diff --git a/Surface_mesh_parameterization/include/CGAL/ARAP_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/ARAP_parameterizer_3.h
index 6b8c65da728..2dbe3c8f22b 100644
--- a/Surface_mesh_parameterization/include/CGAL/ARAP_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/ARAP_parameterizer_3.h
@@ -1245,6 +1245,20 @@ private:
 
 // Public operations
 public:
+  /// Check if the 3D -> 2D mapping is one-to-one.
+  template <typename VertexUVMap>
+  bool is_one_to_one_mapping(const TriangleMesh& mesh,
+                             const VertexUVMap uvmap) const
+  {
+    /// Theorem: A one-to-one mapping is guaranteed if all w_ij coefficients
+    ///          are > 0 (for j vertex neighbor of i) and if the surface
+    ///          border is mapped onto a 2D convex polygon.
+    /// Here, all w_ij coefficients = 1 (for j vertex neighbor of i), thus a
+    /// valid embedding is guaranteed if the surface border is mapped
+    /// onto a 2D convex polygon.
+    return internal::Parameterization::is_one_to_one_mapping(mesh, uvmap);
+  }
+
   /// Compute a mapping from a triangular 3D surface mesh to a piece of the 2D space.
   /// The mapping is piecewise linear (linear in each triangle).
   /// The result is the (u,v) pair image of each vertex of the 3D surface.
diff --git a/Surface_mesh_parameterization/include/CGAL/Barycentric_mapping_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/Barycentric_mapping_parameterizer_3.h
index e8ba50c6597..e745e614e0c 100644
--- a/Surface_mesh_parameterization/include/CGAL/Barycentric_mapping_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Barycentric_mapping_parameterizer_3.h
@@ -24,6 +24,7 @@
 
 #include <CGAL/license/Surface_mesh_parameterization.h>
 
+#include <CGAL/internal/Surface_mesh_parameterization/validity.h>
 #include <CGAL/Circular_border_parameterizer_3.h>
 #include <CGAL/Fixed_border_parameterizer_3.h>
 #include <CGAL/Eigen_solver_traits.h>
@@ -118,6 +119,21 @@ public:
 
   // Default copy constructor and operator =() are fine
 
+  /// Check if the 3D -> 2D mapping is one-to-one.
+  template <typename VertexUVMap>
+  bool is_one_to_one_mapping(const TriangleMesh& mesh,
+                             const VertexUVMap uvmap) const
+  {
+    /// Theorem: A one-to-one mapping is guaranteed if all w_ij coefficients
+    ///          are > 0 (for j vertex neighbor of i) and if the surface
+    ///          border is mapped onto a 2D convex polygon.
+    /// Here, all w_ij coefficients = 1 (for j vertex neighbor of i), thus a
+    /// valid embedding is guaranteed if the surface border is mapped
+    /// onto a 2D convex polygon.
+    return (Base::get_border_parameterizer().is_border_convex() ||
+            internal::Parameterization::is_one_to_one_mapping(mesh, uvmap));
+  }
+
 // Protected operations
 protected:
   /// Compute w_ij = (i,j), coefficient of matrix A for j neighbor vertex of i.
diff --git a/Surface_mesh_parameterization/include/CGAL/LSCM_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/LSCM_parameterizer_3.h
index 8e7a9b5d39e..5929830566d 100644
--- a/Surface_mesh_parameterization/include/CGAL/LSCM_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/LSCM_parameterizer_3.h
@@ -24,6 +24,7 @@
 #include <CGAL/license/Surface_mesh_parameterization.h>
 
 #include <CGAL/internal/Surface_mesh_parameterization/Containers_filler.h>
+#include <CGAL/internal/Surface_mesh_parameterization/validity.h>
 
 #include <CGAL/Parameterizer_traits_3.h>
 #include <CGAL/Two_vertices_parameterizer_3.h>
@@ -147,6 +148,14 @@ public:
 
   // Default copy constructor and operator =() are fine
 
+  /// Check if the 3D -> 2D mapping is one-to-one.
+  template <typename VertexUVMap>
+  bool is_one_to_one_mapping(const TriangleMesh& mesh,
+                             const VertexUVMap uvmap) const
+  {
+    return internal::Parameterization::is_one_to_one_mapping(mesh, uvmap);
+  }
+
   /// Compute a one-to-one mapping from a triangular 3D surface mesh
   /// to a piece of the 2D space.
   /// The mapping is piecewise linear (linear in each triangle).
diff --git a/Surface_mesh_parameterization/include/CGAL/Mean_value_coordinates_parameterizer_3.h b/Surface_mesh_parameterization/include/CGAL/Mean_value_coordinates_parameterizer_3.h
index f9e78fac610..2573a8d2e6f 100644
--- a/Surface_mesh_parameterization/include/CGAL/Mean_value_coordinates_parameterizer_3.h
+++ b/Surface_mesh_parameterization/include/CGAL/Mean_value_coordinates_parameterizer_3.h
@@ -23,7 +23,7 @@
 
 #include <CGAL/license/Surface_mesh_parameterization.h>
 
-
+#include <CGAL/internal/Surface_mesh_parameterization/validity.h>
 #include <CGAL/Fixed_border_parameterizer_3.h>
 #include <CGAL/Eigen_solver_traits.h>
 
@@ -129,6 +129,21 @@ public:
 
     // Default copy constructor and operator =() are fine
 
+  /// Check if the 3D -> 2D mapping is one-to-one.
+  template <typename VertexUVMap>
+  bool is_one_to_one_mapping(const TriangleMesh& mesh,
+                             const VertexUVMap uvmap) const
+  {
+    /// Theorem: A one-to-one mapping is guaranteed if all w_ij coefficients
+    ///          are > 0 (for j vertex neighbor of i) and if the surface
+    ///          border is mapped onto a 2D convex polygon.
+    /// Here, all w_ij coefficients are positive (for j vertex neighbor of i), thus a
+    /// valid embedding is guaranteed if the surface border is mapped
+    /// onto a 2D convex polygon.
+    return (Base::get_border_parameterizer().is_border_convex() ||
+            internal::Parameterization::is_one_to_one_mapping(mesh, uvmap));
+  }
+
 // Protected operations
 protected:
   /// Compute w_ij = (i, j) coefficient of matrix A for j neighbor vertex of i.