diff --git a/Arrangement_on_surface_2/include/CGAL/Arr_overlay_2.h b/Arrangement_on_surface_2/include/CGAL/Arr_overlay_2.h
index a383d8602f8..9b45e51dafc 100644
--- a/Arrangement_on_surface_2/include/CGAL/Arr_overlay_2.h
+++ b/Arrangement_on_surface_2/include/CGAL/Arr_overlay_2.h
@@ -59,12 +59,20 @@ public:
 
   std::size_t min_end_index (const Curve& c) const
   {
-    return reinterpret_cast<std::size_t>(halfedge(c)->target()->inc());
+    if (c.red_halfedge_handle() != typename Curve::HH_red())
+      return reinterpret_cast<std::size_t>(c.red_halfedge_handle()->target()->inc());
+    // else
+    CGAL_assertion (c.blue_halfedge_handle() != typename Curve::HH_blue());
+    return reinterpret_cast<std::size_t>(c.blue_halfedge_handle()->target()->inc());
   }
 
   std::size_t max_end_index (const Curve& c) const
   {
-    return reinterpret_cast<std::size_t>(halfedge(c)->source()->inc());
+    if (c.red_halfedge_handle() != typename Curve::HH_red())
+      return reinterpret_cast<std::size_t>(c.red_halfedge_handle()->source()->inc());
+    // else
+    CGAL_assertion (c.blue_halfedge_handle() != typename Curve::HH_blue());
+    return reinterpret_cast<std::size_t>(c.blue_halfedge_handle()->source()->inc());
   }
 
   const Curve& curve (const Curve& c) const
@@ -103,16 +111,6 @@ public:
 
 private:
 
-  typename Curve::Halfedge_handle halfedge (const Curve& c) const
-  {
-    if (c.red_halfedge_handle() == typename Curve::Halfedge_handle())
-    {
-      CGAL_assertion (c.blue_halfedge_handle() != typename Curve::Halfedge_handle());
-      return c.blue_halfedge_handle();
-    }
-    // else
-    return c.red_halfedge_handle();
-  }
 };
 
 /*! Compute the overlay of two input arrangements.
diff --git a/Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_overlay_traits_2.h b/Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_overlay_traits_2.h
index 4d10d82a005..eff640e5443 100644
--- a/Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_overlay_traits_2.h
+++ b/Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_overlay_traits_2.h
@@ -134,7 +134,8 @@ public:
   class Ex_x_monotone_curve_2 {
   public:
     typedef Base_x_monotone_curve_2             Base;
-    typedef Halfedge_handle_blue                Halfedge_handle;
+    typedef Halfedge_handle_red HH_red;
+    typedef Halfedge_handle_blue HH_blue;
 
   protected:
     Base m_base_xcv;                              // The base curve.