diff --git a/Circulator/include/CGAL/circulator.h b/Circulator/include/CGAL/circulator.h
index c622fb615ad..aa069651668 100644
--- a/Circulator/include/CGAL/circulator.h
+++ b/Circulator/include/CGAL/circulator.h
@@ -22,10 +22,11 @@
 #include <CGAL/assertions.h>
 #include <CGAL/use.h>
 #include <CGAL/tags.h>
+#include <CGAL/type_traits.h>
 
 #include <cstddef>
-#include <functional>
 #include <iterator>
+#include <utility>
 
 // These are name redefinitions for backwards compatibility
 // with the pre iterator-traits style adaptors.
@@ -640,6 +641,34 @@ operator+( Dist n, const Iterator_from_circulator<C,Ref,Ptr>& circ) {
     return tmp += n;
 }
 
+template <class Circ>
+class Range_from_circulator {
+private:
+  Circ anchor;
+public:
+  using pointer = CGAL::cpp20::remove_cvref_t<decltype(anchor.operator->())>;
+  using const_pointer = CGAL::cpp20::remove_cvref_t<decltype(std::as_const(anchor).operator->())>;
+  using reference = decltype(*anchor);
+  using const_reference = decltype(*std::as_const(anchor));
+  using iterator = Iterator_from_circulator<Circ, reference, pointer>;
+  using const_iterator = Iterator_from_circulator<Circ, const_reference, const_pointer>;
+
+  iterator begin() {
+    return iterator(&anchor, 0);
+  }
+  const_iterator begin() const {
+    return const_iterator(&anchor, 0);
+  }
+  iterator end() {
+    return anchor == nullptr ? iterator(&anchor, 0) : iterator(&anchor, 1);
+  }
+  const_iterator end() const {
+    return anchor == nullptr ? const_iterator(&anchor, 0) : const_iterator(&anchor, 1);
+  }
+  Range_from_circulator() = default;
+  Range_from_circulator(const Circ& c) : anchor(get_min_circulator(c)) {}
+};
+
 template < class  C >
 class Container_from_circulator {
 private:
diff --git a/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_3.h b/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_3.h
index 1a070a77585..f42a6adab1d 100644
--- a/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_3.h
+++ b/Constrained_triangulation_3/include/CGAL/Conforming_constrained_Delaunay_triangulation_3.h
@@ -29,6 +29,7 @@
 #include <CGAL/Constrained_triangulation_plus_2.h>
 #include <CGAL/Projection_traits_3.h>
 #include <CGAL/Union_find.h>
+#include <CGAL/circulator.h>
 #include <CGAL/intersection_3.h>
 #include <CGAL/iterator.h>
 #include <CGAL/Iterator_range.h>
@@ -898,83 +899,6 @@ public:
   }
   /// @} // end triangulation section
 
-  /// \cond SKIP_IN_MANUAL
-  Conforming_constrained_Delaunay_triangulation_3 convert_for_remeshing() &&
-  {
-    auto& tr = cdt_impl;
-
-    for(auto v : tr.all_vertex_handles()) {
-      switch(v->ccdt_3_data().vertex_type()) {
-      case CDT_3_vertex_type::CORNER:
-        v->set_dimension(0);
-        v->set_index(0);
-        break;
-      case CDT_3_vertex_type::STEINER_ON_EDGE:
-        v->set_dimension(1);
-        v->set_index(static_cast<int>(v->ccdt_3_data().constrained_polyline_id(tr).index()));
-        break;
-      case CDT_3_vertex_type::STEINER_IN_FACE:
-        v->set_dimension(2);
-        v->set_index(v->ccdt_3_data().face_index());
-        break;
-      case CDT_3_vertex_type::FREE:
-        v->set_dimension(3);
-        v->set_index(1);
-        break;
-      default:
-        CGAL_error();
-        break;
-      }
-    }
-
-    if(tr.dimension() < 3) {
-      for(auto ch : tr.all_cell_handles()) {
-        ch->set_subdomain_index(0);
-      }
-    } else {
-      for(auto ch : tr.all_cell_handles()) {
-        ch->set_subdomain_index(1);
-      }
-
-      std::stack<decltype(tr.infinite_cell())> stack;
-      stack.push(tr.infinite_cell());
-      while(!stack.empty()) {
-        auto ch = stack.top();
-        stack.pop();
-        ch->set_subdomain_index(0);
-        for(int i = 0; i < 4; ++i) {
-          if(ch->ccdt_3_data().is_facet_constrained(i))
-            continue;
-          auto n = ch->neighbor(i);
-          if(n->subdomain_index() == 1) {
-            stack.push(n);
-          }
-        }
-      }
-
-      for(auto f : tr.finite_facets())
-      {
-        const auto& mf = tr.mirror_facet(f);
-        if(f.first->ccdt_3_data().is_facet_constrained(f.second) ||
-           mf.first->ccdt_3_data().is_facet_constrained(mf.second))
-        {
-          const auto& patch = f.first->ccdt_3_data().face_constraint_index(f.second);
-          f.first->set_surface_patch_index(f.second, patch);
-          mf.first->set_surface_patch_index(mf.second, patch);
-        }
-      }
-    }
-    Conforming_constrained_Delaunay_triangulation_3 result{std::move(*this)};
-    static_assert(CGAL::cdt_3_msvc_2019_or_older() ||
-                  CGAL::is_nothrow_movable_v<Triangulation> == false ||
-                  CGAL::is_nothrow_movable_v<Conforming_constrained_Delaunay_triangulation_3>);
-    static_assert(std::is_same_v<std::remove_reference_t<decltype(*this)>, Conforming_constrained_Delaunay_triangulation_3>);
-    *this = Conforming_constrained_Delaunay_triangulation_3{};
-    return result;
-  }
-  /// \endcond
-  // end SKIP_IN_MANUAL for convert_for_remeshing
-
   /**
    * A bidirectional iterator for visiting all constrained facets of the triangulation.
    * The value type of this iterator is `Triangulation::Facet`.
@@ -4098,27 +4022,103 @@ auto convert_to_triangulation_3(Conforming_constrained_Delaunay_triangulation_3<
                                  Triangulation_data_structure>
 
 {
-  constexpr bool has_ecmap =
+  for(auto v : ccdt.triangulation().all_vertex_handles()) {
+    switch(v->ccdt_3_data().vertex_type()) {
+    case CDT_3_vertex_type::CORNER:
+      v->set_dimension(0);
+      v->set_index(0);
+      break;
+    case CDT_3_vertex_type::STEINER_ON_EDGE:
+      v->set_dimension(1);
+      v->set_index(static_cast<int>(v->ccdt_3_data().constrained_polyline_id(ccdt).index()));
+      break;
+    case CDT_3_vertex_type::STEINER_IN_FACE:
+      v->set_dimension(2);
+      v->set_index(v->ccdt_3_data().face_index());
+      break;
+    case CDT_3_vertex_type::FREE:
+      v->set_dimension(3);
+      v->set_index(1);
+      break;
+    default:
+      CGAL_error();
+      break;
+    }
+  }
+
+  constexpr bool has_edge_is_constrained_map =
       !parameters::is_default_parameter<CGAL_NP_CLASS, internal_np::edge_is_constrained_t>::value;
-  if constexpr (has_ecmap)
+  if constexpr (has_edge_is_constrained_map)
   {
     const auto& tr = ccdt.triangulation();
-    auto ecmap = parameters::get_parameter(np, internal_np::edge_is_constrained);
+    auto edge_is_constrained_map = parameters::get_parameter(np, internal_np::edge_is_constrained);
     for(auto e : tr.finite_edges())
     {
       auto [v1, v2] = tr.vertices(e);
-      if(  v1->ccdt_3_data().constrained_polyline_id(ccdt).index()
-        == v2->ccdt_3_data().constrained_polyline_id(ccdt).index())
+      auto v1_dim = v1->in_dimension();
+      auto v2_dim = v2->in_dimension();
+
+      if(v1_dim > 1 || v2_dim > 1) continue;
+
+      if(v1_dim == 1 && v2_dim == 1 &&
+           v1->ccdt_3_data().constrained_polyline_id(ccdt).index()
+        != v2->ccdt_3_data().constrained_polyline_id(ccdt).index()) continue;
+
+      auto incident_facets = Range_from_circulator(tr.incident_facets(e));
+      if(std::any_of(incident_facets.begin(), incident_facets.end(),
+                     [&](const auto& f) { return ccdt.is_facet_constrained(f); }))
       {
         if(v2 > v1)
           std::swap(v1, v2);
-        put(ecmap, std::make_pair(v1, v2), true);
+        put(edge_is_constrained_map, std::make_pair(v1, v2), true);
       }
     }
   }
 
-  auto tmp = std::move(ccdt).convert_for_remeshing();
-  return std::move(tmp).triangulation();
+  auto tr = std::move(ccdt).triangulation();
+
+  using CDT_3 = Conforming_constrained_Delaunay_triangulation_3<Traits, Tr>;
+  using TDS = typename CDT_3::Triangulation::Triangulation_data_structure;
+  static_assert(std::is_same_v<decltype(tr), Triangulation_3<Traits, TDS>>);
+
+  if(tr.dimension() < 3) {
+    for(auto ch : tr.all_cell_handles()) {
+      ch->set_subdomain_index(0);
+    }
+  } else {
+    for(auto ch : tr.all_cell_handles()) {
+      ch->set_subdomain_index(1);
+    }
+
+    std::stack<decltype(tr.infinite_cell())> stack;
+    stack.push(tr.infinite_cell());
+    while(!stack.empty()) {
+      auto ch = stack.top();
+      stack.pop();
+      ch->set_subdomain_index(0);
+      for(int i = 0; i < 4; ++i) {
+        if(ch->ccdt_3_data().is_facet_constrained(i))
+          continue;
+        auto n = ch->neighbor(i);
+        if(n->subdomain_index() == 1) {
+          stack.push(n);
+        }
+      }
+    }
+
+    for(auto f : tr.finite_facets())
+    {
+      auto mf = tr.mirror_facet(f);
+      if(f.first->ccdt_3_data().is_facet_constrained(f.second) ||
+          mf.first->ccdt_3_data().is_facet_constrained(mf.second))
+      {
+        auto patch = f.first->ccdt_3_data().face_constraint_index(f.second);
+        f.first->set_surface_patch_index(f.second, patch);
+        mf.first->set_surface_patch_index(mf.second, patch);
+      }
+    }
+  }
+  return tr;
 }
 
 } // end CGAL
diff --git a/TDS_3/include/CGAL/TDS_3/internal/Triangulation_ds_circulators_3.h b/TDS_3/include/CGAL/TDS_3/internal/Triangulation_ds_circulators_3.h
index 3568d783b53..33bd73b2fcd 100644
--- a/TDS_3/include/CGAL/TDS_3/internal/Triangulation_ds_circulators_3.h
+++ b/TDS_3/include/CGAL/TDS_3/internal/Triangulation_ds_circulators_3.h
@@ -184,8 +184,6 @@ operator!=(typename Tds_::Cell_handle ch, Triangulation_ds_cell_circulator_3<Tds
 
 template < class Tds_ >
 class Triangulation_ds_facet_circulator_3
-  : public Bidirectional_circulator_base<typename Tds_::Facet,
-                                         std::ptrdiff_t, std::size_t>
 {
   // circulates on facets around a given edge
 
@@ -199,6 +197,14 @@ class Triangulation_ds_facet_circulator_3
   typedef Triangulation_ds_facet_circulator_3<Tds> Facet_circulator;
 
 public:
+  struct Proxy_Facet;
+  typedef Bidirectional_circulator_tag iterator_category;
+  typedef Facet                        value_type;
+  typedef std::ptrdiff_t               difference_type;
+  typedef Proxy_Facet                  pointer;
+  typedef Facet                        reference;
+  typedef std::size_t                  size_type;
+
 
   Triangulation_ds_facet_circulator_3()
     : _s(), _t(), pos()