Merge pull request #6408 from efifogel/Aos_2-geodesic_traits-efif

Aos 2 geodesic traits efif

# Conflicts:
#	Installation/CHANGES.md

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Arrangement_on_surface_2.txt

@@ -3820,12 +3820,14 @@ asymmetry is also reflected in the predicates listed below. They help
 determining the order of curve-ends lying on the boundary of the
 parameter space with respect to regular points and among each other.
 
-The concept `ArrangementVerticalSideTraits_2` requires the following
+The concepts `ArrangementClosedLeftTraits_2`,
+`ArrangementClosedRightTraits_2`, and
+`ArrangementIdentifiedVerticalTraits_2` require the following
 additional predicate:
 
 <DL>
 
-<DT>\link ArrangementVerticalSideTraits_2::Compare_y_on_boundary_2 <B>`Compare_y_on_boundary_2`</B>\endlink:
+<DT>\link ArrangementIdentifiedVerticalTraits_2::Compare_y_on_boundary_2 <B>`Compare_y_on_boundary_2`</B>\endlink:
 
 <DD>Given two points \f$p_1=(x_{p_1},y_{p_1})\f$ and
 \f$p_2=(x_{p_2},y_{p_2})\f$, such that at least one of them lies on a
@@ -3839,6 +3841,12 @@ this overloaded version is required only for the concepts
 `ArrangementClosedLeftTraits_2`, `ArrangementClosedRightTraits_2`, and
 `ArrangementIdentifiedVerticalTraits_2`.
 
+</DL>
+
+The concept `ArrangementVerticalSideTraits_2` requires the following additional predicate:
+
+<DL>
+
 <DT>\link ArrangementVerticalSideTraits_2::Compare_y_near_boundary_2 <B>`Compare_y_near_boundary_2`</B>\endlink:
 
 <DD>
@@ -3971,7 +3979,7 @@ boundary-side, \f$p\f$ is located far to the top in a similar manner.
 </DL>
 
 The concept `ArrangementIdentifiedVerticalTraits_2` requires the following
-predicate:
+additional predicate:
 
 <DL>
 
@@ -3991,7 +3999,7 @@ boundary.
 </DL>
 
 Similarly, the concept `ArrangementIdentifiedHorizontalTraits_2`
-requires the following predicate:
+requires the following additional predicate:
 
 <DL>
 

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrTraits--CompareXOnBoundaryOfCurveEnd_2.h

@@ -0,0 +1,74 @@
+namespace ArrTraits {
+
+/*! \ingroup PkgArrangementOnSurface2ConceptsFunctionObjects
+ * \cgalConcept
+ *
+ * \cgalRefines AdaptableFunctor
+ *
+ * \cgalHasModel ArrangementHorizontalSideTraits_2::Compare_x_on_boundary_2
+ * \cgalHasModel ArrangementOpenBoundaryTraits_2::Compare_x_on_boundary_2
+ * \cgalHasModel ArrangementSphericalBoundaryTraits_2::Compare_x_on_boundary_2
+ */
+class CompareXOnBoundaryOfCurveEnd_2 {
+public:
+
+  /// \name Operations
+  /// A model of this concept must provide:
+  /// @{
+
+  /*! Given a point `p`, an \f$x\f$-monotone curve `xcv`, and an
+   * enumeration `ce` that specifies either the minimum or the maximum end of
+   * the curve where the curve has a vertical asymptote, compares the \f$
+   * x\f$-coordinate of `p` and the \f$x\f$-coordinate of the limit of the
+   * curve at its specificed end. The variable `xcv` identifies the parametric
+   * curve \f$c(t) = (x(t), y(t))\f$ defined over an open or half-open interval
+   * with endpoints \f$ 0\f$ and \f$ 1\f$. The enumeration `ce` identifies an
+   * open end \f$d \in\{0,1\}\f$ of \f$c\f$.  Formally, compares the \f$
+   * x\f$-coordinate of `p` and \f$ \lim_{t \rightarrow d} x(t)\f$. Returns
+   * `CGAL::SMALLER`, `CGAL::EQUAL`, or `CGAL::LARGER` accordingly.
+   *
+   * \pre \link ArrangementHorizontalSideTraits_2::Parameter_space_in_y_2 `Parameter_space_in_y_2`\endlink (`xcv`, `ce`) \f$\neq\f$ `CGAL::ARR_INTERIOR`.
+   *
+   * \pre If the parameter space is unbounded, \f$c\f$ has a vertical asymptote
+   *      at its \f$ d\f$-end; that is,
+   *      \link ArrangementVerticalSideTraits_2::Parameter_space_in_x_2 `Parameter_space_in_x_2`\endlink(`xcv`, `ce`) = `CGAL::ARR_INTERIOR`.
+   */
+  Comparison_result operator()(const ArrTraits::Point_2& p,
+                               const ArrTraits::X_monotone_curve_2& xcv,
+                               CGAL::Arr_curve_end ce);
+
+/*! Given two \f$ x\f$-monotone curves `xcv1` and `xcv2` and two indices `ce1`
+ * and `ce2` that specify either the minimum or the maximum ends of `xcv1` and
+ * `xcv2`, respectively, where the curves have vertical asymptotes, compares the
+ * \f$ x\f$-coordinates of the limits of the curves at their specificed
+ * ends. The variables `xcv1` and `xcv2` identify the parametric curves \f$
+ * c_1(t) = (x_1(t),y_1(t))\f$ and \f$ c_2(t) = (x_2(t),y_2(t))\f$,
+ * respectively, defined over open or half-open intervals with endpoints \f$
+ * 0\f$ and \f$ 1\f$. The indices `ce1` and `ce2` identify open ends \f$ d_1
+ * \in\{0,1\}\f$ and \f$ d_2 \in\{0,1\}\f$ of \f$ c_1\f$ and \f$ c_2\f$,
+ * respectively. Formally, compares \f$ \lim_{t \rightarrow d_1} x_1(t)\f$ and
+ * \f$\lim_{t \rightarrow d_2} x_2(t)\f$. Returns `CGAL::SMALLER`,
+ * `CGAL::EQUAL`, or `CGAL::LARGER` accordingly.
+ *
+ * \pre \link ArrangementHorizontalSideTraits_2::Parameter_space_in_y_2 `Parameter_space_in_y_2`\endlink(`xcv1`, `ce1`) \f$\neq\f$ `CGAL::ARR_INTERIOR`.
+ *
+ * \pre \link ArrangementHorizontalSideTraits_2::Parameter_space_in_y_2 `Parameter_space_in_y_2`\endlink(`xcv2`, `ce2`) \f$\neq\f$ `CGAL::ARR_INTERIOR`.
+ *
+ * \pre If the parameter space is unbounded, \f$c_1\f$ has a vertical
+ *      asymptote at its respective end; that is,
+ *      \link ArrangementVerticalSideTraits_2::Parameter_space_in_x_2 `Parameter_space_in_x_2`\endlink(`xcv1`, `ce1`) = `CGAL::ARR_INTERIOR`.
+ *
+ * \pre If the parameter space is unbounded, \f$c_2\f$ has a vertical asymptote
+ *      at its respective end; that is,
+ *      \link ArrangementVerticalSideTraits_2::Parameter_space_in_x_2 `Parameter_space_in_x_2`\endlink(`xcv2`, `ce2`) = `CGAL::ARR_INTERIOR`.
+ */
+Comparison_result operator()(const ArrTraits::X_monotone_curve_2& xcv1,
+                             CGAL::Arr_curve_end ce1,
+                             const ArrTraits::X_monotone_curve_2& xcv2,
+                             CGAL::Arr_curve_end ce2);
+
+/// @}
+
+}; /* end ArrTraits::CompareXOnBoundaryOfCurveEnd_2 */
+
+}

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrTraits--CompareXOnBoundary_2.h

@@ -5,9 +5,9 @@ namespace ArrTraits {
  *
  * \cgalRefines AdaptableFunctor
  *
- * \cgalHasModel ArrangementHorizontalSideTraits_2::Compare_x_on_boundary_2
+ * \cgalHasModel ArrangementClosedBottomTraits_2::Compare_x_on_boundary_2
- * \cgalHasModel ArrangementOpenBoundaryTraits_2::Compare_x_on_boundary_2
+ * \cgalHasModel ArrangementClosedTopTraits_2::Compare_x_on_boundary_2
- * \cgalHasModel ArrangementSphericalBoundaryTraits_2::Compare_x_on_boundary_2
+ * \cgalHasModel ArrangementIdentifiedHorizontalTraits_2::Compare_x_on_boundary_2
  */
 class CompareXOnBoundary_2 {
 public:
@@ -16,59 +16,21 @@ public:
   /// A model of this concept must provide:
   /// @{
 
-  /*! Given a point `p`, an \f$x\f$-monotone curve `xcv`, and an
+  /*! Given two points `p1` and `p2`, such that either `p1` or `p2` (or both)
-   * enumeration `ce` that specifies either the minimum or the maximum end of
+   * lie on the bottom or top boundary of the parameter space, compares the \f$
-   * the curve where the curve has a vertical asymptote, compares the \f$
+   * x\f$-coordinate of `p1` and the \f$x\f$-coordinate of `p2`. Returns
-   * x\f$-coordinate of `p` and the \f$x\f$-coordinate of the limit of the
-   * curve at its specificed end. The variable `xcv` identifies the parametric
-   * curve \f$c(t) = (x(t), y(t))\f$ defined over an open or half-open interval
-   * with endpoints \f$ 0\f$ and \f$ 1\f$. The enumeration `ce` identifies an
-   * open end \f$d \in\{0,1\}\f$ of \f$c\f$.  Formally, compares the \f$
-   * x\f$-coordinate of `p` and \f$ \lim_{t \rightarrow d} x(t)\f$. Returns
    * `CGAL::SMALLER`, `CGAL::EQUAL`, or `CGAL::LARGER` accordingly.
    *
-   * \pre \link ArrangementHorizontalSideTraits_2::Parameter_space_in_y_2 `Parameter_space_in_y_2`\endlink (`xcv`, `ce`) \f$\neq\f$ `CGAL::ARR_INTERIOR`.
+   * \pre \link ArrangementHorizontalSideTraits_2::Parameter_space_in_y_2
-   *
+   * `Parameter_space_in_y_2`\endlink (`p1`) \f$\neq\f$ `CGAL::ARR_INTERIOR` or
-   * \pre If the parameter space is unbounded, \f$c\f$ has a vertical asymptote
+   * \link ArrangementHorizontalSideTraits_2::Parameter_space_in_y_2
-   *      at its \f$ d\f$-end; that is,
+   * `Parameter_space_in_y_2`\endlink (`p2`) \f$\neq\f$ `CGAL::ARR_INTERIOR`.
-   *      \link ArrangementVerticalSideTraits_2::Parameter_space_in_x_2 `Parameter_space_in_x_2`\endlink(`xcv`, `ce`) = `CGAL::ARR_INTERIOR`.
    */
-  Comparison_result operator()(const ArrTraits::Point_2& p,
+  Comparison_result operator()(const ArrTraits::Point_2& p1,
-                               const ArrTraits::X_monotone_curve_2& xcv,
+                               const ArrTraits::Point_2& p2);
-                               CGAL::Arr_curve_end ce);
-
-/*! Given two \f$ x\f$-monotone curves `xcv1` and `xcv2` and two indices `ce1`
- * and `ce2` that specify either the minimum or the maximum ends of `xcv1` and
- * `xcv2`, respectively, where the curves have vertical asymptotes, compares the
- * \f$ x\f$-coordinates of the limits of the curves at their specificed
- * ends. The variables `xcv1` and `xcv2` identify the parametric curves \f$
- * c_1(t) = (x_1(t),y_1(t))\f$ and \f$ c_2(t) = (x_2(t),y_2(t))\f$,
- * respectively, defined over open or half-open intervals with endpoints \f$
- * 0\f$ and \f$ 1\f$. The indices `ce1` and `ce2` identify open ends \f$ d_1
- * \in\{0,1\}\f$ and \f$ d_2 \in\{0,1\}\f$ of \f$ c_1\f$ and \f$ c_2\f$,
- * respectively. Formally, compares \f$ \lim_{t \rightarrow d_1} x_1(t)\f$ and
- * \f$\lim_{t \rightarrow d_2} x_2(t)\f$. Returns `CGAL::SMALLER`,
- * `CGAL::EQUAL`, or `CGAL::LARGER` accordingly.
- *
- * \pre \link ArrangementHorizontalSideTraits_2::Parameter_space_in_y_2 `Parameter_space_in_y_2`\endlink(`xcv1`, `ce1`) \f$\neq\f$ `CGAL::ARR_INTERIOR`.
- *
- * \pre \link ArrangementHorizontalSideTraits_2::Parameter_space_in_y_2 `Parameter_space_in_y_2`\endlink(`xcv2`, `ce2`) \f$\neq\f$ `CGAL::ARR_INTERIOR`.
- *
- * \pre If the parameter space is unbounded, \f$c_1\f$ has a vertical
- *      asymptote at its respective end; that is,
- *      \link ArrangementVerticalSideTraits_2::Parameter_space_in_x_2 `Parameter_space_in_x_2`\endlink(`xcv1`, `ce1`) = `CGAL::ARR_INTERIOR`.
- *
- * \pre If the parameter space is unbounded, \f$c_2\f$ has a vertical asymptote
- *      at its respective end; that is,
- *      \link ArrangementVerticalSideTraits_2::Parameter_space_in_x_2 `Parameter_space_in_x_2`\endlink(`xcv2`, `ce2`) = `CGAL::ARR_INTERIOR`.
- */
-Comparison_result operator()(const ArrTraits::X_monotone_curve_2& xcv1,
-                             CGAL::Arr_curve_end ce1,
-                             const ArrTraits::X_monotone_curve_2& xcv2,
-                             CGAL::Arr_curve_end ce2);
 
 /// @}
 
-}; /* end ArrTraits::CompareXOnBoundary_2 */
+}; /* end ArrTraits::CompareXOnBoundaryOfCurveEnd_2 */
 
 }

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrTraits--CompareYOnBoundary_2.h

@@ -8,6 +8,8 @@ namespace ArrTraits {
  * \cgalHasModel ArrangementClosedLeftTraits_2::Compare_y_on_boundary_2
  * \cgalHasModel ArrangementClosedRightTraits_2::Compare_y_on_boundary_2
  * \cgalHasModel ArrangementIdentifiedVerticalTraits_2::Compare_y_on_boundary_2
+ * \cgalHasModel ArrangementOpenBoundaryTraits_2::Compare_y_on_boundary_2
+ * \cgalHasModel ArrangementSphericalBoundaryTraits_2::Compare_y_on_boundary_2
  */
 class CompareYOnBoundary_2 {
 public:
@@ -16,9 +18,10 @@ public:
   /// A model of this concept must provide:
   /// @{
 
-  /*! Given two points `p1` and `p2` returns `CGAL::SMALLER`, `CGAL::EQUAL`, or
+  /*! Given two points `p1` and `p2`, such that either `p1` or `p2` (or both)
-   * `CGAL::LARGER` according to the lexicographic \f$xy\f$-order of the points
+   * lie on the bottom or top boundary of the parameter space, compares the \f$
-   * `p1` and `p2`.
+   * y\f$-coordinate of `p1` and the \f$y\f$-coordinate of `p2`. Returns
+   * `CGAL::SMALLER`, `CGAL::EQUAL`, or `CGAL::LARGER` accordingly.
    *
    * \pre \link ArrangementVerticalSideTraits_2::Parameter_space_in_x_2
    * `Parameter_space_in_x_2`\endlink (`p1`) \f$\neq\f$ `CGAL::ARR_INTERIOR` or

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrangementClosedBottomTraits_2.h

@@ -34,7 +34,13 @@ public:
   /// \name Functor Types
   /// @{
 
+  /// models the concept `ArrTraits::CompareXOnBoundary_2`.
+  typedef unspecified_type Compare_x_on_boundary_2;
+
+  /// @}
+
   /// \name Accessing Functor Objects
   /// @{
+  Compare_x_on_boundary_2 compare_x_on_boundary_2_object() const;
   /// @}
 }

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrangementClosedIdentifiedVerticalTraits_2.h

@@ -1,42 +0,0 @@
-/*! \ingroup PkgArrangementOnSurface2ConceptsTraits
- * \cgalConcept
- *
- * `ArrangementClosedIdentifiedVerticalTraits_2` is an abstract concept. It
- * generalizes the concepts `ArrangementClosedLeftTraits_2`,
- * `ArrangementClosedRightTraits_2`, and
- * `ArrangementIdentifiedVerticalTraits_2`. (An "abstract" concept is a concept
- * that is useless on its own.) Only a combination of this concept and one or
- * more concepts that handle curves that either reach or approach the remaining
- * boundary sides (that is, borrom and top) are purposeful, and can have models.
- *
- * \cgalRefines `ArrangementBasicTraits_2`
- *
- * \cgalHasModel `CGAL::Arr_geodesic_arc_on_sphere_traits_2<Kernel, X, Y>`
- *
- * \sa `ArrangementVerticalSideTraits_2`
- */
-class ArrangementVerticalSideTraits_2 {
-public:
-
-  /// \name Categories
-  /// @{
-  /// @}
-
-  /// \name Types
-  /// @{
-  /// @}
-
-  /// \name Functor Types
-  /// @{
-
-  /// models the concept `ArrTraits::CompareXOnBoundary_2`.
-  typedef unspecified_type Compare_y_on_boundary_2;
-
-  /// @}
-
-  /// \name Accessing Functor Objects
-  /// @{
-  Compare_y_on_boundary_2 compare_y_on_boundary_2_object() const;
-  /// @}
-
-}; /* end ArrangementHorizontalSideTraits_2 */

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrangementClosedLeftTraits_2.h

@@ -35,7 +35,13 @@ public:
   /// \name Functor Types
   /// @{
 
+  /// models the concept `ArrTraits::CompareYOnBoundary_2`.
+  typedef unspecified_type Compare_y_on_boundary_2;
+
+  /// @}
+
   /// \name Accessing Functor Objects
   /// @{
+  Compare_y_on_boundary_2 compare_y_on_boundary_2_object() const;
   /// @}
-}
+};

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrangementClosedRightTraits_2.h

@@ -34,7 +34,13 @@ public:
   /// \name Functor Types
   /// @{
 
+  /// models the concept `ArrTraits::CompareYOnBoundary_2`.
+  typedef unspecified_type Compare_y_on_boundary_2;
+
+  /// @}
+
   /// \name Accessing Functor Objects
   /// @{
+  Compare_y_on_boundary_2 compare_y_on_boundary_2_object() const;
   /// @}
-}
+};

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrangementClosedTopTraits_2.h

@@ -34,7 +34,13 @@ public:
   /// \name Functor Types
   /// @{
 
+  /// models the concept `ArrTraits::CompareXOnBoundary_2`.
+  typedef unspecified_type Compare_x_on_boundary_2;
+
+  /// @}
+
   /// \name Accessing Functor Objects
   /// @{
+  Compare_x_on_boundary_2 compare_x_on_boundary_2_object() const;
   /// @}
 }

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrangementHorizontalSideTraits_2.h

@@ -36,7 +36,7 @@ public:
   /// models the concept `ArrTraits::ParameterSpaceInY_2`.
   typedef unspecified_type Parameter_space_in_y_2;
 
-  /// models the concept `ArrTraits::CompareXOnBoundary_2`.
+  /// models the concept `ArrTraits::CompareXOnBoundaryOfCurveEnd_2`.
   typedef unspecified_type Compare_x_on_boundary_2;
 
   /// models the concept `ArrTraits::CompareXNearBoundary_2`.

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrangementIdentifiedHorizontalTraits_2.h

@@ -34,6 +34,9 @@ public:
   /// \name Functor Types
   /// @{
 
+  /// models the concept `ArrTraits::CompareXOnBoundary_2`.
+  typedef unspecified_type Compare_x_on_boundary_2;
+
   /// models the concept `ArrTraits::IsOnXIdentification_2`.
   typedef unspecified_type Is_on_x_identification_2;
 
@@ -42,5 +45,6 @@ public:
   /// \name Accessing Functor Objects
   /// @{
   Is_on_x_identification_2 is_on_x_identification_2_object() const;
+  Compare_x_on_boundary_2 compare_x_on_boundary_2_object() const;
   /// @}
 }

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrangementIdentifiedVerticalTraits_2.h

@@ -34,6 +34,9 @@ public:
   /// \name Functor Types
   /// @{
 
+  /// models the concept `ArrTraits::CompareYOnBoundary_2`.
+  typedef unspecified_type Compare_y_on_boundary_2;
+
   /// models the concept `ArrTraits::IsOnYIdentification_2`.
   typedef unspecified_type Is_on_y_identification_2;
 
@@ -41,6 +44,7 @@ public:
 
   /// \name Accessing Functor Objects
   /// @{
+  Compare_y_on_boundary_2 compare_y_on_boundary_2_object() const;
   Is_on_y_identification_2 is_on_y_identification_2_object() const;
   /// @}
 }

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrangementOpenBoundaryTraits_2.h

@@ -96,10 +96,10 @@ public:
    */
   typedef unspecified_type Parameter_space_in_y_2;
 
-  /*! models the concept `ArrTraits::CompareXOnBoundary_2`.  Required only if
+  /*! models the concept `ArrTraits::CompareXOnBoundaryOfCurveEnd_2`.  Required
-   * the traits class supports unbounded curves that approach the bottom or the
+   * only if the traits class supports unbounded curves that approach the bottom
-   * top sides (the `Bottom_side_category` or the `Top_side_category` categories
+   * or the top sides (the `Bottom_side_category` or the `Top_side_category`
-   * are convertible to `CGAL::Arr_open_side_tag`).
+   * categories are convertible to `CGAL::Arr_open_side_tag`).
    */
   typedef unspecified_type Compare_x_on_boundary_2;
 

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/Concepts/ArrangementSphericalBoundaryTraits_2.h

@@ -51,7 +51,7 @@ public:
   /// models the concept `ArrTraits::ParameterSpaceInX_2`.
   typedef unspecified_type Parameter_space_in_x_2;
 
-  /// models the concept `ArrTraits::CompareXOnBoundary_2`.
+  /// models the concept `ArrTraits::CompareXOnBoundaryOfCurveEnd_2`.
   typedef unspecified_type Compare_x_on_boundary_2;
 
   /// models the concept `ArrTraits::CompareXNearBoundary_2`.

Arrangement_on_surface_2/doc/Arrangement_on_surface_2/PackageDescription.txt

@@ -123,7 +123,6 @@ implemented as peripheral classes or as free (global) functions.
 - `ArrangementTraits_2`
 - `ArrangementHorizontalSideTraits_2`
 - `ArrangementVerticalSideTraits_2`
-- `ArrangementClosedIdentifiedVerticalTraits_2`
 - `ArrangementIdentifiedVerticalTraits_2`
 - `ArrangementIdentifiedHorizontalTraits_2`
 - `ArrangementOpenBoundaryTraits_2`
@@ -177,6 +176,7 @@ implemented as peripheral classes or as free (global) functions.
 - `ArrTraits::CompareYOnBoundary_2`
 - `ArrTraits::CompareXNearBoundary_2`
 - `ArrTraits::CompareYNearBoundary_2`
+- `ArrTraits::CompareXOnBoundaryOfCurveEnd_2`
 - `ArrTraits::Intersect_2`
 - `ArrTraits::Split_2`
 - `ArrTraits::AreMergeable_2`

Arrangement_on_surface_2/include/CGAL/Arr_geodesic_arc_on_sphere_traits_2.h

@@ -1194,11 +1194,10 @@ public:
      */
     Comparison_result operator()(const X_monotone_curve_2& xc1,
                                  const X_monotone_curve_2& xc2,
-                                 const Point_2&
+                                 const Point_2& p) const
-                                 CGAL_precondition_code(p)) const
     {
-      CGAL_precondition(!xc1.is_degenerate());
+      CGAL_precondition(! xc1.is_degenerate());
-      CGAL_precondition(!xc2.is_degenerate());
+      CGAL_precondition(! xc2.is_degenerate());
       CGAL_precondition(p == xc1.right());
       CGAL_precondition(p == xc2.right());
 
@@ -1212,6 +1211,7 @@ public:
       // Compare the y-coord. at the x-coord of the most right left-endpoint.
       const Point_2& l1 = xc1.left();
       const Point_2& l2 = xc2.left();
+
       if (!l1.is_no_boundary()) {
         // use l2 and xc1:
         Oriented_side os = m_traits.oriented_side(xc1.normal(), l2);
@@ -1220,7 +1220,32 @@ public:
           ((os == ON_NEGATIVE_SIDE) ? LARGER : SMALLER) :
           ((os == ON_NEGATIVE_SIDE) ? SMALLER : LARGER);
       }
-      if (!l2.is_no_boundary()) {
+
+      // if p and r1 are antipodal, compare the plane normals
+      const Kernel& kernel = m_traits;
+      auto opposite_3 = kernel.construct_opposite_direction_3_object();
+      Direction_3 opposite_p = opposite_3(p);
+      if (kernel.equal_3_object()(opposite_p, Direction_3(l1)) ||
+          kernel.equal_3_object()(opposite_p, Direction_3(l2)))
+      {
+        Sign xsign = Traits::x_sign(p);
+        Sign ysign = Traits::y_sign(p);
+        Project project = (xsign == ZERO) ?
+          ((ysign == POSITIVE) ? Traits::project_minus_xz : Traits::project_xz) :
+          ((xsign == POSITIVE) ? Traits::project_yz : Traits::project_minus_yz);
+
+        Direction_2 n1 = project(xc1.normal());
+        Direction_2 n2 = project(xc2.normal());
+        auto opposite_2 = kernel.construct_opposite_direction_2_object();
+        if (! xc1.is_directed_right()) n1 = opposite_2(n1);
+        if (! xc2.is_directed_right()) n2 = opposite_2(n2);
+        if (kernel.equal_2_object()(n1, n2)) return EQUAL;
+        const Direction_2 d(1, 0);
+        return (kernel.counterclockwise_in_between_2_object()(n1, d, n2)) ?
+          LARGER: SMALLER;
+      }
+
+      if (! l2.is_no_boundary()) {
         // use l1 and xc2:
         Oriented_side os = m_traits.oriented_side(xc2.normal(), l1);
         return (os == ON_ORIENTED_BOUNDARY) ? EQUAL :
@@ -1229,7 +1254,8 @@ public:
           ((os == ON_NEGATIVE_SIDE) ? LARGER : SMALLER);
       }
 
-      if (m_traits.compare_xy(l1, l2) == SMALLER) {
+      Comparison_result res = m_traits.compare_xy(l1, l2);
+      if (res == SMALLER) {
         // use l2 and xc1:
         Oriented_side os = m_traits.oriented_side(xc1.normal(), l2);
         return (os == ON_ORIENTED_BOUNDARY) ? EQUAL :
@@ -1237,6 +1263,7 @@ public:
           ((os == ON_NEGATIVE_SIDE) ? LARGER : SMALLER) :
           ((os == ON_NEGATIVE_SIDE) ? SMALLER : LARGER);
       }
+      if (res == LARGER) {
         // use l1 and xc2:
         Oriented_side os = m_traits.oriented_side(xc2.normal(), l1);
         return (os == ON_ORIENTED_BOUNDARY) ? EQUAL :
@@ -1244,6 +1271,9 @@ public:
           ((os == ON_NEGATIVE_SIDE) ? SMALLER : LARGER) :
           ((os == ON_NEGATIVE_SIDE) ? LARGER : SMALLER);
       }
+      // res == equal
+      return EQUAL;
+    }
   };
 
   /*! Obtain a Compare_y_at_x_left_2 function object */
@@ -1283,8 +1313,8 @@ public:
                                  const X_monotone_curve_2& xc2,
                                  const Point_2& p) const
     {
-      CGAL_precondition(!xc1.is_degenerate());
+      CGAL_precondition(! xc1.is_degenerate());
-      CGAL_precondition(!xc2.is_degenerate());
+      CGAL_precondition(! xc2.is_degenerate());
 
       // CGAL_precondition(p == xc1.left());
       // CGAL_precondition(p == xc2.left());
@@ -1296,10 +1326,35 @@ public:
 
       // Non of the arcs is verticel. Thus, non of the endpoints coincide with
       // a pole.
-      // Compare the y-coord. at the x-coord of the most left right-endpoint.
       const Point_2& r1 = xc1.right();
       const Point_2& r2 = xc2.right();
-      if (!r1.is_no_boundary()) {
+
+      // if p and r1 are antipodal, compare the plane normals
+      const Kernel& kernel = m_traits;
+      auto opposite_3 = kernel.construct_opposite_direction_3_object();
+      Direction_3 opposite_p = opposite_3(p);
+      if (kernel.equal_3_object()(opposite_p, Direction_3(r1)) ||
+          kernel.equal_3_object()(opposite_p, Direction_3(r2)))
+      {
+        Sign xsign = Traits::x_sign(p);
+        Sign ysign = Traits::y_sign(p);
+        Project project = (xsign == ZERO) ?
+          ((ysign == POSITIVE) ? Traits::project_minus_xz : Traits::project_xz) :
+          ((xsign == POSITIVE) ? Traits::project_yz : Traits::project_minus_yz);
+
+        Direction_2 n1 = project(xc1.normal());
+        Direction_2 n2 = project(xc2.normal());
+        auto opposite_2 = kernel.construct_opposite_direction_2_object();
+        if (! xc1.is_directed_right()) n1 = opposite_2(n1);
+        if (! xc2.is_directed_right()) n2 = opposite_2(n2);
+        if (kernel.equal_2_object()(n1, n2)) return EQUAL;
+        const Direction_2 d(1, 0);
+        return (kernel.counterclockwise_in_between_2_object()(n1, d, n2)) ?
+          SMALLER : LARGER;
+      }
+
+      // Compare the y-coord. at the x-coord of the most left right-endpoint.
+      if (! r1.is_no_boundary()) {
         // use r2 and xc1:
         Oriented_side os = m_traits.oriented_side(xc1.normal(), r2);
         return (os == ON_ORIENTED_BOUNDARY) ? EQUAL :
@@ -1307,7 +1362,7 @@ public:
           ((os == ON_NEGATIVE_SIDE) ? LARGER : SMALLER) :
           ((os == ON_NEGATIVE_SIDE) ? SMALLER : LARGER);
       }
-      if (!r2.is_no_boundary()) {
+      if (! r2.is_no_boundary()) {
         // use r1 and xc2:
         Oriented_side os = m_traits.oriented_side(xc2.normal(), r1);
         return (os == ON_ORIENTED_BOUNDARY) ? EQUAL :
@@ -1333,32 +1388,7 @@ public:
           ((os == ON_NEGATIVE_SIDE) ? LARGER : SMALLER);
       }
       // res == equal
-      // if p and r1 are antipodal, compare the plane normals
-      const Kernel& kernel = m_traits;
-      typename Kernel::Construct_opposite_direction_3 opposite_3 =
-        kernel.construct_opposite_direction_3_object();
-      // VC 10 does not like the following:
-      // if (!kernel.equal_3_object()(opposite_3(p), r1)) return EQUAL;
-      Direction_3 tmp1 = opposite_3(p);     // pacify msvc 10
-      if (!kernel.equal_3_object()(tmp1, Direction_3(r1)))
       return EQUAL;
-
-      Sign xsign = Traits::x_sign(p);
-      Sign ysign = Traits::y_sign(p);
-      Project project = (xsign == ZERO) ?
-        ((ysign == POSITIVE) ? Traits::project_minus_xz : Traits::project_xz) :
-        ((xsign == POSITIVE) ? Traits::project_yz : Traits::project_minus_yz);
-
-      Direction_2 n1 = project(xc1.normal());
-      Direction_2 n2 = project(xc2.normal());
-      typename Kernel::Construct_opposite_direction_2 opposite_2 =
-        kernel.construct_opposite_direction_2_object();
-      if (!xc1.is_directed_right()) n1 = opposite_2(n1);
-      if (!xc2.is_directed_right()) n2 = opposite_2(n2);
-      if (kernel.equal_2_object()(n1, n2)) return EQUAL;
-      const Direction_2 d(1, 0);
-      return (kernel.counterclockwise_in_between_2_object()(n1, d, n2)) ?
-        SMALLER : LARGER;
     }
   };
 
@@ -1397,8 +1427,7 @@ public:
       typename Kernel::Equal_3 equal_3 = kernel.equal_3_object();
       if (xc1.is_full() || xc2.is_full()) {
         if (!xc1.is_full() || !xc2.is_full()) return false;
-        typename Kernel::Construct_opposite_direction_3 opposite_3 =
+        auto opposite_3 = kernel.construct_opposite_direction_3_object();
-          kernel.construct_opposite_direction_3_object();
         return (equal_3(xc1.normal(), xc2.normal()) ||
                 equal_3(opposite_3(xc1.normal()), xc2.normal()));
       }
@@ -1764,113 +1793,118 @@ public:
      * \param ce the arc end indicator.
      * \return the second comparison result.
      * \pre the ce ends of the arcs xcv1 and xcv2 lie either on the left
-     *      boundary or on the right boundary of the parameter space (implying
+     *      boundary or on the right boundary of the parameter space.
-     *      that they cannot be vertical).
+     * \pre the curves cannot reach a pole
      * There is no horizontal identification curve!
      */
     Comparison_result operator()(const X_monotone_curve_2& xcv1,
                                  const X_monotone_curve_2& xcv2,
                                  Arr_curve_end ce) const
     {
-      CGAL_precondition(!xcv1.is_degenerate());
+      CGAL_precondition(! xcv1.is_degenerate());
-      CGAL_precondition(!xcv2.is_degenerate());
+      CGAL_precondition(! xcv2.is_degenerate());
 
+      CGAL_precondition((ce != ARR_MIN_END) ||
+                        (xcv1.left().is_mid_boundary() &&
+                         xcv2.left().is_mid_boundary()));
+      CGAL_precondition((ce != ARR_MAX_END) ||
+                        (xcv1.right().is_mid_boundary() &&
+                         xcv2.right().is_mid_boundary()));
+
+      // If the curves lie on the same plane return EQUAL.
+      const Kernel& kernel = m_traits;
+      const Direction_3& n1 = xcv1.normal();
+      const Direction_3& n2 = xcv2.normal();
+      if (xcv1.is_directed_right() == xcv2.is_directed_right()) {
+        if (kernel.equal_3_object()(n1, n2)) return EQUAL;
+      }
+      else {
+        auto opposite_3 = kernel.construct_opposite_direction_3_object();
+        auto opposite_n2 = opposite_3(n2);
+        if (kernel.equal_3_object()(n1, opposite_n2)) return EQUAL;
+      }
+
+      // The curves do not lie on the same plane!
       const Point_2& l1 = xcv1.left();
-      const Point_2& r1 = xcv1.right();
       const Point_2& l2 = xcv2.left();
+      const Point_2& r1 = xcv1.right();
       const Point_2& r2 = xcv2.right();
 
-      // If xcv1 is vertical, xcv1 coincides with the discontinuity arc:
-      if (xcv1.is_vertical()) {
-        CGAL_precondition(!l1.is_no_boundary());
-        CGAL_precondition(!r1.is_no_boundary());
-      }
-
-      // If xcv2 is vertical, xcv2 coincides with the discontinuity arc:
-      if (xcv2.is_vertical()) {
-        CGAL_precondition(!l2.is_no_boundary());
-        CGAL_precondition(!r2.is_no_boundary());
-      }
-
       if (ce == ARR_MIN_END) {
-        // Handle the south pole. It has the smallest y coords:
-        if (l1.is_min_boundary())
-          return (l2.is_min_boundary()) ? EQUAL : SMALLER;
-        if (l2.is_min_boundary()) return LARGER;
-
         // None of xcv1 and xcv2 endpoints coincide with a pole:
         Comparison_result cr = m_traits.compare_y(l1, l2);
         if (cr != EQUAL) return cr;
 
         // If Both arcs are vertical, they overlap:
-        if (xcv1.is_vertical() && xcv2.is_vertical()) return EQUAL;
         if (xcv1.is_vertical()) return LARGER;
         if (xcv2.is_vertical()) return SMALLER;
 
-        // Non of the arcs is verticel. Thus, non of the endpoints coincide
+        // There are 4 cases based on the sign of the z component of the normals
-        // with a pole.
+        // Compute the sign of the x-component of the normal cross product.
-        // Compare the y-coord. at the x-coord of the most left right-endpoint.
+        // There is no point computing the intermediate cross product:
-        CGAL_assertion(r1.is_no_boundary());
+        // auto cross_prod = kernel.construct_cross_product_vector_3_object();
-        CGAL_assertion(r2.is_no_boundary());
+        // Vector_3 v = cross_prod(n1.vector(), n2.vector());
+        // CGAL::Sign xsign = CGAL::sign(v.x());
+        // This predicate is not yet supported; thus, compute directly:
+        CGAL::Sign xsign = CGAL::sign(n1.dy() * n2.dz() - n1.dz() * n2.dy());
 
-        if (m_traits.compare_xy(r1, r2) == LARGER) {
+        // std::cout << "sign(n1.z): " << CGAL::sign(n1.dz()) << std::endl;
-          // use r2 and xcv1:
+        // std::cout << "sign(n2.z): " << CGAL::sign(n2.dz()) << std::endl;
-          Oriented_side os =
+        // std::cout << "x sign: " << xsign << std::endl;
-            m_traits.oriented_side(xcv1.normal(), r2);
+
-          return (os == ON_ORIENTED_BOUNDARY) ? EQUAL :
+        if (CGAL::sign(n1.dz()) == POSITIVE) {
-            (xcv1.is_directed_right()) ?
+          if (CGAL::sign(n2.dz()) == POSITIVE) {
-            ((os == ON_NEGATIVE_SIDE) ? LARGER : SMALLER) :
+            // pos pos
-            ((os == ON_NEGATIVE_SIDE) ? SMALLER : LARGER);
+            return (xsign == POSITIVE) ? LARGER : SMALLER;
           }
-        // use r1 and xcv2:
+          // pos neg
-        Oriented_side os = m_traits.oriented_side(xcv2.normal(), r1);
+          return (xsign == POSITIVE) ? SMALLER : LARGER;
-        return (os == ON_ORIENTED_BOUNDARY) ? EQUAL :
+        }
-          (xcv2.is_directed_right()) ?
+        if (CGAL::sign(n2.dz()) == POSITIVE) {
-          ((os == ON_NEGATIVE_SIDE) ? SMALLER : LARGER) :
+          // neg pos
-          ((os == ON_NEGATIVE_SIDE) ? LARGER : SMALLER);
+          return (xsign == POSITIVE) ? SMALLER : LARGER;
+        }
+        // neg neg
+        return (xsign == POSITIVE) ? LARGER : SMALLER;
       }
 
       // ce == ARR_MAX_END
 
-      // Handle the north pole. It has the largest y coords:
-      if (r1.is_max_boundary()) return (r2.is_max_boundary()) ? EQUAL : LARGER;
-      if (r2.is_max_boundary()) return SMALLER;
-
       // None of xcv1 and xcv2 endpoints coincide with a pole:
-      Direction_2 r1_xy = Traits::project_xy(r1);
       Comparison_result cr = m_traits.compare_y(r1, r2);
       if (cr != EQUAL) return cr;
 
       // If Both arcs are vertical, they overlap:
-      if (xcv1.is_vertical() && xcv2.is_vertical()) return EQUAL;
       if (xcv1.is_vertical()) return LARGER;
       if (xcv2.is_vertical()) return SMALLER;
 
-      // Compare to the left:
+      // There are 4 cases based on the sign of the z component of the normals
-      Direction_2 p_r1 = Traits::project_xy(r1);
+      // Compute the sign of the x-component of the normal cross product.
-      cr = m_traits.compare_y(r1, r2);
+      // There is no point computing the intermediate cross product:
-      if (cr != EQUAL) return cr;
+      // auto cross_prod = kernel.construct_cross_product_vector_3_object();
+      // Vector_3 v = cross_prod(n1.vector(), n2.vector());
+      // CGAL::Sign xsign = CGAL::sign(v.x());
+      // This predicate is not yet supported; thus, compute directly:
+      CGAL::Sign xsign = CGAL::sign(n1.dy() * n2.dz() - n1.dz() * n2.dy());
 
-      // Non of the arcs is verticel. Thus, non of the endpoints coincide with
+      // std::cout << "sign(n1.z): " << CGAL::sign(n1.dz()) << std::endl;
-      // a pole.
+      // std::cout << "sign(n2.z): " << CGAL::sign(n2.dz()) << std::endl;
-      // Compare the y-coord. at the x-coord of the most right left-endpoint.
+      // std::cout << "x sign: " << xsign << std::endl;
-      CGAL_assertion(l1.is_no_boundary());
-      CGAL_assertion(l2.is_no_boundary());
 
-      if (m_traits.compare_xy(l1, l2) == SMALLER) {
+      if (CGAL::sign(n1.dz()) == POSITIVE) {
-        // use l2 and xcv1:
+        if (CGAL::sign(n2.dz()) == POSITIVE) {
-        Oriented_side os = m_traits.oriented_side(xcv1.normal(), l2);
+          // pos pos
-        return (os == ON_ORIENTED_BOUNDARY) ? EQUAL :
+          return (xsign == POSITIVE) ? SMALLER : LARGER;
-          (xcv1.is_directed_right()) ?
-          ((os == ON_NEGATIVE_SIDE) ? LARGER : SMALLER) :
-          ((os == ON_NEGATIVE_SIDE) ? SMALLER : LARGER);
         }
-      // use l1 and xcv2:
+        // pos neg
-      Oriented_side os = m_traits.oriented_side(xcv2.normal(), l1);
+        return (xsign == POSITIVE) ? LARGER: SMALLER;
-      return (os == ON_ORIENTED_BOUNDARY) ? EQUAL :
+      }
-        (xcv2.is_directed_right()) ?
+      if (CGAL::sign(n2.dz()) == POSITIVE) {
-        ((os == ON_NEGATIVE_SIDE) ? SMALLER : LARGER) :
+        // neg pos
-        ((os == ON_NEGATIVE_SIDE) ? LARGER : SMALLER);
+        return (xsign == POSITIVE) ? LARGER : SMALLER;
+      }
+      // neg neg
+      return (xsign == POSITIVE) ? SMALLER : LARGER;
     }
   };
 
@@ -2318,31 +2352,22 @@ public:
       if (equal(l1, r2)) {
         // 1. l1 = r2 < r1 < l2 < l1 | One intersection
         // 2. l1 = r2 < r1 = l2 < l1 | Two intersections
-        // 3. l1 = r2 < l2 < r1 < l1 | One intersection and one overlap
+        // 3. l1 = r2 < l2 < r1 < l1 | One overlap
         // 4. l1 = r2 < l2 < r1 = l1 | One overlap (handled above)
+        // 5. l1 = r2 < r1 < l2 = l1 | One overlap (handled above)
         if (in_between(r1, r2, l2)) {
-          // If the intersection point lies on the identification curve, this
+          // Case 1.
-          // intersection point should be ignored (see below). However, it
-          // cannot occur.
           *oi++ = Intersection_result(Intersection_point(l1_3, 1));
           return oi;
         }
         if (equal(r1, l2)) {
-          // Case 2. This can happen only if one of the intersection points
+          // Case 2.
-          // lies on the identification curve, and this points should be
-          // ignored.
-          if (! m_traits.is_on_y_identification_2_object()(l1_3)) {
           *oi++ = Intersection_result(Intersection_point(l1_3, 1));
-            return oi;
-          }
-          CGAL_assertion(! m_traits.is_on_y_identification_2_object()(l2_3));
           *oi++ = Intersection_result(Intersection_point(l2_3, 1));
           return oi;
         }
         CGAL_assertion(in_between(r1, l2, r2));
-        // Case 3. This can happen only the intersection point lies on the
+        // Case 3.
-        // identification curve; in this case this point should be ignored.
-        CGAL_assertion(m_traits.is_on_y_identification_2_object()(l1_3));
         X_monotone_curve_2 xc(l2_3, r1_3, normal, vertical, true);
         *oi++ = Intersection_result(xc);
         return oi;
@@ -2352,19 +2377,15 @@ public:
       if (equal(r1, l2)) {
         // 1. l1 < r1 = l2 < r2 < l1 | One intersection
         // 2. l1 < r1 = l2 < r2 = l1 | Two intersections (handled above)
-        // 3. l1 < r2 < r1 = l2 < l1 | One intersection and one overlap
+        // 3. l1 < r2 < r1 = l2 < l1 | One overlap
         // 4. l1 < r2 = r1 = l2 < l1 | One overlap (handled above)
+        // 5. l1 < l1 = r1 = l2 < r2 | One overlap (handled above)
         if (in_between(r2, r1, l1)) {
-          // If the intersection point lies on the identification curve, this
+          // Case 1.
-          // intersection point should be ignored (see below). However, it
-          // cannot occur.
           *oi++ = Intersection_result(Intersection_point(l2_3, 1));
           return oi;
         }
-        // Case 3. This can happen only the intersection point lies on the
+        // Case 3.
-        // identification curve; in this case this point should be ignored.
-        CGAL_assertion(in_between(r2, l1, r1));
-        CGAL_assertion(m_traits.is_on_y_identification_2_object()(l2_3));
         X_monotone_curve_2 xc(l1_3, r2_3, normal, vertical, true);
         *oi++ = Intersection_result(xc);
         return oi;
@@ -2552,10 +2573,8 @@ public:
 
       if (equal_3(normal1, normal2) || equal_3(opposite_normal1, normal2)) {
         // The underlying planes are the same
-        Counterclockwise_in_between_2 ccib =
+        Counterclockwise_in_between_2 ccib = kernel.counterclockwise_in_between_2_object();
-          kernel.counterclockwise_in_between_2_object();
+        auto cib = m_traits.clockwise_in_between_2_object();
-        typename Traits::Clockwise_in_between_2 cib =
-          m_traits.clockwise_in_between_2_object();
 
         if (xc1.is_vertical()) {
           // Both arcs are vertical
@@ -2620,23 +2639,19 @@ public:
                                     normal, false, ccib, Traits::project_xy, oi);
       }
 
-      Vector_3 v =
+      auto cross_prod = kernel.construct_cross_product_vector_3_object();
-        kernel.construct_cross_product_vector_3_object()(xc1.normal().vector(),
+      Vector_3 v = cross_prod(xc1.normal().vector(), xc2.normal().vector());
-                                                         xc2.normal().vector());
 
-      // Determine which one of the two directions:
+      // Observe that xc1 and xc2 may share two endpoints.
       Point_2 ed = m_traits.construct_point_2_object()(v.direction());
-      if (is_in_between(ed, xc1) && is_in_between(ed, xc2)) {
+      if (is_in_between(ed, xc1) && is_in_between(ed, xc2))
         *oi++ = Intersection_result(Intersection_point(ed, 1));
-        return oi;
-      }
 
       Vector_3 vo(kernel.construct_opposite_vector_3_object()(v));
       Point_2 edo = m_traits.construct_point_2_object()(vo.direction());
-      if (is_in_between(edo, xc1) && is_in_between(edo, xc2)) {
+      if (is_in_between(edo, xc1) && is_in_between(edo, xc2))
         *oi++ = Intersection_result(Intersection_point(edo, 1));
-        return oi;
+
-      }
       return oi;
     }
   };

Arrangement_on_surface_2/include/CGAL/Arr_tracing_traits_2.h

@@ -535,12 +535,12 @@ public:
 
       size_t i = 0;
       for (auto it = container.begin(); it != container.end(); ++it) {
-        if (const auto* xcv = boost::get<X_monotone_curve_2>(*it)) {
+        if (const auto* xcv = boost::get<X_monotone_curve_2>(&*it)) {
           std::cout << "  result[" << i++ << "]: xcv: " << *xcv << std::endl;
           continue;
         }
 
-        if (const Point_2* p = boost::get<Point_2>(*it)) {
+        if (const auto* p = boost::get<Point_2>(&*it)) {
           std::cout << "  result[" << i++ << "]: p: " << *p << std::endl;
           continue;
         }

Arrangement_on_surface_2/test/Arrangement_on_surface_2/data/spherical_arcs/compare_y_at_x

@@ -55,3 +55,17 @@ compare_y_at_x_right 33 30 0 LARGER
 compare_y_at_x_right 32 31 0 LARGER
 compare_y_at_x_right 33 31 0 LARGER
 compare_y_at_x_right 35 31 0 LARGER
+#
+compare_y_at_x_right 36 37 3 LARGER
+compare_y_at_x_right 37 36 3 SMALLER
+compare_y_at_x_right 38 39 3 SMALLER
+compare_y_at_x_right 39 38 3 LARGER
+compare_y_at_x_right 36 39 3 EQUAL
+compare_y_at_x_right 38 37 3 EQUAL
+#
+compare_y_at_x_left 40 41 3 LARGER
+compare_y_at_x_left 41 40 3 SMALLER
+compare_y_at_x_left 42 43 3 SMALLER
+compare_y_at_x_left 43 42 3 LARGER
+compare_y_at_x_left 40 43 3 EQUAL
+compare_y_at_x_left 42 41 3 EQUAL

Arrangement_on_surface_2/test/Arrangement_on_surface_2/data/spherical_arcs/compare_y_at_x.pt

@@ -1,3 +1,4 @@
-0/1 -1/1 0/1
+0 -1 0
-1/1 0/1 0/1
+1 0 0
-0/1 1/1 0/1
+0 1 0
+-1 0 0

Arrangement_on_surface_2/test/Arrangement_on_surface_2/data/spherical_arcs/compare_y_at_x.xcv

@@ -36,3 +36,13 @@
 0 1 -1 1 0 -1 0
 0 0 0 1 0 -1 0
 0 0 -1 1 0 -1 0
+#
+1 -1 0 0 1 0 0 0 1 1
+1 -1 0 0 1 0 0 0 -1 1
+1 1 0 0 -1 0 0 0 1 -1
+1 1 0 0 -1 0 0 0 -1 -1
+#
+1 -1 0 0 1 0 0 0 1 -1
+1 -1 0 0 1 0 0 0 -1 -1
+1 1 0 0 -1 0 0 0 1 1
+1 1 0 0 -1 0 0 0 -1 1

Arrangement_on_surface_2/test/Arrangement_on_surface_2/data/spherical_arcs/compare_y_near_boundary

@@ -1,78 +1,5 @@
-compare_y_near_boundary 0 1 MIN_END EQUAL
-compare_y_near_boundary 0 2 MIN_END EQUAL
-compare_y_near_boundary 0 3 MIN_END EQUAL
-compare_y_near_boundary 0 4 MIN_END EQUAL
-compare_y_near_boundary 0 5 MIN_END EQUAL
-compare_y_near_boundary 0 6 MIN_END EQUAL
-compare_y_near_boundary 0 7 MIN_END EQUAL
-compare_y_near_boundary 0 8 MIN_END EQUAL
-compare_y_near_boundary 0 9 MIN_END EQUAL
-compare_y_near_boundary 0 10 MIN_END EQUAL
-compare_y_near_boundary 0 11 MIN_END EQUAL
-compare_y_near_boundary 0 14 MIN_END EQUAL
-compare_y_near_boundary 0 15 MIN_END EQUAL
-compare_y_near_boundary 0 16 MIN_END EQUAL
-compare_y_near_boundary 0 17 MIN_END EQUAL
-compare_y_near_boundary 0 18 MIN_END EQUAL
-compare_y_near_boundary 0 19 MIN_END EQUAL
-compare_y_near_boundary 0 20 MIN_END EQUAL
-compare_y_near_boundary 0 21 MIN_END EQUAL
-compare_y_near_boundary 0 22 MIN_END EQUAL
-compare_y_near_boundary 0 23 MIN_END EQUAL
-compare_y_near_boundary 0 24 MIN_END EQUAL
-compare_y_near_boundary 0 25 MIN_END EQUAL
-compare_y_near_boundary 0 26 MIN_END EQUAL
-compare_y_near_boundary 0 27 MIN_END EQUAL
-compare_y_near_boundary 0 28 MIN_END EQUAL
-compare_y_near_boundary 0 29 MIN_END EQUAL
-compare_y_near_boundary 0 30 MIN_END EQUAL
-compare_y_near_boundary 0 31 MIN_END EQUAL
-compare_y_near_boundary 0 48 MIN_END SMALLER
-compare_y_near_boundary 0 49 MIN_END SMALLER
-compare_y_near_boundary 0 50 MIN_END SMALLER
-compare_y_near_boundary 0 51 MIN_END SMALLER
-compare_y_near_boundary 0 52 MIN_END LARGER
-compare_y_near_boundary 0 53 MIN_END LARGER
-compare_y_near_boundary 0 54 MIN_END LARGER
-compare_y_near_boundary 0 55 MIN_END LARGER
-compare_y_near_boundary 0 56 MIN_END SMALLER
-compare_y_near_boundary 0 57 MIN_END SMALLER
-compare_y_near_boundary 0 58 MIN_END SMALLER
-compare_y_near_boundary 0 59 MIN_END SMALLER
-compare_y_near_boundary 0 60 MIN_END LARGER
-compare_y_near_boundary 0 61 MIN_END LARGER
-compare_y_near_boundary 0 62 MIN_END LARGER
-compare_y_near_boundary 0 63 MIN_END LARGER#
-compare_y_near_boundary 0 0 MIN_END EQUAL
-compare_y_near_boundary 1 1 MIN_END EQUAL
-compare_y_near_boundary 2 2 MIN_END EQUAL
-compare_y_near_boundary 3 3 MIN_END EQUAL
-compare_y_near_boundary 4 4 MIN_END EQUAL
-compare_y_near_boundary 5 5 MIN_END EQUAL
-compare_y_near_boundary 6 6 MIN_END EQUAL
-compare_y_near_boundary 7 7 MIN_END EQUAL
 compare_y_near_boundary 8 8 MIN_END EQUAL
 compare_y_near_boundary 9 9 MIN_END EQUAL
-compare_y_near_boundary 10 10 MIN_END EQUAL
-compare_y_near_boundary 11 11 MIN_END EQUAL
-compare_y_near_boundary 14 14 MIN_END EQUAL
-compare_y_near_boundary 15 15 MIN_END EQUAL
-compare_y_near_boundary 16 16 MIN_END EQUAL
-compare_y_near_boundary 17 17 MIN_END EQUAL
-compare_y_near_boundary 18 18 MIN_END EQUAL
-compare_y_near_boundary 19 19 MIN_END EQUAL
-compare_y_near_boundary 20 20 MIN_END EQUAL
-compare_y_near_boundary 21 21 MIN_END EQUAL
-compare_y_near_boundary 22 22 MIN_END EQUAL
-compare_y_near_boundary 23 23 MIN_END EQUAL
-compare_y_near_boundary 24 24 MIN_END EQUAL
-compare_y_near_boundary 25 25 MIN_END EQUAL
-compare_y_near_boundary 26 26 MIN_END EQUAL
-compare_y_near_boundary 27 27 MIN_END EQUAL
-compare_y_near_boundary 28 28 MIN_END EQUAL
-compare_y_near_boundary 29 29 MIN_END EQUAL
-compare_y_near_boundary 30 30 MIN_END EQUAL
-compare_y_near_boundary 31 31 MIN_END EQUAL
 compare_y_near_boundary 48 48 MIN_END EQUAL
 compare_y_near_boundary 49 49 MIN_END EQUAL
 compare_y_near_boundary 50 50 MIN_END EQUAL
@@ -85,42 +12,11 @@ compare_y_near_boundary 56 56 MIN_END EQUAL
 compare_y_near_boundary 57 57 MIN_END EQUAL
 compare_y_near_boundary 58 58 MIN_END EQUAL
 compare_y_near_boundary 59 59 MIN_END EQUAL
-compare_y_near_boundary 60 60 MIN_END EQUAL
-compare_y_near_boundary 61 61 MIN_END EQUAL
 compare_y_near_boundary 62 62 MIN_END EQUAL
-compare_y_near_boundary 63 63 MIN_END EQUAL#
+compare_y_near_boundary 63 63 MIN_END EQUAL
-compare_y_near_boundary 0 0 MAX_END EQUAL
+#
-compare_y_near_boundary 1 1 MAX_END EQUAL
-compare_y_near_boundary 2 2 MAX_END EQUAL
-compare_y_near_boundary 3 3 MAX_END EQUAL
-compare_y_near_boundary 4 4 MAX_END EQUAL
-compare_y_near_boundary 5 5 MAX_END EQUAL
-compare_y_near_boundary 6 6 MAX_END EQUAL
-compare_y_near_boundary 7 7 MAX_END EQUAL
-compare_y_near_boundary 10 10 MAX_END EQUAL
-compare_y_near_boundary 11 11 MAX_END EQUAL
 compare_y_near_boundary 12 12 MAX_END EQUAL
 compare_y_near_boundary 13 13 MAX_END EQUAL
-compare_y_near_boundary 14 14 MAX_END EQUAL
-compare_y_near_boundary 15 15 MAX_END EQUAL
-compare_y_near_boundary 16 16 MAX_END EQUAL
-compare_y_near_boundary 17 17 MAX_END EQUAL
-compare_y_near_boundary 18 18 MAX_END EQUAL
-compare_y_near_boundary 19 19 MAX_END EQUAL
-compare_y_near_boundary 20 20 MAX_END EQUAL
-compare_y_near_boundary 21 21 MAX_END EQUAL
-compare_y_near_boundary 22 22 MAX_END EQUAL
-compare_y_near_boundary 23 23 MAX_END EQUAL
-compare_y_near_boundary 24 24 MAX_END EQUAL
-compare_y_near_boundary 25 25 MAX_END EQUAL
-compare_y_near_boundary 26 26 MAX_END EQUAL
-compare_y_near_boundary 27 27 MAX_END EQUAL
-compare_y_near_boundary 28 28 MAX_END EQUAL
-compare_y_near_boundary 29 29 MAX_END EQUAL
-compare_y_near_boundary 30 30 MAX_END EQUAL
-compare_y_near_boundary 31 31 MAX_END EQUAL
-compare_y_near_boundary 48 48 MAX_END EQUAL
-compare_y_near_boundary 49 49 MAX_END EQUAL
 compare_y_near_boundary 50 50 MAX_END EQUAL
 compare_y_near_boundary 51 51 MAX_END EQUAL
 compare_y_near_boundary 52 52 MAX_END EQUAL
@@ -135,3 +31,23 @@ compare_y_near_boundary 60 60 MAX_END EQUAL
 compare_y_near_boundary 61 61 MAX_END EQUAL
 compare_y_near_boundary 62 62 MAX_END EQUAL
 compare_y_near_boundary 63 63 MAX_END EQUAL
+#
+compare_y_near_boundary 96 102 MIN_END EQUAL
+compare_y_near_boundary 97 103 MIN_END EQUAL
+compare_y_near_boundary 98 100 MAX_END EQUAL
+compare_y_near_boundary 99 101 MAX_END EQUAL
+#
+compare_y_near_boundary 96 97 MIN_END LARGER
+compare_y_near_boundary 97 96 MIN_END SMALLER
+compare_y_near_boundary 98 99 MAX_END SMALLER
+compare_y_near_boundary 99 98 MAX_END LARGER
+#
+compare_y_near_boundary 100 101 MAX_END SMALLER
+compare_y_near_boundary 101 100 MAX_END LARGER
+compare_y_near_boundary 102 103 MIN_END LARGER
+compare_y_near_boundary 103 102 MIN_END SMALLER
+#
+compare_y_near_boundary 104 105 MIN_END LARGER
+compare_y_near_boundary 105 104 MIN_END SMALLER
+compare_y_near_boundary 106 107 MAX_END LARGER
+compare_y_near_boundary 107 106 MAX_END SMALLER

Arrangement_on_surface_2/test/Arrangement_on_surface_2/data/spherical_arcs/intersect

@@ -288,3 +288,22 @@ intersect 89 90 1 0 22 1
 intersect 90 89 1 0 22 1
 intersect 93 94 1 0 23 1
 intersect 94 93 1 0 23 1
+#
+intersect 96 96 1 1 96
+intersect 97 97 1 1 97
+intersect 98 98 1 1 98
+intersect 99 99 1 1 99
+#
+intersect 96 97 2 0 24 1 0 25 1
+intersect 96 98 2 0 25 1 0 24 1
+intersect 96 99 2 0 24 1 0 25 1
+intersect 97 98 2 0 25 1 0 24 1
+intersect 97 99 2 0 25 1 0 24 1
+intersect 98 99 2 0 24 1 0 25 1
+#
+intersect 97 96 2 0 25 1 0 24 1
+intersect 98 96 2 0 24 1 0 25 1
+intersect 99 96 2 0 25 1 0 24 1
+intersect 98 97 2 0 24 1 0 25 1
+intersect 99 97 2 0 24 1 0 25 1
+intersect 99 98 2 0 25 1 0 24 1

Arrangement_on_surface_2/test/Arrangement_on_surface_2/data/spherical_arcs/points

@@ -1,26 +1,29 @@
-1/4 -3/4 0/1
+1/4 -3/4 0
--1/4 -3/4 0/1
+-1/4 -3/4 0
--3/4 -1/4 0/1
+-3/4 -1/4 0
--3/4 1/4 0/1
+-3/4 1/4 0
-1/4 3/4 0/1
+1/4 3/4 0
--1/4 3/4 0/1
+-1/4 3/4 0
-3/4 -1/4 0/1
+3/4 -1/4 0
-3/4 1/4 0/1
+3/4 1/4 0
 ##############
-1/4 0/1 -3/4
+1/4 0 -3/4
-3/4 0/1 -1/4
+3/4 0 -1/4
-3/4 0/1 1/4
+3/4 0 1/4
-1/4 0/1 3/4
+1/4 0 3/4
--1/4 0/1 3/4
+-1/4 0 3/4
--3/4 0/1 -1/4
+-3/4 0 -1/4
--3/4 0/1 1/4
+-3/4 0 1/4
--1/4 0/1 -3/4
+-1/4 0 -3/4
 ##############
-0/1 1/4 -3/4
+0 1/4 -3/4
-0/1 3/4 -1/4
+0 3/4 -1/4
-0/1 3/4 1/4
+0 3/4 1/4
-0/1 1/4 3/4
+0 1/4 3/4
-0/1 -1/4 3/4
+0 -1/4 3/4
-0/1 -3/4 -1/4
+0 -3/4 -1/4
-0/1 -3/4 1/4
+0 -3/4 1/4
-0/1 -1/4 -3/4
+0 -1/4 -3/4
+#
+1 0 0
+-1 0 0

Arrangement_on_surface_2/test/Arrangement_on_surface_2/data/spherical_arcs/xcurves

@@ -1,99 +1,114 @@
 # xy plane
-0 0/1 -1/1 0/1 1/2 -1/2 0/1
+0 0 -1 0 1/2 -1/2 0
-0 0/1 -1/1 0/1 1/4 -3/4 0/1
+0 0 -1 0 1/4 -3/4 0
-0 1/4 -3/4 0/1 1/2 -1/2 0/1
+0 1/4 -3/4 0 1/2 -1/2 0
-0 1/8 -7/8 0/1 3/8 -5/8 0/1
+0 1/8 -7/8 0 3/8 -5/8 0
-0 0/1 -1/1 0/1 -1/2 -1/2 0/1
+0 0 -1 0 -1/2 -1/2 0
-0 0/1 -1/1 0/1 -1/4 -3/4 0/1
+0 0 -1 0 -1/4 -3/4 0
-0 -1/4 -3/4 0/1 -1/2 -1/2 0/1
+0 -1/4 -3/4 0 -1/2 -1/2 0
-0 -1/8 -7/8 0/1 -3/8 -5/8 0/1
+0 -1/8 -7/8 0 -3/8 -5/8 0
-0 -1/1 0/1 0/1 -1/2 -1/2 0/1
+0 -1 0 0 -1 -1 0
-0 -1/1 0/1 0/1 -3/4 -1/4 0/1
+0 -1 0 0 -3 -1 0
-0 -3/4 -1/4 0/1 -1/2 -1/2 0/1
+0 -3/4 -1/4 0 -1/2 -1/2 0
-0 -7/8 -1/8 0/1 -5/8 -3/8 0/1
+0 -7/8 -1/8 0 -5/8 -3/8 0
-0 -1/1 0/1 0/1 -1/2 1/2 0/1
+0 -1 0 0 -1 1 0
-0 -1/1 0/1 0/1 -3/4 1/4 0/1
+0 -1 0 0 -3 1 0
-0 -3/4 1/4 0/1 -1/2 1/2 0/1
+0 -3/4 1/4 0 -1/2 1/2 0
-0 -7/8 1/8 0/1 -5/8 3/8 0/1
+0 -7/8 1/8 0 -5/8 3/8 0
-0 0/1 1/1 0/1 1/2 1/2 0/1
+0 0 1 0 1/2 1/2 0
-0 0/1 1/1 0/1 1/4 3/4 0/1
+0 0 1 0 1/4 3/4 0
-0 1/4 3/4 0/1 1/2 1/2 0/1
+0 1/4 3/4 0 1/2 1/2 0
-0 1/8 7/8 0/1 3/8 5/8 0/1
+0 1/8 7/8 0 3/8 5/8 0
-0 0/1 1/1 0/1 -1/2 1/2 0/1
+0 0 1 0 -1/2 1/2 0
-0 0/1 1/1 0/1 -1/4 3/4 0/1
+0 0 1 0 -1/4 3/4 0
-0 -1/4 3/4 0/1 -1/2 1/2 0/1
+0 -1/4 3/4 0 -1/2 1/2 0
-0 -1/8 7/8 0/1 -3/8 5/8 0/1
+0 -1/8 7/8 0 -3/8 5/8 0
-0 1/1 0/1 0/1 1/2 -1/2 0/1
+0 1 0 0 1/2 -1/2 0
-0 1/1 0/1 0/1 3/4 -1/4 0/1
+0 1 0 0 3/4 -1/4 0
-0 3/4 -1/4 0/1 1/2 -1/2 0/1
+0 3/4 -1/4 0 1/2 -1/2 0
-0 7/8 -1/8 0/1 5/8 -3/8 0/1
+0 7/8 -1/8 0 5/8 -3/8 0
-0 1/1 0/1 0/1 1/2 1/2 0/1
+0 1 0 0 1/2 1/2 0
-0 1/1 0/1 0/1 3/4 1/4 0/1
+0 1 0 0 3/4 1/4 0
-0 3/4 1/4 0/1 1/2 1/2 0/1
+0 3/4 1/4 0 1/2 1/2 0
-0 7/8 1/8 0/1 5/8 3/8 0/1
+0 7/8 1/8 0 5/8 3/8 0
 # xz plane
-0 0/1 0/1 -1/1 1/2 0/1 -1/2
+0 0 0 -1 1/2 0 -1/2
-0 0/1 0/1 -1/1 1/4 0/1 -3/4
+0 0 0 -1 1/4 0 -3/4
-0 1/4 0/1 -3/4 1/2 0/1 -1/2
+0 1/4 0 -3/4 1/2 0 -1/2
-0 1/8 0/1 -7/8 3/8 0/1 -5/8
+0 1/8 0 -7/8 3/8 0 -5/8
-0 1/1 0/1 0/1 1/2 0/1 -1/2
+0 1 0 0 1/2 0 -1/2
-0 1/1 0/1 0/1 3/4 0/1 -1/4
+0 1 0 0 3/4 0 -1/4
-0 3/4 0/1 -1/4 1/2 0/1 -1/2
+0 3/4 0 -1/4 1/2 0 -1/2
-0 5/8 0/1 -1/8 7/8 0/1 -3/8
+0 5/8 0 -1/8 7/8 0 -3/8
-0 1/1 0/1 0/1 1/2 0/1 1/2
+0 1 0 0 1/2 0 1/2
-0 1/1 0/1 0/1 3/4 0/1 1/4
+0 1 0 0 3/4 0 1/4
-0 3/4 0/1 1/4 1/2 0/1 1/2
+0 3/4 0 1/4 1/2 0 1/2
-0 5/8 0/1 1/8 7/8 0/1 3/8
+0 5/8 0 1/8 7/8 0 3/8
-0 0/1 0/1 1/1 1/2 0/1 1/2
+0 0 0 1 1/2 0 1/2
-0 0/1 0/1 1/1 1/4 0/1 3/4
+0 0 0 1 1/4 0 3/4
-0 1/4 0/1 3/4 1/2 0/1 1/2
+0 1/4 0 3/4 1/2 0 1/2
-0 1/8 0/1 7/8 3/8 0/1 5/8
+0 1/8 0 7/8 3/8 0 5/8
-0 0/1 0/1 1/1 -1/2 0/1 1/2
+0 0 0 1 -1/2 0 1/2
-0 0/1 0/1 1/1 -1/4 0/1 3/4
+0 0 0 1 -1/4 0 3/4
-0 -1/4 0/1 3/4 -1/2 0/1 1/2
+0 -1/4 0 3/4 -1/2 0 1/2
-0 -1/8 0/1 7/8 -3/8 0/1 5/8
+0 -1/8 0 7/8 -3/8 0 5/8
-0 -1/1 0/1 0/1 -1/2 0/1 -1/2
+0 -1 0 0 -1 0 -1
-0 -1/1 0/1 0/1 -3/4 0/1 -1/4
+0 -1 0 0 -3 0 -1
-0 -3/4 0/1 -1/4 -1/2 0/1 -1/2
+0 -3/4 0 -1/4 -1/2 0 -1/2
-0 -5/8 0/1 -1/8 -7/8 0/1 -3/8
+0 -5/8 0 -1/8 -7/8 0 -3/8
-0 -1/1 0/1 0/1 -1/2 0/1 1/2
+0 -1 0 0 -1 0 1
-0 -1/1 0/1 0/1 -3/4 0/1 1/4
+0 -1 0 0 -3 0 1
-0 -3/4 0/1 1/4 -1/2 0/1 1/2
+0 -3/4 0 1/4 -1/2 0 1/2
-0 -5/8 0/1 1/8 -7/8 0/1 3/8
+0 -5/8 0 1/8 -7/8 0 3/8
-0 0/1 0/1 -1/1 -1/2 0/1 -1/2
+0 0 0 -1 -1/2 0 -1/2
-0 0/1 0/1 -1/1 -1/4 0/1 -3/4
+0 0 0 -1 -1/4 0 -3/4
-0 -1/4 0/1 -3/4 -1/2 0/1 -1/2
+0 -1/4 0 -3/4 -1/2 0 -1/2
-0 -1/8 0/1 -7/8 -1/8 0/1 -5/8
+0 -1/8 0 -7/8 -1/8 0 -5/8
 # yz plane
-0 0/1 0/1 -1/1 0/1 1/2 -1/2
+0 0 0 -1 0 1/2 -1/2
-0 0/1 0/1 -1/1 0/1 1/4 -3/4
+0 0 0 -1 0 1/4 -3/4
-0 0/1 1/4 -3/4 0/1 1/2 -1/2
+0 0 1/4 -3/4 0 1/2 -1/2
-0 0/1 1/8 -7/8 0/1 3/8 -5/8
+0 0 1/8 -7/8 0 3/8 -5/8
-0 0/1 1/1 0/1 0/1 1/2 -1/2
+0 0 1 0 0 1/2 -1/2
-0 0/1 1/1 0/1 0/1 3/4 -1/4
+0 0 1 0 0 3/4 -1/4
-0 0/1 3/4 -1/4 0/1 1/2 -1/2
+0 0 3/4 -1/4 0 1/2 -1/2
-0 0/1 5/8 -1/8 0/1 7/8 -3/8
+0 0 5/8 -1/8 0 7/8 -3/8
-0 0/1 1/1 0/1 0/1 1/2 1/2
+0 0 1 0 0 1/2 1/2
-0 0/1 1/1 0/1 0/1 3/4 1/4
+0 0 1 0 0 3/4 1/4
-0 0/1 3/4 1/4 0/1 1/2 1/2
+0 0 3/4 1/4 0 1/2 1/2
-0 0/1 5/8 1/8 0/1 7/8 3/8
+0 0 5/8 1/8 0 7/8 3/8
-0 0/1 0/1 1/1 0/1 1/2 1/2
+0 0 0 1 0 1/2 1/2
-0 0/1 0/1 1/1 0/1 1/4 3/4
+0 0 0 1 0 1/4 3/4
-0 0/1 1/4 3/4 0/1 1/2 1/2
+0 0 1/4 3/4 0 1/2 1/2
-0 0/1 1/8 7/8 0/1 3/8 5/8
+0 0 1/8 7/8 0 3/8 5/8
-0 0/1 0/1 1/1 0/1 -1/2 1/2
+0 0 0 1 0 -1/2 1/2
-0 0/1 0/1 1/1 0/1 -1/4 3/4
+0 0 0 1 0 -1/4 3/4
-0 0/1 -1/4 3/4 0/1 -1/2 1/2
+0 0 -1/4 3/4 0 -1/2 1/2
-0 0/1 -1/8 7/8 0/1 -3/8 5/8
+0 0 -1/8 7/8 0 -3/8 5/8
-0 0/1 -1/1 0/1 0/1 -1/2 -1/2
+0 0 -1 0 0 -1/2 -1/2
-0 0/1 -1/1 0/1 0/1 -3/4 -1/4
+0 0 -1 0 0 -3/4 -1/4
-0 0/1 -3/4 -1/4 0/1 -1/2 -1/2
+0 0 -3/4 -1/4 0 -1/2 -1/2
-0 0/1 -5/8 -1/8 0/1 -7/8 -3/8
+0 0 -5/8 -1/8 0 -7/8 -3/8
-0 0/1 -1/1 0/1 0/1 -1/2 1/2
+0 0 -1 0 0 -1/2 1/2
-0 0/1 -1/1 0/1 0/1 -3/4 1/4
+0 0 -1 0 0 -3/4 1/4
-0 0/1 -3/4 1/4 0/1 -1/2 1/2
+0 0 -3/4 1/4 0 -1/2 1/2
-0 0/1 -5/8 1/8 0/1 -7/8 3/8
+0 0 -5/8 1/8 0 -7/8 3/8
-0 0/1 0/1 -1/1 0/1 -1/2 -1/2
+0 0 0 -1 0 -1/2 -1/2
-0 0/1 0/1 -1/1 0/1 -1/4 -3/4
+0 0 0 -1 0 -1/4 -3/4
-0 0/1 -1/4 -3/4 0/1 -1/2 -1/2
+0 0 -1/4 -3/4 0 -1/2 -1/2
-0 0/1 -1/8 -7/8 0/1 -1/8 -5/8
+0 0 -1/8 -7/8 0 -1/8 -5/8
+#
+1 -1 0 0 1 0 0 0 1 1
+1 -1 0 0 1 0 0 0 -1 1
+1 1 0 0 -1 0 0 0 1 1
+1 1 0 0 -1 0 0 0 -1 1
+#
+1 -1 0 0 1 0 0 0 -1 -1
+1 -1 0 0 1 0 0 0 1 -1
+1 1 0 0 -1 0 0 0 -1 -1
+1 1 0 0 -1 0 0 0 1 -1
+#
+0 -1 0 0 0 -1 1
+0 -1 0 0 0 -1 -1
+0 -1 0 0 0 1 1
+0 -1 0 0 0 1 -1

Arrangement_on_surface_2/test/Arrangement_on_surface_2/data/test_construction/geodesic_arcs_on_sphere/test39.txt

@@ -0,0 +1,10 @@
+3
+1 2 0 0 -2 0 0 0 2 2
+1 2 0 0 -2 0 0 0 -2 2
+1 2 0 0 -2 0 0 0 2 2
+0
+2 2 2
+-2 0 0
+2 0 0
+1 2 0 0 -2 0 0 0 2 2 2
+1 2 0 0 -2 0 0 0 -2 2 1

Arrangement_on_surface_2/test/Arrangement_on_surface_2/test_construction.geodesic_arcs_on_sphere.cmd

@@ -36,4 +36,4 @@ data/test_construction/geodesic_arcs_on_sphere/test35.txt
 data/test_construction/geodesic_arcs_on_sphere/test36.txt
 data/test_construction/geodesic_arcs_on_sphere/test37.txt
 data/test_construction/geodesic_arcs_on_sphere/test38.txt
-
+data/test_construction/geodesic_arcs_on_sphere/test39.txt

Installation/CHANGES.md

@@ -48,6 +48,10 @@ Release date: June 2022
 ### [2D Regularized Boolean Set-Operations](https://doc.cgal.org/5.5/Manual/packages.html#PkgBooleanSetOperations2)
 -   The concept `GeneralPolygonSetTraits_2` now requires the nested type `Construct_polygon_with_holes_2` instead of `Construct_general_polygon_with_holes_2`.
 
+### [2D Arrangements](https://doc.cgal.org/5.5/Manual/packages.html#PkgArrangementOnSurface2)
+-   Fixed the intersect_2, compare_y_at_x_right, and compare_y_at_x_left function objects of the traits class template that handles geodesic arcs on sphere and applied a small syntactical fix to the tracing traits.
+
+
 [Release 5.4](https://github.com/CGAL/cgal/releases/tag/v5.4)
 -----------