diff --git a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_base_3.h b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_base_3.h
index 78a8a967c0d..4d88637ab5b 100644
--- a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_base_3.h
+++ b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_base_3.h
@@ -95,8 +95,11 @@ public:
   void reset_cache_validity() const { sliver_cache_validity_ = false; }
 
   static std::string io_signature() {
-    return Get_io_signature<Base>()() + "+(" + Get_io_signature<int>()()
-      + ")[4]";
+    static_assert(
+        std::is_same_v<
+            decltype(std::declval<Constrained_Delaunay_triangulation_cell_data_3>().face_constraint_index(0)), int>);
+
+    return Get_io_signature<Base>()() + "+(" + Get_io_signature<int>()() + ")[4]";
   }
 
   friend std::ostream&
@@ -106,9 +109,9 @@ public:
     os << static_cast<const Base&>(c);
     for( unsigned li = 0; li < 4; ++li ) {
       if(IO::is_ascii(os)) {
-        os << " " << c.cdt_3_data().face_id[li];
+        os << " " << c.cdt_3_data().face_constraint_index(li);
       } else {
-        CGAL::write(os, c.cdt_3_data().face_id[li]);
+        CGAL::write(os, c.cdt_3_data().face_constraint_index(li));
       }
     }
     return os;
diff --git a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_data_3.h b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_data_3.h
index b029587d545..cfc3e20f91e 100644
--- a/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_data_3.h
+++ b/Constrained_triangulation_3/include/CGAL/Constrained_Delaunay_triangulation_cell_data_3.h
@@ -29,18 +29,6 @@
 #include <bitset>
 
 namespace CGAL {
-#ifdef DOXYGEN_RUNNING
-/*!
- * @brief Internal per-cell data for \cgal 3D constrained Delaunay triangulations
- *
- * This class is an internal detail of the implementation of \cgal 3D constrained Delaunay triangulations.
- *
- * Any model of the `ConstrainedDelaunayTriangulationCellBase_3` concept must include one object of this type
- * as a non-static data member.
- */
-struct Constrained_Delaunay_triangulation_cell_data_3 {};
-#else // DOXYGEN_RUNNING
-
 enum class CDT_3_cell_marker {
   CLEAR = 0,
   IN_REGION = 1,
@@ -49,7 +37,19 @@ enum class CDT_3_cell_marker {
   nb_of_markers
 };
 
-struct Constrained_Delaunay_triangulation_cell_data_3 {
+/*!
+ * @brief Internal per-cell data for \cgal 3D constrained Delaunay triangulations
+ *
+ * This class is an internal detail of the implementation of \cgal 3D constrained Delaunay triangulations.
+ *
+ * Any model of the `ConstrainedDelaunayTriangulationCellBase_3` concept must include one object of this type
+ * as a non-static data member.
+ */
+class Constrained_Delaunay_triangulation_cell_data_3 {
+  /// @cond SKIP_IN_MANUAL
+  template <typename Tr> friend class Constrained_Delaunay_triangulation_3_impl;
+  /// @endcond
+
   std::array<CDT_3_face_index, 4> face_id = { -1, -1, -1, -1 };
   std::array<void*, 4> facet_2d = {nullptr, nullptr, nullptr, nullptr};
   std::bitset<static_cast<unsigned>(CDT_3_cell_marker::nb_of_markers)> markers;
@@ -60,8 +60,6 @@ struct Constrained_Delaunay_triangulation_cell_data_3 {
   void clear_mark(CDT_3_cell_marker m) { markers.reset(static_cast<unsigned>(m)); }
   void clear_marks() { markers.reset(); }
 
-  bool is_facet_constrained(int i) const { return face_id[unsigned(i)] >= 0; }
-
   template <typename Facet_handle>
   void set_facet_constraint(int i, CDT_3_face_index face_id,
                             Facet_handle facet_2d)
@@ -70,19 +68,24 @@ struct Constrained_Delaunay_triangulation_cell_data_3 {
     this->facet_2d[unsigned(i)] = static_cast<void*>(facet_2d == Facet_handle{} ?  nullptr : std::addressof(*facet_2d));
   }
 
-  CDT_3_face_index face_constraint_index(int i) const {
-    return face_id[unsigned(i)];
-  }
-
   template <typename CDT_2>
   auto face_2 (const CDT_2& cdt, int i) const {
     using Face = typename CDT_2::Face;
     auto ptr = static_cast<Face*>(facet_2d[unsigned(i)]);
     return cdt.tds().faces().iterator_to(*ptr);
   }
+public:
+  /// @brief Returns if the i-th facet of the cell is constrained.
+  bool is_facet_constrained(int i) const { return face_id[unsigned(i)] >= 0; }
+
+  /// @brief Returns the index of the constraint that constrains the
+  /// i-th facet of the cell.
+  /// @pre `is_facet_constrained(i)`
+  CDT_3_face_index face_constraint_index(int i) const {
+    return face_id[unsigned(i)];
+  }
 };
 
-#endif // DOXYGEN_RUNNING
 } // namespace CGAL
 
 #endif // CGAL_CONSTRAINED_DELAUNAY_TRIANGULATION_CELL_DATA_3_H
diff --git a/Constrained_triangulation_3/include/CGAL/Constrained_triangulation_3/internal/config.h b/Constrained_triangulation_3/include/CGAL/Constrained_triangulation_3/internal/config.h
index a566ed1bab7..a7c1a3a0acc 100644
--- a/Constrained_triangulation_3/include/CGAL/Constrained_triangulation_3/internal/config.h
+++ b/Constrained_triangulation_3/include/CGAL/Constrained_triangulation_3/internal/config.h
@@ -25,6 +25,8 @@
 
 #include <CGAL/config.h>
 
+#include <CGAL/Constrained_triangulation_3_types.h>
+
 #if CGAL_CAN_USE_CXX20_FORMAT
 #  define CGAL_CDT_3_CAN_USE_CXX20_FORMAT 1
 #  include <format>
@@ -32,8 +34,6 @@
 
 namespace CGAL {
 
-using CDT_3_face_index = int; // must be signed
-
 #if CGAL_CDT_3_CAN_USE_CXX20_FORMAT
 
 constexpr bool cdt_3_can_use_cxx20_format() {
diff --git a/Constrained_triangulation_3/include/CGAL/Constrained_triangulation_3_types.h b/Constrained_triangulation_3/include/CGAL/Constrained_triangulation_3_types.h
new file mode 100644
index 00000000000..94d4a4f6bc8
--- /dev/null
+++ b/Constrained_triangulation_3/include/CGAL/Constrained_triangulation_3_types.h
@@ -0,0 +1,42 @@
+// Copyright (c) 2024  GeometryFactory Sarl (France).
+// All rights reserved.
+//
+// This file is part of CGAL (www.cgal.org).
+// You can redistribute it and/or modify it under the terms of the GNU
+// General Public License as published by the Free Software Foundation,
+// either version 3 of the License, or (at your option) any later version.
+//
+// Licensees holding a valid commercial license may use this file in
+// accordance with the commercial license agreement provided with the software.
+//
+// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
+// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
+//
+// $URL$
+// $Id$
+// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
+//
+// Author(s)     : Laurent Rineau
+
+#ifndef CGAL_CT_3_TYPES_H
+#define CGAL_CT_3_TYPES_H
+
+#include <CGAL/license/Triangulation_3.h>
+
+#include <CGAL/config.h>
+
+namespace CGAL {
+
+/**
+ * @addtogroup PkgCT_3Classes
+ * @typedef CDT_3_face_index
+ * Integral type to store the index of constraints.
+ * @see `Constrained_Delaunay_triangulation_cell_data_3`
+ * @see `Constrained_Delaunay_triangulation_vertex_base_3`
+ *
+ */
+using CDT_3_face_index = int; // must be signed
+
+} // namespace CGAL
+
+#endif // CGAL_CT_3_TYPES_H