diff --git a/Envelope_2/doc/Envelope_2/CGAL/envelope_2.h b/Envelope_2/doc/Envelope_2/CGAL/envelope_2.h
index 5dd0d75a15f..7a07ab96e3b 100644
--- a/Envelope_2/doc/Envelope_2/CGAL/envelope_2.h
+++ b/Envelope_2/doc/Envelope_2/CGAL/envelope_2.h
@@ -40,6 +40,28 @@ namespace CGAL {
 /*!
 \ingroup PkgEnvelope2Ref
 
+Computes the lower envelope of a set of \f$ x\f$-monotone curves in
+\f$ \mathbb{R}^2\f$, as given by the range `[begin, end)` with the help
+of the arrangement traits object `traits` responsible for their creation.
+Reusing the same traits object improves speed if the traits class caches data.
+The lower envelope is represented using the output minimization diagram `diag`.
+
+\tparam InputIterator must be an input iterator with value type `EnvelopeDiagram::X_monotone_curve_2`.
+\tparam EnvelopeDiagram must be a model of the concept `EnvelopeDiagram_1`.
+\tparam Traits must be a model of the concept `ArrangementXMonotoneTraits_2`.
+*/
+template<class InputIterator, class EnvelopeDiagram, class Traits>
+void lower_envelope_x_monotone_2
+(InputIterator begin, InputIterator end,
+EnvelopeDiagram& diag, const Traits& traits);
+
+} /* namespace CGAL */
+
+namespace CGAL {
+
+/*!
+\ingroup PkgEnvelope2Ref
+
 Computes the upper envelope of a set of curves in \f$ \mathbb{R}^2\f$,
 as given by the range `[begin, end)`. The upper envelope is
 represented using the output maximization diagram `diag`.
@@ -71,3 +93,25 @@ void upper_envelope_x_monotone_2
 EnvelopeDiagram& diag);
 
 } /* namespace CGAL */
+
+namespace CGAL {
+
+/*!
+\ingroup PkgEnvelope2Ref
+
+Computes the upper envelope of a set of \f$ x\f$-monotone curves in
+\f$ \mathbb{R}^2\f$, as given by the range `[begin, end)` with the help
+of the arrangement traits object `traits` responsbile for their creation.
+Reusing the same traits object improves speed if the traits class caches data.
+The upper envelope is represented using the output maximization diagram `diag`.
+
+\tparam InputIterator must be an input iterator with value type `EnvelopeDiagram::X_monotone_curve_2`.
+\tparam EnvelopeDiagram must be a model of the concept `EnvelopeDiagram_1`.
+\tparam Traits must be a model of the concept `ArrangementXMonotoneTraits_2`.
+*/
+template<class InputIterator, class EnvelopeDiagram, class Traits>
+void upper_envelope_x_monotone_2
+(InputIterator begin, InputIterator end,
+EnvelopeDiagram& diag, const Traits& traits);
+
+} /* namespace CGAL */
diff --git a/Envelope_2/include/CGAL/Envelope_2/Env_divide_and_conquer_2.h b/Envelope_2/include/CGAL/Envelope_2/Env_divide_and_conquer_2.h
index bc72cace257..fc2f04a27ec 100644
--- a/Envelope_2/include/CGAL/Envelope_2/Env_divide_and_conquer_2.h
+++ b/Envelope_2/include/CGAL/Envelope_2/Env_divide_and_conquer_2.h
@@ -62,9 +62,9 @@ protected:
   typedef Arr_traits_adaptor_2<Traits_2>           Traits_adaptor_2;
 
   // Data members:
-  Traits_adaptor_2  *traits;        // The traits object.
-  bool               own_traits;    // Whether we own the traits object.
-  Envelope_type      env_type;      // Either LOWER or UPPER.
+  const Traits_adaptor_2  *traits;        // The traits object.
+  bool                     own_traits;    // Whether we own the traits object.
+  Envelope_type            env_type;      // Either LOWER or UPPER.
 
   // Copy constructor and assignment operator - not supported.
   Envelope_divide_and_conquer_2 (const Self& );
diff --git a/Envelope_2/include/CGAL/envelope_2.h b/Envelope_2/include/CGAL/envelope_2.h
index 7eb007edc94..6d23f724393 100644
--- a/Envelope_2/include/CGAL/envelope_2.h
+++ b/Envelope_2/include/CGAL/envelope_2.h
@@ -93,6 +93,30 @@ void lower_envelope_x_monotone_2 (InputIterator begin, InputIterator end,
   return;
 }
 
+/*!
+ * Compute the lower envelope of a range of x-monotone curves.
+ * \param begin An iterator for the first x-monotone curve.
+ * \param end A past-the-end iterator for the x-monotone curves.
+ * \param diag Output: The minimization diagram.
+ * \param traits The arrangement traits responsible for the x-monotone curves.
+ * \pre The value-type of the iterator is Traits::X_monotone_curve_2.
+ */
+template <class InputIterator, class EnvelopeDiagram, class Traits>
+void lower_envelope_x_monotone_2 (InputIterator begin, InputIterator end,
+                                  EnvelopeDiagram& diag, const Traits& traits)
+{
+  typedef typename EnvelopeDiagram::Traits_2                       Traits_2;
+  typedef Envelope_divide_and_conquer_2<Traits_2, EnvelopeDiagram> Envelope_2;
+
+  Envelope_2      env{&traits};
+
+  env.insert_x_monotone_curves (begin, end,
+                                true,              // Lower envelope.
+                                diag);
+
+  return;
+}
+
 /*!
  * Compute the upper envelope of a range of x-monotone curves.
  * \param begin An iterator for the first x-monotone curve.
@@ -116,6 +140,30 @@ void upper_envelope_x_monotone_2 (InputIterator begin, InputIterator end,
   return;
 }
 
+/*!
+ * Compute the upper envelope of a range of x-monotone curves.
+ * \param begin An iterator for the first x-monotone curve.
+ * \param end A past-the-end iterator for the x-monotone curves.
+ * \param diag Output: The maximization diagram.
+ * \param traits The arrangement traits responsible for the x-monotone curves.
+ * \pre The value-type of the iterator is Traits::X_monotone_curve_2.
+ */
+template <class InputIterator, class EnvelopeDiagram, class Traits>
+void upper_envelope_x_monotone_2 (InputIterator begin, InputIterator end,
+                                  EnvelopeDiagram& diag, const Traits& traits)
+{
+  typedef typename EnvelopeDiagram::Traits_2                       Traits_2;
+  typedef Envelope_divide_and_conquer_2<Traits_2, EnvelopeDiagram> Envelope_2;
+
+  Envelope_2      env{&traits};
+
+  env.insert_x_monotone_curves (begin, end,
+                                false,          // Upper envelope.
+                                diag);
+
+  return;
+}
+
 } //namespace CGAL
 
 #endif