a given output iterator.
+     * \param cv the curve.
+     * \param oi an output iterator for the result. Its value type is a variant
+     *           that wraps Point_2 or X_monotone_curve_2 objects.
+     * \return the output iterator.
      */
-    template<typename OutputIterator>
+    template <typename OutputIterator>
     OutputIterator operator()(const Curve_2 & cv, OutputIterator oi) const
     {
-      if (!m_enabled) return m_object(cv, oi);
+      if (! m_enabled) return m_object(cv, oi);
       std::cout << "make_x_monotone" << std::endl
                 << "  cv: " << cv << std::endl;
-      std::list<CGAL::Object> container;
+
+      typedef boost::variant<Point_2, X_monotone_curve_2>
+        Make_x_monotone_result;
+
+      std::list<Make_x_monotone_result> container;
       m_object(cv, std::back_inserter(container));
       if (container.empty()) return oi;
 
-      std::list<CGAL::Object>::iterator it;
-      unsigned int i = 0;
-      for (it = container.begin(); it != container.end(); ++it) {
-        X_monotone_curve_2 xcv;
-        if (assign (xcv, *it)) {
-          std::cout << "  result[" << i++ << "]: xcv: " << xcv << std::endl;
+      size_t i = 0;
+      for (auto it = container.begin(); it != container.end(); ++it) {
+        if (const auto* xcv = boost::get<X_monotone_curve_2>(*it)) {
+          std::cout << "  result[" << i++ << "]: xcv: " << *xcv << std::endl;
           continue;
         }
 
-        Point_2 p;
-        if (assign (p, *it)) {
-          std::cout << "  result[" << i++ << "]: p: " << p << std::endl;
+        if (const Point_2* p = boost::get<Point_2>(*it)) {
+          std::cout << "  result[" << i++ << "]: p: " << *p << std::endl;
           continue;
         }
+
+        CGAL_error();
       }
 
-      for (it = container.begin(); it != container.end(); ++it) *oi++ = *it;
+      for (auto it = container.begin(); it != container.end(); ++it) *oi++ = *it;
       container.clear();
       return oi;
     }
@@ -597,14 +606,14 @@ public:
     Intersect_2(const Base* base, bool enabled = true) :
       m_object(base->intersect_2_object()), m_enabled(enabled) {}
 
-    /*! Operate
+    /*! Compute the intersections of the two given curves and insert them into
+     * a given output iterator.
      * \param xcv1 the first curve
      * \param xcv2 the ssecond curve
-     * \param oi an output iterator that contains the result. It's value
-     * type is CGAL::Object, which wraps either an x-monotone overlapping
-     * curve or pair that consists of an intersection point and its
-     * multiplicity
-     * \return the output iterator
+     * \param oi the output iterator for the result. It value type is a variant
+     *           that wraps an x-monotone overlapping curve or a pair that
+     *           consists of the intersection point and its multiplicity
+     * \return the past-the-end output iterator.
      */
     template <typename OutputIterator>
     OutputIterator operator()(const X_monotone_curve_2 & xcv1,

Arrangement_on_surface_2/include/CGAL/Arr_unb_planar_topology_traits_2.h

@@ -7,9 +7,8 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-//
-// Author(s)     : Ron Wein   <wein@post.tau.ac.il>
-//                 Efi Fogel  <efif@post.tau.ac.il>
+// Author(s): Ron Wein   <wein@post.tau.ac.il>
+//            Efi Fogel  <efif@post.tau.ac.il>
 
 #ifndef CGAL_ARR_UNB_PLANAR_TOPOLOGY_TRAITS_2_H
 #define CGAL_ARR_UNB_PLANAR_TOPOLOGY_TRAITS_2_H
@@ -308,11 +307,12 @@ public:
    * \return An object that contains the curve end.
    *         In our case this object always wraps a fictitious edge.
    */
-  CGAL::Object place_boundary_vertex(Face* f,
-                                     const X_monotone_curve_2& cv,
-                                     Arr_curve_end ind,
-                                     Arr_parameter_space ps_x,
-                                     Arr_parameter_space ps_y);
+  boost::optional<boost::variant<Vertex*, Halfedge*> >
+  place_boundary_vertex(Face* f,
+                        const X_monotone_curve_2& cv,
+                        Arr_curve_end ind,
+                        Arr_parameter_space ps_x,
+                        Arr_parameter_space ps_y);
 
   /*! Locate the predecessor halfedge for the given curve around a given
    * vertex with boundary conditions.
@@ -346,10 +346,11 @@ public:
    *         In our case this object may either wrap an unbounded face,
    *         or an edge with an end-vertex at infinity (in case of an overlap).
    */
-  CGAL::Object locate_curve_end(const X_monotone_curve_2& cv,
-                                Arr_curve_end ind,
-                                Arr_parameter_space ps_x,
-                                Arr_parameter_space ps_y);
+  boost::variant<Vertex*, Halfedge*, Face*>
+  locate_curve_end(const X_monotone_curve_2& cv,
+                   Arr_curve_end ind,
+                   Arr_parameter_space ps_x,
+                   Arr_parameter_space ps_y);
 
   /*! Split a fictitious edge using the given vertex.
    * \param e The edge to split (one of the pair of halfedges).

Arrangement_on_surface_2/include/CGAL/Arr_vertical_decomposition_2.h

@@ -7,7 +7,7 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-// Author(s)     : Ron Wein <wein@post.tau.ac.il>
+// Author(s): Ron Wein <wein@post.tau.ac.il>
 
 #ifndef CGAL_ARR_VERTICAL_DECOMPOSITION_2_H
 #define CGAL_ARR_VERTICAL_DECOMPOSITION_2_H
@@ -34,14 +34,13 @@ namespace Ss2 = Surface_sweep_2;
 /*! Perform a vertical decomposition of an arrangement, by performing a
  * "batched vertical ray-shooting" query from all arrangement vertices.
  * \param arr The arrangement.
- * \param oi Output: An output iterator of the vertices, each paired with
- *                   a pair of arrangement features that lie below and above
- *                   it, respectively.
- *                   The vertices are sorted by increasing xy-order.
+ * \param oi An output iterator of the vertices, each paired with a pair of
+ *           arrangement features that lie below and above it, respectively.
+ *           The vertices are sorted by increasing xy-order.
+ *           The OutputIterator dereferences the type \c
+ *           pair<Vertex_const_handle, pair<Vert_type, Vert_type> >, where
+ *           \c Vert_type is an optional handle to an arrangement feature.
  * \return A past-the-end iterator for the ordered arrangement vertices.
- * \pre The value-type of OutputIterator is
- *      pair<Vertex_const_handle, pair<Object, Object> >, where
- *      the Object represents a handle to an arrangement feature.
  */
 template <typename GeometryTraits_2, typename TopologyTraits,
           typename OutputIterator>

Arrangement_on_surface_2/include/CGAL/Arrangement_2/Arr_compute_zone_visitor.h

@@ -8,27 +8,29 @@
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
 //
-// Author(s)     : Ophir Setter      <ophirset@post.tau.ac.il>
+// Author(s) : Ophir Setter      <ophirset@post.tau.ac.il>
+//             Efi Fogel         <efifogel@gmail.com>
 //
 #ifndef CGAL_ARR_COMPUTE_ZONE_VISITOR_H
 #define CGAL_ARR_COMPUTE_ZONE_VISITOR_H
 
 #include <CGAL/license/Arrangement_on_surface_2.h>
 
-
 /*! \file
  * Definition of the Arr_compute_zone_visitor class.
  */
 
+#include <boost/variant.hpp>
+
 namespace CGAL {
 
 /*! \class
- * A visitor class for Arrangement_zone_2, which outputs the
- * zone of an x-monotone curve. Meaning, it output the arrangment's
- * vertices, edges and faces that the x-monotone curve intersects.
+ * A visitor class for Arrangement_zone_2 that outputs the zone of an
+ * x-monotone curve. Specifically, it outputs handles to the the arrangment
+ * cells that the x-monotone curve intersects.
  * The class should be templated by an Arrangement_2 class, and by an
- * output iterator of CGAL Objects, where we store all arrangement
- * features the x-monotone curve intersects.
+ * output iterator of a variant of types of handles to the arrangement cells
+ * that appear in the zone, namely, vertex, halfedge, and face handles.
  */
 template <class Arrangement_, class OutputIterator_>
 class Arr_compute_zone_visitor
@@ -53,7 +55,7 @@ private:
   const Vertex_handle        invalid_v;    // Invalid vertex.
 
   OutputIterator&            out_iter;     // for outputing the zone objects.
-                                           // Its value type is CGAL::Object.
+                                           // Its value type is boost::variant.
   bool                       output_left;  // Determines wheter we should
                                            // output the left end point of a
                                            // subcurve (to avoid outputing
@@ -65,8 +67,8 @@ public:
   Arr_compute_zone_visitor (OutputIterator& oi) :
     invalid_he(),
     invalid_v(),
-    out_iter (oi),
-    output_left (true)
+    out_iter(oi),
+    output_left(true)
   {}
 
   /*! Initialize the visitor. */
@@ -90,50 +92,35 @@ public:
    * \return A handle to the halfedge obtained from the insertion of the
    *         subcurve into the arrangement.
    */
-  Result found_subcurve (const X_monotone_curve_2&,
-                         Face_handle face,
-                         Vertex_handle left_v, Halfedge_handle left_he,
-                         Vertex_handle right_v, Halfedge_handle right_he)
+  Result found_subcurve(const X_monotone_curve_2&,
+                        Face_handle face,
+                        Vertex_handle left_v, Halfedge_handle left_he,
+                        Vertex_handle right_v, Halfedge_handle right_he)
   {
-    if (output_left)
-    {
+    typedef boost::variant<Vertex_handle, Halfedge_handle, Face_handle>
+                                                                Zone_result;
+
+    if (output_left) {
       // Only the first subcurve should output the arrangement feature incident
       // to its left endpoint. This way we avoid reporting the same feature
       // twice.
-      if (left_v != invalid_v)
-      {
-        *out_iter = CGAL::make_object (left_v);
-        ++out_iter;
-      }
-      else if (left_he != invalid_he)
-      {
-        *out_iter = CGAL::make_object (left_he);
-        ++out_iter;
-      }
+      if (left_v != invalid_v) *out_iter++ = Zone_result(left_v);
+      else if (left_he != invalid_he) *out_iter++ = Zone_result(left_he);
 
       output_left = false;
     }
 
     // Report the face that contains the interior of the subcurve.
-    *out_iter = CGAL::make_object (face);
-    ++out_iter;
+    *out_iter++ = Zone_result(face);
 
     // If the right endpoint of the subcurve is incident to an arrangement
     // vertex or an arrangement edge, report this feature.
-    if (right_v != invalid_v)
-    {
-      *out_iter = CGAL::make_object(right_v);
-      ++out_iter;
-    }
-    else if (right_he != invalid_he)
-    {
-      *out_iter = CGAL::make_object(right_he);
-      ++out_iter;
-    }
+    if (right_v != invalid_v) *out_iter++ = Zone_result(right_v);
+    else if (right_he != invalid_he) *out_iter++ = Zone_result(right_he);
 
     // We did not modify the arrangement, so we return an invalid handle
     // and a flag indicating that the zone-computation process should continue.
-    return (Result (invalid_he, false));
+    return Result(invalid_he, false);
   }
 
   /*!
@@ -151,35 +138,28 @@ public:
                         Halfedge_handle he,
                         Vertex_handle left_v, Vertex_handle right_v)
   {
-    if (output_left)
-    {
+    typedef boost::variant<Vertex_handle, Halfedge_handle, Face_handle>
+                                                                Zone_result;
+
+    if (output_left) {
       // Only the first subcurve should output the arrangement feature incident
       // to its left endpoint. This way we avoid reporting the same feature
       // twice.
-      if (left_v != invalid_v)
-      {
-        *out_iter = CGAL::make_object (left_v);
-        ++out_iter;
-      }
+      if (left_v != invalid_v) *out_iter++ = Zone_result(left_v);
 
       output_left = false;
     }
 
     // Report the arrangement edge the curve currently overlaps.
-    *out_iter = CGAL::make_object(he);
-    ++out_iter;
+    *out_iter++ = Zone_result(he);
 
     // If the right endpoint of the overlapping subcurve is incident to an
     // arrangement vertex, report this vertex as well.
-    if (right_v != invalid_v)
-    {
-      *out_iter = CGAL::make_object (right_v);
-      ++out_iter;
-    }
+    if (right_v != invalid_v) *out_iter++ = Zone_result(right_v);
 
     // We did not modify the arrangement, so we return an invalid handle
     // and a flag indicating that the zone-computation process should continue.
-    return (Result (invalid_he, false));
+    return Result(invalid_he, false);
   }
 };
 

Arrangement_on_surface_2/include/CGAL/Arrangement_2/Arr_traits_adaptor_2.h

@@ -8,7 +8,6 @@
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 // $Date$
 //
-//
 // Author(s): Ron Wein             <wein@post.tau.ac.il>s
 //            Efi Fogel            <efif@post.tau.ac.il>
 //            Eric Berberich       <eric@mpi-inf.mpg.de>
@@ -24,7 +23,6 @@
 
 #include <CGAL/license/Arrangement_on_surface_2.h>
 
-
 /*! \file
  * Definitions of the adaptor classes for the arrangement traits class.
  */
@@ -50,6 +48,7 @@ public:
   typedef Arr_traits_basic_adaptor_2<Base>          Self;
   typedef typename Base::X_monotone_curve_2         X_monotone_curve_2;
   typedef typename Base::Point_2                    Point_2;
+  typedef typename Base::Multiplicity               Multiplicity;
 
   // Categories
   typedef typename Base::Has_left_category          Has_left_category;
@@ -320,9 +319,12 @@ public:
                            const X_monotone_curve_2& xcv2,
                            Tag_false) const
     {
-      std::list<CGAL::Object> intersections;
+      typedef std::pair<Point_2, Multiplicity>          Intersection_point;
+      typedef boost::variant<Intersection_point, X_monotone_curve_2>
+                                                        Intersection_result;
+      std::list<Intersection_result> intersections;
       m_self->intersect_2_object()(xcv1, xcv2, back_inserter(intersections));
-      return !intersections.empty();
+      return ! intersections.empty();
     }
   };
 
@@ -665,7 +667,7 @@ public:
       return parameter_space_in_y(xcv, ind, Psy_2_curve_end_tag());
     }
 
-      /*!
+    /*!
      * Obtain the location of the given curve end in y.
      * \param xcv The curve.
      * \return The location of the curve end in y direction.
@@ -2276,6 +2278,7 @@ public:
   typedef typename Base_traits_2::Curve_2              Curve_2;
   typedef typename Base::X_monotone_curve_2            X_monotone_curve_2;
   typedef typename Base::Point_2                       Point_2;
+  typedef typename Base::Multiplicity                  Multiplicity;
 
   // Categories.
   typedef typename Base::Has_left_category             Has_left_category;
@@ -2324,6 +2327,10 @@ public:
    * endpoint, then by the graphs, and finally by their right-most endpoint.
    */
   class Compare_xy_2 {
+      typedef std::pair<Point_2, Multiplicity>          Intersection_point;
+      typedef boost::variant<Intersection_point, X_monotone_curve_2>
+                                                        Intersection_result;
+
   public:
     /*! Compare two points lexigoraphically: by x, then by y.
      * \param p1 the first point.
@@ -2360,7 +2367,7 @@ public:
     Comparison_result operator()(const X_monotone_curve_2& c1,
                                  const X_monotone_curve_2& c2) const
     {
-      std::list<CGAL::Object> intersections;
+      std::list<Intersection_result> intersections;
       return operator()(c1, c2, intersections,
                         Are_all_sides_oblivious_category());
     }
@@ -2586,7 +2593,7 @@ public:
     Comparison_result
     compare_remainder(const X_monotone_curve_2& c1,
                       const X_monotone_curve_2& c2,
-                      std::list<CGAL::Object>& intersections) const
+                      std::list<Intersection_result>& intersections) const
     {
       // Right-most sections are equal.
       // Advance to the next respective sections:
@@ -2597,9 +2604,9 @@ public:
       }
       // Verify the first intersection is an overlap, remove it, and
       // recursively call.
-      CGAL::Object first = intersections.front();
-      X_monotone_curve_2 xcv;
-      if (!assign(xcv, first)) {
+      const X_monotone_curve_2* xcv =
+        boost::get<X_monotone_curve_2>(&(intersections.front()));
+      if (! xcv) {
         CGAL_error_msg("The first intersection is not an overlap!");
         return SMALLER;
       }
@@ -2611,10 +2618,9 @@ public:
       typedef typename Self::Split_2        Split_2;
       Split_2 split = m_self.split_2_object();
       X_monotone_curve_2 c11, c12, c21, c22;
-      Construct_max_vertex_2 ctr_max =
-        m_self.construct_max_vertex_2_object();
-      const Point_2& p1 = ctr_max(xcv);
-      const Point_2& p2 = ctr_max(xcv);
+      Construct_max_vertex_2 ctr_max = m_self.construct_max_vertex_2_object();
+      const Point_2& p1 = ctr_max(*xcv);
+      const Point_2& p2 = ctr_max(*xcv);
       split(c1, p1, c11, c12);
       split(c2, p2, c21, c22);
       return operator()(c12, c22, intersections,
@@ -2625,7 +2631,7 @@ public:
      */
     Comparison_result operator()(const X_monotone_curve_2& c1,
                                  const X_monotone_curve_2& c2,
-                                 std::list<CGAL::Object>& intersections,
+                                 std::list<Intersection_result>& intersections,
                                  Arr_all_sides_oblivious_tag) const
     {
       const Point_2& c1_min = m_self.construct_min_vertex_2_object()(c1);
@@ -2646,7 +2652,7 @@ public:
      */
     Comparison_result operator()(const X_monotone_curve_2& c1,
                                  const X_monotone_curve_2& c2,
-                                 std::list<CGAL::Object>& intersections,
+                                 std::list<Intersection_result>& intersections,
                                  Arr_not_all_sides_oblivious_tag) const
     {
       typedef typename Base::Parameter_space_in_x_2     Parameter_space_in_x_2;

Arrangement_on_surface_2/include/CGAL/Arrangement_2/Arrangement_on_surface_2_global.h

@@ -8,9 +8,9 @@
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
 //
-// Author(s)     : Ron Wein          <wein@post.tau.ac.il>
-//                 Baruch Zukerman   <baruchzu@post.tau.ac.il>
-//                 Efi Fogel         <efif@post.tau.ac.il>
+// Author(s): Ron Wein          <wein@post.tau.ac.il>
+//            Baruch Zukerman   <baruchzu@post.tau.ac.il>
+//            Efi Fogel         <efif@post.tau.ac.il>
 //
 #ifndef CGAL_ARRANGEMENT_ON_SURFACE_2_GLOBAL_H
 #define CGAL_ARRANGEMENT_ON_SURFACE_2_GLOBAL_H
@@ -32,6 +32,7 @@
 #include <CGAL/Arrangement_2/Arr_compute_zone_visitor.h>
 #include <CGAL/Arrangement_2/Arr_do_intersect_zone_visitor.h>
 #include <CGAL/Arrangement_2/Arr_traits_adaptor_2.h>
+#include <CGAL/Arr_point_location_result.h>
 #include <CGAL/No_intersection_surface_sweep_2.h>
 #include <CGAL/Surface_sweep_2/Arr_insertion_ss_visitor.h>
 #include <CGAL/Surface_sweep_2/Arr_no_intersection_insertion_ss_visitor.h>
@@ -80,6 +81,10 @@ void insert(Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr,
   typedef Arrangement_on_surface_2<Gt2, Tt>             Arr;
   typedef ZoneVisitor                                   Zone_visitor;
 
+  typedef typename Gt2::Point_2                         Point_2;
+  typedef typename Gt2::X_monotone_curve_2              X_monotone_curve_2;
+  typedef boost::variant<Point_2, X_monotone_curve_2>   Make_x_monotone_result;
+
   // Obtain an arrangement accessor.
   Arr_accessor<Arr> arr_access(arr);
 
@@ -88,19 +93,14 @@ void insert(Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr,
   Arrangement_zone_2<Arr, Zone_visitor> arr_zone(arr, &visitor);
 
   // Break the input curve into x-monotone subcurves and isolated points.
-  std::list<CGAL::Object> x_objects;
-  std::list<CGAL::Object>::const_iterator obj_iter;
-  const typename Gt2::X_monotone_curve_2* x_curve;
-  const typename Gt2::Point_2* iso_p;
-
-  arr.geometry_traits()->
-    make_x_monotone_2_object()(c, std::back_inserter(x_objects));
+  std::list<Make_x_monotone_result> x_objects;
+  const auto* traits = arr.geometry_traits();
+  traits->make_x_monotone_2_object()(c, std::back_inserter(x_objects));
 
   // Insert each x-monotone curve into the arrangement.
-  for (obj_iter = x_objects.begin(); obj_iter != x_objects.end(); ++obj_iter) {
+  for (const auto& x_obj : x_objects) {
     // Act according to the type of the current object.
-    x_curve = object_cast<typename Gt2::X_monotone_curve_2>(&(*obj_iter));
-
+    const auto* x_curve = boost::get<X_monotone_curve_2>(&x_obj);
     if (x_curve != nullptr) {
       // Inserting an x-monotone curve:
       // Initialize the zone-computation object with the given curve.
@@ -116,14 +116,13 @@ void insert(Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr,
       // Notify the arrangement observers that the global operation has been
       // completed.
       arr_access.notify_after_global_change();
+      continue;
     }
-    else {
-      iso_p = object_cast<typename Gt2::Point_2>(&(*obj_iter));
-      CGAL_assertion(iso_p != nullptr);
+    const auto* iso_p = boost::get<Point_2>(&x_obj);
+    CGAL_assertion(iso_p != nullptr);
 
-      // Inserting a point into the arrangement:
-      insert_point(arr, *iso_p, pl);
-    }
+    // Inserting a point into the arrangement:
+    insert_point(arr, *iso_p, pl);
   }
 }
 
@@ -529,7 +528,9 @@ void insert(Arrangement_on_surface_2<GeometryTraits_2,TopologyTraits>& arr,
 template <typename GeometryTraits_2, typename TopologyTraits>
 void insert(Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr,
             const typename GeometryTraits_2::X_monotone_curve_2& c,
-            const Object& obj)
+            typename Arr_point_location_result<
+              Arrangement_on_surface_2<GeometryTraits_2,
+                                       TopologyTraits> >::type obj)
 {
   typedef GeometryTraits_2                              Gt2;
   typedef TopologyTraits                                Tt;
@@ -597,7 +598,8 @@ template <typename GeometryTraits_2, typename TopologyTraits>
 CGAL_DEPRECATED void insert_x_monotone_curve
 (Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr,
  const typename GeometryTraits_2::X_monotone_curve_2& c,
- const Object& obj)
+ typename Arr_point_location_result<
+   Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits> >::type obj)
 {
   insert(arr, c, obj);
 }
@@ -658,6 +660,7 @@ insert_non_intersecting_curve
                                                         Traits_adaptor_2;
   typedef typename Arr::Vertex_const_handle             Vertex_const_handle;
   typedef typename Arr::Halfedge_const_handle           Halfedge_const_handle;
+  typedef typename Arr::Face_const_handle               Face_const_handle;
   CGAL_USE_TYPE(Halfedge_const_handle);
 
   const Traits_adaptor_2* geom_traits =
@@ -666,13 +669,13 @@ insert_non_intersecting_curve
 
   // Check whether the left end has boundary conditions, and locate it in the
   // arrangement accordingly.
-  const Arr_parameter_space bx1 =
-    geom_traits->parameter_space_in_x_2_object()(c, ARR_MIN_END);
-  const Arr_parameter_space by1 =
-    geom_traits->parameter_space_in_y_2_object()(c, ARR_MIN_END);
-  CGAL::Object obj1;
+  auto bx1 = geom_traits->parameter_space_in_x_2_object()(c, ARR_MIN_END);
+  auto by1 = geom_traits->parameter_space_in_y_2_object()(c, ARR_MIN_END);
   const Vertex_const_handle* vh1 = nullptr;
 
+  typedef Arr_point_location_result<Arr>        Pl_result;
+
+  typename Pl_result::type obj1;
   if ((bx1 == ARR_INTERIOR) && (by1 == ARR_INTERIOR)) {
     // We have a normal left endpoint with no boundary conditions:
     // use a point-location query.
@@ -680,33 +683,26 @@ insert_non_intersecting_curve
 
     // The endpoint must not lie on an existing edge, but may coincide with
     // and existing vertex vh1.
-    CGAL_precondition_msg
-      (object_cast<Halfedge_const_handle>(&obj1) == nullptr,
-       "The curve must not intersect an existing edge.");
+    CGAL_precondition_msg(boost::get<Halfedge_const_handle>(&obj1) == nullptr,
+                          "The curve must not intersect an existing edge.");
 
-    vh1 = object_cast<Vertex_const_handle>(&obj1);
   }
   else {
     // We have a left end with boundary conditions. Use the accessor to locate
     // the feature that contains it.
     obj1 = arr_access.locate_curve_end(c, ARR_MIN_END, bx1, by1);
-
-    CGAL_precondition_msg
-      (object_cast<Halfedge_const_handle>(&obj1) == nullptr,
-       "The curve must not overlap an existing edge.");
-
-    vh1 = object_cast<Vertex_const_handle>(&obj1);
+    CGAL_precondition_msg(boost::get<Halfedge_const_handle>(&obj1) == nullptr,
+                          "The curve must not overlap an existing edge.");
   }
+  vh1 = Pl_result::template assign<Vertex_const_handle>(&obj1);
 
   // Check whether the right end has boundary conditions, and locate it in the
   // arrangement accordingly.
-  const Arr_parameter_space  bx2 =
-    geom_traits->parameter_space_in_x_2_object()(c, ARR_MAX_END);
-  const Arr_parameter_space  by2 =
-    geom_traits->parameter_space_in_y_2_object()(c, ARR_MAX_END);
-  CGAL::Object obj2;
+  auto bx2 = geom_traits->parameter_space_in_x_2_object()(c, ARR_MAX_END);
+  auto by2 = geom_traits->parameter_space_in_y_2_object()(c, ARR_MAX_END);
   const Vertex_const_handle* vh2 = nullptr;
 
+  typename Pl_result::type obj2;
   if ((bx2 == ARR_INTERIOR) && (by2 == ARR_INTERIOR)) {
     // We have a normal right endpoint with no boundary conditions:
     // use a point-location query.
@@ -714,11 +710,8 @@ insert_non_intersecting_curve
 
     // The endpoint must not lie on an existing edge, but may coincide with
     // and existing vertex vh2.
-    CGAL_precondition_msg
-      (object_cast<Halfedge_const_handle>(&obj2) == nullptr,
-       "The curve must not intersect an existing edge.");
-
-    vh2 = object_cast<Vertex_const_handle>(&obj2);
+    CGAL_precondition_msg(boost::get<Halfedge_const_handle>(&obj2) == nullptr,
+                          "The curve must not intersect an existing edge.");
   }
   else {
     // We have a right end with boundary conditions. Use the accessor to locate
@@ -728,13 +721,10 @@ insert_non_intersecting_curve
     //           << ", by2: " << by2
     //           << std::endl;
     obj2 = arr_access.locate_curve_end(c, ARR_MAX_END, bx2, by2);
-
-    CGAL_precondition_msg
-      (object_cast<Halfedge_const_handle>(&obj2) == nullptr,
-       "The curve must not overlap an existing edge.");
-
-    vh2 = object_cast<Vertex_const_handle>(&obj2);
+    CGAL_precondition_msg(boost::get<Halfedge_const_handle>(&obj2) == nullptr,
+                          "The curve must not overlap an existing edge.");
   }
+  vh2 = Pl_result::template assign<Vertex_const_handle>(&obj2);
 
   // Notify the arrangement observers that a global operation is about to
   // take place.
@@ -776,10 +766,8 @@ insert_non_intersecting_curve
       // we must insert the curve in the interior of a face.
       // In this case insert_in_face_interior() already returns a halfedge
       // directed from left to right.
-      const typename Arr::Face_const_handle* fh1 =
-        object_cast<typename Arr::Face_const_handle>(&obj1);
-      const typename Arr::Face_const_handle* fh2 =
-        object_cast<typename Arr::Face_const_handle>(&obj2);
+      const Face_const_handle* fh1 = boost::get<Face_const_handle>(&obj1);
+      const Face_const_handle* fh2 = boost::get<Face_const_handle>(&obj2);
 
       // std::cout << arr << std::endl;
       // std::cout << "(*fh1)->number_of_outer_ccbs(): "
@@ -1140,49 +1128,49 @@ insert_point(Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr,
 
   typedef Arrangement_on_surface_2<Gt2, Tt>             Arr;
 
+  typedef typename Arr::Vertex_const_handle             Vertex_const_handle;
+  typedef typename Arr::Halfedge_const_handle           Halfedge_const_handle;
+  typedef typename Arr::Face_const_handle               Face_const_handle;
+
   // Act according to the type of arrangement feature that contains the point.
-  const typename Arr::Face_const_handle* fh;
-  const typename Arr::Halfedge_const_handle* hh;
-  const typename Arr::Vertex_const_handle* vh;
   typename Arr::Vertex_handle vh_for_p;
 
   // Locate the given point in the arrangement.
-  CGAL::Object obj = pl.locate (p);
+  auto obj = pl.locate(p);
 
   // Notify the arrangement observers that a global operation is about to
   // take place.
-  Arr_accessor<Arr> arr_access (arr);
+  Arr_accessor<Arr> arr_access(arr);
 
   arr_access.notify_before_global_change();
 
-  if ((fh = object_cast<typename Arr::Face_const_handle>(&obj)) != nullptr) {
+  const Face_const_handle* fh = boost::get<Face_const_handle>(&obj);
+  if (fh != nullptr) {
     // p lies inside a face: Insert it as an isolated vertex it the interior of
     // this face.
     vh_for_p = arr.insert_in_face_interior(p, arr.non_const_handle (*fh));
   }
-  else if ((hh = object_cast<typename Arr::Halfedge_const_handle>(&obj)) !=
-           nullptr)
-  {
-    // p lies in the interior of an edge: Split this edge to create a new
-    // vertex associated with p.
-    typename Gt2::X_monotone_curve_2 sub_cv1, sub_cv2;
-    typename Arr::Halfedge_handle split_he;
-
-    arr.geometry_traits()->split_2_object()((*hh)->curve(), p,
-                                            sub_cv1, sub_cv2);
-
-    split_he = arr.split_edge(arr.non_const_handle(*hh), sub_cv1, sub_cv2);
-
-    // The new vertex is the target of the returned halfedge.
-    vh_for_p = split_he->target();
-  }
   else {
-    // In this case p lies on an existing vertex, so we just update this
-    // vertex.
-    vh = object_cast<typename Arr::Vertex_const_handle>(&obj);
-    CGAL_assertion (vh != nullptr);
+    const Halfedge_const_handle* hh = boost::get<Halfedge_const_handle>(&obj);
+    if (hh != nullptr) {
+      // p lies in the interior of an edge: Split this edge to create a new
+      // vertex associated with p.
+      typename Gt2::X_monotone_curve_2 sub_cv1, sub_cv2;
+      typename Arr::Halfedge_handle split_he;
 
-    vh_for_p = arr.modify_vertex (arr.non_const_handle (*vh), p);
+      const auto* gt = arr.geometry_traits();
+      gt->split_2_object()((*hh)->curve(), p, sub_cv1, sub_cv2);
+      split_he = arr.split_edge(arr.non_const_handle(*hh), sub_cv1, sub_cv2);
+
+      // The new vertex is the target of the returned halfedge.
+      vh_for_p = split_he->target();
+    }
+    else {
+      // p lies on an existing vertex, so we just update this vertex.
+      const Vertex_const_handle* vh = boost::get<Vertex_const_handle>(&obj);
+      CGAL_assertion(vh != nullptr);
+      vh_for_p = arr.modify_vertex (arr.non_const_handle (*vh), p);
+    }
   }
 
   // Notify the arrangement observers that the global operation has been
@@ -1381,60 +1369,47 @@ is_valid(const Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr)
 
   // Shoot a vertical ray from each vertex we have collected downward, and
   // check that this vertex is really contained in the proper face.
-  typename Gt2::Compare_y_at_x_right_2 comp_y_at_x_right =
-    traits->compare_y_at_x_right_2_object();
-  typename Gt2::Compare_y_at_x_left_2 comp_y_at_x_left =
-    traits->compare_y_at_x_left_2_object();
+  auto comp_y_at_x_right = traits->compare_y_at_x_right_2_object();
+  auto comp_y_at_x_left = traits->compare_y_at_x_left_2_object();
 
-  typename std::list<std::pair<Vertex_const_handle,
-                               Face_const_handle> >::iterator    vf_iter;
   typename Tt::Default_point_location_strategy def_pl(arr);
-  Vertex_const_handle curr_v;
-  Object obj;
-  Halfedge_const_handle he_below;
-  Vertex_const_handle v_below;
-  Face_const_handle in_face;
-  Halfedge_around_vertex_const_circulator first, circ;
-  bool assign_ok;
   const Halfedge_const_handle invalid_he;
 
-  for (vf_iter = vf_list.begin(); vf_iter != vf_list.end(); ++vf_iter) {
+  Face_const_handle in_face;
+  for (auto vf_iter = vf_list.begin(); vf_iter != vf_list.end(); ++vf_iter) {
     // Perform ray-shooting from the current vertex.
-    curr_v = vf_iter->first;
-    obj = def_pl.ray_shoot_down(curr_v->point());
+    Vertex_const_handle curr_v = vf_iter->first;
+    auto obj = def_pl.ray_shoot_down(curr_v->point());
 
-    if (CGAL::assign(he_below, obj)) {
-      // Hit an edge - take the incident face of the halfedge directed to the
+    // if (CGAL::assign(he_below, obj)) {
+    if (auto* he_below_p = boost::get<Halfedge_const_handle>(&obj)) {
+      // Hit an edge; take the incident face of the halfedge directed to the
       // right.
-      if (he_below->direction() == ARR_RIGHT_TO_LEFT)
-        he_below = he_below->twin();
-
-      in_face = he_below->face();
+      auto he_below = *he_below_p;
+      in_face = (he_below->direction() == ARR_RIGHT_TO_LEFT) ?
+        he_below->twin()->face() : he_below->face();
     }
-    else if (CGAL::assign(v_below, obj)) {
+    else if (auto* v_below_p = boost::get<Vertex_const_handle>(&obj)) {
+      auto v_below = *v_below_p;
       // Hit a vertex.
       if (v_below->is_isolated()) in_face = v_below->face();
       else {
         // Get the first halfedge around v_below that is directed from left to
         // right and the first halfedge that is directed from right to left.
-        first = circ = v_below->incident_halfedges();
+        Halfedge_around_vertex_const_circulator circ =
+          v_below->incident_halfedges();
+        Halfedge_around_vertex_const_circulator first = circ;
         Halfedge_const_handle he_left;  // A halfedge to the left of v_below.
         Halfedge_const_handle he_right; // A halfedge to the right of v_below.
-
         do {
-          if (circ->direction() == ARR_LEFT_TO_RIGHT) {
-            he_left = circ;
-          }
+          if (circ->direction() == ARR_LEFT_TO_RIGHT) he_left = circ;
           else {
             he_right = circ;
-            if (he_left != invalid_he && he_right != invalid_he)
-              break;
+            if ((he_left != invalid_he) && (he_right != invalid_he)) break;
           }
-          ++circ;
+        } while(++circ != first);
 
-        } while(circ != first);
-
-        CGAL_assertion (he_left != invalid_he || he_right != invalid_he);
+        CGAL_assertion((he_left != invalid_he) || (he_right != invalid_he));
 
         if (he_left != invalid_he && he_right != invalid_he) {
           while (he_left->direction() == ARR_LEFT_TO_RIGHT)
@@ -1460,26 +1435,24 @@ is_valid(const Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr)
         else {
           Comparison_result res;
           Halfedge_const_handle he_curr = he_right;
-
-          do // as long as we have he_right halfedge which is below
-          {
+          do {
+            // as long as we have he_right halfedge which is below
             he_right = he_curr;
             he_curr = he_right->next()->twin();
-            res = comp_y_at_x_right (he_curr->curve(),
-                                     he_right->curve(),
-                                     v_below->point());
+            res = comp_y_at_x_right(he_curr->curve(),
+                                    he_right->curve(),
+                                    v_below->point());
           } while(res == SMALLER);
           in_face = he_right->face();
         }
       }
     }
     else {
+      auto* in_face_p = boost::get<Face_const_handle>(&obj);
+      CGAL_assertion(in_face_p);
+      in_face = *in_face_p;
       // Hit nothing (an unbounded face is returned).
-      assign_ok = CGAL::assign(in_face, obj);
-
-      CGAL_assertion (assign_ok && in_face->is_unbounded());
-
-      if (! assign_ok) return false;
+      CGAL_assertion(in_face->is_unbounded());
     }
 
     if (vf_iter->second != in_face) {
@@ -1611,35 +1584,34 @@ do_intersect(Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr,
   // Break the input curve into x-monotone subcurves and isolated points.
   typedef Arr_traits_adaptor_2<Gt2>                     Traits_adaptor_2;
 
+  typedef typename Gt2::Point_2                         Point_2;
+  typedef typename Gt2::X_monotone_curve_2              X_monotone_curve_2;
+  typedef boost::variant<Point_2, X_monotone_curve_2>   Make_x_monotone_result;
+  typedef typename Arr::Face_const_handle               Face_const_handle;
+
   const Traits_adaptor_2* traits =
     static_cast<const Traits_adaptor_2*>(arr.geometry_traits());
 
-  std::list<CGAL::Object> x_objects;
-  std::list<CGAL::Object>::const_iterator obj_iter;
-  const typename Gt2::X_monotone_curve_2* x_curve;
-  const typename Gt2::Point_2* iso_p;
-
+  std::list<Make_x_monotone_result> x_objects;
   traits->make_x_monotone_2_object()(c, std::back_inserter(x_objects));
 
   // Insert each x-monotone curve into the arrangement.
-  for (obj_iter = x_objects.begin(); obj_iter != x_objects.end(); ++obj_iter) {
+  for (const auto& x_obj : x_objects) {
     // Act according to the type of the current object.
-    x_curve = object_cast<typename Gt2::X_monotone_curve_2>(&(*obj_iter));
+    const X_monotone_curve_2* x_curve = boost::get<X_monotone_curve_2>(&x_obj);
     if (x_curve != nullptr) {
       // Check if the x-monotone subcurve intersects the arrangement.
-      if (do_intersect(arr, *x_curve, pl) == true)
-        return true;
+      if (do_intersect(arr, *x_curve, pl) == true) return true;
+      continue;
     }
-    else {
-      iso_p = object_cast<typename Gt2::Point_2>(&(*obj_iter));
-      CGAL_assertion(iso_p != nullptr);
 
-      // Check whether the isolated point lies inside a face (otherwise,
-      // it conincides with a vertex or an edge).
-      CGAL::Object  obj = pl.locate (*iso_p);
+    const Point_2* iso_p = boost::get<Point_2>(&x_obj);
+    CGAL_assertion(iso_p != nullptr);
 
-      return (object_cast<typename Arr::Face_const_handle>(&obj) != nullptr);
-    }
+    // Check whether the isolated point lies inside a face (otherwise,
+    // it conincides with a vertex or an edge).
+    auto obj = pl.locate(*iso_p);
+    if (boost::get<Face_const_handle>(&x_obj) != nullptr) return true;
   }
 
   // If we reached here, the curve does not intersect the arrangement.
@@ -1651,8 +1623,8 @@ do_intersect(Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr,
 template <typename GeometryTraits_2, typename TopologyTraits, typename Curve,
           typename PointLocation>
 bool
-do_intersect (Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr,
-              const Curve& c, const PointLocation& pl)
+do_intersect(Arrangement_on_surface_2<GeometryTraits_2, TopologyTraits>& arr,
+             const Curve& c, const PointLocation& pl)
 {
   typedef GeometryTraits_2                              Gt2;
 

Arrangement_on_surface_2/include/CGAL/Arrangement_2/Arrangement_on_surface_2_impl.h

@@ -7,16 +7,15 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-//
-// Author(s)     : Ron Wein          <wein@post.tau.ac.il>
-//                 Efi Fogel         <efif@post.tau.ac.il>
-//                 Eric Berberich    <eric.berberich@cgal.org>
-//                 (based on old version by: Iddo Hanniel,
-//                                           Eyal Flato,
-//                                           Oren Nechushtan,
-//                                           Ester Ezra,
-//                                           Shai Hirsch,
-//                                           and Eugene Lipovetsky)
+// Author(s): Ron Wein          <wein@post.tau.ac.il>
+//            Efi Fogel         <efif@post.tau.ac.il>
+//            Eric Berberich    <eric.berberich@cgal.org>
+//            (based on old version by: Iddo Hanniel,
+//                                      Eyal Flato,
+//                                      Oren Nechushtan,
+//                                      Ester Ezra,
+//                                      Shai Hirsch,
+//                                      and Eugene Lipovetsky)
 
 #ifndef CGAL_ARRANGEMENT_ON_SURFACE_2_IMPL_H
 #define CGAL_ARRANGEMENT_ON_SURFACE_2_IMPL_H
@@ -32,6 +31,8 @@
  * class-template.
  */
 
+#include <boost/variant.hpp>
+
 #include <CGAL/function_objects.h>
 #include <CGAL/use.h>
 
@@ -2195,16 +2196,30 @@ _place_and_set_curve_end(DFace* f,
 {
   // Use the topology traits to locate the DCEL feature that contains the
   // given curve end.
-  CGAL::Object obj =
-    m_topol_traits.place_boundary_vertex(f, cv, ind, ps_x, ps_y);
-  DVertex* v;
-  DHalfedge* fict_he;
-
+  auto obj = m_topol_traits.place_boundary_vertex(f, cv, ind, ps_x, ps_y);
   // Act according to the result type.
-  if (CGAL::assign(fict_he, obj)) {
+
+  if (! obj) {
+    // We have to create a new vertex that reprsents the given curve end.
+    DVertex* v = _create_boundary_vertex(cv, ind, ps_x, ps_y);
+
+    // Notify the topology traits on the creation of the boundary vertex.
+    m_topol_traits.notify_on_boundary_vertex_creation(v, cv, ind, ps_x, ps_y);
+
+    // There are no edges incident to v, therefore no predecessor halfedge.
+    *p_pred = nullptr;
+
+    // Return the vertex that represents the curve end.
+    return v;
+  }
+
+  DHalfedge** fict_he_p = boost::get<DHalfedge*>(&*obj);
+  if (fict_he_p != nullptr) {
+    DHalfedge* fict_he = *fict_he_p;
+    CGAL_assertion(fict_he != nullptr);
     // The curve end is located on a fictitious edge. We first create a new
     // vertex that corresponds to the curve end.
-    v = _create_boundary_vertex(cv, ind, ps_x, ps_y);
+    DVertex* v = _create_boundary_vertex(cv, ind, ps_x, ps_y);
 
     // Split the fictitious halfedge at the newly created vertex.
     // The returned halfedge is the predecessor for the insertion of the curve
@@ -2216,29 +2231,16 @@ _place_and_set_curve_end(DFace* f,
 
     _notify_after_split_fictitious_edge(Halfedge_handle(*p_pred),
                                         Halfedge_handle((*p_pred)->next()));
+    return v;
   }
-  else if (CGAL::assign(v, obj)) {
-    // In this case we are given an existing vertex that represents the curve
-    // end. We now have to locate the predecessor edge for the insertion of cv
-    // around this vertex.
-    *p_pred =
-      m_topol_traits.locate_around_boundary_vertex(v, cv, ind, ps_x, ps_y);
-  }
-  else {
-    CGAL_assertion(obj.is_empty());
-
-    // In this case we have to create a new vertex that reprsents the given
-    // curve end.
-    v = _create_boundary_vertex(cv, ind, ps_x, ps_y);
-
-    // Notify the topology traits on the creation of the boundary vertex.
-    m_topol_traits.notify_on_boundary_vertex_creation(v, cv, ind, ps_x, ps_y);
-
-    // There are no edges incident to v, therefore no predecessor halfedge.
-    *p_pred = nullptr;
-  }
-
-  // Return the vertex that represents the curve end.
+  DVertex** v_p = boost::get<DVertex*>(&*obj);
+  CGAL_assertion(v_p != nullptr);
+  DVertex* v = *v_p;
+  CGAL_assertion(v != nullptr);
+  // In this case we are given an existing vertex that represents the curve
+  // end. We now have to locate the predecessor edge for the insertion of cv
+  // around this vertex.
+  *p_pred = m_topol_traits.locate_around_boundary_vertex(v, cv, ind, ps_x, ps_y);
   return v;
 }
 

Arrangement_on_surface_2/include/CGAL/Arrangement_2/Arrangement_zone_2_impl.h

@@ -29,7 +29,7 @@ namespace CGAL {
 //
 template <typename Arrangement, typename ZoneVisitor>
 void Arrangement_zone_2<Arrangement, ZoneVisitor>::
-init_with_hint(const X_monotone_curve_2& cv, const Object& obj)
+init_with_hint(const X_monotone_curve_2& cv, Pl_result_type obj)
 {
   // Set the curve and check whether its ends are bounded, therefore
   // associated with valid endpoints.
@@ -95,9 +95,8 @@ void Arrangement_zone_2<Arrangement, ZoneVisitor>::compute_zone()
   // curve (currently m_obj stores the object containing it).
   const Vertex_const_handle* vh;
   const Halfedge_const_handle* hh;
-  const Face_const_handle* fh;
 
-  if ((vh = object_cast<Vertex_const_handle>(&m_obj)) != nullptr) {
+  if ((vh = boost::get<Vertex_const_handle>(&m_obj)) != nullptr) {
     CGAL_assertion(m_has_left_pt);
 
     // The left endpoint coincides with an existing vertex:
@@ -119,7 +118,7 @@ void Arrangement_zone_2<Arrangement, ZoneVisitor>::compute_zone()
 #endif
 
   }
-  else if ((hh = object_cast<Halfedge_const_handle>(&m_obj)) != nullptr) {
+  else if ((hh = boost::get<Halfedge_const_handle>(&m_obj)) != nullptr) {
     if (m_has_left_pt) {
       // Obtain the right halfedge from the halfedge-pair containing m_left_pt
       // in their interior.
@@ -160,7 +159,7 @@ void Arrangement_zone_2<Arrangement, ZoneVisitor>::compute_zone()
   }
   else {
     // The left endpoint lies inside a face.
-    fh = object_cast<Face_const_handle>(&m_obj);
+    const Face_const_handle* fh = boost::get<Face_const_handle>(&m_obj);
 
     CGAL_assertion_msg(fh != nullptr,
                        "Invalid object returned by the point-location query.");

Arrangement_on_surface_2/include/CGAL/Arrangement_on_surface_2.h

@@ -2832,7 +2832,8 @@ void insert(Arrangement_on_surface_2<GeomTraits, TopTraits>& arr,
 template <typename GeomTraits, typename TopTraits>
 void insert(Arrangement_on_surface_2<GeomTraits, TopTraits>& arr,
             const typename GeomTraits::X_monotone_curve_2& c,
-            const Object& obj);
+            typename Arr_point_location_result<
+              Arrangement_on_surface_2<GeomTraits, TopTraits> >::type obj);
 
 /*!
  * Insert an x-monotone curve into the arrangement, such that the curve
@@ -2948,16 +2949,16 @@ remove_vertex(Arrangement_on_surface_2<GeomTraits, TopTraits>& arr,
 template <typename GeomTraits, typename TopTraits>
 bool is_valid(const Arrangement_on_surface_2<GeomTraits, TopTraits>& arr);
 
-/*!
- * Compute the zone of the given x-monotone curve in the existing arrangement.
+/*! Compute the zone of the given x-monotone curve in the existing arrangement.
  * Meaning, it output the arrangment's vertices, edges and faces that the
  * x-monotone curve intersects.
  * \param arr The arrangement.
- * \param c The x-monotone curve that its zone was computed.
- * \param oi Output iterator of CGAL::Object to insert the zone elements to.
- * \param pi The point location strategy that is used to locate the starting
- * point.
- * \return The output iterator that the curves were inserted to.
+ * \param c the x-monotone curve that its zone is computed.
+ * \param oi the output iterator for the resulting zone elements. Its
+ *           dereference type is a variant that wraps a \c Vertex_handle, a
+ *           \c Halfedge_handle, or a \c Face_handle.
+ * \param pl the point location strategy used to locate the starting point.
+ * \return the past-the-end output iterator.
  */
 template <typename GeomTraits, typename TopTraits,
           typename OutputIterator, typename PointLocation>
@@ -2971,9 +2972,11 @@ OutputIterator zone(Arrangement_on_surface_2<GeomTraits, TopTraits>& arr,
  * Overloaded version with no point location object - the walk point-location
  * strategy is used as default.
  * \param arr The arrangement.
- * \param c The x-monotone curve that its zone was computed.
- * \param oi Output iterator of CGAL::Object to insert the zone elements to.
- * \return The output iterator that the curves were inserted to.
+ * \param c the x-monotone curve that its zone was computed.
+ * \param oi the output iterator for the resulting zone elements. Its
+ *           dereference type is a variant that wraps a \c Vertex_handle, a
+ *           \c Halfedge_handle, or a \c Face_handle.
+ * \return the past-the-end output iterator.
  */
 template <typename GeomTraits, typename TopTraits, typename OutputIterator>
 OutputIterator zone(Arrangement_on_surface_2<GeomTraits, TopTraits>& arr,

Arrangement_on_surface_2/include/CGAL/Arrangement_zone_2.h

@@ -28,6 +28,7 @@
 #include <CGAL/Arr_tags.h>
 #include <CGAL/Arr_accessor.h>
 #include <CGAL/Arrangement_2/Arr_traits_adaptor_2.h>
+#include <CGAL/Arr_point_location_result.h>
 
 #include <list>
 #include <map>
@@ -113,6 +114,9 @@ protected:
   typedef std::set<const X_monotone_curve_2*>           Curves_set;
   typedef typename Curves_set::iterator                 Curves_set_iterator;
 
+  typedef Arr_point_location_result<Arrangement_2>      Pl_result;
+  typedef typename Pl_result::Type                      Pl_result_type;
+
   // Data members:
   Arrangement_2& m_arr;                 // The associated arrangement.
   const Traits_adaptor_2* m_geom_traits; // Its associated geometry traits.
@@ -127,7 +131,7 @@ protected:
 
   X_monotone_curve_2 m_cv;              // The current portion of the
                                         // inserted curve.
-  CGAL::Object m_obj;                   // The location of the left endpoint.
+  Pl_result_type m_obj;                 // The location of the left endpoint.
   bool m_has_left_pt;                   // Is the left end of the curve bounded.
   bool m_left_on_boundary;              // Is the left point on the boundary.
   Point_2 m_left_pt;                    // Its current left endpoint.
@@ -247,7 +251,7 @@ public:
    * \param obj An object that represents the location of the left end of the
    *            curve.
    */
-  void init_with_hint(const X_monotone_curve_2& cv, const Object& obj);
+  void init_with_hint(const X_monotone_curve_2& cv, Pl_result_type obj);
 
   /*! Compute the zone of the given curve and issue the apporpriate
    * notifications for the visitor.

Arrangement_on_surface_2/include/CGAL/Curved_kernel_via_analysis_2/Curved_kernel_via_analysis_2_functors.h

@@ -1410,9 +1410,9 @@ public:
  * Functor that computes the intersections of two arcs
  */
 template < class CurvedKernelViaAnalysis_2 >
-class Intersect_2 : public
-Curved_kernel_via_analysis_2_functor_base< CurvedKernelViaAnalysis_2 > {
-
+class Intersect_2 :
+    public Curved_kernel_via_analysis_2_functor_base<CurvedKernelViaAnalysis_2>
+{
 public:
     //! this instance' first template parameter
     typedef CurvedKernelViaAnalysis_2 Curved_kernel_via_analysis_2;
@@ -1420,13 +1420,17 @@ public:
     //! the base type
     typedef
     Curved_kernel_via_analysis_2_functor_base< Curved_kernel_via_analysis_2 >
-    Base;
+      Base;
 
     CGAL_CKvA_2_GRAB_BASE_FUNCTOR_TYPES
 
+    typedef unsigned int                              Multiplicity;
+    typedef std::pair<Point_2, Multiplicity>          Intersection_point;
+    typedef boost::variant<Intersection_point, Arc_2> Intersection_result;
+
     //! the result type
-    typedef CGAL::cpp98::iterator< std::output_iterator_tag, CGAL::Object >
-    result_type;
+    typedef CGAL::cpp98::iterator<std::output_iterator_tag, Intersection_result>
+      result_type;
 
     //! the arity of the functor
 
@@ -1458,11 +1462,6 @@ public:
     template < class OutputIterator >
     OutputIterator operator()(const Arc_2& cv1, const Arc_2& cv2,
                               OutputIterator oi) const {
-
-      typedef unsigned int                              Multiplicity;
-      typedef std::pair<Point_2, Multiplicity>          Intersection_point;
-      typedef boost::variant<Intersection_point, Arc_2> Intersection_result;
-
         CERR("\nintersect; cv1: " << cv1
              << ";\n cv2:" << cv2 << "");
 
@@ -1860,100 +1859,110 @@ public:
 /*!\brief
  * Functor that decomposes curve into x-monotone arcs and isolated points
  */
-template < class CurvedKernelViaAnalysis_2 >
+template <class CurvedKernelViaAnalysis_2>
 class Make_x_monotone_2 : public
-Curved_kernel_via_analysis_2_functor_base< CurvedKernelViaAnalysis_2 > {
+Curved_kernel_via_analysis_2_functor_base<CurvedKernelViaAnalysis_2> {
 
 public:
-    //! this instance' first template parameter
-    typedef CurvedKernelViaAnalysis_2 Curved_kernel_via_analysis_2;
+  //! this instance' first template parameter
+  typedef CurvedKernelViaAnalysis_2 Curved_kernel_via_analysis_2;
 
-    //! the base type
-    typedef
-    Curved_kernel_via_analysis_2_functor_base< Curved_kernel_via_analysis_2 >
+  //! the base type
+  typedef
+  Curved_kernel_via_analysis_2_functor_base< Curved_kernel_via_analysis_2>
     Base;
 
-    CGAL_CKvA_2_GRAB_BASE_FUNCTOR_TYPES
+  CGAL_CKvA_2_GRAB_BASE_FUNCTOR_TYPES
 
-    //! the result type
-    typedef CGAL::cpp98::iterator< std::output_iterator_tag, CGAL::Object >
+  //! the result type
+  typedef CGAL::cpp98::iterator< std::output_iterator_tag, CGAL::Object>
     result_type;
 
-    //! the arity of the functor
+  //! the arity of the functor
 
-    /*!\brief
-     * Standard constructor
-     *
-     * \param kernel The kernel
-     */
-    Make_x_monotone_2(Curved_kernel_via_analysis_2 *kernel) :
-        Base(kernel) {
-    }
+  /*!\brief
+   * Standard constructor
+   *
+   * \param kernel The kernel
+   */
+  Make_x_monotone_2(Curved_kernel_via_analysis_2* kernel) :
+    Base(kernel)
+  {}
 
-    /*!\brief
-     * Decomposes a given arc into list of x-monotone arcs
-     * (subcurves) and insert them to the output iterator. Since \c Arc_2
-     * is by definition x-monotone, an input arc is passed to the
-     * output iterator directly.
-     *
-     * \param cv The arc
-     * \param oi The output iterator, whose value-type is Object
-     * The returned objects are all wrappers Arc_2 objects
-     * \return The past-the-end iterator
-     */
-    template < class OutputIterator >
-    OutputIterator operator()(const Arc_2& cv, OutputIterator oi) const {
+  /*!\brief
+   * Decomposes a given arc into list of x-monotone arcs
+   * (subcurves) and insert them to the output iterator. Since \c Arc_2
+   * is by definition x-monotone, an input arc is passed to the
+   * output iterator directly.
+   *
+   * \param cv The arc
+   * \param oi The output iterator, whose value-type is Object
+   * The returned objects are all wrappers Arc_2 objects
+   * \return The past-the-end iterator
+   */
+  template <class OutputIterator>
+  OutputIterator operator()(const Arc_2& cv, OutputIterator oi) const
+  {
+    typedef typename Curved_kernel_via_analysis_2::Point_2      Point_2;
+    typedef typename Curved_kernel_via_analysis_2::Arc_2        Arc_2;
+    typedef boost::variant<Point_2, Arc_2>      Make_x_monotone_result;
+    *oi++ = Make_x_monotone_result(cv);
+    return oi;
+  }
 
-        *oi++ = CGAL::make_object(cv);
-        return oi;
-    }
+  /*!\brief
+   * Decomposes a given curve into list of x-monotone arcs
+   * (subcurves) and isolated points and insert them to the output iterator.
+   *
+   * \param cv The curve
+   * \param oi The output iterator, whose value-type is Object.
+   * The returned objects either wrapper Arc_2 or Point_2 objects
+   * \return The past-the-end iterator
+   */
+  template <class OutputIterator>
+  OutputIterator operator()(const Curve_analysis_2& cv, OutputIterator oi)
+    const
+  {
+    CGAL::internal::Make_x_monotone_2< Curved_kernel_via_analysis_2 >
+      make_x_monotone(Base::_ckva());
 
-    /*!\brief
-     * Decomposes a given curve into list of x-monotone arcs
-     * (subcurves) and isolated points and insert them to the output iterator.
-     *
-     * \param cv The curve
-     * \param oi The output iterator, whose value-type is Object.
-     * The returned objects either wrapper Arc_2 or Point_2 objects
-     * \return The past-the-end iterator
-     */
-    template < class OutputIterator >
-    OutputIterator operator()(const Curve_analysis_2& cv, OutputIterator oi)
-         const {
+    return make_x_monotone(cv, oi);
+  }
 
-        CGAL::internal::Make_x_monotone_2< Curved_kernel_via_analysis_2 >
-            make_x_monotone(Base::_ckva());
+  /*!\brief
+   * Splits an input object \c obj into x-monotone arcs and isolated points
+   *
+   * \param obj the polymorph input object: can represet \c Point_2,
+   * \c Arc_2, \c Non_x_monotone_arc_2 or \c Curve_analysis_2
+   * \param oi Output iterator that stores CGAL::Object, which either
+   *           encapsulates \c Point_2 or \c Arc_2
+   * \return Past-the-end iterator of \c oi
+   */
+  template <class OutputIterator>
+  OutputIterator operator()(CGAL::Object obj, OutputIterator oi)
+  {
+    typedef typename Curved_kernel_via_analysis_2::Point_2      Point_2;
+    typedef typename Curved_kernel_via_analysis_2::Arc_2        Arc_2;
+    typedef typename Curved_kernel_via_analysis_2::Non_x_monotone_arc_2
+      Non_x_monotone_arc_2;
+    typedef boost::variant<Point_2, Arc_2>
+      Make_x_monotone_result;
 
-        return make_x_monotone(cv, oi);
-    }
+    Curve_analysis_2 curve;
+    Point_2 p;
+    Arc_2 xcv;
+    Non_x_monotone_arc_2 nxarc;
 
-    /*!\brief
-     * Splits an input object \c obj into x-monotone arcs and isolated points
-     *
-     * \param obj the polymorph input object: can represet \c Point_2,
-     * \c Arc_2, \c Non_x_monotone_arc_2 or \c Curve_analysis_2
-     * \param oi Output iterator that stores CGAL::Object, which either
-     *           encapsulates \c Point_2 or \c Arc_2
-     * \return Past-the-end iterator of \c oi
-     */
-    template <class OutputIterator>
-    OutputIterator operator()(CGAL::Object obj, OutputIterator oi) {
-
-        typedef typename Curved_kernel_via_analysis_2::Non_x_monotone_arc_2
-            Non_x_monotone_arc_2;
-        Curve_analysis_2 curve;
-        Non_x_monotone_arc_2 nxarc;
-
-        if(CGAL::assign(curve, obj))
-            oi = (*this)(curve, oi);
-        else if(CGAL::assign(nxarc, obj))
-          std::cerr << "AU BACKE" << std::endl;
-          //oi = std::transform(nxarc.begin(), nxarc.end(), oi,
-          //      std::ptr_fun(CGAL::make_object<Arc_2>));
-        else // allow the remaining objects to pass through
-            *oi++ = obj;
-        return oi;
-    }
+    if (CGAL::assign(curve, obj)) oi = (*this)(curve, oi);
+    else if (CGAL::assign(nxarc, obj))
+      std::cerr << "AU BACKE" << std::endl;
+    //oi = std::transform(nxarc.begin(), nxarc.end(), oi,
+    //      std::ptr_fun(CGAL::make_object<Arc_2>));
+    else if (CGAL::assign(xcv, obj)) *oi++ = Make_x_monotone_result(xcv);
+    else if (CGAL::assign(p, obj)) *oi++ = Make_x_monotone_result(p);
+    else CGAL_error();
+    return oi;
+  }
 };
 
 

Arrangement_on_surface_2/include/CGAL/Curved_kernel_via_analysis_2/Filtered_curved_kernel_via_analysis_2_impl.h

@@ -169,12 +169,10 @@ public:
         Coordinate_1 asym_info1, asym_info2;
         CGAL::Arr_parameter_space ps1, ps2;
 
-        obj1 = cv1.curve().asymptotic_value_of_arc(
-                cv1.location(ce), cv1.arcno()
-        );
-        obj2 = cv2.curve().asymptotic_value_of_arc(
-                cv2.location(ce), cv2.arcno()
-        );
+        obj1 =
+          cv1.curve().asymptotic_value_of_arc(cv1.location(ce), cv1.arcno());
+        obj2 =
+          cv2.curve().asymptotic_value_of_arc(cv2.location(ce), cv2.arcno());
 
         CGAL::Comparison_result filter_res = CGAL::EQUAL;
 

Arrangement_on_surface_2/include/CGAL/Curved_kernel_via_analysis_2/Make_x_monotone_2.h

@@ -42,15 +42,26 @@ namespace internal {
  * Isolated points are stored as \c CurvedKernelViaAnalysis_2::Point_2 objects.
  *
  * The resulting arcs and points are written to the output iterator as
- * polymorphic \c CGAL::Object. Past-the-end value of the iterator is returned.
+ * polymorphic \c variant. Past-the-end value of the iterator is returned.
+ *
+ * EF: I believe that the inheritance from binary_function is not exploited,
+ *     and thus redundant, but I keep it anyway.
  */
 template < class CurvedKernelViaAnalysis_2,
            class ConstructArc_2 =
            typename CurvedKernelViaAnalysis_2::Construct_arc_2 >
 struct Make_x_monotone_2 :
-    public CGAL::cpp98::binary_function< typename CurvedKernelViaAnalysis_2::Curve_2,
-            CGAL::cpp98::iterator<std::output_iterator_tag, CGAL::Object>,
-            CGAL::cpp98::iterator<std::output_iterator_tag, CGAL::Object> > {
+    public CGAL::cpp98::binary_function<
+        typename CurvedKernelViaAnalysis_2::Curve_2,
+        CGAL::cpp98::iterator<std::output_iterator_tag,
+                              boost::variant<
+                                typename CurvedKernelViaAnalysis_2::Point_2,
+                                typename CurvedKernelViaAnalysis_2::Arc_2> >,
+        CGAL::cpp98::iterator<std::output_iterator_tag,
+                              boost::variant<
+                                typename CurvedKernelViaAnalysis_2::Point_2,
+                                typename CurvedKernelViaAnalysis_2::Arc_2> > >
+{
 
     //!\name Public types
     //!@{
@@ -87,242 +98,250 @@ struct Make_x_monotone_2 :
     typedef typename Curved_kernel_via_analysis_2::Non_x_monotone_arc_2
         Non_x_monotone_arc_2;
 
-    //!@}
+  //!@}
 
-    //!\name Constructors
-    //!@{
+  //!\name Constructors
+  //!@{
 
-    /*!\brief
-     * Standard constructor
-     *
-     * \param kernel The kernel instance to use
-     */
-    Make_x_monotone_2(Curved_kernel_via_analysis_2 *kernel) :
-        _m_curved_kernel(kernel) {
-        CGAL_assertion(kernel != nullptr);
+  /*!\brief
+   * Standard constructor
+   *
+   * \param kernel The kernel instance to use
+   */
+  Make_x_monotone_2(Curved_kernel_via_analysis_2 *kernel) :
+    _m_curved_kernel(kernel)
+  {
+    CGAL_assertion(kernel != nullptr);
+  }
+
+  //!@}
+
+  //!\name Functor invokation
+  //!@{
+
+  // TODO add operator for non-x-monotone arc
+
+  /*!\brief
+   * Splits a curve into x-monotone arcs and isolated points
+   *
+   * \param curve The input curve
+   * \param oi the output iterator for the result. Its dereference type is a
+   *           variant that wraps a Point_2 or an X_monotone_curve_2 objects.
+   * \return Past-the-end iterator of \c oi
+   */
+  template <typename OutputIterator>
+  OutputIterator operator()(Curve_analysis_2 curve, OutputIterator oi)
+  {
+    typedef boost::variant<Point_2, Arc_2>      Make_x_monotone_result;
+
+    Construct_arc_2 construct_arc_2 =
+      _m_curved_kernel->construct_arc_2_object();
+    // use CGAL::Total_degree ?
+    if (typename CGAL::Polynomial_traits_d<
+        typename Curve_analysis_2::Polynomial_2 >::
+        Total_degree()(curve.polynomial_2()) < 1)
+    {
+      return oi;
     }
 
-    //!@}
+    Status_line_1 evt_line1, evt_line2,
+      int_line = curve.status_line_of_interval(0);
+    int total_events = curve.number_of_status_lines_with_event();
+    // handle special case of a curve without any events
+    if(total_events == 0) {
+      for (int k = 0; k < int_line.number_of_events(); k++)
+        *oi++ = Make_x_monotone_result(construct_arc_2(curve, k));
+      return oi;
+    }
+    _m_curve = curve;
+    typedef typename Curved_kernel_via_analysis_2::
+      Curve_interval_arcno_cache CIA_cache;
+    const CIA_cache& map_interval_arcno =
+      _m_curved_kernel->interval_arcno_cache();
 
-    //!\name Functor invokation
-    //!@{
+    typename Curved_kernel_via_analysis_2::Construct_point_2
+      construct_point =
+      _m_curved_kernel->construct_point_2_object();
 
-    // TODO add operator for non-x-monotone arc
+    typename CIA_cache::result_type info1, info2;
+    std::vector<Point_2> min_pts, max_pts;
+    Coordinate_1 min_x, max_x;
+    int i, k, n;
+    Arc_2 arc;
+    // first handle segments before first event
+    evt_line1 = curve.status_line_at_event(0);
+    max_x = evt_line1.x();
 
-    /*!\brief
-     * Splits a curve into x-monotone arcs and isolated points
-     *
-     * \param curve The input curve
-     * \param oi Output iterator that stores CGAL::Object, which either
-     *           encapsulates Point_2 or Arc_2
-     * \return Past-the-end iterator of \c oi
-     */
-    template <class OutputIterator>
-    OutputIterator operator()(Curve_analysis_2 curve, OutputIterator oi) {
+    for (k = 0; k < evt_line1.number_of_events(); k++)
+      max_pts.push_back(construct_point(max_x, curve, k));
 
-        Construct_arc_2 construct_arc_2 =
-            _m_curved_kernel->construct_arc_2_object();
-        // use CGAL::Total_degree ?
-        if (typename CGAL::Polynomial_traits_d<
-            typename Curve_analysis_2::Polynomial_2 >::
-            Total_degree()(curve.polynomial_2()) < 1) {
-            return oi;
-        }
+    //std::cout << "handling events over the 1st interval\n";
+    for (k = 0; k < int_line.number_of_events(); k++) {
 
-        Status_line_1 evt_line1, evt_line2,
-            int_line = curve.status_line_of_interval(0);
-        int total_events = curve.number_of_status_lines_with_event();
-        // handle special case of a curve without any events
-        if(total_events == 0) {
-            for(int k = 0; k < int_line.number_of_events(); k++)
-                *oi++ = CGAL::make_object(construct_arc_2(curve, k));
-            return oi;
-        }
-        _m_curve = curve;
-        typedef typename Curved_kernel_via_analysis_2::
-            Curve_interval_arcno_cache CIA_cache;
-        const CIA_cache& map_interval_arcno =
-            _m_curved_kernel->interval_arcno_cache();
+      info1 = map_interval_arcno(evt_line1, 1, k);
+      if (info1.second != CGAL::ARR_INTERIOR) {
+        arc = construct_arc_2(CGAL::ARR_MIN_END, max_x,
+                              (info1.second ==
+                               CGAL::ARR_BOTTOM_BOUNDARY ?
+                               CGAL::ARR_MIN_END : CGAL::ARR_MAX_END),
+                              curve, k);
+      }
+      else {
+        arc = construct_arc_2(max_pts[info1.first], CGAL::ARR_MIN_END,
+                              curve, k, info1.first);
+      }
+      *oi++ = Make_x_monotone_result(arc);
+    }
+    min_pts = max_pts;
+    max_pts.clear();
+    min_x = max_x;
 
-        typename Curved_kernel_via_analysis_2::Construct_point_2
-            construct_point =
-            _m_curved_kernel->construct_point_2_object();
+    // next handle arcs between events, including isolated points
+    for (i = 0; i < total_events-1; i++) {
+      evt_line1 = curve.status_line_at_event(i);
+      evt_line2 = curve.status_line_at_event(i+1);
+      max_x = evt_line2.x();
+      oi = _handle_vertical_and_isolated(evt_line1, min_x, min_pts, oi);
 
-        typename CIA_cache::result_type info1, info2;
-        std::vector<Point_2> min_pts, max_pts;
-        Coordinate_1 min_x, max_x;
-        int i, k, n;
-        Arc_2 arc;
-        // first handle segments before first event
-        evt_line1 = curve.status_line_at_event(0);
-        max_x = evt_line1.x();
+      n = evt_line2.number_of_events();
+      for (k = 0; k < n; k++)
+        max_pts.push_back(construct_point(max_x, curve, k));
 
-        for(k = 0; k < evt_line1.number_of_events(); k++)
-            max_pts.push_back(construct_point(max_x, curve, k));
-
-        //std::cout << "handling events over the 1st interval\n";
-        for(k = 0; k < int_line.number_of_events(); k++) {
-
-            info1 = map_interval_arcno(evt_line1, 1, k);
-            if (info1.second != CGAL::ARR_INTERIOR) {
-                arc = construct_arc_2(CGAL::ARR_MIN_END, max_x,
-                                      (info1.second ==
-                                       CGAL::ARR_BOTTOM_BOUNDARY ?
-                                       CGAL::ARR_MIN_END : CGAL::ARR_MAX_END),
-                                      curve, k);
-            } else {
-                arc = construct_arc_2(max_pts[info1.first], CGAL::ARR_MIN_END,
-                                      curve, k, info1.first
-                );
-            }
-            *oi++ = CGAL::make_object(arc);
-        }
-        min_pts = max_pts;
-        max_pts.clear();
-        min_x = max_x;
-
-        // next handle arcs between events, including isolated points
-        for (i = 0; i < total_events-1; i++) {
-            evt_line1 = curve.status_line_at_event(i);
-            evt_line2 = curve.status_line_at_event(i+1);
-            max_x = evt_line2.x();
-            oi = _handle_vertical_and_isolated(evt_line1, min_x, min_pts, oi);
-
-            n = evt_line2.number_of_events();
-            for(k = 0; k < n; k++)
-                max_pts.push_back(construct_point(max_x, curve, k));
-
-            n = curve.status_line_of_interval(i+1).number_of_events();
-            CGAL::Arr_curve_end inf1_end, inf2_end;
-            for (k = 0; k < n; k++) {
-
-                info1 = map_interval_arcno(evt_line1, 0, k);
-                info2 = map_interval_arcno(evt_line2, 1, k);
-                inf2_end = (info2.second == CGAL::ARR_BOTTOM_BOUNDARY ?
+      n = curve.status_line_of_interval(i+1).number_of_events();
+      CGAL::Arr_curve_end inf1_end, inf2_end;
+      for (k = 0; k < n; k++) {
+        info1 = map_interval_arcno(evt_line1, 0, k);
+        info2 = map_interval_arcno(evt_line2, 1, k);
+        inf2_end = (info2.second == CGAL::ARR_BOTTOM_BOUNDARY ?
                     CGAL::ARR_MIN_END : CGAL::ARR_MAX_END);
 
-                if (info1.second != CGAL::ARR_INTERIOR) {
-                    inf1_end = (info1.second == CGAL::ARR_BOTTOM_BOUNDARY ?
-                        CGAL::ARR_MIN_END : CGAL::ARR_MAX_END);
-                    if (info2.second != CGAL::ARR_INTERIOR) {
-                        arc = construct_arc_2(min_x, inf1_end, max_x, inf2_end,
-                                              curve, k);
-                    } else {
-                        arc = construct_arc_2(max_pts[info2.first], min_x,
-                                              inf1_end, curve, k, info2.first);
-                    }
-                } else if (info2.second != CGAL::ARR_INTERIOR) {
-                    arc = construct_arc_2(min_pts[info1.first],  max_x,
-                                          inf2_end, curve, k, info1.first);
-                } else {
-                    arc = construct_arc_2(min_pts[info1.first],
-                                          max_pts[info2.first],
-                                          curve, k, info1.first, info2.first);
-                }
-                *oi++ = CGAL::make_object(arc);
-            }
-            min_pts = max_pts;
-            max_pts.clear();
-            min_x = max_x;
+        if (info1.second != CGAL::ARR_INTERIOR) {
+          inf1_end = (info1.second == CGAL::ARR_BOTTOM_BOUNDARY ?
+                      CGAL::ARR_MIN_END : CGAL::ARR_MAX_END);
+          if (info2.second != CGAL::ARR_INTERIOR) {
+            arc = construct_arc_2(min_x, inf1_end, max_x, inf2_end,
+                                  curve, k);
+          }
+          else {
+            arc = construct_arc_2(max_pts[info2.first], min_x,
+                                  inf1_end, curve, k, info2.first);
+          }
         }
-
-        // here: min_x/min_pts hold information about the last event line
-        // event_line2 - points to the last event line
-        // vertical line or isolated points at last event?
-        evt_line2 = curve.status_line_at_event(total_events-1);
-        min_x = evt_line2.x();
-        oi = _handle_vertical_and_isolated(evt_line2, min_x, min_pts, oi);
-
-        n = curve.status_line_of_interval(total_events).number_of_events();
-        for (k = 0; k < n; k++) {
-
-            info1 = map_interval_arcno(evt_line2, 0, k);
-            if (info1.second != CGAL::ARR_INTERIOR) {
-                arc = construct_arc_2(
-                        CGAL::ARR_MAX_END, min_x,
-                        (info1.second == CGAL::ARR_BOTTOM_BOUNDARY ?
-                         CGAL::ARR_MIN_END : CGAL::ARR_MAX_END), curve, k
-                );
-            } else {
-                arc = construct_arc_2(min_pts[info1.first],
-                                      CGAL::ARR_MAX_END, curve, k,
-                                      info1.first);
-            }
-            *oi++ = CGAL::make_object(arc);
+        else if (info2.second != CGAL::ARR_INTERIOR) {
+          arc = construct_arc_2(min_pts[info1.first],  max_x,
+                                inf2_end, curve, k, info1.first);
         }
-        return oi;
+        else {
+          arc = construct_arc_2(min_pts[info1.first],
+                                max_pts[info2.first],
+                                curve, k, info1.first, info2.first);
+        }
+        *oi++ = Make_x_monotone_result(arc);
+      }
+      min_pts = max_pts;
+      max_pts.clear();
+      min_x = max_x;
     }
-    //!@}
+
+    // here: min_x/min_pts hold information about the last event line
+    // event_line2 - points to the last event line
+    // vertical line or isolated points at last event?
+    evt_line2 = curve.status_line_at_event(total_events-1);
+    min_x = evt_line2.x();
+    oi = _handle_vertical_and_isolated(evt_line2, min_x, min_pts, oi);
+
+    n = curve.status_line_of_interval(total_events).number_of_events();
+    for (k = 0; k < n; k++) {
+      info1 = map_interval_arcno(evt_line2, 0, k);
+      if (info1.second != CGAL::ARR_INTERIOR) {
+        arc = construct_arc_2(CGAL::ARR_MAX_END, min_x,
+                              (info1.second == CGAL::ARR_BOTTOM_BOUNDARY ?
+                               CGAL::ARR_MIN_END : CGAL::ARR_MAX_END), curve, k
+                              );
+      }
+      else {
+        arc = construct_arc_2(min_pts[info1.first],
+                              CGAL::ARR_MAX_END, curve, k,
+                              info1.first);
+      }
+      *oi++ = Make_x_monotone_result(arc);
+    }
+    return oi;
+  }
+  //!@}
 
 private:
-    //!\name Private members
-    //!@{
+  //!\name Private members
+  //!@{
 
-    /*!\brief
-     * Constructs vertical arcs and isolated points at event line
-     *
-     * \param cv_line The event line in focus
-     * \param x x-coordinate of event
-     * \param pts Points at event line
-     * \param oi Output iterator that stores CGAL::Object, which either
-     *           encapsulates Point_2 or Arc_2
-     * \return Past-the-end iterator of \c oi
-     */
-    template <class OutputIterator>
-    OutputIterator _handle_vertical_and_isolated(
-            Status_line_1 cv_line,
-            Coordinate_1 x, std::vector<Point_2> pts,
-            OutputIterator oi) const {
+  /*!\brief
+   * Constructs vertical arcs and isolated points at event line
+   *
+   * \param cv_line The event line in focus
+   * \param x x-coordinate of event
+   * \param pts Points at event line
+   * \param oi the output iterator for the result. Its dereference type is a
+   *           variant that wraps a Point_2 or an X_monotone_curve_2 objects.
+   * \return Past-the-end iterator of \c oi
+   */
+  template <class OutputIterator>
+  OutputIterator
+  _handle_vertical_and_isolated(Status_line_1 cv_line,
+                                Coordinate_1 x, std::vector<Point_2> pts,
+                                OutputIterator oi) const
+  {
+    typedef boost::variant<Point_2, Arc_2>      Make_x_monotone_result;
 
-        Construct_arc_2 construct_arc_2 =
-            _m_curved_kernel->construct_arc_2_object();
+    Construct_arc_2 construct_arc_2 =
+      _m_curved_kernel->construct_arc_2_object();
 
-        int n = cv_line.number_of_events(), j;
-        if(cv_line.covers_line()) { // look for vertical arcs
-            if(n > 0) {
-                // the first vertical ray
-                *oi++ = CGAL::make_object(
-                        construct_arc_2(pts[0], CGAL::ARR_MIN_END, _m_curve)
-                );
-                for(j = 0; j < n-1; j++)  // interior bounded arcs
-                    *oi++ = CGAL::make_object(construct_arc_2(pts[j], pts[j+1],
-                                                              _m_curve));
-                // the last vertical ray
-                *oi++ = CGAL::make_object(construct_arc_2(pts[n-1],
-                    CGAL::ARR_MAX_END, _m_curve));
-            } else // unbounded vertical line
-                *oi++ = CGAL::make_object(construct_arc_2(x, _m_curve));
-            return oi;
-        }
-        // look for isolated points
-        std::pair<int, int> ipair;
-        for(j = 0; j < n; j++) {
-            ipair = cv_line.number_of_incident_branches(j);
-            if(ipair.first == 0&&ipair.second == 0) {
-                //std::cout << "isolated point found\n";
-                typename Curved_kernel_via_analysis_2::Construct_point_2
-                    construct_point =
-                    _m_curved_kernel->construct_point_2_object();
-
-                *oi++ = CGAL::make_object(construct_point(x, _m_curve, j));
-            }
-        }
-        return oi;
+    int n = cv_line.number_of_events(), j;
+    if (cv_line.covers_line()) { // look for vertical arcs
+      if (n > 0) {
+        // the first vertical ray
+        *oi++ =
+          Make_x_monotone_result(construct_arc_2(pts[0], CGAL::ARR_MIN_END,
+                                                 _m_curve));
+        for (j = 0; j < n-1; j++)  // interior bounded arcs
+          *oi++ = Make_x_monotone_result(construct_arc_2(pts[j], pts[j+1],
+                                                    _m_curve));
+        // the last vertical ray
+        *oi++ =
+          Make_x_monotone_result(construct_arc_2(pts[n-1], CGAL::ARR_MAX_END,
+                                                 _m_curve));
+      }
+      else // unbounded vertical line
+        *oi++ = Make_x_monotone_result(construct_arc_2(x, _m_curve));
+      return oi;
     }
+    // look for isolated points
+    std::pair<int, int> ipair;
+    for (j = 0; j < n; j++) {
+      ipair = cv_line.number_of_incident_branches(j);
+      if (ipair.first == 0&&ipair.second == 0) {
+        //std::cout << "isolated point found\n";
+        typename Curved_kernel_via_analysis_2::Construct_point_2
+          construct_point = _m_curved_kernel->construct_point_2_object();
 
-    //!@}
+        *oi++ = Make_x_monotone_result(construct_point(x, _m_curve, j));
+      }
+    }
+    return oi;
+  }
 
-    //!\name Private data
-    //!@{
+  //!@}
 
-    //! pointer to \c Curved_kernel_via_analysis_2
-    Curved_kernel_via_analysis_2 *_m_curved_kernel;
+  //!\name Private data
+  //!@{
 
-    //! to avoid passing curve as a parameter
-    Curve_analysis_2 _m_curve;
+  //! pointer to \c Curved_kernel_via_analysis_2
+  Curved_kernel_via_analysis_2 *_m_curved_kernel;
 
-    //!@}
-};  // struct Make_x_monotone
+  //! to avoid passing curve as a parameter
+  Curve_analysis_2 _m_curve;
+
+  //!@}
+}; // struct Make_x_monotone
 
 } // namespace internal
 

Arrangement_on_surface_2/include/CGAL/Curved_kernel_via_analysis_2/Sweep_curves_adapter_2.h

@@ -699,13 +699,14 @@ public:
      * is by definition x-monotone, an input arc is passed to the
      * output iterator directly.
      * \param cv The curve.
-     * \param oi The output iterator, whose value-type is Object.
-     * The returned objects are all wrappers X_monotone_curve_2 objects.
-     * \return The past-the-end iterator.
+     * \param oi The output iteratorfor the result. Its dereference type is a
+     *        variant that wraps a \c Point_2 or an \c X_monotone_curve_2
+     *        objects..
+     * \return The past-the-end output iterator.
      */
     template<class OutputIterator>
-    OutputIterator operator()(const Generic_arc_2& cv,
-            OutputIterator oi) const {
+    OutputIterator operator()(const Generic_arc_2& cv, OutputIterator oi) const
+    {
         *oi++ = cv;
         return oi;
     }
@@ -714,32 +715,32 @@ public:
      * decompose a given curve into list of x-monotone pieces
      * (subcurves) and insert them to the output iterator.
      * \param cv The curve.
-     * \param oi The output iterator, whose value-type is Object.
-     * The returned objects are all wrappers X_monotone_curve_2 objects.
-     * \return The past-the-end iterator.
+     * \param oi the output iterator for the result. Its dereference type is a
+     *           variant that wraps a \c Point_2 or an \c X_monotone_curve_2
+     *           objects.
+     * \return The past-the-end output iterator.
      */
     template<class OutputIterator>
     OutputIterator operator()(const Curve_2& cv, OutputIterator oi) const {
 
-        typedef typename SweepCurvesAdapter_2::Native_arc_2 Native_arc_2;
-        typedef typename SweepCurvesAdapter_2::Native_point_2 Native_point_2;
-        typedef typename SweepCurvesAdapter_2::Generic_point_2 Generic_point_2;
+        typedef typename SweepCurvesAdapter_2::Native_arc_2     Native_arc_2;
+        typedef typename SweepCurvesAdapter_2::Native_point_2   Native_point_2;
+        typedef typename SweepCurvesAdapter_2::Generic_point_2  Generic_point_2;
+        typedef boost::variant<Native_arc_2, Native_point_2>
+          Make_x_monotone_result;
 
-        typedef std::vector<CGAL::Object> Objects;
-        Objects objs;
-        _m_adapter->kernel().make_x_monotone_2_object()(cv,
-            std::back_inserter(objs));
+        std::vector<Make_x_monotone_result> objs;
+        auto make_x_monotone = _m_adapter->kernel().make_x_monotone_2_object();
+        make_x_monotone(cv, std::back_inserter(objs));
         // sort out normal and degenerate arcs
-        for(typename Objects::const_iterator it = objs.begin();
-                it != objs.end(); it++) {
-            Native_arc_2 arc;
-            Native_point_2 pt;
-            if(CGAL::assign(arc, *it))
-                *oi++ = Generic_arc_2(arc);
-            else if(CGAL::assign(pt, *it))
-                *oi++ = Generic_arc_2(Generic_point_2(pt));
-            else
-                CGAL_error_msg("Bogus object..\n");
+        for (auto& obj : objs) {
+          if (auto* arc = boost::get<Native_arc_2>(&obj)) {
+            *oi++ = Generic_arc_2(*arc);
+            continue;
+          }
+          auto* pt = boost::get<Native_point_2>(&obj);
+          CGAL_assertion(pt);
+          *oi++ = Generic_arc_2(Generic_point_2(*pt));
         }
         return oi;
     }

Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_basic_insertion_traits_2.h

@@ -23,7 +23,6 @@
  * Defintion of the Arr_basic_insertion_traits_2<Traits,Arrangement> class.
  */
 
-#include <CGAL/Object.h>
 #include <CGAL/Arr_tags.h>
 
 #include <list>

Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_batched_pl_ss_visitor.h

@@ -23,7 +23,6 @@
  */
 
 #include <CGAL/Arr_point_location_result.h>
-#include <CGAL/Object.h>
 #include <CGAL/Surface_sweep_2/Default_visitor_base.h>
 #include <CGAL/Default.h>
 

Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_no_intersection_insertion_ss_visitor.h

@@ -8,9 +8,9 @@
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
 //
-// Author(s)     : Baruch Zukerman <baruchzu@post.tau.ac.il>
-//                 Ron Wein <wein@post.tau.ac.il>
-//                 Efi Fogel <efif@post.tau.ac.il>
+// Author(s): Baruch Zukerman <baruchzu@post.tau.ac.il>
+//            Ron Wein <wein@post.tau.ac.il>
+//            Efi Fogel <efif@post.tau.ac.il>
 
 #ifndef CGAL_ARR_NO_INTERSECTION_INSERTION_SS_VISITOR_H
 #define CGAL_ARR_NO_INTERSECTION_INSERTION_SS_VISITOR_H

Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_overlay_traits_2.h

@@ -26,7 +26,6 @@
 #include <boost/optional.hpp>
 
 #include <CGAL/Arr_tags.h>
-#include <CGAL/Object.h>
 
 namespace CGAL {
 

Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_vert_decomp_ss_visitor.h

@@ -7,9 +7,8 @@
 // $Id$
 // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
 //
-//
-// Author(s)     : Ron Wein <wein@post.tau.ac.il>
-//                 Efi Fogel <efifogel@gmail.com>
+// Author(s): Ron Wein <wein@post.tau.ac.il>
+//            Efi Fogel <efifogel@gmail.com>
 
 #ifndef CGAL_ARR_VERT_DECOMP_SS_VISITOR_H
 #define CGAL_ARR_VERT_DECOMP_SS_VISITOR_H
@@ -21,10 +20,11 @@
  * Definition of the Arr_vert_decomp_ss_visitor class-template.
  */
 
+#include <boost/variant.hpp>
+
 namespace CGAL {
 
 #include <CGAL/Surface_sweep_2/Default_visitor_base.h>
-#include <CGAL/Object.h>
 #include <CGAL/Default.h>
 
 /*! \class Arr_vert_decomp_ss_visitor
@@ -63,13 +63,18 @@ private:
 public:
   typedef typename Helper::Arrangement_2                Arrangement_2;
   typedef typename Arrangement_2::Vertex_const_handle   Vertex_const_handle;
+  typedef typename Arrangement_2::Halfedge_const_handle Halfedge_const_handle;
+  typedef typename Arrangement_2::Face_const_handle     Face_const_handle;
 
-  typedef std::pair<CGAL::Object, CGAL::Object>         Vert_pair;
+  typedef boost::variant<Vertex_const_handle, Halfedge_const_handle,
+                         Face_const_handle>
+    Cell_type;
+  typedef boost::optional<Cell_type>                    Vert_type;
+  typedef std::pair<Vert_type, Vert_type>               Vert_pair;
   typedef std::pair<Vertex_const_handle, Vert_pair>     Vert_entry;
 
 protected:
   typedef typename Base::Status_line_iterator           Status_line_iterator;
-  typedef typename Arrangement_2::Halfedge_const_handle Halfedge_const_handle;
   //typedef typename Arrangement_2::Vertex_const_handle   Vertex_const_handle;
   typedef typename Arrangement_2::Halfedge_around_vertex_const_circulator
     Halfedge_around_vertex_const_circulator;
@@ -84,8 +89,8 @@ protected:
                                         // An invalid vertex handle.
 
   Vertex_const_handle m_prev_vh;        // The previous vertex.
-  CGAL::Object m_prev_obj_below;        // The object this vertex sees below it.
-  CGAL::Object m_prev_obj_above;        // The object this vertex sees above it.
+  Vert_type m_prev_obj_below;           // The object this vertex sees below it.
+  Vert_type m_prev_obj_above;           // The object this vertex sees above it.
 
   Output_iterator* m_out;               // An output iterator for the result.
 
@@ -155,7 +160,7 @@ after_handle_event(Event* event,
   // Get the vertex handle associated with the current event (stored with
   // the point).
   Vertex_const_handle vh = event->point().vertex_handle();
-  CGAL::Object obj_above, obj_below;
+  Vert_type obj_above, obj_below;
 
   // Check the feature from above.
   if (above == this->status_line_end()) {
@@ -166,8 +171,7 @@ after_handle_event(Event* event,
   else {
     // We have a valid subcurve above the event: get its halfedge handle
     // and associate it with the vertex.
-    obj_above =
-      CGAL::make_object((*above)->last_curve().halfedge_handle());
+    obj_above = Vert_type((*above)->last_curve().halfedge_handle());
   }
 
   // Check if the previous vertex we handled has the same x-coordinate
@@ -206,12 +210,12 @@ after_handle_event(Event* event,
       }
 
       if (! vert_connected) {
-        obj_below = CGAL::make_object(m_prev_vh);
-        m_prev_obj_above = CGAL::make_object(vh);
+        obj_below = Vert_type(m_prev_vh);
+        m_prev_obj_above = Vert_type(vh);
       }
       else {
-        obj_below = CGAL::Object();
-        m_prev_obj_above = CGAL::Object();
+        obj_below = Vert_type();
+        m_prev_obj_above = Vert_type();
       }
     }
     else {
@@ -260,19 +264,18 @@ after_handle_event(Event* event,
         }
 
         if (! vert_connected) {
-          obj_below = CGAL::make_object(m_prev_vh);
-          m_prev_obj_above = CGAL::make_object(vh);
+          obj_below = Vert_type(m_prev_vh);
+          m_prev_obj_above = Vert_type(vh);
         }
         else {
-          obj_below = CGAL::Object();
-          m_prev_obj_above = CGAL::Object();
+          obj_below = Vert_type();
+          m_prev_obj_above = Vert_type();
         }
       }
       else {
         // Get the halfedge handle of the subcurve below the current event and
         // associate it with its vertex.
-        obj_below =
-          CGAL::make_object((*below)->last_curve().halfedge_handle());
+        obj_below = Vert_type((*below)->last_curve().halfedge_handle());
       }
     }
 

Arrangement_on_surface_2/test/Arrangement_on_surface_2/Batched_point_location_test.h

@@ -208,8 +208,7 @@ verify(InputIterator begin, InputIterator end)
   typename TopolTraits::Default_point_location_strategy pl(m_arr);
   for (InputIterator it = begin; it != end; ++it) {
 
-    if (m_verbose_level > 1)
-      print(*it);
+    if (m_verbose_level > 1) print(*it);
 
     // Perform (single) point location.
     Result_type obj = pl.locate(it->first);

Arrangement_on_surface_2/test/Arrangement_on_surface_2/IO_base_test.h

@@ -71,10 +71,10 @@ read_xcurve(InputStream_& is, typename GeomTraits_::X_monotone_curve_2& xcv)
   typedef GeomTraits_   Geom_traits;
   Basic_number_type x1, y1, x2, y2;
   is >> x1 >> y1 >> x2 >> y2;
-  CGAL_assertion(!is.fail());
+  assert(!is.fail());
   Point_2 p1(x1, y1);
   Point_2 p2(x2, y2);
-  CGAL_assertion(p1 != p2);
+  assert(p1 != p2);
   xcv = typename Geom_traits::X_monotone_curve_2(p1, p2);
   return true;
 }
@@ -89,7 +89,7 @@ read_curve(InputStream_& is, typename GeomTraits_::Curve_2& cv)
   is >> x1 >> y1 >> x2 >> y2;
   Point_2 p1(x1, y1);
   Point_2 p2(x2, y2);
-  CGAL_assertion(p1 != p2);
+  assert(p1 != p2);
   cv = typename Geom_traits::Curve_2(p1, p2);
   return true;
 }
@@ -690,21 +690,20 @@ bool IO_base_test<Base_geom_traits>::read_xsegment(InputStream_& is,
     point_vector.push_back(Control_point_2(point_x, point_y));
   }
   //get the non x-monotone bezier segment
-  Subcurve_2 seg (point_vector.begin(), point_vector.end());
+  Subcurve_2 seg(point_vector.begin(), point_vector.end());
 
   //convert it into x-monotone bezier segment.
-  std::vector<CGAL::Object> obj_vector;
-  bezier_traits.make_x_monotone_2_object()(seg,
-                                           std::back_inserter(obj_vector));
-  X_monotone_subcurve_2 x_segment =
-    CGAL::object_cast<X_monotone_subcurve_2>((obj_vector[0]));
-
-  xseg = x_segment;
+  typedef boost::variant<Point_2, X_monotone_subcurve_2> Make_x_monotone_result;
+  std::vector<Make_x_monotone_result> objs;
+  bezier_traits.make_x_monotone_2_object()(seg, std::back_inserter(objs));
+  assert(! objs.empty());
+  const auto* x_seg_p = boost::get<X_monotone_subcurve_2>(&(objs[0]));
+  assert(x_seg_p);
+  xseg = *x_seg_p;
 
   return true;
 }
 
-
 template <>
 template <typename InputStream_>
 bool IO_base_test<Base_geom_traits>::read_xcurve(InputStream_& is,
@@ -721,6 +720,8 @@ bool IO_base_test<Base_geom_traits>::read_xcurve(InputStream_& is,
   unsigned int number_of_segments;
   is >> number_of_segments;
 
+  typedef boost::variant<Point_2, X_monotone_subcurve_2> Make_x_monotone_result;
+  auto make_x_monotone = bezier_traits.make_x_monotone_2_object();
   if ((type == 'x') || (type == 'X')) {
     for (unsigned int i=0; i<number_of_segments; ++i) {
       unsigned int num_control_points;
@@ -737,13 +738,12 @@ bool IO_base_test<Base_geom_traits>::read_xcurve(InputStream_& is,
       Subcurve_2 seg(point_vector.begin(), point_vector.end());
 
       //convert it into x-monotone bezier segment.
-      std::vector<CGAL::Object> obj_vector;
-      bezier_traits.make_x_monotone_2_object()(seg,
-                                               std::back_inserter(obj_vector));
-      X_monotone_subcurve_2 x_seg =
-        CGAL::object_cast<X_monotone_subcurve_2>((obj_vector[0]));
-
-      x_segments.push_back(x_seg);
+      std::vector<Make_x_monotone_result> objs;
+      make_x_monotone(seg, std::back_inserter(objs));
+      assert(! objs.empty());
+      const auto* x_seg_p = boost::get<X_monotone_subcurve_2>(&(objs[0]));
+      assert(x_seg_p);
+      x_segments.push_back(*x_seg_p);
 
     } //for loop (number of segments)
   }
@@ -1470,8 +1470,8 @@ bool IO_base_test<Base_geom_traits>::read_xcurve(InputStream_& is,
                                                  X_monotone_curve_2& xcv)
 {
   std::cout << std::endl;
-  std::list<CGAL::Object> x_objs;
-  std::list<CGAL::Object>::const_iterator xoit;
+  typedef boost::variant<Point_2, X_monotone_curve_2> Make_x_monotone_result;
+  std::list<Make_x_monotone_result> x_objs;
   Curve_2 tmp_cv;
   is >> tmp_cv;
   Rational B_psx = Rational(tmp_cv.control_point(0).x());
@@ -1482,15 +1482,15 @@ bool IO_base_test<Base_geom_traits>::read_xcurve(InputStream_& is,
     Rational(tmp_cv.control_point(tmp_cv.number_of_control_points()-1).y());
   Point_2 B_ps(B_psx, B_psy);
   Point_2 B_pt(B_ptx, B_pty);
-  Base_geom_traits::Make_x_monotone_2 make_x_monotone =
-    this->m_geom_traits.make_x_monotone_2_object();
-  make_x_monotone(tmp_cv, std::front_inserter (x_objs));
-  xoit = x_objs.begin();
+  auto make_x_monotone = this->m_geom_traits.make_x_monotone_2_object();
+  make_x_monotone(tmp_cv, std::front_inserter(x_objs));
+  auto xoit = x_objs.begin();
   size_t id(0);
-  if (!is.eof()) is >> id;
+  if (! is.eof()) is >> id;
   std::advance(xoit, id);
-  if (CGAL::assign(xcv, *xoit))
-    return true;
+  const auto* xcv_p = boost::get<X_monotone_curve_2>(&*xoit);
+  assert(xcv_p);
+  xcv = *xcv_p;
   return false;
 }
 
@@ -1546,9 +1546,8 @@ bool IO_base_test<Base_geom_traits>::read_point(InputStream_& is, Point_2& p) {
      is >> x;
      Base_geom_traits::X_monotone_curve_2 xcv;
      CGAL::swallow(is, '(');
-     CGAL_assertion_code(bool check = )
-     read_xcurve(is, xcv);
-     CGAL_assertion(check);
+     bool check = read_xcurve(is, xcv);
+     assert(check);
 
      CGAL::swallow(is, ')');
      p = construct_point_2(x, xcv);
@@ -1558,9 +1557,8 @@ bool IO_base_test<Base_geom_traits>::read_point(InputStream_& is, Point_2& p) {
      Base_geom_traits::Algebraic_real_1 x;
      is >> x;
      Base_geom_traits::Curve_2 c;
-     CGAL_assertion_code(bool check = )
-     read_curve(is, c);
-     CGAL_assertion(check);
+     bool check = read_curve(is, c);
+     assert(check);
      int arcno = 0;
      is >> arcno;
      p = construct_point_2(x, c, arcno);
@@ -1589,36 +1587,31 @@ bool IO_base_test<Base_geom_traits>::read_xcurve(InputStream_& is,
   switch (type) {
    case '1': {
      Curve_2 cv;
-     Point_2 end_left,end_right;
-     CGAL_assertion_code(bool check = )
-     read_curve(is,cv);
-     CGAL_assertion(check);
+     Point_2 end_left, end_right;
+     bool check = read_curve(is,cv);
+     assert(check);
      CGAL::swallow(is, '(');
-     CGAL_assertion_code(check = )
-     read_point(is,end_left);
-     CGAL_assertion(check);
+     check = read_point(is,end_left);
+     assert(check);
      CGAL::swallow(is, ')');
      CGAL::swallow(is, '(');
-     CGAL_assertion_code(check = )
-     read_point(is,end_right);
-     CGAL_assertion(check);
+     check = read_point(is,end_right);
+     assert(check);
      CGAL::swallow(is, ')');
      std::vector<Base_geom_traits::X_monotone_curve_2> xcvs;
      construct_segment_2(cv, end_left, end_right, std::back_inserter(xcvs));
-     CGAL_assertion(xcvs.size() == 1);
+     assert(xcvs.size() == 1);
      xcv = xcvs[0];
      break;
     }
    case '2': {
      Curve_2 cv;
      Point_2 p;
-     CGAL_assertion_code(bool check = )
-     read_curve(is, cv);
-     CGAL_assertion(check);
+     bool check = read_curve(is, cv);
+     assert(check);
      CGAL::swallow(is, '(');
-     CGAL_assertion_code(check = )
-     read_point(is, p);
-     CGAL_assertion(check);
+     check = read_point(is, p);
+     assert(check);
      CGAL::swallow(is, ')');
      std::string site_of_p_string;
      Base_geom_traits::Site_of_point site_of_p;
@@ -1630,12 +1623,12 @@ bool IO_base_test<Base_geom_traits>::read_xcurve(InputStream_& is,
        site_of_p = Base_geom_traits::MAX_ENDPOINT;
      }
      else {
-       CGAL_assertion(site_of_p_string == "POINT_IN_INTERIOR");
+       assert(site_of_p_string == "POINT_IN_INTERIOR");
        site_of_p = Base_geom_traits::POINT_IN_INTERIOR;
      }
      std::vector<Base_geom_traits::X_monotone_curve_2> xcvs;
      construct_segment_2(cv, p, site_of_p, std::back_inserter(xcvs));
-     CGAL_assertion(xcvs.size() == 1);
+     assert(xcvs.size() == 1);
      xcv = xcvs[0];
      break;
     }
@@ -1685,7 +1678,7 @@ bool IO_base_test<Base_geom_traits>::read_xcurve(InputStream_& is,
   Point_2 p1, p2;
   read_point(is, p1);
   read_point(is, p2);
-  CGAL_assertion(p1 != p2);
+  assert(p1 != p2);
 
   unsigned int flag;
   is >> flag;
@@ -1707,7 +1700,7 @@ bool IO_base_test<Base_geom_traits>::read_curve(InputStream_& is, Curve_2& cv)
   Point_2 p1, p2;
   read_point(is, p1);
   read_point(is, p2);
-  CGAL_assertion(p1 != p2);
+  assert(p1 != p2);
   unsigned int flag;
   is >> flag;
   if (flag == 1) {

Arrangement_on_surface_2/test/Arrangement_on_surface_2/Point_location_test.h

@@ -262,7 +262,7 @@ private:
   void allocate_pl()
   {
     Strategy* locator = new Strategy();
-    CGAL_assertion(locator);
+    assert(locator);
     m_locators[id].m_variant = locator;
   }
 

Arrangement_on_surface_2/test/Arrangement_on_surface_2/Traits_base_test.h

@@ -13,7 +13,6 @@
 #include <boost/lexical_cast.hpp>
 
 #include <CGAL/exceptions.h>
-#include <CGAL/Object.h>
 #include <CGAL/Arr_tags.h>
 #include <CGAL/Arr_enums.h>
 #include <CGAL/use.h>

Arrangement_on_surface_2/test/Arrangement_on_surface_2/Traits_test.h

@@ -13,7 +13,6 @@
 #include <boost/lexical_cast.hpp>
 
 #include <CGAL/exceptions.h>
-#include <CGAL/Object.h>
 #include <CGAL/Arr_tags.h>
 #include <CGAL/Arrangement_2/Arr_traits_adaptor_2_dispatching.h>
 #include <CGAL/use.h>
@@ -931,22 +930,22 @@ template <typename Geom_traits_T>
 bool Traits_test<Geom_traits_T>::
 make_x_monotone_wrapper(std::istringstream& str_stream)
 {
-  typedef Geom_traits_T                         Traits;
-  typedef typename Traits::Point_2              Point_2;
-  typedef typename Traits::X_monotone_curve_2   X_monotone_curve_2;
+  typedef Geom_traits_T                                 Traits;
+  typedef typename Traits::Point_2                      Point_2;
+  typedef typename Traits::X_monotone_curve_2           X_monotone_curve_2;
+  typedef boost::variant<Point_2, X_monotone_curve_2>   Make_x_monotone_result;
   CGAL_USE_TYPE(typename Traits::Curve_2);
 
   unsigned int id;
   str_stream >> id;
-  std::cout << "Test: make_x_monotone( " << this->m_curves[id]
-            << " ) ? ";
-  std::vector<CGAL::Object> object_vec;
+  std::cout << "Test: make_x_monotone( " << this->m_curves[id] << " ) ? ";
+  std::vector<Make_x_monotone_result> objs;
   this->m_geom_traits.make_x_monotone_2_object()(this->m_curves[id],
-                                                 std::back_inserter(object_vec));
+                                                 std::back_inserter(objs));
 
   size_t num;
   str_stream >> num;
-  if (!this->compare(num, object_vec.size(), "size")) return false;
+  if (!this->compare(num, objs.size(), "size")) return false;
 
   for (size_t i = 0; i < num; ++i) {
     unsigned int type;                  // 0 - point, 1 - x-monotone curve
@@ -955,20 +954,19 @@ make_x_monotone_wrapper(std::istringstream& str_stream)
     unsigned int id;                    // The id of the point or x-monotone
     str_stream >> id;                   // ... curve respectively
 
-    const X_monotone_curve_2 * xcv_ptr =
-      CGAL::object_cast<X_monotone_curve_2> (&(object_vec[i]));
-    if (xcv_ptr != NULL) {
+    const auto* xcv_ptr = boost::get<X_monotone_curve_2>(&(objs[i]));
+    if (xcv_ptr != nullptr) {
       if (!this->compare(type, 1u, "type")) return false;
       if (!this->compare_curves(this->m_xcurves[id], *xcv_ptr)) return false;
       continue;
     }
 
-    const Point_2 * pt_ptr = CGAL::object_cast<Point_2> (&(object_vec[i]));
-    assert (pt_ptr != NULL);
+    const auto* pt_ptr = boost::get<Point_2>(&(objs[i]));
+    assert(pt_ptr != nullptr);
     if (!this->compare(type, 0u, "type")) return false;
     if (!this->compare_points(this->m_points[id], *pt_ptr)) return false;
   }
-  object_vec.clear();
+  objs.clear();
   return true;
 }
 

Arrangement_on_surface_2/test/Arrangement_on_surface_2/Vertical_decomposition_test.h

@@ -12,11 +12,11 @@
 #include "IO_test.h"
 
 /*! Point location test */
-template <typename GeomTraits_T, typename TopolTraits_T>
-class Vertical_decomposition_test : public IO_test<GeomTraits_T> {
+template <typename GeomTraits_2, typename TopolTraits>
+class Vertical_decomposition_test : public IO_test<GeomTraits_2> {
 public:
-  typedef GeomTraits_T                                    Geom_traits;
-  typedef TopolTraits_T                                   Topol_traits;
+  typedef GeomTraits_2                                    Geom_traits;
+  typedef TopolTraits                                   Topol_traits;
 
 private:
   typedef IO_test<Geom_traits>                            Base;
@@ -44,8 +44,11 @@ public:
   typedef typename Arrangement::Edge_const_iterator       Edge_const_iterator;
   typedef typename Arrangement::Vertex_const_iterator     Vertex_const_iterator;
 
-  typedef typename std::pair<CGAL::Object, CGAL::Object>  Object_pair;
-  typedef typename std::pair<Vertex_const_handle, Object_pair>
+  typedef boost::variant<Vertex_const_handle, Halfedge_const_handle,
+                         Face_const_handle>               Cell_type;
+  typedef boost::optional<Cell_type>                      Vert_type;
+  typedef typename std::pair<Vert_type, Vert_type>        Vert_pair;
+  typedef typename std::pair<Vertex_const_handle, Vert_pair>
                                                           Vert_decomp_entry;
   typedef typename std::list<Vert_decomp_entry>           Vert_decomp_list;
 
@@ -69,7 +72,7 @@ protected:
 
   /*! Compare the results.
    */
-  bool compare(const Result_type& expected, const CGAL::Object& actual);
+  bool compare(const Result_type& expected, Vert_type actual);
 
   /*! print the results.
    */
@@ -104,8 +107,8 @@ public:
 /*!
  * Constructor from a geometry traits object.
  */
-template <typename GeomTraits_T, typename TopolTraits_T>
-Vertical_decomposition_test<GeomTraits_T, TopolTraits_T>::
+template <typename GeomTraits_2, typename TopolTraits>
+Vertical_decomposition_test<GeomTraits_2, TopolTraits>::
 Vertical_decomposition_test(const Geom_traits& geom_traits) :
   Base(geom_traits),
   m_geom_traits(geom_traits),
@@ -113,24 +116,24 @@ Vertical_decomposition_test(const Geom_traits& geom_traits) :
 {}
 
 //! \brief sets the verbosity level.
-template <typename GeomTraits_T, typename TopolTraits_T>
-void Vertical_decomposition_test<GeomTraits_T, TopolTraits_T>::
+template <typename GeomTraits_2, typename TopolTraits>
+void Vertical_decomposition_test<GeomTraits_2, TopolTraits>::
 set_verbose_level(size_t verbose_level)
 { m_verbose_level = verbose_level; }
 
 /*! Clear the data structures */
-template <typename GeomTraits_T, typename TopolTraits_T>
-void Vertical_decomposition_test<GeomTraits_T, TopolTraits_T>::clear()
+template <typename GeomTraits_2, typename TopolTraits>
+void Vertical_decomposition_test<GeomTraits_2, TopolTraits>::clear()
 {
   m_arr.clear();
   Base::clear();
 }
 
-template <typename GeomTraits_T, typename TopolTraits_T>
-bool Vertical_decomposition_test<GeomTraits_T, TopolTraits_T>::init()
+template <typename GeomTraits_2, typename TopolTraits>
+bool Vertical_decomposition_test<GeomTraits_2, TopolTraits>::init()
 {
   // Initialize the input.
-  if (!Base::init()) return false;
+  if (! Base::init()) return false;
 
   // Insert all into the arrangement
   CGAL::insert(m_arr, this->m_xcurves.begin(), this->m_xcurves.end());
@@ -147,8 +150,8 @@ bool Vertical_decomposition_test<GeomTraits_T, TopolTraits_T>::init()
 }
 
 //! \brief performs the test.
-template <typename GeomTraits_T, typename TopolTraits_T>
-bool Vertical_decomposition_test<GeomTraits_T, TopolTraits_T>::perform()
+template <typename GeomTraits_2, typename TopolTraits>
+bool Vertical_decomposition_test<GeomTraits_2, TopolTraits>::perform()
 {
   // Apply vertical decomposition.
   Vert_decomp_list results;
@@ -159,51 +162,48 @@ bool Vertical_decomposition_test<GeomTraits_T, TopolTraits_T>::perform()
 }
 
 //! \brief verifies the results.
-template <typename GeomTraits_T, typename TopolTraits_T>
+template <typename GeomTraits_2, typename TopolTraits>
 template <typename InputIterator>
-bool Vertical_decomposition_test<GeomTraits_T, TopolTraits_T>::
+bool Vertical_decomposition_test<GeomTraits_2, TopolTraits>::
 verify(InputIterator begin, InputIterator end)
 {
-  typedef TopolTraits_T TopolTraits;
-
-  InputIterator it;
-  if (m_verbose_level > 1) for (it = begin; it != end; ++it) print(*it);
+  if (m_verbose_level > 1) for (auto it = begin; it != end; ++it) print(*it);
 
   // Compare the results.
   typename TopolTraits::Default_vertical_ray_shooting_strategy vs(m_arr);
-  for (it = begin; it != end; ++it) {
+  for (auto it = begin; it != end; ++it) {
     Vertex_const_handle vh = it->first;
-    const Object_pair& res = it->second;
-    const CGAL::Object& obj_below_actual = res.first;
-    const CGAL::Object& obj_above_actual = res.second;
+    const auto& res = it->second;
+    auto obj_below_actual = res.first;
+    auto obj_above_actual = res.second;
 
     Result_type obj_below_expected = vs.ray_shoot_down(vh->point());
     Result_type obj_above_expected = vs.ray_shoot_up(vh->point());
 
-    if (!compare(obj_below_expected, obj_below_actual)) return false;
-    if (!compare(obj_above_expected, obj_above_actual)) return false;
+    if (! compare(obj_below_expected, obj_below_actual)) return false;
+    if (! compare(obj_above_expected, obj_above_actual)) return false;
   }
 
   return true;
 }
 
-template <typename GeomTraits_T, typename TopolTraits_T>
-bool Vertical_decomposition_test<GeomTraits_T, TopolTraits_T>::
-compare(const Result_type& expected, const CGAL::Object& actual)
+template <typename GeomTraits_2, typename TopolTraits>
+bool Vertical_decomposition_test<GeomTraits_2, TopolTraits>::
+compare(const Result_type& expected, Vert_type actual)
 {
+  // This test does not test the case where the result is empty!
+  if (! actual) return false;
+
+  auto obj = *actual;
   // Assign object to a fase.
-  const Face_const_handle* fh_expected =
-    boost::get<Face_const_handle>(&(expected));
-  if (fh_expected) {
-    Vertex_const_handle vh_actual;
-    if (CGAL::assign(vh_actual, actual)) {
+  if (const auto* fh_expected = boost::get<Face_const_handle>(&(expected))) {
+    if (boost::get<Vertex_const_handle>(&obj)) {
       std::cout << "Error: vertical decomposition!" << std::endl;
       std::cout << "Expected: a face."  << std::endl;
       std::cout << "Actual: a vertex." << std::endl;
       return false;
     }
-    Halfedge_const_handle hh_actual;
-    if (CGAL::assign(hh_actual, actual)) {
+    if (boost::get<Halfedge_const_handle>(&obj)) {
       std::cout << "Error: vertical decomposition!" << std::endl;
       std::cout << "Expected: a face."  << std::endl;
       std::cout << "Actual: a halfedge." << std::endl;
@@ -213,17 +213,15 @@ compare(const Result_type& expected, const CGAL::Object& actual)
   }
 
   // Assign object to a halfedge.
-  const Halfedge_const_handle* hh_expected =
-    boost::get<Halfedge_const_handle>(&(expected));
+  const auto* hh_expected = boost::get<Halfedge_const_handle>(&(expected));
   if (hh_expected) {
-    Halfedge_const_handle hh_actual;
-    if (CGAL::assign(hh_actual, actual)) {
-      if (*hh_expected == hh_actual) return true;
+    if (const auto* hh_actual = boost::get<Halfedge_const_handle>(&obj)) {
+      if (*hh_expected == *hh_actual) return true;
 
       std::cout << "Error: vertical decomposition!" << std::endl;
       std::cout << "Expected: a halfedge, " << (*hh_expected)->curve()
                 << std::endl;
-      std::cout << "Actual: a different halfedge." << hh_actual->curve()
+      std::cout << "Actual: a different halfedge." << (*hh_actual)->curve()
                 << std::endl;
       return false;
     }
@@ -232,14 +230,13 @@ compare(const Result_type& expected, const CGAL::Object& actual)
     std::cout << "Expected: a halfedge, " << (*hh_expected)->curve()
               << std::endl;
 
-    Vertex_const_handle vh_actual;
-    if (CGAL::assign(vh_actual, actual)) {
-      std::cout << "Actual: a vertex, " << vh_actual->point() << std::endl;
+    if (const auto* vh_actual = boost::get<Vertex_const_handle>(&obj)) {
+      std::cout << "Actual: a vertex, " << (*vh_actual)->point() << std::endl;
       return false;
     }
 
     Face_const_handle fh_actual;
-    if (CGAL::assign(fh_actual, actual)) {
+    if (boost::get<Face_const_handle>(&obj)) {
       std::cout << "Actual: a face." << std::endl;
       return false;
     }
@@ -248,17 +245,15 @@ compare(const Result_type& expected, const CGAL::Object& actual)
   }
 
   // Assign object to a vertex.
-  const Vertex_const_handle* vh_expected =
-    boost::get<Vertex_const_handle>(&(expected));
+  const auto* vh_expected = boost::get<Vertex_const_handle>(&(expected));
   if (vh_expected) {
-    Vertex_const_handle vh_actual;
-    if (CGAL::assign(vh_actual, actual)) {
-      if (*vh_expected == vh_actual) return true;
+    if (const auto* vh_actual = boost::get<Vertex_const_handle>(&obj)) {
+      if (*vh_expected == *vh_actual) return true;
 
       std::cout << "Error: vertical decomposition!" << std::endl;
       std::cout << "Expected: a vertex, " << (*vh_expected)->point()
                 << std::endl;
-      std::cout << "Actual: a different vertex, " << vh_actual->point()
+      std::cout << "Actual: a different vertex, " << (*vh_actual)->point()
                 << std::endl;
       return false;
     }
@@ -266,14 +261,12 @@ compare(const Result_type& expected, const CGAL::Object& actual)
     std::cout << "Error: vertical decomposition!" << std::endl;
     std::cout << "Expected: a vertex, " << (*vh_expected)->point() << std::endl;
 
-    Halfedge_const_handle hh_actual;
-    if (CGAL::assign(hh_actual, actual)) {
-      std::cout << "Actual: a halfedge, " << hh_actual->curve() << std::endl;
+    if (const auto* hh_actual = boost::get<Halfedge_const_handle>(&obj)) {
+      std::cout << "Actual: a halfedge, " << (*hh_actual)->curve() << std::endl;
       return false;
     }
 
-    Face_const_handle fh_actual;
-    if (CGAL::assign(fh_actual, actual)) {
+    if (boost::get<Face_const_handle>(&obj)) {
       std::cout << "Actual: a face." << std::endl;
       return false;
     }
@@ -285,28 +278,36 @@ compare(const Result_type& expected, const CGAL::Object& actual)
 }
 
 //! \brief prints the results.
-template <typename GeomTraits_T, typename TopolTraits_T>
-void Vertical_decomposition_test<GeomTraits_T, TopolTraits_T>::
+template <typename GeomTraits_2, typename TopolTraits>
+void Vertical_decomposition_test<GeomTraits_2, TopolTraits>::
 print(const Vert_decomp_entry& result)
 {
-  // Print the result.
-  Vertex_const_handle vh;
-  Halfedge_const_handle hh;
-  Face_const_handle fh;
+  const auto& res = result.second;
+  auto obj_below = res.first;
+  auto obj_above = res.second;
 
-  const Object_pair& res = result.second;
   std::cout << "Vertex (" << result.first->point() << ") : ";
 
+  assert(obj_below);
+  auto obj = *obj_below;
   std::cout << "  feature below: ";
-  if (CGAL::assign(hh, res.first)) std::cout << '[' << hh->curve() << ']';
-  else if (CGAL::assign(vh, res.first)) std::cout << '(' << vh->point() << ')';
-  else if (CGAL::assign(fh, res.first)) std::cout << "NONE";
+  if (const auto* hh  = boost::get<Halfedge_const_handle>(&obj))
+    std::cout << '[' << (*hh)->curve() << ']';
+  else if (const auto* vh = boost::get<Vertex_const_handle>(&obj))
+    std::cout << '(' << (*vh)->point() << ')';
+  else if (const auto* fh = boost::get<Face_const_handle>(&obj))
+    std::cout << "NONE";
   else std::cout << "EMPTY";
 
+  assert(obj_above);
+  obj = *obj_above;
   std::cout << "  feature above: ";
-  if (CGAL::assign(hh, res.second)) std::cout << '[' << hh->curve() << ']';
-  else if (CGAL::assign(vh, res.second)) std::cout << '(' << vh->point() << ')';
-  else if (CGAL::assign(fh, res.second)) std::cout << "NONE";
+  if (const auto* hh  = boost::get<Halfedge_const_handle>(&obj))
+    std::cout << '[' << (*hh)->curve() << ']';
+  else if (const auto* vh = boost::get<Vertex_const_handle>(&obj))
+    std::cout << '(' << (*vh)->point() << ')';
+  else if (const auto* vh = boost::get<Face_const_handle>(&obj))
+    std::cout << "NONE";
   else std::cout << "EMPTY";
 
   std::cout << std::endl;

Arrangement_on_surface_2/test/Arrangement_on_surface_2/test_arc_polycurve.cpp

@@ -1,64 +1,48 @@
-
-// Constructing an arrangement of polycurves.
-
-#include <CGAL/config.h>
-#ifndef CGAL_USE_CORE
-#include <iostream>
-int main()
-{
-  std::cout << "Sorry, this example needs CORE ..." << std::endl;
-  return 0;
-}
-
-#else
-
-#include <CGAL/Cartesian.h>
-#include <CGAL/Quotient.h>
-#include <CGAL/MP_Float.h>
-#include <CGAL/CORE_algebraic_number_traits.h>
 #include <vector>
 #include <list>
 
+#include <CGAL/config.h>
+#include <CGAL/Cartesian.h>
+#include <CGAL/Quotient.h>
+#include <CGAL/MP_Float.h>
 #include <CGAL/Arr_polyline_traits_2.h>
 #include <CGAL/Arr_circle_segment_traits_2.h>
-#include <CGAL/Arrangement_2.h>
-
 
 ///////////////
 //circle segment traits
 //////////////
 typedef CGAL::Quotient<CGAL::MP_Float>               Number_type;
 typedef CGAL::Cartesian<Number_type>                 Kernel;
-typedef CGAL::Arr_circle_segment_traits_2<Kernel>    Arc_traits_2;
-typedef CGAL::Arr_polyline_traits_2<Arc_traits_2>    Polycurve_arc_traits_2;
-typedef Arc_traits_2::CoordNT                        CoordNT;
-typedef Arc_traits_2::Point_2                        Arc_point_2;
-typedef Arc_traits_2::Curve_2                        Arc_section_2;
-typedef Arc_traits_2::X_monotone_curve_2             Arc_section_x_monotone_2;
-typedef CGAL::Arrangement_2<Polycurve_arc_traits_2>  Arc_arrangment_2;
+typedef CGAL::Arr_circle_segment_traits_2<Kernel>    Sub_traits_2;
+typedef CGAL::Arr_polyline_traits_2<Sub_traits_2>    Traits_2;
+typedef Sub_traits_2::CoordNT                        CoordNT;
+typedef Sub_traits_2::Point_2                        Point_2;
+typedef Sub_traits_2::Curve_2                        Subcurve_2;
+typedef Sub_traits_2::X_monotone_curve_2             Subcurve_x_monotone_2;
+typedef boost::variant<Point_2, Subcurve_x_monotone_2>
+  Make_sub_x_monotone_result;
 
 void check_equal()
 {
-  Polycurve_arc_traits_2 Polycurve_traits_2;
-  Polycurve_arc_traits_2::Equal_2 equal_2 = Polycurve_traits_2.equal_2_object();
-  Polycurve_arc_traits_2::Construct_x_monotone_curve_2
-    construct_x_monotone_curve_2 =
+  Traits_2 Polycurve_traits_2;
+  auto equal_2 = Polycurve_traits_2.equal_2_object();
+  auto construct_x_monotone_curve_2 =
     Polycurve_traits_2.construct_x_monotone_curve_2_object();
 
-  Arc_section_2 curve1, curve2, curve3;
+  Subcurve_2 curve1, curve2, curve3;
 
   Kernel::Point_2 p1 = Kernel::Point_2(-5, 0);
   Kernel::Point_2 mid = Kernel::Point_2(0, 5);
   Kernel::Point_2 p2 = Kernel::Point_2(5, 0);
-  curve1= Arc_section_2(p1, mid, p2);
-  curve2= Arc_section_2(p1, mid, p2);
+  curve1= Subcurve_2(p1, mid, p2);
+  curve2= Subcurve_2(p1, mid, p2);
 
   // //make x_monotone
-  // Polycurve_arc_traits_2::X_monotone_curve_2 xmc1 =
+  // Traits_2::X_monotone_curve_2 xmc1 =
   //   construct_x_monotone_curve_2(curve1);
-  // Polycurve_arc_traits_2::X_monotone_curve_2 xmc2 =
+  // Traits_2::X_monotone_curve_2 xmc2 =
   //   construct_x_monotone_curve_2(curve2);
-  // Polycurve_arc_traits_2::X_monotone_curve_2 xmc3 =
+  // Traits_2::X_monotone_curve_2 xmc3 =
   //   construct_x_monotone_curve_2(curve3);
 
   // //simple equal
@@ -71,11 +55,11 @@ void check_equal()
   //           << ((Are_equal) ? "Equal" : "Not equal") << std::endl;
 }
 
-void check_intersect(Polycurve_arc_traits_2::Make_x_monotone_2
+void check_intersect(Traits_2::Make_x_monotone_2
                      make_x_monotone_2,
-                     Polycurve_arc_traits_2::Intersect_2 intersect_2)
+                     Traits_2::Intersect_2 intersect_2)
 {
-  Arc_section_2 curve1, curve2, curve3;
+  Subcurve_2 curve1, curve2, curve3;
 
   // Create a circular arc that correspond to the upper half of the
   // circle centered at (1,1) with squared radius 3. We create the
@@ -85,24 +69,22 @@ void check_intersect(Polycurve_arc_traits_2::Make_x_monotone_2
   Kernel::Circle_2 circ1 = Kernel::Circle_2(c1, 3, CGAL::CLOCKWISE);
   CoordNT one_minus_sqrt_3 = CoordNT(1, -1, 3);
   CoordNT one_plus_sqrt_3 = CoordNT(1, 1, 3);
-  Arc_point_2 s1 = Arc_point_2(one_minus_sqrt_3, CoordNT(1));
-  Arc_point_2 t1 = Arc_point_2(one_plus_sqrt_3, CoordNT(1));
-  curve1 = Arc_section_2(circ1, s1, t1);
-  curve2 = Arc_section_2(circ1, s1, t1);
+  Point_2 s1 = Point_2(one_minus_sqrt_3, CoordNT(1));
+  Point_2 t1 = Point_2(one_plus_sqrt_3, CoordNT(1));
+  curve1 = Subcurve_2(circ1, s1, t1);
+  curve2 = Subcurve_2(circ1, s1, t1);
   //push the same semi circle again
-  //curves.push_back(Arc_section_2(circ1, s1, t1));
+  //curves.push_back(Subcurve_2(circ1, s1, t1));
 
   //make x_monotone
-  std::vector<CGAL::Object> X_monotone_curves;
-  make_x_monotone_2(curve1, std::back_inserter(X_monotone_curves));
-  make_x_monotone_2(curve2, std::back_inserter(X_monotone_curves));
+  std::vector<Make_sub_x_monotone_result> x_monotone_curves;
+  make_x_monotone_2(curve1, std::back_inserter(x_monotone_curves));
+  make_x_monotone_2(curve2, std::back_inserter(x_monotone_curves));
 
-  Arc_section_x_monotone_2 X_monotone_curve1, X_monotone_curve2,
-    X_monotone_curve3 ;
-  CGAL::assign(X_monotone_curve1, X_monotone_curves[0]);
-  CGAL::assign(X_monotone_curve2, X_monotone_curves[1]);
+  const auto* x_curve1 = boost::get<Subcurve_x_monotone_2>(curves[0]);
+  const auto* x_curve2 = boost::get<Subcurve_x_monotone_2>(curves[1]);
 
-  std::vector<CGAL::Object> Points_of_intersection;
+  std::vector<Intersection_result> Points_of_intersection;
 
   //intersect_2(X_monotone_curve1, X_monotone_curve2,
   //            std::back_inserter(Points_of_intersection));
@@ -113,12 +95,12 @@ void check_intersect(Polycurve_arc_traits_2::Make_x_monotone_2
   Kernel::Point_2 c6 = Kernel::Point_2(0, 0);
   CoordNT sqrt_3_div_2 =
     CoordNT(Number_type(0), Number_type(1,2), Number_type(3));
-  Arc_point_2 s6 = Arc_point_2(Number_type(-1, 2), sqrt_3_div_2);
-  Arc_point_2 t6 = Arc_point_2(Number_type(1, 2), sqrt_3_div_2);
+  Point_2 s6 = Point_2(Number_type(-1, 2), sqrt_3_div_2);
+  Point_2 t6 = Point_2(Number_type(1, 2), sqrt_3_div_2);
 
-  curve3 = Arc_section_2(c6, 1, CGAL::CLOCKWISE, s6, t6);
-  make_x_monotone_2(curve3, std::back_inserter(X_monotone_curves));
-  CGAL::assign(X_monotone_curve2, X_monotone_curves[2]);
+  curve3 = Subcurve_2(c6, 1, CGAL::CLOCKWISE, s6, t6);
+  make_x_monotone_2(curve3, std::back_inserter(x_curves));
+  const auto* x_curve2 = boost::get<>(&x_curves[2]);
 
   Points_of_intersection.clear();
   //intersect_2(X_monotone_curve1, X_monotone_curve2,
@@ -126,12 +108,12 @@ void check_intersect(Polycurve_arc_traits_2::Make_x_monotone_2
 }
 
 void
-check_compare_end_points_xy_2(Polycurve_arc_traits_2::Compare_endpoints_xy_2
+check_compare_end_points_xy_2(Traits_2::Compare_endpoints_xy_2
                               compare_endpoints_xy_2,
-                              Polycurve_arc_traits_2::Make_x_monotone_2
+                              Traits_2::Make_x_monotone_2
                               make_x_monotone_2)
 {
-  Arc_section_2 curve1, curve2;
+  Subcurve_2 curve1, curve2;
 
   // Create a circular arc that correspond to the upper half of the
   // circle centered at (1,1) with squared radius 3. We create the
@@ -141,39 +123,37 @@ check_compare_end_points_xy_2(Polycurve_arc_traits_2::Compare_endpoints_xy_2
   Kernel::Circle_2 circ1 = Kernel::Circle_2(c1, 3, CGAL::CLOCKWISE);
   CoordNT one_minus_sqrt_3 = CoordNT(1, -1, 3);
   CoordNT one_plus_sqrt_3 = CoordNT(1, 1, 3);
-  Arc_point_2 s1 = Arc_point_2(one_minus_sqrt_3, CoordNT(1));
-  Arc_point_2 t1 = Arc_point_2(one_plus_sqrt_3, CoordNT(1));
-  curve1= Arc_section_2(circ1, s1, t1);
+  Point_2 s1 = Point_2(one_minus_sqrt_3, CoordNT(1));
+  Point_2 t1 = Point_2(one_plus_sqrt_3, CoordNT(1));
+  curve1= Subcurve_2(circ1, s1, t1);
 
   //make x_monotone
-  std::vector<CGAL::Object> X_monotone_curves;
-  make_x_monotone_2(curve1, std::back_inserter(X_monotone_curves));
+  std::vector<Make_sub_x_monotone_result> x_curves;
+  make_x_monotone_2(curve1, std::back_inserter(x_curves));
 
-  Arc_section_x_monotone_2 X_monotone_curve1, X_monotone_curve2 ;
-  CGAL::assign(X_monotone_curve1, X_monotone_curves[ 0 ]);
-
-  int res = compare_endpoints_xy_2(X_monotone_curve1);
+  const auto* x_curve1 = boost::get<Subcurve_x_monotone_2>(&x_curves[0]);
+  const auto* x_curve2 = boost::get<Subcurve_x_monotone_2>(&x_curves[1]);
 
+  auto res = compare_endpoints_xy_2(x_curve1);
   std::cout<< "The first result is: " << res << std::endl;
 
   Kernel::Point_2 c2 = Kernel::Point_2(1, 1);
   Kernel::Circle_2 circ2 = Kernel::Circle_2(c2, 3, CGAL::COUNTERCLOCKWISE);
-  Arc_point_2 t2 = Arc_point_2(one_minus_sqrt_3, CoordNT(1));
-  Arc_point_2 s2 = Arc_point_2(one_plus_sqrt_3, CoordNT(1));
-  curve2= Arc_section_2(circ2, s1, t1);
+  Point_2 t2 = Point_2(one_minus_sqrt_3, CoordNT(1));
+  Point_2 s2 = Point_2(one_plus_sqrt_3, CoordNT(1));
+  curve2 = Subcurve_2(circ2, s1, t1);
+  make_x_monotone_2(curve2, std::back_inserter(x_curves));
+  const auto* x_curve2 = boost::get<Subcurve_x_monotone_2>(&x_curves[1]);
 
-  make_x_monotone_2(curve2, std::back_inserter(X_monotone_curves));
-  CGAL::assign(X_monotone_curve2, X_monotone_curves[ 1 ]);
-
-  res = compare_endpoints_xy_2(X_monotone_curve2);
+  res = compare_endpoints_xy_2(x_curve2);
 
   std::cout<< "The second result is: " << res << std::endl;
 }
 
-void check_split(Polycurve_arc_traits_2::Split_2  split_2,
-                 Polycurve_arc_traits_2::Make_x_monotone_2 make_x_monotone_2)
+void check_split(Traits_2::Split_2  split_2,
+                 Traits_2::Make_x_monotone_2 make_x_monotone_2)
 {
-  Arc_section_2 curve;
+  Subcurve_2 curve;
 
   // Create a circular arc that correspond to the upper half of the
   // circle centered at (1,1) with squared radius 3. We create the
@@ -183,64 +163,59 @@ void check_split(Polycurve_arc_traits_2::Split_2  split_2,
   Kernel::Circle_2 circ1 = Kernel::Circle_2(c1, 3, CGAL::CLOCKWISE);
   CoordNT one_minus_sqrt_3 = CoordNT(1, -1, 3);
   CoordNT one_plus_sqrt_3 = CoordNT(1, 1, 3);
-  Arc_point_2 s1 = Arc_point_2(one_minus_sqrt_3, CoordNT(1));
-  Arc_point_2 t1 = Arc_point_2(one_plus_sqrt_3, CoordNT(1));
-  curve= Arc_section_2(circ1, s1, t1);
+  Point_2 s1 = Point_2(one_minus_sqrt_3, CoordNT(1));
+  Point_2 t1 = Point_2(one_plus_sqrt_3, CoordNT(1));
+  curve= Subcurve_2(circ1, s1, t1);
 
   //make x_monotone
-  std::vector<CGAL::Object> X_monotone_curves;
-  make_x_monotone_2(curve, std::back_inserter(X_monotone_curves));
-
-  Arc_section_x_monotone_2 X_monotone_curve, split_x_monotone_curve1,
-    split_x_monotone_curve2 ;
-  CGAL::assign(X_monotone_curve, X_monotone_curves[0]);
+  std::vector<Make_sub_x_monotone_result> x_curves;
+  make_x_monotone_2(curve, std::back_inserter(x_curves));
+  const auto* x_curve1 = boost::get<Subcurve_x_monotone_2>(&x_curves[0]);
 
+  // Subcurve_x_monotone_2 split_x_monotone_curve1, split_x_monotone_curve2 ;
   //split_2(X_monotone_curve, Kernel::Point_2::Kernel::Point_2(1, 4),
   //        split_x_monotone_curve1, split_x_monotone_curve2);
 }
 
-void check_is_vertical(Polycurve_arc_traits_2::Make_x_monotone_2
+void check_is_vertical(Traits_2::Make_x_monotone_2
                        make_x_monotone_2,
-                       Polycurve_arc_traits_2::Is_vertical_2 is_vertical)
+                       Traits_2::Is_vertical_2 is_vertical)
 {
-  std::vector<Arc_section_2> curves;
+  std::vector<Subcurve_2> curves;
 
   // Create a circular arc defined by two endpoints and a midpoint,
   // all having rational coordinates. This arc is the upper-right
   // quarter of a circle centered at the origin with radius 5.
-  Kernel::Point_2  p1 = Kernel::Point_2(0, 5);
-  Kernel::Point_2  mid = Kernel::Point_2(3, 4);
-  Kernel::Point_2  p2 = Kernel::Point_2(5, 0);
-  Kernel::Point_2  p3 = Kernel::Point_2(0, -5);
-  curves.push_back(Arc_section_2(p1, mid, p2)); //quarter of a circle
-  curves.push_back(Arc_section_2(p1, mid, p3));  //semi-circle
+  Kernel::Point_2 p1 = Kernel::Point_2(0, 5);
+  Kernel::Point_2 mid = Kernel::Point_2(3, 4);
+  Kernel::Point_2 p2 = Kernel::Point_2(5, 0);
+  Kernel::Point_2 p3 = Kernel::Point_2(0, -5);
+  curves.push_back(Subcurve_2(p1, mid, p2)); //quarter of a circle
+  curves.push_back(Subcurve_2(p1, mid, p3));  //semi-circle
 
   //convert all curves to x-monotone curves
-  std::vector<CGAL::Object> X_monotone_curves;
-  for(int i = 0; i < curves.size(); ++i)
-    make_x_monotone_2(curves[i], std::back_inserter(X_monotone_curves));
+  std::vector<Make_sub_x_monotone_result> x_curves;
+  for (const auto& cv : curves)
+    make_x_monotone_2(cv, std::back_inserter(x_curves));
 
-  //std::vector<Arc_section_x_monotone_2> x_monotone_polycurves;
+  //std::vector<Subcurve_x_monotone_2> x_monotone_polycurves;
 
-  Arc_section_x_monotone_2 x_monotone_polycurve1, x_monotone_polycurve2;
-  CGAL::assign(x_monotone_polycurve1, X_monotone_curves[0]);
-  CGAL::assign(x_monotone_polycurve2, X_monotone_curves[1]);
+  const auto* x_polycurve1 = boost::get<Subcurve_x_monotone_2>(&x_curves[0]);
+  const auto* x_polycurve2 = boost::get<Subcurve_x_monotone_2>(&x_curves[1]);
 
-  bool res = is_vertical(x_monotone_polycurve1);
+  bool res = is_vertical(x_polycurve1);
   std::cout << "Is_verticle:: The xmonotone curve (quarter circle) is : "
             << ((res)? "vertical" : "not vertical") << std::endl;
 
-  res = is_vertical(x_monotone_polycurve2);
+  res = is_vertical(x_polycurve2);
   std::cout << "Is_verticle:: The xmonotone curve (Smi-circle) is : "
             << ((res)? "vertical" : "not vertical") << std::endl;
 }
 
-void check_compare_y_at_x_2(Polycurve_arc_traits_2::Make_x_monotone_2
-                            make_x_monotone_2,
-                            Polycurve_arc_traits_2::Compare_y_at_x_2
-                            cmp_y_at_x_2)
+void check_compare_y_at_x_2(Traits_2::Make_x_monotone_2 make_x_monotone_2,
+                            Traits_2::Compare_y_at_x_2 cmp_y_at_x_2)
 {
-  std::vector<Arc_section_2> curves;
+  std::vector<Subcurve_2> curves;
 
   // Create a circular arc defined by two endpoints and a midpoint,
   // all having rational coordinates. This arc is the upper-right
@@ -249,30 +224,29 @@ void check_compare_y_at_x_2(Polycurve_arc_traits_2::Make_x_monotone_2
   Kernel::Point_2 mid = Kernel::Point_2(4, 4);
   Kernel::Point_2 p2 = Kernel::Point_2(7, 1);
   Kernel::Point_2 p3 = Kernel::Point_2(1, 4);
-  curves.push_back(Arc_section_2(p1, mid, p2)); //quarter of a circle
-  curves.push_back(Arc_section_2(p1, mid, p3));  //semi-circle
+  curves.push_back(Subcurve_2(p1, mid, p2)); //quarter of a circle
+  curves.push_back(Subcurve_2(p1, mid, p3));  //semi-circle
 
   //convert all curves to x-monotone curves
-  std::vector<CGAL::Object> X_monotone_curves;
-  for (int i = 0; i < curves.size(); ++i)
-    make_x_monotone_2(curves[i], std::back_inserter(X_monotone_curves));
+  std::vector<Make_sub_x_monotone_result> x_curves;
+  for (const auto& cv : curves)
+    make_x_monotone_2(cv, std::back_inserter(x_curves));
 
-  Arc_section_x_monotone_2 x_monotone_polycurve1, x_monotone_polycurve2;
-  CGAL::assign(x_monotone_polycurve1, X_monotone_curves[ 0 ]);
-  CGAL::assign(x_monotone_polycurve2, X_monotone_curves[ 1 ]);
+  const auto* x_polycurve1 = boost::get<Subcurve_x_monotone_2>(&x_curves[0]);
+  const auto* x_polycurve2 = boost::get<Subcurve_x_monotone_2>(&x_curves[1]);
 
-  Kernel::Point_2  p_test = Kernel::Point_2(3, 1);
+  Kernel::Point_2 p_test = Kernel::Point_2(3, 1);
 
   // int res = cmp_y_at_x_2(p_test, x_monotone_polycurve1);
   // cmp_y_at_x_2(x_monotone_polycurve1, CGAL::ARR_MIN_END,
   //              x_monotone_polycurve2);
 }
 
-void check_push_back(Polycurve_arc_traits_2::Make_x_monotone_2
+void check_push_back(Traits_2::Make_x_monotone_2
                      make_x_monotone_2,
-                     Polycurve_arc_traits_2::Push_back_2  push_back_2)
+                     Traits_2::Push_back_2  push_back_2)
 {
-  std::vector<Arc_section_2> curves;
+  std::vector<Subcurve_2> curves;
 
   //check if segment is pushed in empty curve.
   Kernel::Point_2 p1 = Kernel::Point_2(1, 1);
@@ -282,10 +256,10 @@ void check_push_back(Polycurve_arc_traits_2::Make_x_monotone_2
   Kernel::Point_2 mid2 = Kernel::Point_2(10, 3);
   Kernel::Point_2 p3 = Kernel::Point_2(7, 7);
 
-  curves.push_back(Arc_section_2(p1, mid, p2));
-  curves.push_back(Arc_section_2(p2, mid2, p3));
+  curves.push_back(Subcurve_2(p1, mid, p2));
+  curves.push_back(Subcurve_2(p2, mid2, p3));
 
-  CGAL::internal::Polycurve_2<Arc_section_2, Arc_point_2> polycurve;
+  CGAL::internal::Polycurve_2<Subcurve_2, Point_2> polycurve;
 
   ////pushing segments in polycurve
   push_back_2(polycurve, curves[0]);
@@ -300,87 +274,70 @@ void check_push_back(Polycurve_arc_traits_2::Make_x_monotone_2
 
 int main()
 {
-  Polycurve_arc_traits_2 Polycurve_traits_2;
+  Traits_2 Polycurve_traits_2;
 
   // Compare_x_2
-  //  Polycurve_arc_traits_2::Compare_x_2 compare_x_2 =
+  //  Traits_2::Compare_x_2 compare_x_2 =
   //   Polycurve_traits_2.compare_xy_2_object();
 
   // number of points
-  // Polycurve_arc_traits_2::Number_of_points_2 num_of_points =
+  // Traits_2::Number_of_points_2 num_of_points =
   //   Polycurve_traits_2.number_of_points_2_object();
 
   //construct min vertex
-  Polycurve_arc_traits_2::Construct_min_vertex_2 cnst_min_vertex =
-    Polycurve_traits_2.construct_min_vertex_2_object();
+  auto cnst_min_vertex = Polycurve_traits_2.construct_min_vertex_2_object();
 
   //construct max vertex
-  Polycurve_arc_traits_2::Construct_max_vertex_2 cnst_max_vertex_2 =
-    Polycurve_traits_2.construct_max_vertex_2_object();
+  auto cnst_max_vertex_2 = Polycurve_traits_2.construct_max_vertex_2_object();
 
   //is vertical (return bool)
-  Polycurve_arc_traits_2::Is_vertical_2 is_vertical =
-    Polycurve_traits_2.is_vertical_2_object();
+  auto is_vertical = Polycurve_traits_2.is_vertical_2_object();
 
   //Compare y at x 2 (return comparison_result)
-  Polycurve_arc_traits_2::Compare_y_at_x_2 cmp_y_at_x_2 =
-    Polycurve_traits_2.compare_y_at_x_2_object();
+  auto cmp_y_at_x_2 = Polycurve_traits_2.compare_y_at_x_2_object();
 
   //compare y at x left
-  Polycurve_arc_traits_2::Compare_y_at_x_left_2 cmp_y_at_x_left_2 =
-    Polycurve_traits_2.compare_y_at_x_left_2_object();
+  auto cmp_y_at_x_left_2 = Polycurve_traits_2.compare_y_at_x_left_2_object();
 
   //compare y at x right
-  Polycurve_arc_traits_2::Compare_y_at_x_right_2 cmp_y_at_x_right_2 =
-    Polycurve_traits_2.compare_y_at_x_right_2_object();
+  auto cmp_y_at_x_right_2 = Polycurve_traits_2.compare_y_at_x_right_2_object();
 
   //equal_2
-  Polycurve_arc_traits_2::Equal_2 equal_2 =
-    Polycurve_traits_2.equal_2_object();
+  auto equal_2 = Polycurve_traits_2.equal_2_object();
 
   //compare end points xy_2
-  Polycurve_arc_traits_2::Compare_endpoints_xy_2 compare_endpoints_xy_2 =
+  auto compare_endpoints_xy_2 =
     Polycurve_traits_2.compare_endpoints_xy_2_object();
 
   //construct opposite
-  Polycurve_arc_traits_2::Construct_opposite_2 construct_opposite_2 =
-    Polycurve_traits_2.construct_opposite_2_object();
+  auto construct_opposite_2 = Polycurve_traits_2.construct_opposite_2_object();
 
   //make x_monotone
-  Polycurve_arc_traits_2::Make_x_monotone_2 make_x_monotone_2 =
-    Polycurve_traits_2.make_x_monotone_2_object();
+  auto make_x_monotone_2 = Polycurve_traits_2.make_x_monotone_2_object();
 
   //push back
-  Polycurve_arc_traits_2::Push_back_2 push_back_2 =
-    Polycurve_traits_2.push_back_2_object();
+  auto push_back_2 = Polycurve_traits_2.push_back_2_object();
 
   //push front
-  Polycurve_arc_traits_2::Push_front_2 push_front_2 =
-    Polycurve_traits_2.push_front_2_object();
+  auto push_front_2 = Polycurve_traits_2.push_front_2_object();
 
   //split_2
-  Polycurve_arc_traits_2::Split_2 split_2 =
-    Polycurve_traits_2.split_2_object();
+  auto split_2 = Polycurve_traits_2.split_2_object();
 
   //Intersect_2
-  Polycurve_arc_traits_2::Intersect_2 intersect_2 =
-    Polycurve_traits_2.intersect_2_object();
+  auto intersect_2 = Polycurve_traits_2.intersect_2_object();
 
   //Are_mergable
-  Polycurve_arc_traits_2::Are_mergeable_2 are_mergeable_2 =
-    Polycurve_traits_2.are_mergeable_2_object();
+  auto are_mergeable_2 = Polycurve_traits_2.are_mergeable_2_object();
 
   //Merge_2
-  Polycurve_arc_traits_2::Merge_2 merge_2 =
-    Polycurve_traits_2.merge_2_object();
+  auto merge_2 = Polycurve_traits_2.merge_2_object();
 
   //construct_curve_2
-  Polycurve_arc_traits_2::Construct_curve_2 construct_curve_2 =
-    Polycurve_traits_2.construct_curve_2_object();
+  auto construct_curve_2 = Polycurve_traits_2.construct_curve_2_object();
 
   //construct x_monotone curve_2
-  Polycurve_arc_traits_2::Construct_x_monotone_curve_2
-    construct_x_monotone_curve_2 =
+  auto construct_x_monotone_curve_2 =
     Polycurve_traits_2.construct_x_monotone_curve_2_object();
 
   //check_equal();
@@ -393,4 +350,3 @@ int main()
 
   return 0;
 }
-#endif

Arrangement_on_surface_2/test/Arrangement_on_surface_2/test_bezier_polycurve.cpp

@@ -23,15 +23,19 @@ typedef CGAL::Cartesian<Algebraic>                      Alg_kernel;
 typedef Rat_kernel::Point_2                             Rat_point_2;
 
 //bezier traits
-typedef CGAL::Arr_Bezier_curve_traits_2<Rat_kernel, Alg_kernel, Nt_traits> Bezier_traits_2;
-typedef Bezier_traits_2::Curve_2                               Bezier_curve;
-typedef Bezier_traits_2::X_monotone_curve_2                    Bezier_x_monotone_curve;
+typedef CGAL::Arr_Bezier_curve_traits_2<Rat_kernel, Alg_kernel, Nt_traits>
+                                                        Bezier_traits_2;
+typedef Bezier_traits_2::Point_2                        Bezier_point;
+typedef Bezier_traits_2::Curve_2                        Bezier_curve;
+typedef Bezier_traits_2::X_monotone_curve_2             Bezier_x_monotone_curve;
 
 //polyline traits
-typedef CGAL::Arr_polyline_traits_2<Bezier_traits_2>            Bezier_polycurve_traits;
-typedef Bezier_polycurve_traits::Curve_2                        Polycurve_bezier;
-typedef Bezier_polycurve_traits::X_monotone_curve_2             X_polycurve_bezier;
+typedef CGAL::Arr_polyline_traits_2<Bezier_traits_2>    Bezier_polycurve_traits;
+typedef Bezier_polycurve_traits::Curve_2                Polycurve_bezier;
+typedef Bezier_polycurve_traits::X_monotone_curve_2     X_polycurve_bezier;
 
+typedef boost::variant<Bezier_point, Bezier_x_monotone_curve>
+                                                        Make_x_monotone_result;
 
 int main (int argc, char *argv[])
 {
@@ -51,35 +55,40 @@ int main (int argc, char *argv[])
   points_vector.push_back( Rat_point_2(900,0) );
   Bezier_curve curve_1 (points_vector.begin(), points_vector.end());
 
-  std::vector<CGAL::Object> obj_vector;
+  std::vector<Make_x_monotone_result> objs;
 
   //creating x-mono bezier
-  bezier_traits.make_x_monotone_2_object()( curve_1, std::back_inserter(obj_vector));
+  bezier_traits.make_x_monotone_2_object()(curve_1, std::back_inserter(objs));
   //std::cout << "number of x_curves: " << obj_vector.size() << std::endl;
-  Bezier_x_monotone_curve x_curve_1 = CGAL::object_cast<Bezier_x_monotone_curve>( (obj_vector[0]) );
+  const auto* x_curve_1 = boost::get<Bezier_x_monotone_curve>(&(obj_vector[0]));
+  assert(x_curve_1);
 
   //std::cout << x_curve << std::endl;
 
   points_vector.clear();
-  points_vector.push_back( Rat_point_2(900,0) );
-  points_vector.push_back( Rat_point_2(1000,400) );
-  points_vector.push_back( Rat_point_2(1500,200) );
-  points_vector.push_back( Rat_point_2(2000,0) );
-  Bezier_curve curve_2 (points_vector.begin(), points_vector.end());
+  points_vector.push_back(Rat_point_2(900,0));
+  points_vector.push_back(Rat_point_2(1000,400));
+  points_vector.push_back(Rat_point_2(1500,200));
+  points_vector.push_back(Rat_point_2(2000,0));
+  Bezier_curve curve_2(points_vector.begin(), points_vector.end());
   //creating x-monotne
   obj_vector.clear();
-  bezier_traits.make_x_monotone_2_object()( curve_2, std::back_inserter(obj_vector));
-  Bezier_x_monotone_curve x_curve_2 = CGAL::object_cast<Bezier_x_monotone_curve>( (obj_vector[0]) );
+  bezier_traits.make_x_monotone_2_object()( curve_2, std::back_inserter(objs));
+  const auto* x_curve_2 = boost::get<Bezier_x_monotone_curve>(&(obj_vector[0]));
 
   //push curves into polyline vectors
   curves_vector.push_back(curve_1);
   curves_vector.push_back(curve_2);
-  x_curves_vector.push_back(x_curve_1);
-  x_curves_vector.push_back(x_curve_2);
+  x_curves_vector.push_back(*x_curve_1);
+  x_curves_vector.push_back(*x_curve_2);
 
   //create polycurves
-  Polycurve_bezier polycurve_1 = poly_traits.construct_curve_2_object()(curves_vector.begin(), curves_vector.end());
-  X_polycurve_bezier x_polycurve_1 = poly_traits.construct_x_monotone_curve_2_object()(x_curves_vector.begin(), x_curves_vector.end());
+  auto ctr_curve = poly_traits.construct_curve_2_object();
+  Polycurve_bezier
+    polycurve_1 = ctr_curve(curves_vector.begin(), curves_vector.end());
+  auto ctr_x_monotone_curve = poly_traits.construct_x_monotone_curve_2_object();
+  X_polycurve_bezier x_polycurve_1 =
+    ctr_x_monotone_curve(x_curves_vector.begin(), x_curves_vector.end());
 
   std::cout << "polycurve : " << polycurve_1 << std::endl;
   std::cout << "x_polycurve : " << x_polycurve_1 << std::endl;

Arrangement_on_surface_2/test/Arrangement_on_surface_2/test_circular_arc_polycurve.cpp

@@ -73,25 +73,31 @@ bool check_compare_y_at_x_2(Curve& cv)
   return true;
 }
 
-template<typename Curve>
-void check_intersect(const Curve& cv1, const Curve& cv2)
+template <typename GeometryTraits_2>
+void check_intersect(const typename GeometryTraits_2::X_monotone_curve_2& cv1,
+                     const typename GeometryTraits_2::X_monotone_curve_2& cv2,
+                     const GeometryTraits_2& traits)
 {
   // Polycurve_arc_traits_2 traits;
-
-  // std::vector<CGAL::Object> object_vec;
-  // traits.intersect_2_object()(cv1, cv2, object_vec);
-  // std::cout<< "number of intersections is: " << object_vec.size();
+  // std::vector<Intersection_result> objs;
+  // traits.intersect_2_object()(cv1, cv2, std::back_inserter(objs));
+  // std::cout<< "number of intersections is: " << objs.size();
 }
 
-template<typename Curve>
-void check_make_x_monotone(Curve cv)
+template <typename GeometryTraits_2>
+void check_make_x_monotone(typename GeometryTraits::Curve_2 cv,
+                           const GeometryTraits_2& traits)
 {
-   Polycurve_arc_traits_2 traits;
-   std::vector<CGAL::Object> object_vec;
+  typedef GeometryTraits_2                                  Geometry_traits_2;
+  typedef typename Geometry_traits_2::Point_2               Point_2;
+  typedef typename Geometry_traits_2::X_monotone_traits_2   X_monotone_traits_2;
 
-   traits.make_x_monotone_2_object()(cv, std::back_inserter(object_vec));
-   std::cout << "Number of x-monotone curves: "
-             << object_vec.size() << std::endl;
+  typedef boost::variant<Point_2, X_monotone_traits_2>
+    Make_x_monotone_result;
+
+  std::vector<Make_x_monotone_result> objs;
+  traits.make_x_monotone_2_object()(cv, std::back_inserter(objs));
+  std::cout << "Number of x-monotone curves: " << objs.size() << std::endl;
 }
 
 template<typename Curve>
@@ -198,8 +204,8 @@ int main(int argc, char* argv[])
   //////////////////////
 
   // check_compare_y_at_x_2(x_polycurve_1);
-  // check_intersect(x_polycurve_1, x_polycurve_2);
-  //check_make_x_monotone(curve_1);
+  // check_intersect(x_polycurve_1, x_polycurve_2, traits);
+  //check_make_x_monotone<Polycurve_arc_traits_2>(curve_1, traits);
 
   //checking if the cgal_assertion for curve construction for two points work
   //or not.

Arrangement_on_surface_2/test/Arrangement_on_surface_2/test_conic_polycurve.cpp

@@ -11,19 +11,19 @@ int main()
 
 #else
 
+#include <vector>
+#include <list>
+
+#include <boost/type_traits/is_same.hpp>
+
 #include <CGAL/Cartesian.h>
 #include <CGAL/Quotient.h>
 #include <CGAL/MP_Float.h>
 #include <CGAL/CORE_algebraic_number_traits.h>
-#include <vector>
-#include <list>
-
 #include <CGAL/Arr_polyline_traits_2.h>
 #include <CGAL/Arr_conic_traits_2.h>
-#include <CGAL/Arrangement_2.h>
 #include <CGAL/tags.h>
 #include <CGAL/Arr_tags.h>
-#include <boost/type_traits/is_same.hpp>
 
 ////////////////////
 //conic traits
@@ -43,7 +43,7 @@ typedef Conic_traits_2::Curve_2                       Conic_curve_2;
 typedef Conic_traits_2::X_monotone_curve_2            Conic_x_monotone_curve_2;
 typedef CGAL::Arr_polyline_traits_2<Conic_traits_2>   Polycurve_conic_traits_2;
 typedef Polycurve_conic_traits_2::X_monotone_curve_2  Pc_x_monotone_curve_2;
-// typedef Polycurve_conic_traits_2::Point_2          polypoint;
+typedef Polycurve_conic_traits_2::Point_2             Pc_point_2;
 
 // typedef CGAL::Arr_polyline_traits_2<
 //                CGAL::Arr_conic_traits_2<CGAL::Cartesian<BigRat>,
@@ -102,27 +102,36 @@ void check_equal()
             << ((are_equal) ? "equal" : "Not equal") << std::endl;
  }
 
- template <typename curve_type>
- void check_intersect(curve_type &xcv1, curve_type &xcv2)
+ template <typename Traits>
+ void check_intersect(typename Traits::X_monotone_curve_2& xcv1,
+                      typename Traits::X_monotone_curve_2& xcv2,
+                      const Traits& traits)
  {
-  Polycurve_conic_traits_2 traits;
-  std::vector<CGAL::Object> intersection_points;
-  traits.intersect_2_object()(xcv1, xcv2, std::back_inserter(intersection_points));
-  std::cout<< "Number of intersection Points: " << intersection_points.size()
-           << std::endl;
+   typedef typename Traits::Multiplicity                Multiplicity;
+   typedef typename Traits::Point_2                     Point_2;
+   typedef typename Traits::X_monotone_curve_2          X_monotone_curve_2;
+   typedef std::pair<Multiplicity, Point_2>             Intersection_point;
+   typedef boost::variant<Intersection_point, X_monotone_curve_2>
+     Intersection_result;
 
-  //dynamic cast the cgal_objects
-  // std::vector< std::pair<Polycurve_conic_traits_2::Point_2,
-  //                        Polycurve_conic_traits_2::Multiplicity> > pm_vector;
-  // for(int i=0; i<intersection_points.size(); i++)
-  // {
-  //   std::pair<Polycurve_conic_traits_2::Point_2,
-  //             Polycurve_conic_traits_2::Multiplicity> pm =
-  //   CGAL::object_cast<std::pair<Polycurve_conic_traits_2::Point_2,
-  //                               Polycurve_conic_traits_2::Multiplicity> >
-  //                              (&(intersection_points[i]));
-  //   pm_vector.push_back(pm);
-  // }
+   std::vector<Intersection_result> intersection_points;
+   traits.intersect_2_object()(xcv1, xcv2,
+                               std::back_inserter(intersection_points));
+   std::cout<< "Number of intersection Points: " << intersection_points.size()
+            << std::endl;
+
+   //dynamic cast the cgal_objects
+   // std::vector< std::pair<Polycurve_conic_traits_2::Point_2,
+   //                        Polycurve_conic_traits_2::Multiplicity> > pm_vector;
+   // for(int i=0; i<intersection_points.size(); i++)
+   // {
+   //   std::pair<Polycurve_conic_traits_2::Point_2,
+   //             Polycurve_conic_traits_2::Multiplicity> pm =
+   //   CGAL::object_cast<std::pair<Polycurve_conic_traits_2::Point_2,
+   //                               Polycurve_conic_traits_2::Multiplicity> >
+   //                              (&(intersection_points[i]));
+   //   pm_vector.push_back(pm);
+   // }
  }
 
 void check_compare_end_points_xy_2()
@@ -151,7 +160,7 @@ void check_compare_end_points_xy_2()
                        0, 0, 0, 0, 1, 3,         // The line: y = -3.
                        Conic_point_2(1.41, -2),  // Approximation of the target.
                        0, 0, 0, 0, 1, 2);        // The line: y = -2.
-  CGAL_assertion(c2.is_valid());
+  assert(c2.is_valid());
 
   //make polyline x-monotone curves
   Polycurve_conic_traits_2::X_monotone_curve_2 polyline_xmc1 =
@@ -631,22 +640,23 @@ void check_compare_y_at_x_left()
                (result == CGAL::LARGER ? "Larger" : "equal")) << std::endl;
 }
 
-template <typename Curve_type>
-void check_make_x_monotne_curve(Curve_type c1)
+template <typename GeometryTraits>
+void check_make_x_monotne_curve(const typename GeometryTraits::Curve_2& c1)
 {
+  typename GeometryTraits::Point_2                      Point_2;
+  typename GeometryTraits::X_monotone_curve_2           X_monotone_curve_2;
+  typedef boost::variant<Point_2, X_monotone_curve_2>   Make_x_monotone_result;
   Polycurve_conic_traits_2 traits;
-  std::vector<CGAL::Object> obj_vec;
-  traits.make_x_monotone_2_object()(c1, std::back_inserter(obj_vec));
-
+  std::vector<Make_x_monotone_result> objs;
+  traits.make_x_monotone_2_object()(c1, std::back_inserter(objs));
   std::cout << "The polycurve is: " << c1 << std::endl;
-
-  std::cout<< "The poly curve have been split into " << obj_vec.size()
+  std::cout<< "The poly curve have been split into " << objs.size()
            << " polycurves" << std::endl;
 
   //const Pc_x_monotone_curve_2 *split_curve_1 =
-  //  CGAL::object_cast<Pc_x_monotone_curve_2> (&(obj_vec[0]));
+  //  CGAL::object_cast<Pc_x_monotone_curve_2> (&(objs[0]));
   //const Pc_x_monotone_curve_2 *split_curve_2 =
-  //  CGAL::object_cast<Pc_x_monotone_curve_2> (&(obj_vec[1]));
+  //  CGAL::object_cast<Pc_x_monotone_curve_2> (&(objs[1]));
 
   //std::cout << "The split curve 1 is: " << *split_curve_1 << std::endl;
   //std::cout << "The split curve 2 is: " << *split_curve_2 << std::endl;
@@ -726,10 +736,8 @@ int main(int argc, char* argv[])
 {
   Polycurve_conic_traits_2 traits;
     //polycurve constructors
-  Polycurve_conic_traits_2::Construct_x_monotone_curve_2
-    construct_x_mono_polycurve = traits.construct_x_monotone_curve_2_object();
-  Polycurve_conic_traits_2::Construct_curve_2  construct_polycurve =
-    traits.construct_curve_2_object();
+  auto construct_x_mono_polycurve = traits.construct_x_monotone_curve_2_object();
+  auto construct_polycurve = traits.construct_curve_2_object();
 
    //create a curve
 
@@ -886,7 +894,7 @@ int main(int argc, char* argv[])
   //check_split(conic_x_mono_polycurve_1, conic_x_mono_polycurve_2);
   // std::cout<< std::endl;
 
-  //check_make_x_monotne_curve(conic_polycurve_2);
+  //check_make_x_monotne_curve<Point_2, Curve_2>(conic_polycurve_2);
    //std::cout<< std::endl;
 
   // check_is_vertical();

Arrangement_on_surface_2/test/Arrangement_on_surface_2/test_vert_ray_shoot_vert_segments.cpp

@@ -1,146 +1,131 @@
-#include <CGAL/Cartesian.h>
+#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
 #include <CGAL/Arr_segment_traits_2.h>
 #include <CGAL/Arrangement_2.h>
 #include <CGAL/Arr_simple_point_location.h>
 #include <CGAL/Arr_walk_along_line_point_location.h>
 #include <CGAL/Arr_trapezoid_ric_point_location.h>
 
-template <class VerticalRayShoot>
+template <typename VerticalRayShoot>
 void vertical_ray_shooting_query
-    (const VerticalRayShoot& vrs, bool shoot_up,
-     const typename VerticalRayShoot::Arrangement_2::Point_2& q)
+(const VerticalRayShoot& vrs, bool shoot_up,
+ const typename VerticalRayShoot::Arrangement_2::Point_2& q)
 {
   // Perform the point-location query.
-  CGAL::Object    obj = (shoot_up) ? vrs.ray_shoot_up (q) :
-                                     vrs.ray_shoot_down (q);
+  auto obj = (shoot_up) ? vrs.ray_shoot_up(q) : vrs.ray_shoot_down(q);
 
   // Print the result.
-  typedef typename VerticalRayShoot::Arrangement_2  Arrangement_2;
+  typedef typename VerticalRayShoot::Arrangement_2      Arrangement_2;
+  typedef typename Arrangement_2::Vertex_const_handle   Vertex_const_handle;
+  typedef typename Arrangement_2::Halfedge_const_handle Halfedge_const_handle;
+  typedef typename Arrangement_2::Face_const_handle     Face_const_handle;
 
-  typename Arrangement_2::Vertex_const_handle    v;
-  typename Arrangement_2::Halfedge_const_handle  e;
-  typename Arrangement_2::Face_const_handle      f;
+  if (shoot_up) std::cout << "  Shooting up from (" << q << ") : ";
+  else std::cout << "  Shooting down from (" << q << ") : ";
 
-  if (shoot_up)
-    std::cout << "  Shooting up from (" << q << ") : ";
-  else
-    std::cout << "  Shooting down from (" << q << ") : ";
-
-  if (CGAL::assign (e, obj))
-  {
+  if (auto* e_p = boost::get<Halfedge_const_handle>(&obj)) {
     // We hit an edge:
-    std::cout << "hit an edge: " << e->curve() << std::endl;
+    std::cout << "hit an edge: " << (*e_p)->curve() << std::endl;
   }
-  else if (CGAL::assign (v, obj))
-  {
+  else if (auto* v_p = boost::get<Vertex_const_handle>(&obj)) {
     // We hit a vertex:
-    if (v->is_isolated())
-      std::cout << "hit an isolated vertex: " << v->point() << std::endl;
-    else
-      std::cout << "hit a vertex: " << v->point() << std::endl;
+    if ((*v_p)->is_isolated())
+      std::cout << "hit an isolated vertex: " << (*v_p)->point() << std::endl;
+    else std::cout << "hit a vertex: " << (*v_p)->point() << std::endl;
   }
-  else if (CGAL::assign (f, obj))
-  {
+  else if (auto* f_p = boost::get<Face_const_handle>(&obj)) {
     // We did not hit anything:
-    assert (f->is_unbounded());
+    assert((*f_p)->is_unbounded());
 
     std::cout << "hit nothing." << std::endl;
   }
-  else
-  {
-    CGAL_error_msg( "Invalid object.");
-  }
-
-  return;
+  else CGAL_error_msg( "Invalid object.");
 }
 
+typedef CGAL::Exact_predicates_exact_constructions_kernel       Kernel;
+typedef Kernel::FT                                              Number_type;
+typedef CGAL::Arr_segment_traits_2<Kernel>                      Traits_2;
+typedef Traits_2::Point_2                                       Point_2;
+typedef Traits_2::X_monotone_curve_2                            Segment_2;
+typedef CGAL::Arrangement_2<Traits_2>                           Arrangement_2;
+typedef CGAL::Arr_simple_point_location<Arrangement_2>          Simple_pl;
+typedef CGAL::Arr_walk_along_line_point_location<Arrangement_2> Walk_pl;
+typedef CGAL::Arr_trapezoid_ric_point_location<Arrangement_2>   RIC_pl;
 
-typedef int                                           Number_type;
-typedef CGAL::Cartesian<Number_type>                  Kernel;
-typedef CGAL::Arr_segment_traits_2<Kernel>            Traits_2;
-typedef Traits_2::Point_2                             Point_2;
-typedef Traits_2::X_monotone_curve_2                  Segment_2;
-typedef CGAL::Arrangement_2<Traits_2>                 Arrangement_2;
-typedef CGAL::Arr_simple_point_location<Arrangement_2>           Simple_pl;
-typedef CGAL::Arr_walk_along_line_point_location<Arrangement_2>  Walk_pl;
-typedef CGAL::Arr_trapezoid_ric_point_location<Arrangement_2>    RIC_pl;
-
-int main ()
+int main()
 {
   // Construct the arrangement.
-  Arrangement_2    arr;
-  Simple_pl        simple_pl (arr);
-  Walk_pl          walk_pl (arr);
-  RIC_pl           ric_pl (arr);
+  Arrangement_2 arr;
+  Simple_pl simple_pl(arr);
+  Walk_pl walk_pl(arr);
+  RIC_pl ric_pl(arr);
 
-  insert (arr, Segment_2 (Point_2 (0, 0), Point_2 (0, 1)));
-  insert (arr, Segment_2 (Point_2 (0, 3), Point_2 (0, 4)));
-  insert (arr, Segment_2 (Point_2 (-1, -1), Point_2 (1, -1)));
-  insert (arr, Segment_2 (Point_2 (-1, 5), Point_2 (1, 5)));
+  insert(arr, Segment_2(Point_2(0, 0), Point_2(0, 1)));
+  insert(arr, Segment_2(Point_2(0, 3), Point_2(0, 4)));
+  insert(arr, Segment_2(Point_2(-1, -1), Point_2(1, -1)));
+  insert(arr, Segment_2(Point_2(-1, 5), Point_2(1, 5)));
 
   // Perform the vertical ray-shooting queries.
   std::cout << "First round of queries: " << std::endl;
 
-  Point_2          q1 (0, 2);
+  Point_2 q1(0, 2);
 
-  vertical_ray_shooting_query (simple_pl, true, q1);
-  vertical_ray_shooting_query (walk_pl, true, q1);
-  vertical_ray_shooting_query (ric_pl, true, q1);
+  vertical_ray_shooting_query(simple_pl, true, q1);
+  vertical_ray_shooting_query(walk_pl, true, q1);
+  vertical_ray_shooting_query(ric_pl, true, q1);
 
-  vertical_ray_shooting_query (simple_pl, false, q1);
-  vertical_ray_shooting_query (walk_pl, false, q1);
-  vertical_ray_shooting_query (ric_pl, false, q1);
+  vertical_ray_shooting_query(simple_pl, false, q1);
+  vertical_ray_shooting_query(walk_pl, false, q1);
+  vertical_ray_shooting_query(ric_pl, false, q1);
 
-  Point_2          q2 (0, 1);
+  Point_2 q2(0, 1);
 
-  vertical_ray_shooting_query (simple_pl, true, q2);
-  vertical_ray_shooting_query (walk_pl, true, q2);
-  vertical_ray_shooting_query (ric_pl, true, q2);
+  vertical_ray_shooting_query(simple_pl, true, q2);
+  vertical_ray_shooting_query(walk_pl, true, q2);
+  vertical_ray_shooting_query(ric_pl, true, q2);
 
-  vertical_ray_shooting_query (simple_pl, false, q2);
-  vertical_ray_shooting_query (walk_pl, false, q2);
-  vertical_ray_shooting_query (ric_pl, false, q2);
+  vertical_ray_shooting_query(simple_pl, false, q2);
+  vertical_ray_shooting_query(walk_pl, false, q2);
+  vertical_ray_shooting_query(ric_pl, false, q2);
 
-  Point_2          q3 (0, 3);
+  Point_2 q3(0, 3);
 
-  vertical_ray_shooting_query (simple_pl, true, q3);
-  vertical_ray_shooting_query (walk_pl, true, q3);
-  vertical_ray_shooting_query (ric_pl, true, q3);
+  vertical_ray_shooting_query(simple_pl, true, q3);
+  vertical_ray_shooting_query(walk_pl, true, q3);
+  vertical_ray_shooting_query(ric_pl, true, q3);
 
-  vertical_ray_shooting_query (simple_pl, false, q3);
-  vertical_ray_shooting_query (walk_pl, false, q3);
-  vertical_ray_shooting_query (ric_pl, false, q3);
+  vertical_ray_shooting_query(simple_pl, false, q3);
+  vertical_ray_shooting_query(walk_pl, false, q3);
+  vertical_ray_shooting_query(ric_pl, false, q3);
 
   // Insert additional curves and perform the ray-shooting queries again.
-  insert (arr, Segment_2 (Point_2 (-1, 0), Point_2 (1, 0)));
-  insert (arr, Segment_2 (Point_2 (-1, 4), Point_2 (1, 4)));
+  insert (arr, Segment_2(Point_2(-1, 0), Point_2(1, 0)));
+  insert (arr, Segment_2(Point_2(-1, 4), Point_2(1, 4)));
 
   std::cout << "Second round of queries: " << std::endl;
 
-  vertical_ray_shooting_query (simple_pl, true, q1);
-  vertical_ray_shooting_query (walk_pl, true, q1);
-  vertical_ray_shooting_query (ric_pl, true, q1);
+  vertical_ray_shooting_query(simple_pl, true, q1);
+  vertical_ray_shooting_query(walk_pl, true, q1);
+  vertical_ray_shooting_query(ric_pl, true, q1);
 
-  vertical_ray_shooting_query (simple_pl, false, q1);
-  vertical_ray_shooting_query (walk_pl, false, q1);
-  vertical_ray_shooting_query (ric_pl, false, q1);
+  vertical_ray_shooting_query(simple_pl, false, q1);
+  vertical_ray_shooting_query(walk_pl, false, q1);
+  vertical_ray_shooting_query(ric_pl, false, q1);
 
-  vertical_ray_shooting_query (simple_pl, true, q2);
-  vertical_ray_shooting_query (walk_pl, true, q2);
-  vertical_ray_shooting_query (ric_pl, true, q2);
+  vertical_ray_shooting_query(simple_pl, true, q2);
+  vertical_ray_shooting_query(walk_pl, true, q2);
+  vertical_ray_shooting_query(ric_pl, true, q2);
 
-  vertical_ray_shooting_query (simple_pl, false, q2);
-  vertical_ray_shooting_query (walk_pl, false, q2);
-  vertical_ray_shooting_query (ric_pl, false, q2);
+  vertical_ray_shooting_query(simple_pl, false, q2);
+  vertical_ray_shooting_query(walk_pl, false, q2);
+  vertical_ray_shooting_query(ric_pl, false, q2);
 
-  vertical_ray_shooting_query (simple_pl, true, q3);
-  vertical_ray_shooting_query (walk_pl, true, q3);
-  vertical_ray_shooting_query (ric_pl, true, q3);
+  vertical_ray_shooting_query(simple_pl, true, q3);
+  vertical_ray_shooting_query(walk_pl, true, q3);
+  vertical_ray_shooting_query(ric_pl, true, q3);
 
-  vertical_ray_shooting_query (simple_pl, false, q3);
-  vertical_ray_shooting_query (walk_pl, false, q3);
-  vertical_ray_shooting_query (ric_pl, false, q3);
+  vertical_ray_shooting_query(simple_pl, false, q3);
+  vertical_ray_shooting_query(walk_pl, false, q3);
+  vertical_ray_shooting_query(ric_pl, false, q3);
 
   return (0);
 }
-

Arrangement_on_surface_2/test/Arrangement_on_surface_2/test_zone.cpp

@@ -8,13 +8,17 @@
 #include <CGAL/Arr_segment_traits_2.h>
 #include <CGAL/Arrangement_2.h>
 
-typedef CGAL::Quotient<int>                           Number_type;
-typedef CGAL::Simple_cartesian<Number_type>           Kernel;
-typedef CGAL::Arr_segment_traits_2<Kernel>            Traits_2;
-typedef Traits_2::Point_2                             Point_2;
-typedef Traits_2::X_monotone_curve_2                  Segment_2;
-typedef CGAL::Arrangement_2<Traits_2>                 Arrangement_2;
-typedef Arrangement_2::Halfedge_handle                Halfedge_handle;
+typedef CGAL::Quotient<int>                             Number_type;
+typedef CGAL::Simple_cartesian<Number_type>             Kernel;
+typedef CGAL::Arr_segment_traits_2<Kernel>              Traits_2;
+typedef Traits_2::Point_2                               Point_2;
+typedef Traits_2::X_monotone_curve_2                    Segment_2;
+typedef CGAL::Arrangement_2<Traits_2>                   Arrangement_2;
+typedef Arrangement_2::Vertex_handle                    Vertex_handle;
+typedef Arrangement_2::Halfedge_handle                  Halfedge_handle;
+typedef Arrangement_2::Face_handle                      Face_handle;
+typedef boost::variant<Vertex_handle, Halfedge_handle, Face_handle>
+                                                        Zone_result;
 
 #define N_SEGMENTS 3
 
@@ -40,7 +44,7 @@ int main ()
 
   for (k = 0; k < N_SEGMENTS; k++)
   {
-    std::list<CGAL::Object> zone_elems;
+    std::list<Zone_result> zone_elems;
     zone(arr, segs[k], std::back_inserter(zone_elems));
     std::size_t zone_actual_comp = zone_elems.size();
 

BGL/doc/BGL/NamedParameters.txt

@@ -38,8 +38,9 @@ CGAL::copy_face_graph(g1, g2);
 
 // specifying named parameters for the second graph
 CGAL::copy_face_graph(g1, g2,
-                      CGAL::parameters::all_default(),
-                      CGAL::parameters::vertex_point_map(vpm_2));
+                      CGAL::parameters::vertex_point_map(vpm)  //parameter for g1
+                      .vertex_to_vertex_map(v2v),              //other parameter for g1
+                      CGAL::parameters::all_default());        //parameter for g2
 \endcode
 */
 ˛

BGL/doc/BGL/PackageDescription.txt

@@ -443,7 +443,7 @@ the requirement for traversal of all faces in a graph.
 /// \defgroup PkgBGLPropertiesDynamic Dynamic Properties
 /// \ingroup PkgBGLRef
 
-/// \defgroup PkgBGLGraphExternalIndices External Indices
+/// \defgroup BGLGraphExternalIndices External Indices
 /// \ingroup PkgBGLRef
 
 /// \defgroup PkgBGLHelperFct Helper Functions

BGL/examples/BGL_LCC/normals_lcc.cpp

@@ -77,7 +77,7 @@ int main(int argc, char** argv)
   // http://www.boost.org/libs/property_map/doc/vector_property_map.html
   // for details.
   boost::vector_property_map<Vector, Face_index_map>
-    normals(get(CGAL::face_index, lcc));
+    normals(static_cast<unsigned>(num_faces(lcc)), get(CGAL::face_index, lcc));
 
   calculate_face_normals(
     lcc // Graph

BGL/examples/BGL_polyhedron_3/normals.cpp

@@ -82,7 +82,7 @@ int main(int argc, char** argv)
   // http://www.boost.org/libs/property_map/doc/vector_property_map.html
   // for details.
   boost::vector_property_map<Vector, Face_index_map>
-    normals(get(CGAL::face_index, P));
+    normals(static_cast<unsigned>(num_faces(P)), get(CGAL::face_index, P));
 
   calculate_face_normals(
     P // Graph

BGL/include/CGAL/boost/graph/Euler_operations.h

@@ -13,6 +13,8 @@
 #define CGAL_EULER_OPERATIONS_H
 
 #include <stdexcept>
+#include <algorithm>
+#include <vector>
 
 #include <boost/graph/graph_traits.hpp>
 #include <CGAL/boost/graph/properties.h>
@@ -558,6 +560,127 @@ add_edge(typename boost::graph_traits<Graph>::vertex_descriptor s,
   return e;
 }
 
+
+/**
+* checks whether a new face defined by a range of vertices (identified by their descriptors,
+* `boost::graph_traits<Graph>::%vertex_descriptor`) can be added.
+*/
+template <typename VertexRange,typename PMesh>
+bool can_add_face(const VertexRange& vrange, const PMesh& sm)
+{
+  typedef typename boost::graph_traits<PMesh>::vertex_descriptor vertex_descriptor;
+  typedef typename boost::graph_traits<PMesh>::halfedge_descriptor halfedge_descriptor;
+
+  std::vector<typename boost::graph_traits<PMesh>::vertex_descriptor> face(vrange.begin(), vrange.end());
+
+  std::size_t N = face.size();
+  std::vector<vertex_descriptor> f2(face);
+  std::sort(f2.begin(), f2.end());
+
+  typename std::vector<vertex_descriptor>::iterator it = std::unique(f2.begin(),f2.end());
+
+  if((N > 0) && (it != f2.end())){
+    return false;
+  }
+
+  if(N < 3){
+    return false;
+  }
+
+  face.push_back(face.front());
+
+  for(std::size_t i=0; i < N; ++i){
+    halfedge_descriptor hd;
+    bool found;
+    boost::tie(hd,found) = halfedge(face[i],face[i+1],sm);
+    if(found && (! is_border(hd,sm))){
+      return false;
+    }
+  }
+
+  for(std::size_t i=0; i < N; ++i){
+    if(halfedge(face[i],sm) == boost::graph_traits<PMesh>::null_halfedge()){
+      continue;
+    }
+
+    if(! is_border(face[i],sm)){
+      return false;
+    }
+  }
+  //Test if all halfedges of the new face
+  //are possibly consecutive border halfedges in the HDS.
+  //Possibly because it may be not directly encoded in the HDS
+  //(using next() function ). This situation can occur when one or
+  //more facets share only a vertex: For example, the new facet we try to add
+  //would make the vertex indices[i] a manifold but this should be forbidden
+  //if a facet only incident to that vertex has already been inserted.
+  //We check this for each vertex of the sequence.
+  for(std::size_t i = 0; i < N; ++i) {
+    std::size_t prev_index= (i-1+N)%N;
+    std::size_t next_index= (i+1)%N;
+    vertex_descriptor   previous_vertex = face[ prev_index ];
+    vertex_descriptor   next_vertex     = face[ next_index ];
+
+    halfedge_descriptor halfedge_around_vertex = halfedge(face[i],sm);
+
+    if ( halfedge_around_vertex == boost::graph_traits<PMesh>::null_halfedge() ||
+         halfedge(previous_vertex,sm) == boost::graph_traits<PMesh>::null_halfedge()||
+         halfedge(next_vertex,sm) == boost::graph_traits<PMesh>::null_halfedge()
+         ) continue;
+
+    halfedge_descriptor start=halfedge_around_vertex;
+    //halfedges pointing to/running out from vertex indices[i]
+    //and that need to be possibly consecutive
+    halfedge_descriptor prev_hd= boost::graph_traits<PMesh>::null_halfedge(),next_hd= boost::graph_traits<PMesh>::null_halfedge();
+
+    halfedge_around_vertex = opposite(next(halfedge_around_vertex,sm),sm);
+    //look for a halfedge incident to vertex indices[i]
+    //and which opposite is incident to previous_vertex
+    do{
+      if(target(opposite(halfedge_around_vertex,sm),sm)==previous_vertex){
+        prev_hd=halfedge_around_vertex;
+        CGAL_precondition(is_border(prev_hd,sm));
+        break;
+      }
+      halfedge_around_vertex = opposite(next(halfedge_around_vertex,sm),sm);
+    }
+    while (halfedge_around_vertex!=start);
+
+    if (prev_hd != boost::graph_traits<PMesh>::null_halfedge()){
+      halfedge_around_vertex = opposite(next(halfedge_around_vertex,sm),sm);
+      //prev_hd and next are already consecutive in the HDS
+      if (target(opposite(halfedge_around_vertex,sm),sm)==next_vertex) continue;
+
+      //look for a border halfedge which opposite is
+      //incident to next_vertex: set next halfedge
+      do
+        {
+          if (target(opposite(halfedge_around_vertex,sm),sm)==next_vertex){
+            next_hd = opposite(halfedge_around_vertex,sm);
+            break;
+          }
+          halfedge_around_vertex = opposite(next(halfedge_around_vertex,sm),sm);
+        }
+      while(halfedge_around_vertex != prev_hd);
+      if (next_hd==boost::graph_traits<PMesh>::null_halfedge()) continue;
+
+      //check if no constraint prevents
+      //prev_hd and next_hd to be adjacent:
+      do{
+        halfedge_around_vertex = opposite(next(halfedge_around_vertex, sm),sm);
+        if ( is_border(opposite(halfedge_around_vertex,sm),sm) ) break;
+      }
+      while (halfedge_around_vertex != prev_hd);
+      if (halfedge_around_vertex == prev_hd) return false;
+      start = halfedge_around_vertex;
+    }
+  }
+
+  return true;
+}
+
+
+
 /**
 * adds a new face defined by a range of vertices (identified by their descriptors,
 * `boost::graph_traits<Graph>::%vertex_descriptor`).

BGL/include/CGAL/boost/graph/Face_filtered_graph.h

@@ -104,6 +104,66 @@ struct Face_filtered_graph
 
   typedef Face_filtered_graph<Graph, FIMap, VIMap, HIMap> Self;
 
+  /*!
+   * \brief constructs an empty face filtered graph (no face is selected)
+   *
+   * \tparam NamedParameters a sequence of named parameters
+   *
+   * \param graph the underlying graph.
+   *
+   * \param np optional sequence of named parameters among the ones listed below
+   *
+   * \cgalNamedParamsBegin
+   *   \cgalParamNBegin{vertex_index_map}
+   *     \cgalParamDescription{a property map associating to each vertex of `graph` a unique index between `0` and `num_vertices(graph) - 1`}
+   *     \cgalParamType{a class model of `ReadablePropertyMap` with `boost::graph_traits<Graph>::%vertex_descriptor`
+   *                    as key type and `std::size_t` as value type}
+   *     \cgalParamDefault{an automatically indexed internal map}
+   *     \cgalParamExtra{If this parameter is not passed, internal machinery will create and initialize
+   *                     a face index property map, either using the internal property map if it exists
+   *                     or using an external map. The latter might result in  - slightly - worsened performance
+   *                     in case of non-constant complexity for index access.}
+   *   \cgalParamNEnd
+   *
+   *   \cgalParamNBegin{halfedge_index_map}
+   *     \cgalParamDescription{a property map associating to each halfedge of `graph` a unique index between `0` and `num_halfedges(graph) - 1`}
+   *     \cgalParamType{a class model of `ReadablePropertyMap` with `boost::graph_traits<Graph>::%halfedge_descriptor`
+   *                    as key type and `std::size_t` as value type}
+   *     \cgalParamDefault{an automatically indexed internal map}
+   *     \cgalParamExtra{If this parameter is not passed, internal machinery will create and initialize
+   *                     a face index property map, either using the internal property map if it exists
+   *                     or using an external map. The latter might result in  - slightly - worsened performance
+   *                     in case of non-constant complexity for index access.}
+   *   \cgalParamNEnd
+   *
+   *   \cgalParamNBegin{face_index_map}
+   *     \cgalParamDescription{a property map associating to each face of `graph` a unique index between `0` and `num_faces(graph) - 1`}
+   *     \cgalParamType{a class model of `ReadablePropertyMap` with `boost::graph_traits<Graph>::%face_descriptor`
+   *                    as key type and `std::size_t` as value type}
+   *     \cgalParamDefault{an automatically indexed internal map}
+   *     \cgalParamExtra{If this parameter is not passed, internal machinery will create and initialize
+   *                     a face index property map, either using the internal property map if it exists
+   *                     or using an external map. The latter might result in  - slightly - worsened performance
+   *                     in case of non-constant complexity for index access.}
+   *   \cgalParamNEnd
+   * \cgalNamedParamsEnd
+   */
+  template <class CGAL_BGL_NP_TEMPLATE_PARAMETERS>
+  Face_filtered_graph(const Graph& graph,
+                      const CGAL_BGL_NP_CLASS& np)
+    : _graph(const_cast<Graph&>(graph))
+    , fimap(CGAL::get_initialized_face_index_map(graph, np))
+    , vimap(CGAL::get_initialized_vertex_index_map(graph, np))
+    , himap(CGAL::get_initialized_halfedge_index_map(graph, np))
+    , selected_faces(num_faces(graph), 0)
+    , selected_vertices(num_vertices(graph), 0)
+    , selected_halfedges(num_halfedges(graph), 0)
+  {}
+
+  Face_filtered_graph(const Graph& graph)
+    :Face_filtered_graph(graph, parameters::all_default())
+  {}
+
   /*!
    * \brief Constructor where the set of selected faces is specified as a range of patch ids.
    *

BGL/include/CGAL/boost/graph/alpha_expansion_graphcut.h

@@ -1,4 +1,3 @@
-#ifndef CGAL_BOOST_GRAPH_ALPHA_EXPANSION_GRAPHCUT_H
 // Copyright (c) 2014  GeometryFactory (France).  All rights reserved.
 //
 // This file is part of CGAL (www.cgal.org)
@@ -10,6 +9,7 @@
 //
 // Author(s)     : Ilker O. Yaz, Simon Giraudot
 
+#ifndef CGAL_BOOST_GRAPH_ALPHA_EXPANSION_GRAPHCUT_H
 #define CGAL_BOOST_GRAPH_ALPHA_EXPANSION_GRAPHCUT_H
 
 #include <CGAL/Iterator_range.h>

BGL/include/CGAL/boost/graph/dijkstra_shortest_paths.h

@@ -15,21 +15,6 @@
 
 // This will push/pop a VC15 warning
 #include <CGAL/boost/graph/Named_function_parameters.h>
-
-#include <boost/version.hpp>
-#include <climits>
-
-#if BOOST_VERSION == 105400
-  #ifdef BOOST_GRAPH_DIJKSTRA_HPP
-  #    pragma message \
-      "Warning: the header file boost/graph/dijkstra_shortest_paths.hpp "       \
-      "of boost 1.54 contains a bug that may impact some functions in CGAL. "   \
-      "Please consider including CGAL/boost/graph/dijkstra_shortest_paths.hpp "  \
-      "before boost header"
-  #endif
-  #include <CGAL/boost/graph/dijkstra_shortest_paths.hpp>
-#else
-  #include <boost/graph/dijkstra_shortest_paths.hpp>
-#endif
+#include <boost/graph/dijkstra_shortest_paths.hpp>
 
 #endif // CGAL_BOOST_GRAPH_DIJKSTRA_SHORTEST_PATHS_H

BGL/include/CGAL/boost/graph/dijkstra_shortest_paths.hpp

@@ -1,633 +0,0 @@
-// This file is a copy of the file distributed with boost 1.55
-// It is distributed with CGAL to work around a bug in boost 1.54
-// and must be used only with boost 1.54 by including
-// CGAL/boost/graph/dijkstra_shortest_paths.hpp
-
-
-//=======================================================================
-// Copyright 1997, 1998, 1999, 2000 University of Notre Dame.
-// Authors: Andrew Lumsdaine, Lie-Quan Lee, Jeremy G. Siek
-//
-// Distributed under the Boost Software License, Version 1.0. (See
-// accompanying file LICENSE_1_0.txt or copy at
-// https://www.boost.org/LICENSE_1_0.txt)
-//=======================================================================
-//
-// $URL$
-// $Id$
-// SPDX-License-Identifier: BSL-1.0
-//
-// Revision History:
-//   04 April 2001: Added named parameter variant. (Jeremy Siek)
-//   01 April 2001: Modified to use new <boost/limits.hpp> header. (JMaddock)
-//
-#ifndef BOOST_GRAPH_DIJKSTRA_HPP
-#define BOOST_GRAPH_DIJKSTRA_HPP
-
-#include <functional>
-#include <boost/limits.hpp>
-#include <boost/graph/named_function_params.hpp>
-#include <boost/graph/breadth_first_search.hpp>
-#include <boost/graph/relax.hpp>
-#include <boost/pending/indirect_cmp.hpp>
-#include <boost/graph/exception.hpp>
-#include <boost/pending/relaxed_heap.hpp>
-#include <boost/graph/overloading.hpp>
-#include <boost/smart_ptr.hpp>
-#include <boost/graph/detail/d_ary_heap.hpp>
-#include <boost/graph/two_bit_color_map.hpp>
-#include <boost/property_map/property_map.hpp>
-#include <boost/property_map/vector_property_map.hpp>
-#include <boost/type_traits.hpp>
-#include <boost/concept/assert.hpp>
-
-#ifdef BOOST_GRAPH_DIJKSTRA_TESTING
-#  include <boost/pending/mutable_queue.hpp>
-#endif // BOOST_GRAPH_DIJKSTRA_TESTING
-
-namespace boost {
-
-  /**
-   * @brief Updates a particular value in a queue used by Dijkstra's
-   * algorithm.
-   *
-   * This routine is called by Dijkstra's algorithm after it has
-   * decreased the distance from the source vertex to the given @p
-   * vertex. By default, this routine will just call @c
-   * Q.update(vertex). However, other queues may provide more
-   * specialized versions of this routine.
-   *
-   * @param Q             the queue that will be updated.
-   * @param vertex        the vertex whose distance has been updated
-   * @param old_distance  the previous distance to @p vertex
-   */
-  template<typename Buffer, typename Vertex, typename DistanceType>
-  inline void
-  dijkstra_queue_update(Buffer& Q, Vertex vertex, DistanceType old_distance)
-  {
-    (void)old_distance;
-    Q.update(vertex);
-  }
-
-#ifdef BOOST_GRAPH_DIJKSTRA_TESTING
-  // This is a misnomer now: it now just refers to the "default heap", which is
-  // currently d-ary (d=4) but can be changed by a #define.
-  static bool dijkstra_relaxed_heap = true;
-#endif
-
-  template <class Visitor, class Graph>
-  struct DijkstraVisitorConcept {
-    void constraints() {
-//      commented to avoid an unused vaiable warning
-//      BOOST_CONCEPT_ASSERT(( CopyConstructibleConcept<Visitor> ));
-      vis.initialize_vertex(u, g);
-      vis.discover_vertex(u, g);
-      vis.examine_vertex(u, g);
-      vis.examine_edge(e, g);
-      vis.edge_relaxed(e, g);
-      vis.edge_not_relaxed(e, g);
-      vis.finish_vertex(u, g);
-    }
-    Visitor vis;
-    Graph g;
-    typename graph_traits<Graph>::vertex_descriptor u;
-    typename graph_traits<Graph>::edge_descriptor e;
-  };
-
-  template <class Visitors = null_visitor>
-  class dijkstra_visitor : public bfs_visitor<Visitors> {
-  public:
-    dijkstra_visitor() { }
-    dijkstra_visitor(Visitors vis)
-      : bfs_visitor<Visitors>(vis) { }
-
-    template <class Edge, class Graph>
-    void edge_relaxed(Edge e, Graph& g) {
-      invoke_visitors(this->m_vis, e, g, on_edge_relaxed());
-    }
-    template <class Edge, class Graph>
-    void edge_not_relaxed(Edge e, Graph& g) {
-      invoke_visitors(this->m_vis, e, g, on_edge_not_relaxed());
-    }
-  private:
-    template <class Edge, class Graph>
-    void tree_edge(Edge /* u */, Graph& /* g */) { }
-  };
-  template <class Visitors>
-  dijkstra_visitor<Visitors>
-  make_dijkstra_visitor(Visitors vis) {
-    return dijkstra_visitor<Visitors>(vis);
-  }
-  typedef dijkstra_visitor<> default_dijkstra_visitor;
-
-  namespace detail {
-
-    template <class UniformCostVisitor, class UpdatableQueue,
-      class WeightMap, class PredecessorMap, class DistanceMap,
-      class BinaryFunction, class BinaryPredicate>
-    struct dijkstra_bfs_visitor
-    {
-      typedef typename property_traits<DistanceMap>::value_type D;
-      typedef typename property_traits<WeightMap>::value_type W;
-
-      dijkstra_bfs_visitor(UniformCostVisitor vis, UpdatableQueue& Q,
-                           WeightMap w, PredecessorMap p, DistanceMap d,
-                           BinaryFunction combine, BinaryPredicate compare,
-                           D zero)
-        : m_vis(vis), m_Q(Q), m_weight(w), m_predecessor(p), m_distance(d),
-          m_combine(combine), m_compare(compare), m_zero(zero)  { }
-
-      template <class Edge, class Graph>
-      void tree_edge(Edge e, Graph& g) {
-        bool decreased = relax(e, g, m_weight, m_predecessor, m_distance,
-                               m_combine, m_compare);
-        if (decreased)
-          m_vis.edge_relaxed(e, g);
-        else
-          m_vis.edge_not_relaxed(e, g);
-      }
-      template <class Edge, class Graph>
-      void gray_target(Edge e, Graph& g) {
-        D old_distance = get(m_distance, target(e, g));
-
-        bool decreased = relax(e, g, m_weight, m_predecessor, m_distance,
-                               m_combine, m_compare);
-        if (decreased) {
-          dijkstra_queue_update(m_Q, target(e, g), old_distance);
-          m_vis.edge_relaxed(e, g);
-        } else
-          m_vis.edge_not_relaxed(e, g);
-      }
-
-      template <class Vertex, class Graph>
-      void initialize_vertex(Vertex u, Graph& g)
-        { m_vis.initialize_vertex(u, g); }
-      template <class Edge, class Graph>
-      void non_tree_edge(Edge, Graph&) { }
-      template <class Vertex, class Graph>
-      void discover_vertex(Vertex u, Graph& g) { m_vis.discover_vertex(u, g); }
-      template <class Vertex, class Graph>
-      void examine_vertex(Vertex u, Graph& g) { m_vis.examine_vertex(u, g); }
-      template <class Edge, class Graph>
-      void examine_edge(Edge e, Graph& g) {
-        // Test for negative-weight edges:
-        //
-        // Reasons that other comparisons do not work:
-        //
-        // m_compare(e_weight, D(0)):
-        //    m_compare only needs to work on distances, not weights, and those
-        //    types do not need to be the same (bug 8398,
-        //    https://svn.boost.org/trac/boost/ticket/8398).
-        // m_compare(m_combine(source_dist, e_weight), source_dist):
-        //    if m_combine is project2nd (as in prim_minimum_spanning_tree),
-        //    this test will claim that the edge weight is negative whenever
-        //    the edge weight is less than source_dist, even if both of those
-        //    are positive (bug 9012,
-        //    https://svn.boost.org/trac/boost/ticket/9012).
-        // m_compare(m_combine(e_weight, source_dist), source_dist):
-        //    would fix project2nd issue, but documentation only requires that
-        //    m_combine be able to take a distance and a weight (in that order)
-        //    and return a distance.
-
-        // W e_weight = get(m_weight, e);
-        // sd_plus_ew = source_dist + e_weight.
-        // D sd_plus_ew = m_combine(source_dist, e_weight);
-        // sd_plus_2ew = source_dist + 2 * e_weight.
-        // D sd_plus_2ew = m_combine(sd_plus_ew, e_weight);
-        // The test here is equivalent to e_weight < 0 if m_combine has a
-        // cancellation law, but always returns false when m_combine is a
-        // projection operator.
-        if (m_compare(m_combine(m_zero, get(m_weight, e)), m_zero))
-            boost::throw_exception(negative_edge());
-        // End of test for negative-weight edges.
-
-        m_vis.examine_edge(e, g);
-
-      }
-      template <class Edge, class Graph>
-      void black_target(Edge, Graph&) { }
-      template <class Vertex, class Graph>
-      void finish_vertex(Vertex u, Graph& g) { m_vis.finish_vertex(u, g); }
-
-      UniformCostVisitor m_vis;
-      UpdatableQueue& m_Q;
-      WeightMap m_weight;
-      PredecessorMap m_predecessor;
-      DistanceMap m_distance;
-      BinaryFunction m_combine;
-      BinaryPredicate m_compare;
-      D m_zero;
-    };
-
-  } // namespace detail
-
-  namespace detail {
-    template <class Graph, class IndexMap, class Value, bool KnownNumVertices>
-    struct vertex_property_map_generator_helper {};
-
-    template <class Graph, class IndexMap, class Value>
-    struct vertex_property_map_generator_helper<Graph, IndexMap, Value, true> {
-      typedef boost::iterator_property_map<Value*, IndexMap> type;
-      static type build(const Graph& g, const IndexMap& index, boost::scoped_array<Value>& array_holder) {
-        array_holder.reset(new Value[num_vertices(g)]);
-        std::fill(array_holder.get(), array_holder.get() + num_vertices(g), Value());
-        return make_iterator_property_map(array_holder.get(), index);
-      }
-    };
-
-    template <class Graph, class IndexMap, class Value>
-    struct vertex_property_map_generator_helper<Graph, IndexMap, Value, false> {
-      typedef boost::vector_property_map<Value, IndexMap> type;
-      static type build(const Graph& /* g */, const IndexMap& index, boost::scoped_array<Value>& /* array_holder */) {
-        return boost::make_vector_property_map<Value>(index);
-      }
-    };
-
-    template <class Graph, class IndexMap, class Value>
-    struct vertex_property_map_generator {
-      typedef boost::is_base_and_derived<
-                boost::vertex_list_graph_tag,
-                typename boost::graph_traits<Graph>::traversal_category>
-              known_num_vertices;
-      typedef vertex_property_map_generator_helper<Graph, IndexMap, Value, known_num_vertices::value> helper;
-      typedef typename helper::type type;
-      static type build(const Graph& g, const IndexMap& index, boost::scoped_array<Value>& array_holder) {
-        return helper::build(g, index, array_holder);
-      }
-    };
-  }
-
-  namespace detail {
-    template <class Graph, class IndexMap, bool KnownNumVertices>
-    struct default_color_map_generator_helper {};
-
-    template <class Graph, class IndexMap>
-    struct default_color_map_generator_helper<Graph, IndexMap, true> {
-      typedef boost::two_bit_color_map<IndexMap> type;
-      static type build(const Graph& g, const IndexMap& index) {
-        size_t nv = num_vertices(g);
-        return boost::two_bit_color_map<IndexMap>(nv, index);
-      }
-    };
-
-    template <class Graph, class IndexMap>
-    struct default_color_map_generator_helper<Graph, IndexMap, false> {
-      typedef boost::vector_property_map<boost::two_bit_color_type, IndexMap> type;
-      static type build(const Graph& /* g */, const IndexMap& index) {
-        return boost::make_vector_property_map<boost::two_bit_color_type>(index);
-      }
-    };
-
-    template <class Graph, class IndexMap>
-    struct default_color_map_generator {
-      typedef boost::is_base_and_derived<
-                boost::vertex_list_graph_tag,
-                typename boost::graph_traits<Graph>::traversal_category>
-              known_num_vertices;
-      typedef default_color_map_generator_helper<Graph, IndexMap, known_num_vertices::value> helper;
-      typedef typename helper::type type;
-      static type build(const Graph& g, const IndexMap& index) {
-        return helper::build(g, index);
-      }
-    };
-  }
-
-  // Call breadth first search with default color map.
-  template <class Graph, class SourceInputIter, class DijkstraVisitor,
-            class PredecessorMap, class DistanceMap,
-            class WeightMap, class IndexMap, class Compare, class Combine,
-            class DistZero>
-  inline void
-  dijkstra_shortest_paths_no_init
-    (const Graph& g,
-     SourceInputIter s_begin, SourceInputIter s_end,
-     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
-     IndexMap index_map,
-     Compare compare, Combine combine, DistZero zero,
-     DijkstraVisitor vis)
-  {
-    typedef
-      detail::default_color_map_generator<Graph, IndexMap>
-      ColorMapHelper;
-    typedef typename ColorMapHelper::type ColorMap;
-    ColorMap color =
-      ColorMapHelper::build(g, index_map);
-    dijkstra_shortest_paths_no_init( g, s_begin, s_end, predecessor, distance, weight,
-      index_map, compare, combine, zero, vis,
-        color);
-  }
-
-  // Call breadth first search with default color map.
-  template <class Graph, class DijkstraVisitor,
-            class PredecessorMap, class DistanceMap,
-            class WeightMap, class IndexMap, class Compare, class Combine,
-            class DistZero>
-  inline void
-  dijkstra_shortest_paths_no_init
-    (const Graph& g,
-     typename graph_traits<Graph>::vertex_descriptor s,
-     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
-     IndexMap index_map,
-     Compare compare, Combine combine, DistZero zero,
-     DijkstraVisitor vis)
-  {
-    dijkstra_shortest_paths_no_init(g, &s, &s + 1, predecessor, distance,
-                                    weight, index_map, compare, combine, zero,
-                                    vis);
-  }
-
-  // Call breadth first search
-  template <class Graph, class SourceInputIter, class DijkstraVisitor,
-            class PredecessorMap, class DistanceMap,
-            class WeightMap, class IndexMap, class Compare, class Combine,
-            class DistZero, class ColorMap>
-  inline void
-  dijkstra_shortest_paths_no_init
-    (const Graph& g,
-     SourceInputIter s_begin, SourceInputIter s_end,
-     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
-     IndexMap index_map,
-     Compare compare, Combine combine, DistZero zero,
-     DijkstraVisitor vis, ColorMap color)
-  {
-    typedef indirect_cmp<DistanceMap, Compare> IndirectCmp;
-    IndirectCmp icmp(distance, compare);
-
-    typedef typename graph_traits<Graph>::vertex_descriptor Vertex;
-
-#ifdef BOOST_GRAPH_DIJKSTRA_TESTING
-    if (!dijkstra_relaxed_heap) {
-      typedef mutable_queue<Vertex, std::vector<Vertex>, IndirectCmp, IndexMap>
-        MutableQueue;
-
-      MutableQueue Q(num_vertices(g), icmp, index_map);
-      detail::dijkstra_bfs_visitor<DijkstraVisitor, MutableQueue, WeightMap,
-        PredecessorMap, DistanceMap, Combine, Compare>
-      bfs_vis(vis, Q, weight, predecessor, distance, combine, compare, zero);
-
-      breadth_first_visit(g, s_begin, s_end, Q, bfs_vis, color);
-      return;
-    }
-#endif // BOOST_GRAPH_DIJKSTRA_TESTING
-
-#ifdef BOOST_GRAPH_DIJKSTRA_USE_RELAXED_HEAP
-    typedef relaxed_heap<Vertex, IndirectCmp, IndexMap> MutableQueue;
-    MutableQueue Q(num_vertices(g), icmp, index_map);
-#else // Now the default: use a d-ary heap
-      boost::scoped_array<std::size_t> index_in_heap_map_holder;
-      typedef
-        detail::vertex_property_map_generator<Graph, IndexMap, std::size_t>
-        IndexInHeapMapHelper;
-      typedef typename IndexInHeapMapHelper::type IndexInHeapMap;
-      IndexInHeapMap index_in_heap =
-        IndexInHeapMapHelper::build(g, index_map, index_in_heap_map_holder);
-      typedef d_ary_heap_indirect<Vertex, 4, IndexInHeapMap, DistanceMap, Compare>
-        MutableQueue;
-      MutableQueue Q(distance, index_in_heap, compare);
-#endif // Relaxed heap
-
-    detail::dijkstra_bfs_visitor<DijkstraVisitor, MutableQueue, WeightMap,
-      PredecessorMap, DistanceMap, Combine, Compare>
-        bfs_vis(vis, Q, weight, predecessor, distance, combine, compare, zero);
-
-    breadth_first_visit(g, s_begin, s_end, Q, bfs_vis, color);
-  }
-
-  // Call breadth first search
-  template <class Graph, class DijkstraVisitor,
-            class PredecessorMap, class DistanceMap,
-            class WeightMap, class IndexMap, class Compare, class Combine,
-            class DistZero, class ColorMap>
-  inline void
-  dijkstra_shortest_paths_no_init
-    (const Graph& g,
-     typename graph_traits<Graph>::vertex_descriptor s,
-     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
-     IndexMap index_map,
-     Compare compare, Combine combine, DistZero zero,
-     DijkstraVisitor vis, ColorMap color)
-  {
-    dijkstra_shortest_paths_no_init(g, &s, &s + 1, predecessor, distance,
-                                    weight, index_map, compare, combine,
-                                    zero, vis, color);
-  }
-
-  // Initialize distances and call breadth first search with default color map
-  template <class VertexListGraph, class SourceInputIter, class DijkstraVisitor,
-            class PredecessorMap, class DistanceMap,
-            class WeightMap, class IndexMap, class Compare, class Combine,
-            class DistInf, class DistZero, typename T, typename Tag,
-            typename Base>
-  inline void
-  dijkstra_shortest_paths
-    (const VertexListGraph& g,
-     SourceInputIter s_begin, SourceInputIter s_end,
-     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
-     IndexMap index_map,
-     Compare compare, Combine combine, DistInf inf, DistZero zero,
-     DijkstraVisitor vis,
-     const bgl_named_params<T, Tag, Base>&
-     BOOST_GRAPH_ENABLE_IF_MODELS_PARM(VertexListGraph,vertex_list_graph_tag))
-  {
-    boost::two_bit_color_map<IndexMap> color(num_vertices(g), index_map);
-    dijkstra_shortest_paths(g, s_begin, s_end, predecessor, distance, weight,
-                            index_map, compare, combine, inf, zero, vis,
-                            color);
-  }
-
-  // Initialize distances and call breadth first search with default color map
-  template <class VertexListGraph, class DijkstraVisitor,
-            class PredecessorMap, class DistanceMap,
-            class WeightMap, class IndexMap, class Compare, class Combine,
-            class DistInf, class DistZero, typename T, typename Tag,
-            typename Base>
-  inline void
-  dijkstra_shortest_paths
-    (const VertexListGraph& g,
-     typename graph_traits<VertexListGraph>::vertex_descriptor s,
-     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
-     IndexMap index_map,
-     Compare compare, Combine combine, DistInf inf, DistZero zero,
-     DijkstraVisitor vis,
-     const bgl_named_params<T, Tag, Base>&
-     BOOST_GRAPH_ENABLE_IF_MODELS_PARM(VertexListGraph,vertex_list_graph_tag))
-  {
-    dijkstra_shortest_paths(g, &s, &s + 1, predecessor, distance, weight,
-                            index_map, compare, combine, inf, zero, vis);
-  }
-
-  // Initialize distances and call breadth first search
-  template <class VertexListGraph, class SourceInputIter, class DijkstraVisitor,
-            class PredecessorMap, class DistanceMap,
-            class WeightMap, class IndexMap, class Compare, class Combine,
-            class DistInf, class DistZero, class ColorMap>
-  inline void
-  dijkstra_shortest_paths
-    (const VertexListGraph& g,
-     SourceInputIter s_begin, SourceInputIter s_end,
-     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
-     IndexMap index_map,
-     Compare compare, Combine combine, DistInf inf, DistZero zero,
-     DijkstraVisitor vis, ColorMap color)
-  {
-    typedef typename property_traits<ColorMap>::value_type ColorValue;
-    typedef color_traits<ColorValue> Color;
-    typename graph_traits<VertexListGraph>::vertex_iterator ui, ui_end;
-    for (boost::tie(ui, ui_end) = vertices(g); ui != ui_end; ++ui) {
-      vis.initialize_vertex(*ui, g);
-      put(distance, *ui, inf);
-      put(predecessor, *ui, *ui);
-      put(color, *ui, Color::white());
-    }
-    for (SourceInputIter it = s_begin; it != s_end; ++it) {
-      put(distance, *it, zero);
-    }
-
-    dijkstra_shortest_paths_no_init(g, s_begin, s_end, predecessor, distance,
-                            weight, index_map, compare, combine, zero, vis,
-                            color);
-  }
-
-  // Initialize distances and call breadth first search
-  template <class VertexListGraph, class DijkstraVisitor,
-            class PredecessorMap, class DistanceMap,
-            class WeightMap, class IndexMap, class Compare, class Combine,
-            class DistInf, class DistZero, class ColorMap>
-  inline void
-  dijkstra_shortest_paths
-    (const VertexListGraph& g,
-     typename graph_traits<VertexListGraph>::vertex_descriptor s,
-     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
-     IndexMap index_map,
-     Compare compare, Combine combine, DistInf inf, DistZero zero,
-     DijkstraVisitor vis, ColorMap color)
-  {
-    dijkstra_shortest_paths(g, &s, &s + 1, predecessor, distance, weight,
-                            index_map, compare, combine, inf, zero,
-                            vis, color);
-  }
-
-  // Initialize distances and call breadth first search
-  template <class VertexListGraph, class SourceInputIter,
-            class DijkstraVisitor,
-            class PredecessorMap, class DistanceMap,
-            class WeightMap, class IndexMap, class Compare, class Combine,
-            class DistInf, class DistZero>
-  inline void
-  dijkstra_shortest_paths
-    (const VertexListGraph& g,
-     SourceInputIter s_begin, SourceInputIter s_end,
-     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
-     IndexMap index_map,
-     Compare compare, Combine combine, DistInf inf, DistZero zero,
-     DijkstraVisitor vis)
-  {
-    dijkstra_shortest_paths(g, s_begin, s_end, predecessor, distance,
-                            weight, index_map,
-                            compare, combine, inf, zero, vis,
-                            no_named_parameters());
-  }
-
-  // Initialize distances and call breadth first search
-  template <class VertexListGraph, class DijkstraVisitor,
-            class PredecessorMap, class DistanceMap,
-            class WeightMap, class IndexMap, class Compare, class Combine,
-            class DistInf, class DistZero>
-  inline void
-  dijkstra_shortest_paths
-    (const VertexListGraph& g,
-     typename graph_traits<VertexListGraph>::vertex_descriptor s,
-     PredecessorMap predecessor, DistanceMap distance, WeightMap weight,
-     IndexMap index_map,
-     Compare compare, Combine combine, DistInf inf, DistZero zero,
-     DijkstraVisitor vis)
-  {
-    dijkstra_shortest_paths(g, &s, &s + 1, predecessor, distance,
-                            weight, index_map,
-                            compare, combine, inf, zero, vis);
-  }
-
-  namespace detail {
-
-    // Handle defaults for PredecessorMap and
-    // Distance Compare, Combine, Inf and Zero
-    template <class VertexListGraph, class DistanceMap, class WeightMap,
-              class IndexMap, class Params>
-    inline void
-    dijkstra_dispatch2
-      (const VertexListGraph& g,
-       typename graph_traits<VertexListGraph>::vertex_descriptor s,
-       DistanceMap distance, WeightMap weight, IndexMap index_map,
-       const Params& params)
-    {
-      // Default for predecessor map
-      dummy_property_map p_map;
-
-      typedef typename property_traits<DistanceMap>::value_type D;
-      D inf = choose_param(get_param(params, distance_inf_t()),
-                           (std::numeric_limits<D>::max)());
-
-      dijkstra_shortest_paths
-        (g, s,
-         choose_param(get_param(params, vertex_predecessor), p_map),
-         distance, weight, index_map,
-         choose_param(get_param(params, distance_compare_t()),
-                      std::less<D>()),
-         choose_param(get_param(params, distance_combine_t()),
-                      closed_plus<D>(inf)),
-         inf,
-         choose_param(get_param(params, distance_zero_t()),
-                      D()),
-         choose_param(get_param(params, graph_visitor),
-                      make_dijkstra_visitor(null_visitor())),
-         params);
-    }
-
-    template <class VertexListGraph, class DistanceMap, class WeightMap,
-              class IndexMap, class Params>
-    inline void
-    dijkstra_dispatch1
-      (const VertexListGraph& g,
-       typename graph_traits<VertexListGraph>::vertex_descriptor s,
-       DistanceMap distance, WeightMap weight, IndexMap index_map,
-       const Params& params)
-    {
-      // Default for distance map
-      typedef typename property_traits<WeightMap>::value_type D;
-      typename std::vector<D>::size_type
-        n = is_default_param(distance) ? num_vertices(g) : 1;
-      std::vector<D> distance_map(n);
-
-      detail::dijkstra_dispatch2
-        (g, s, choose_param(distance, make_iterator_property_map
-                            (distance_map.begin(), index_map,
-                             distance_map[0])),
-         weight, index_map, params);
-    }
-  } // namespace detail
-
-  // Named Parameter Variant
-  template <class VertexListGraph, class Param, class Tag, class Rest>
-  inline void
-  dijkstra_shortest_paths
-    (const VertexListGraph& g,
-     typename graph_traits<VertexListGraph>::vertex_descriptor s,
-     const bgl_named_params<Param,Tag,Rest>& params)
-  {
-    // Default for edge weight and vertex index map is to ask for them
-    // from the graph.  Default for the visitor is null_visitor.
-    detail::dijkstra_dispatch1
-      (g, s,
-       get_param(params, vertex_distance),
-       choose_const_pmap(get_param(params, edge_weight), g, edge_weight),
-       choose_const_pmap(get_param(params, vertex_index), g, vertex_index),
-       params);
-  }
-
-} // namespace boost
-
-#ifdef BOOST_GRAPH_USE_MPI
-#  include <boost/graph/distributed/dijkstra_shortest_paths.hpp>
-#endif
-
-#endif // BOOST_GRAPH_DIJKSTRA_HPP

BGL/include/CGAL/boost/graph/io.h

@@ -20,13 +20,6 @@
 #include <CGAL/boost/graph/IO/STL.h>
 #include <CGAL/boost/graph/IO/VTK.h>
 #include <CGAL/boost/graph/IO/WRL.h>
-  \param os the output stream
-  \param g the graph to be written
-  \param os the output stream
-  \param g the graph to be written
- \param is the input stream
- \param g the graph to be read
-
 #include <CGAL/boost/graph/IO/polygon_mesh_io.h>
 
 #endif // CGAL_BOOST_GRAPH_IO_H

BGL/include/CGAL/boost/graph/named_params_helper.h

@@ -1,18 +1,8 @@
-//=======================================================================
-// Copyright 1997, 1998, 1999, 2000 University of Notre Dame.
-// Authors: Andrew Lumsdaine, Lie-Quan Lee, Jeremy G. Siek
-//
-// This file is part of the Boost Graph Library
-//
-// Distributed under the Boost Software License, Version 1.0. (See
-// accompanying file LICENSE_1_0.txt or copy at
-// https://www.boost.org/LICENSE_1_0.txt)
-//=======================================================================
 // Copyright (c) 2007-2015  GeometryFactory (France).  All rights reserved.
 //
 // $URL$
 // $Id$
-// SPDX-License-Identifier: BSL-1.0
+// SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial
 //
 // Author(s)     : Andreas Fabri, Fernando Cacciola, Jane Tournois
 
@@ -299,7 +289,6 @@ CGAL_DEF_GET_INITIALIZED_INDEX_MAP(face, typename boost::graph_traits<Graph>::fa
       > ::type  type;
   };
 
-
   namespace internal {
     BOOST_MPL_HAS_XXX_TRAIT_NAMED_DEF(Has_nested_type_iterator, iterator, false)
   }

BGL/include/CGAL/boost/graph/parameters_interface.h

@@ -15,7 +15,7 @@ CGAL_add_named_parameter(halfedge_index_t, halfedge_index, halfedge_index_map)
 CGAL_add_named_parameter(edge_index_t, edge_index, edge_index_map)
 CGAL_add_named_parameter(face_index_t, face_index, face_index_map)
 CGAL_add_named_parameter(vertex_index_t, vertex_index, vertex_index_map)
-CGAL_add_named_parameter(graph_visitor_t, graph_visitor, visitor)
+CGAL_add_named_parameter(visitor_t, visitor, visitor)
 
 CGAL_add_named_parameter(point_t, point_map, point_map)
 
@@ -186,7 +186,6 @@ CGAL_add_named_parameter(pca_plane_t, pca_plane, pca_plane)
 CGAL_add_named_parameter(remesh_boundaries_t, remesh_boundaries, remesh_boundaries)
 CGAL_add_named_parameter(cell_selector_t, cell_selector, cell_selector)
 CGAL_add_named_parameter(facet_is_constrained_t, facet_is_constrained, facet_is_constrained_map)
-CGAL_add_named_parameter(remeshing_visitor_t, remeshing_visitor, remeshing_visitor)
 CGAL_add_named_parameter(smooth_constrained_edges_t, smooth_constrained_edges, smooth_constrained_edges)
 
 // output parameters

BGL/test/BGL/CMakeLists.txt

@@ -83,6 +83,8 @@ create_single_source_cgal_program( "test_Face_filtered_graph.cpp" )
 
 create_single_source_cgal_program( "test_Euler_operations.cpp" )
 
+create_single_source_cgal_program( "test_test_face.cpp" )
+
 create_single_source_cgal_program( "test_Collapse_edge.cpp" )
 
 create_single_source_cgal_program( "test_graph_traits.cpp" )

BGL/test/BGL/test_cgal_bgl_named_params.cpp

@@ -28,7 +28,7 @@ void test(const NamedParameters& np)
 
     // Named parameters that we use in CGAL
   assert(get_parameter(np, CGAL::internal_np::vertex_index).v == 0);
-  assert(get_parameter(np, CGAL::internal_np::graph_visitor).v == 1);
+  assert(get_parameter(np, CGAL::internal_np::visitor).v == 1);
   assert(get_parameter(np, CGAL::internal_np::vertex_point).v == 2);
   assert(get_parameter(np, CGAL::internal_np::halfedge_index).v == 3);
   assert(get_parameter(np, CGAL::internal_np::edge_index).v == 4);
@@ -130,7 +130,7 @@ void test(const NamedParameters& np)
 
     // Named parameters that we use in CGAL
   check_same_type<0>(get_parameter(np, CGAL::internal_np::vertex_index));
-  check_same_type<1>(get_parameter(np, CGAL::internal_np::graph_visitor));
+  check_same_type<1>(get_parameter(np, CGAL::internal_np::visitor));
   check_same_type<2>(get_parameter(np, CGAL::internal_np::vertex_point));
   check_same_type<3>(get_parameter(np, CGAL::internal_np::halfedge_index));
   check_same_type<4>(get_parameter(np, CGAL::internal_np::edge_index));

BGL/test/BGL/test_test_face.cpp

@@ -0,0 +1,60 @@
+#include <CGAL/Surface_mesh.h>
+#include <CGAL/Simple_cartesian.h>
+#include <CGAL/boost/graph/Euler_operations.h>
+
+#include <vector>
+#include <iostream>
+
+typedef CGAL::Simple_cartesian<double> K;
+typedef K::Point_3 Point_3;
+typedef CGAL::Surface_mesh<Point_3> SM;
+
+typedef boost::graph_traits<SM>::vertex_descriptor vertex_descriptor;
+typedef boost::graph_traits<SM>::vertex_iterator vertex_iterator;
+typedef std::vector<vertex_descriptor> V;
+int main()
+{
+  {
+    SM sm;
+
+    vertex_descriptor vp = CGAL::add_vertex(sm);
+    vertex_descriptor vq = CGAL::add_vertex(sm);
+    vertex_descriptor vr = CGAL::add_vertex(sm);
+    vertex_descriptor vs = CGAL::add_vertex(sm);
+    std::array<vertex_descriptor,0> face0;
+    assert( ! CGAL::Euler::can_add_face(face0,sm) );
+    std::array<vertex_descriptor,1> face1;
+    assert( ! CGAL::Euler::can_add_face(face1,sm) );
+    std::array<vertex_descriptor,2> face2;
+    assert( ! CGAL::Euler::can_add_face(face2,sm) );
+
+    std::array<vertex_descriptor,3> face = { vp, vq, vr };
+    CGAL::Euler::add_face(face, sm);
+
+    assert( ! CGAL::Euler::can_add_face(face,sm) );
+    std::swap(face[0],face[1]);
+    assert( CGAL::Euler::can_add_face(face,sm) );
+
+    face[2] = vs;
+    assert( CGAL::Euler::can_add_face(face,sm) );
+    std::swap(face[0],face[1]);
+    assert( ! CGAL::Euler::can_add_face(face,sm) );
+  }
+
+  {
+    SM sm;
+    Point_3 p(0,0,0), q(1,0,0), r(0,1,0), s(0,0,1);
+    CGAL::make_tetrahedron(p, q, r, s, sm);
+    std::array<vertex_descriptor,3> face;
+    vertex_iterator it = vertices(sm).first;
+    face[0] = *it;
+    ++it;
+    face[1] = *it;
+    face[2] = CGAL::add_vertex(sm);
+    assert( ! CGAL::Euler::can_add_face(face,sm) );
+    std::swap(face[0],face[1]);
+    assert( ! CGAL::Euler::can_add_face(face,sm) );
+  }
+
+  return 0;
+}

Bounding_volumes/doc/Bounding_volumes/Bounding_volumes.txt

@@ -9,37 +9,16 @@ namespace CGAL {
 \authors Kaspar Fischer, Bernd Gärtner, Thomas Herrmann, Michael Hoffmann, and Sven Schönherr
 
 \image html ball.png
-\image latex ball.png
 
 This chapter describes algorithms which for a given point set compute
 the <i>best</i> circumscribing object from a specific
 class. If the class consists of all spheres in \f$ d\f$-dimensional
 Euclidean space and <i>best</i> is defined as having smallest radius,
-then we obtain the smallest enclosing sphere problem already mentioned
-above.
+then we obtain the smallest enclosing sphere.
 
-In the following example a smallest enclosing circle
-(`Min_circle_2<Traits>`) is constructed from points
-on a line and written to standard output. The example
-shows that it is advisable to switch on random shuffling
-in order to deal with a <i>bad</i> order of the input points.
 
-\cgalExample{Min_circle_2/min_circle_2.cpp}
 
-Other classes for which we provide solutions are ellipses
-(`Min_ellipse_2<Traits>`), rectangles
-(`min_rectangle_2()`), parallelograms
-(`min_parallelogram_2()`) and strips (`min_strip_2()`)
-in the plane, with appropriate optimality criteria. For arbitrary
-dimensions we provide smallest enclosing spheres for points
-(`Min_sphere_d<Traits>`) and spheres for spheres
-(`Min_sphere_of_spheres_d<Traits>`), smallest enclosing
-annuli (`Min_annulus_d<Traits>`), and approximate
-minimum-volume enclosing ellipsoid with user-specified
-approximation ratio (`Approximate_min_ellipsoid_d<Traits>`).
-
-\image html annulus.png
-\image latex annulus.png
+\section SectBoundingIntroduction Introduction
 
 Bounding volumes can be used to obtain simple approximations of
 complicated objects. For example, consider the problem of deciding
@@ -60,13 +39,74 @@ geometric properties of objects. For example, the smallest enclosing
 annulus of a point set can be used to test whether a set of points is
 approximately cospherical. Here, the width of the annulus (or its
 area, or still another criterion that we use) is a good measure for
-this property. The largest area triangle is for example used in
-heuristics for matching archaeological aerial photographs. Largest
-perimeter triangles are used in scoring cross country soaring flights,
-where the goal is basically to fly as far as possible, but still
-return to the departure airfield. To score simply based on the total
-distance flown is not a good measure, since circling in thermals
-allows to increase it easily.
+this property.
+
+\section SectBoundingSphere Bounding Spheres in dD
+
+We provide the class `Min_sphere_of_spheres_d<Traits>` for arbitrary dimensions
+to compute the smallest enclosing spheres for points as well as for spheres.
+The dimension as well as the input type depend on the chosen traits class.
+
+The following example is for 2D points
+
+\cgalExample{Min_circle_2/min_circle_2.cpp}
+
+The example for 2D circles as input looks rather similar.
+
+
+\cgalExample{Min_sphere_of_spheres_d/min_sphere_of_spheres_d_2.cpp}
+
+
+
+\subsection  SectBoundingSphereHomogeneous Bounding Spheres for the Homogeneous Kernel
+
+In the previous section we saw that we used `Min_sphere_of_spheres_d`
+to compute the smallest circle for points. This package also provides
+the classes `Min_circle_2` and `Min_sphere_d`, but they are slower,
+and they should only be used in case of homogeneous coordinates which
+are not supported by `Min_sphere_of_spheres_d`.
+
+In the following example a smallest enclosing circle
+(`Min_circle_2<Traits>`) is constructed from points
+on a line and written to standard output. The example
+shows that it is advisable to switch on random shuffling
+in order to deal with a <i>bad</i> order of the input points.
+
+\cgalExample{Min_circle_2/min_circle_homogeneous_2.cpp}
+
+
+\section SectBoundingAnnulus Bounding Annulus in dD
+
+We provide the class `Min_annulus_d<Traits>` for arbitrary dimensions
+to compute the smalles enclosing annulus for a set of points.
+In 2D the annulus consists of two concentric circles, in 3D of
+two concentric spheres.
+
+\image html annulus.png
+
+
+
+
+\section SectBounding2D Various Bounding Areas in 2D
+
+Other classes for which we provide solutions are ellipses
+(`Min_ellipse_2<Traits>`), rectangles
+(`min_rectangle_2()`), parallelograms
+(`min_parallelogram_2()`) and strips (`min_strip_2()`)
+in the plane, with appropriate optimality criteria.
+
+
+\section SectBoundingEllipsoid Approximate Bounding Ellipsoid in dD
+
+While this package provides an exact smallest 2D ellipse, it also
+provides the class `Approximate_min_ellipsoid_d<Traits>` to compute
+an approximate minimum-volume enclosing ellipsoid with user-specified
+approximation ratio.
+
+
+
+
+\section SectBoundingPcenter Rectangular P-Center
 
 Bounding volumes also define geometric "center points" of objects.
 For example, if two objects are to be matched (approximately), one
@@ -81,8 +121,6 @@ planar point set with between two and four minimal boxes
 three boxes; the center points are shown in red.
 
 \image html pcenter.png
-\image latex pcenter.png
 
 */
 } /* namespace CGAL */
-

Bounding_volumes/doc/Bounding_volumes/CGAL/Approximate_min_ellipsoid_d.h

@@ -191,7 +191,7 @@ to iterate over the %Cartesian coordinates of the direction of a fixed
 axis of the computed ellipsoid, see
 `axis_direction_cartesian_begin()`.
 */
-typedef unspecified_type Axis_direction_iterator;
+typedef unspecified_type Axes_direction_coordinate_iterator;
 
 /// @}
 

Bounding_volumes/doc/Bounding_volumes/CGAL/Min_annulus_d.h

@@ -24,17 +24,17 @@ The underlying algorithm can cope with all kinds of input, e.g. \f$ P\f$ may be
 empty or points may occur more than once. The algorithm computes a support
 set \f$ S\f$ which remains fixed until the next set, insert, or clear operation.
 
-\tparam Traits must be a model for `OptimisationDTraits`.
+\tparam Traits must be a model for `MinSphereAnnulusDTraits`.
 
-We provide the models `Optimisation_d_traits_2`,
-`Optimisation_d_traits_3`, and `Optimisation_d_traits_d` using the
+We provide the models `Min_sphere_annulus_d_traits_2`,
+`Min_sphere_annulus_d_traits_3`, and `Min_sphere_annulus_d_traits_d` using the
 two-, three-, and \f$ d\f$-dimensional \cgal kernel, respectively.
 
 \sa `CGAL::Min_sphere_d<Traits>`
-\sa `CGAL::Optimisation_d_traits_2<K,ET,NT>`
-\sa `CGAL::Optimisation_d_traits_3<K,ET,NT>`
-\sa `CGAL::Optimisation_d_traits_d<K,ET,NT>`
-\sa `OptimisationDTraits`
+\sa `CGAL::Min_sphere_annulus_d_traits_2<K,ET,NT>`
+\sa `CGAL::Min_sphere_annulus_d_traits_3<K,ET,NT>`
+\sa `CGAL::Min_sphere_annulus_d_traits_d<K,ET,NT>`
+\sa `MinSphereAnnulusDTraits`
 
 \cgalHeading{Implementation}
 

Bounding_volumes/doc/Bounding_volumes/CGAL/Min_circle_2.h

@@ -23,9 +23,9 @@ The underlying algorithm can cope with all kinds of input, e.g. \f$ P\f$ may be
 empty or points may occur more than once. The algorithm computes a support
 set \f$ S\f$ which remains fixed until the next insert or clear operation.
 
-<B>Please note:</B> This class is (almost) obsolete. The class
+\note This class is (almost) obsolete. The class
 `CGAL::Min_sphere_of_spheres_d<Traits>` solves a more general problem
-and is faster then `Min_circle_2` even if used only for points in two
+and is faster than `Min_circle_2` even if used only for points in two
 dimensions as input. Most importantly,
 `CGAL::Min_sphere_of_spheres_d<Traits>` has
 a specialized implementation for floating-point arithmetic which