Add Compare_xy_2 to TriangulationTraits_2

This enables customization of projection traits for CDT_2s and
facilitates unification with the predicates utilized in
3D triangulations.

Kernel_23/include/CGAL/Kernel_23/internal/Projection_traits_3.h

@@ -971,6 +971,7 @@ public:
   typedef typename Rp::Construct_triangle_3                   Construct_triangle_2;
   typedef typename Rp::Construct_line_3                       Construct_line_2;
 
+  typedef typename Rp::Compare_xyz_3                          Compare_xy_2;
 
   struct Less_xy_2 {
     typedef typename R::Boolean result_type;
@@ -1065,6 +1066,10 @@ public:
   less_x_2_object() const
     { return Less_x_2();}
 
+  Compare_xy_2
+  compare_xy_2_object() const
+    { return Compare_xy_2();}
+
   Less_xy_2
   less_xy_2_object() const
     { return Less_xy_2();}

Kernel_23/include/CGAL/Kernel_23/internal/Projection_traits_base_3.h

@@ -426,6 +426,7 @@ public:
   typedef typename K::Line_3      Line_2;
 
   typedef typename K::Angle_3                                Angle_2;
+  typedef typename K::Compare_xyz_3                          Compare_xy_2;
 
   typedef TriangulationProjectionTraitsCartesianFunctors::
     Compare_along_axis<Self>                                 Compare_x_2;
@@ -524,6 +525,9 @@ public:
   Angle_2  angle_2_object() const
     {return Angle_2();}
 
+  Compare_xy_2  compare_xy_2_object() const
+    {return Compare_xy_2();}
+
   Construct_point_2  construct_point_2_object() const
     {return Construct_point_2();}
 

Triangulation_2/doc/Triangulation_2/Concepts/TriangulationTraits_2.h

@@ -144,6 +144,19 @@ according to the
 */
 typedef unspecified_type Compare_y_2;
 
+/*!
+A function object that lexicographically compares two points by their Cartesian
+coordinates.
+It provides the operator:
+
+`Comparison_result operator()(Point p, Point q)`
+
+which returns
+(`SMALLER, EQUAL` or `LARGER`)
+according to the lexicographical order of points `p` and `q`.
+*/
+typedef unspecified_type Compare_xy_2;
+
 /*!
 A function object to compute the orientation of three points.
 
@@ -245,6 +258,11 @@ Compare_y_2 compare_y_2_object();
 
 /*!
 
+*/
+Compare_xy_2 compare_xy_2_object();
+
+/*!
+
 */
 Orientation_2 orientation_2_object();
 

Triangulation_2/include/CGAL/Triangulation_2.h

@@ -3671,13 +3671,8 @@ Comparison_result
 Triangulation_2<Gt, Tds>::
 compare_xy(const Point& p, const Point& q) const
 {
-  Comparison_result res = geom_traits().compare_x_2_object()(construct_point(p),
+  return geom_traits().compare_xy_2_object()(construct_point(p),
-                                                             construct_point(q));
+                                             construct_point(q));
-  if(res == EQUAL){
-    return geom_traits().compare_y_2_object()(construct_point(p),
-                                              construct_point(q));
-  }
-  return res;
 }
 
 template <class Gt, class Tds >

Triangulation_2/test/Triangulation_2/include/CGAL/_test_traits.h

@@ -188,6 +188,19 @@ public:
     }
 };
 
+class Triangulation_test_Compare_xy_2{
+public:
+  typedef Triangulation_test_point  Point;
+  CGAL::Comparison_result operator()( const Point& p, const Point&  q) const
+    {
+      if (p.test_x() < q.test_x()) return CGAL::SMALLER;
+      if (p.test_x() > q.test_x()) return CGAL::LARGER;
+      if (p.test_y() < q.test_y()) return CGAL::SMALLER;
+      if (p.test_y() > q.test_y()) return CGAL::LARGER;
+      else return CGAL::EQUAL;
+    }
+};
+
 class Triangulation_test_Orientation_2
 {
 public:
@@ -416,6 +429,7 @@ public:
   typedef Triangulation_test_Less_y_2       Less_y_2;
   typedef Triangulation_test_Compare_x_2    Compare_x_2;
   typedef Triangulation_test_Compare_y_2    Compare_y_2;
+  typedef Triangulation_test_Compare_xy_2   Compare_xy_2;
   typedef Triangulation_test_Orientation_2  Orientation_2;
   typedef Triangulation_test_Side_of_oriented_circle_2
                                              Side_of_oriented_circle_2;
@@ -450,6 +464,10 @@ public:
   compare_y_2_object() const
     { return Compare_y_2();}
 
+  Compare_xy_2
+  compare_xy_2_object() const
+    { return Compare_xy_2();}
+
   Orientation_2
   orientation_2_object() const
     { return Orientation_2();}