diff --git a/Frechet_distance/include/CGAL/Frechet_distance.h b/Frechet_distance/include/CGAL/Frechet_distance.h
index 42d46197ff86..531928cf7713 100644
--- a/Frechet_distance/include/CGAL/Frechet_distance.h
+++ b/Frechet_distance/include/CGAL/Frechet_distance.h
@@ -35,7 +35,7 @@ namespace CGAL
  * \param traits the geometric traits object
  *
  * \tparam Traits a model of `FrechetDistanceTraits`
- * \tparam force_filtering if `true` interval arithmetic will be used as a filter
+ * \tparam force_filtering if `true` interval arithmetic combined with exact rational will be used internally
  * \tparam PointRange  a model of the concept `RandomAccessContainer`
  * with `Traits::Point` as value type.
  */
@@ -69,7 +69,7 @@ bool is_Frechet_distance_larger(const PointRange& polyline1,
  *  \param traits the geometric traits object
  *
  * \tparam Traits a model of `FrechetDistanceTraits`
- * \tparam force_filtering if `true` interval arithmetic will be used as a filter
+ * \tparam force_filtering if `true` interval arithmetic combined with exact rational will be used internally
  * \tparam PointRange  a model of the concept `RandomAccessContainer`
  * with `Traits::Point` as value type.
  *
@@ -83,7 +83,7 @@ std::pair<double,double> approximate_Frechet_distance(const PointRange& polyline
                                                       const Traits& traits = Traits())
 {
     constexpr bool filtered = force_filtering ||
-      std::is_same_v<typename decltype(Frechet_distance_::internal::toCurve(polyline1, traits))::FT,
+      std::is_same_v<typename decltype(Frechet_distance_::internal::toCurve<force_filtering>(polyline1, traits))::FT,
                      Interval_nt<false>>;
     Protect_FPU_rounding<filtered> p;
 
diff --git a/Frechet_distance/include/CGAL/Frechet_distance/internal/Frechet_distance.h b/Frechet_distance/include/CGAL/Frechet_distance/internal/Frechet_distance.h
index 9c4a49b62519..b0847959cac1 100644
--- a/Frechet_distance/include/CGAL/Frechet_distance/internal/Frechet_distance.h
+++ b/Frechet_distance/include/CGAL/Frechet_distance/internal/Frechet_distance.h
@@ -61,7 +61,7 @@ auto toCurve(const PointRange& point_range, const Traits& traits)
 
         using Filtered_traits = std::pair<Frechet_distance_traits_2<AK>, Frechet_distance_traits_2<EK>>;
 
-        return Curve<Filtered_traits, true>(point_range);
+        return Curve<Filtered_traits, true>(point_range, traits);
       }
       else if constexpr (Traits::Dimension::value==3)
       {
@@ -70,7 +70,7 @@ auto toCurve(const PointRange& point_range, const Traits& traits)
 
         using Filtered_traits = std::pair<Frechet_distance_traits_3<AK>, Frechet_distance_traits_3<EK>>;
 
-        return Curve<Filtered_traits, true>(point_range);
+        return Curve<Filtered_traits, true>(point_range, traits);
       }
       else
       {
@@ -91,7 +91,7 @@ auto toCurve(const PointRange& point_range, const Traits& traits)
 
         using Filtered_traits = std::tuple<typename Traits::Point_d, Frechet_distance_traits_2<AK>, Frechet_distance_traits_2<EK>>;
 
-        return Curve<Filtered_traits, true>(point_range);
+        return Curve<Filtered_traits, true>(point_range, traits);
       }
       else if constexpr (Traits::Dimension::value==3)
       {
@@ -100,7 +100,7 @@ auto toCurve(const PointRange& point_range, const Traits& traits)
 
         using Filtered_traits = std::tuple<typename Traits::Point_d, Frechet_distance_traits_3<AK>, Frechet_distance_traits_3<EK>>;
 
-        return Curve<Filtered_traits, true>(point_range);
+        return Curve<Filtered_traits, true>(point_range, traits);
       }
       else
       {
@@ -126,7 +126,7 @@ auto toCurve(const PointRange& point_range, const Traits& traits)
 
           using Filtered_traits = std::pair<AT,ET>;
 
-          return Curve<Filtered_traits, true>(point_range);
+          return Curve<Filtered_traits, true>(point_range, traits);
         }
         else if constexpr (Traits::Dimension::value==3)
         {
@@ -135,7 +135,7 @@ auto toCurve(const PointRange& point_range, const Traits& traits)
 
           using Filtered_traits = std::pair<AT,ET>;
 
-          return Curve<Filtered_traits, true>(point_range);
+          return Curve<Filtered_traits, true>(point_range, traits);
         }
         else
         {
@@ -156,7 +156,7 @@ auto toCurve(const PointRange& point_range, const Traits& traits)
 
           using Filtered_traits = std::tuple<typename ET::Point_d,AT,ET>;
 
-          return Curve<Filtered_traits, true>(point_range);
+          return Curve<Filtered_traits, true>(point_range, traits, traits);
         }
         else if constexpr (Traits::Dimension::value==3)
         {
@@ -165,7 +165,7 @@ auto toCurve(const PointRange& point_range, const Traits& traits)
 
           using Filtered_traits = std::tuple<typename ET::Point_d,AT,ET>;
 
-          return Curve<Filtered_traits, true>(point_range);
+          return Curve<Filtered_traits, true>(point_range, traits, traits);
         }
         else
         {
diff --git a/Frechet_distance/include/CGAL/Frechet_distance/internal/curve.h b/Frechet_distance/include/CGAL/Frechet_distance/internal/curve.h
index 5c231835a2df..00262b3482fd 100644
--- a/Frechet_distance/include/CGAL/Frechet_distance/internal/curve.h
+++ b/Frechet_distance/include/CGAL/Frechet_distance/internal/curve.h
@@ -100,17 +100,27 @@ class Curve<std::pair<Approximate_traits,Rational_traits>, true>
     Curve() = default;
 
 
-    template <class PointRange>
-    Curve(const PointRange& point_range)
+    template <class PointRange, class Input_traits>
+    Curve(const PointRange& point_range, const Input_traits& in_traits, const Rational_traits& rt = Rational_traits())
+      : rational_traits_(rt)
     {
-        using IPoint = typename std::iterator_traits<typename PointRange::const_iterator>::value_type;
-        using K2I = Cartesian_converter< typename Kernel_traits<IPoint>::Kernel,
-                                         typename Kernel_traits<Point>::Kernel>;
+        static constexpr int dim = Approximate_traits::Dimension::value;
 
         this->points.reserve(point_range.size());
-        K2I convert;
+        std::array<distance_t, dim> coords;
+        auto ccci = in_traits.construct_cartesian_const_iterator_d_object();
         for (auto const& p : point_range)
-          this->points.push_back(convert(p));
+        {
+          auto itp = ccci(p);
+          for (int i=0;i<dim; ++i)
+            coords[i]=distance_t(to_interval(*itp++));
+          if constexpr (dim==2)
+            this->points.emplace_back(coords[0], coords[1]);
+          else if constexpr (dim==3)
+            this->points.emplace_back(coords[0], coords[1], coords[2]);
+          else
+            this->points.emplace_back(coords.begin(), coords.end());
+        }
 
         this->init();
     }
@@ -120,6 +130,10 @@ class Curve<std::pair<Approximate_traits,Rational_traits>, true>
       I2R convert;
       return convert(this->point(i));
     }
+
+    const Rational_traits& rational_traits() const { return rational_traits_; }
+
+    Rational_traits rational_traits_;
 };
 
 //Exact rational based filtered version
@@ -141,8 +155,9 @@ class Curve<std::tuple<Input_point, Approximate_traits,Rational_traits>, true>
 
     Curve() = default;
 
-    template <class PointRange>
-    Curve(const PointRange& point_range)
+    template <class PointRange, class Input_traits>
+    Curve(const PointRange& point_range, const Input_traits& in_traits, const Rational_traits& rt = Rational_traits())
+      : rational_traits_(rt)
     {
         this->points.reserve(point_range.size());
         rational_points.reserve(point_range.size());
@@ -151,9 +166,23 @@ class Curve<std::tuple<Input_point, Approximate_traits,Rational_traits>, true>
         {
           if constexpr(std::is_same_v<Input_point, Rational_point>)
           {
-            Cartesian_converter<typename Kernel_traits<Input_point>::Kernel,
-                                typename Kernel_traits<typename Approximate_traits::Point_d>::Kernel> convert;
-            this->points.push_back(convert(p));
+            static constexpr int dim = Approximate_traits::Dimension::value;
+
+            this->points.reserve(point_range.size());
+            std::array<distance_t, dim> coords;
+            auto ccci = in_traits.construct_cartesian_const_iterator_d_object();
+            for (auto const& p : point_range)
+            {
+              auto itp = ccci(p);
+              for (int i=0;i<dim; ++i)
+                coords[i]=distance_t(to_interval(*itp++));
+              if constexpr (dim==2)
+                this->points.emplace_back(coords[0], coords[1]);
+              else if constexpr (dim==3)
+                this->points.emplace_back(coords[0], coords[1], coords[2]);
+              else
+                this->points.emplace_back(coords.begin(), coords.end());
+            }
           }
           else
           {
@@ -173,6 +202,9 @@ class Curve<std::tuple<Input_point, Approximate_traits,Rational_traits>, true>
         return exact(rational_points[i]);
     }
 
+    const Rational_traits& rational_traits() const { return rational_traits_; }
+
+    Rational_traits rational_traits_;
     std::vector<Input_point> rational_points;
 };
 
diff --git a/Frechet_distance/include/CGAL/Frechet_distance/internal/geometry_basics.h b/Frechet_distance/include/CGAL/Frechet_distance/internal/geometry_basics.h
index 99b267897cfb..89e173c00552 100644
--- a/Frechet_distance/include/CGAL/Frechet_distance/internal/geometry_basics.h
+++ b/Frechet_distance/include/CGAL/Frechet_distance/internal/geometry_basics.h
@@ -117,13 +117,19 @@ struct Lambda<Curve<FilteredTraits,true>>
         const typename Curve::Rational_point& ls = curve2->rpoint(line_start);
         const typename Curve::Rational_point& le = curve2->rpoint(line_start+1);
         const typename Curve::Rational_point& cc = curve1->rpoint(circle_center);
+
+        auto ccci = curve1->rational_traits().construct_cartesian_const_iterator_d_object();
         Rational a, b, c;
+
+        auto it_le=ccci(le), it_ls=ccci(ls), it_cc=ccci(cc);
+
         for (auto i = 0; i < Curve::dimension; ++i) {
-            Rational start_end_diff = le[i] - ls[i];
+            Rational start_end_diff = *it_le - *it_ls;
             a += CGAL::square(start_end_diff);
-            Rational start_center_diff = ls[i] - cc[i];
+            Rational start_center_diff = *it_ls - *it_cc;
             b -= start_center_diff * start_end_diff;
             c += CGAL::square(start_center_diff);
+            ++it_le; ++it_ls; ++it_cc;
         }
         CGAL_assertion(radius.is_point());
         c -= CGAL::square(Rational(to_double(radius)));
diff --git a/Frechet_distance/test/Frechet_distance/Frechet_distance_minimal_traits_2.cpp b/Frechet_distance/test/Frechet_distance/Frechet_distance_minimal_traits_2.cpp
index 910e942bdf38..06e462a73c56 100644
--- a/Frechet_distance/test/Frechet_distance/Frechet_distance_minimal_traits_2.cpp
+++ b/Frechet_distance/test/Frechet_distance/Frechet_distance_minimal_traits_2.cpp
@@ -1,9 +1,10 @@
 #include <CGAL/Dimension.h>
 #include <CGAL/Bbox_2.h>
 
+template <class FT_>
 struct MinimalFrechetTraits {
   using Dimension = CGAL::Dimension_tag<2>;
-  using FT = double;
+  using FT = FT_;
 
   struct Point_d
   {};
@@ -52,7 +53,19 @@ struct MinimalFrechetTraits {
 
 int main()
 {
-  std::vector<MinimalFrechetTraits::Point_d> curve;
-  /* bool decision = */ CGAL::is_Frechet_distance_larger<MinimalFrechetTraits>(curve, curve, 0.1);
+  {
+    using Traits = MinimalFrechetTraits<double>;
+    std::vector<Traits::Point_d> curve;
+    /* bool decision = */ CGAL::is_Frechet_distance_larger<Traits, true>(curve, curve, 0.1);
+  }
+
+  {
+    using Traits = MinimalFrechetTraits<CGAL::Exact_rational>;
+    std::vector<Traits::Point_d> curve;
+    /* bool decision = */ CGAL::is_Frechet_distance_larger<Traits, true>(curve, curve, 0.1);
+  }
+
+
+
   return 0;
 }